diff --git a/-9E3T4oBgHgl3EQfrwpm/content/tmp_files/2301.04662v1.pdf.txt b/-9E3T4oBgHgl3EQfrwpm/content/tmp_files/2301.04662v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6bc49e44892bb68b50efc081ab48da7dd0fff345 --- /dev/null +++ b/-9E3T4oBgHgl3EQfrwpm/content/tmp_files/2301.04662v1.pdf.txt @@ -0,0 +1,1685 @@ +Draft version January 13, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +SN 2020bio: A Double-peaked Type IIb Supernova with Evidence of Early-time Circumstellar +Interaction +C. Pellegrino,1, 2 D. Hiramatsu,3, 4 I. Arcavi,5, 6 D. A. Howell,1, 2 K. A. Bostroem,7, ∗ P. J. Brown,8, 9 J. Burke,1, 2 +N. Elias-Rosa,10, 11 K. Itagaki,12 H. Kaneda,13 C. McCully,1, 2 M. Modjaz,14 E. Padilla Gonzalez,1, 2 and +T. A. Pritchard15 +1Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA +2Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA +3Center for Astrophysics |Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138-1516, USA +4The NSF AI Institute for Artificial Intelligence and Fundamental Interactions +5The School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel +6CIFAR Azrieli Global Scholars Program, CIFAR, Toronto, Canada +7Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195-0002, USA +8Department of Physics and Astronomy, Texas A&M University, 4242 TAMU, College Station, TX 77843, USA +9George P. and Cynthia Woods Mitchell Institute for Fundamental Physics & Astronomy, College Station, TX 77843, USA +10INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy +11Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Barcelona, Spain +12Itagaki Astronomical Observatory, Yamagata, Yamagata 990-2492, Japan +13Kaneda Astronomical Observatory, Sapporo, Hokkaido 005-0862, Japan +14Department of Astronomy, University of Virginia, Charlottesville, VA 22904 +15Department of Physics, New York University, New York, NY 10003, USA +Submitted to ApJ +ABSTRACT +We present photometric and spectroscopic observations of SN 2020bio, a double-peaked Type IIb +supernova (SN) discovered within a day of explosion, primarily obtained by Las Cumbres Observatory +and Swift. SN 2020bio displays a rapid and long-lasting initial decline throughout the first week of its +light curve, similar to other well-studied Type IIb SNe. This early-time emission is thought to originate +from the cooling of the extended outer envelope of the progenitor star that is shock-heated by the SN +explosion. We compare SN 2020bio to a sample of other double-peaked Type IIb SNe to investigate its +progenitor properties. Analytical model fits to the early-time emission give progenitor radius (≈ 100– +1500 R⊙) and H-rich envelope mass (≈ 0.01–0.5 M⊙) estimates that are consistent with other Type IIb +SNe. However, SN 2020bio displays several peculiarities, including: 1) weak H spectral features and +narrow emission lines indicative of pre-existing circumstellar material; 2) an underluminous secondary +light curve peak which implies a small amount of synthesized 56Ni (MNi ≈ 0.02 M⊙); and 3) low- +luminosity nebular [O I] features. These observations are more consistent with a lower-mass progenitor +(MZAMS ≈ 12 M⊙) that was stripped of most of its H envelope before exploding. This study adds to +the growing diversity in the observed properties of Type IIb SNe and their progenitors. +Keywords: Circumstellar matter(241) — Core-collapse supernovae(304) — Supernovae(1668) +1. INTRODUCTION +Corresponding author: Craig Pellegrino +cpellegrino@lco.global +∗ LSST Catalyst Fellow +While the majority of stars with initial masses ≳ 8 M⊙ +end their lives as H-rich core-collapse supernovae (SNe; +e.g., Janka 2012), some massive stars lose their outer H +and even He envelopes and explode as stripped-envelope +SNe (SESNe; e.g., Filippenko 1997; Gal-Yam 2017). A +small but growing number of SNe have been observed +arXiv:2301.04662v1 [astro-ph.HE] 11 Jan 2023 + +2 +Pellegrino et al. +with spectra that show similarities to both these classes +(Smith et al. 2011). Classified as Type IIb SNe (SNe +IIb), their spectra have H features at early times that +gradually give way to He features, indicating that their +progenitors were partially stripped of their outer en- +velopes before exploding (Woosley et al. 1994). +It is unclear what mechanisms are responsible for this +mass loss. Common hypotheses include stellar winds, +binary interaction, or late-stage stellar instabilities (see +e.g., Smith 2014, for a review). +Recent studies have +shown that mass loss is common during the late stages +of massive star evolution, as inferred from early-time +observations of core-collapse SNe (e.g., Ofek et al. 2014; +Bruch et al. 2021; Strotjohann et al. 2021). A signif- +icant fraction of core-collapse SNe show signatures of +pre-existing circumstellar material (CSM) in their early- +time spectra, obtained days after their estimated explo- +sion epochs. This CSM is the material shed by the pro- +genitor star in the months to years before core-collapse. +As the SN shock breaks out of the expanding ejecta the +resulting X-ray and ultraviolet (UV) flash may ionize +the surrounding CSM, producing narrow spectral fea- +tures as the CSM cools and recombines (e.g., Fassia et +al. 2001; Yaron et al. 2017). +Interaction between the +SN ejecta and CSM can also influence the early-time +light-curve evolution (Morozova et al. 2018). +Some SNe IIb are observed to have double-peaked +light curves, with rapidly-fading luminosities during the +first several days after explosion before the radioactive +decay of 56Ni synthesized during the explosion causes a +re-brightening that lasts for several weeks. The early- +time emission is thought to be the cooling of the ex- +tended envelope of the progenitor star that is heated +by the SN shock (Soderberg et al. 2012). This shock- +cooling emission (SCE) has only been extensively ob- +served in a handful of cases, including SN 1993J (e.g., +Woosley et al. 1994; Richmond et al. 1994), SN 2011dh +(e.g., Arcavi et al. 2011; Ergon et al. 2014), SN2013 df +(e.g., Morales-Garoffolo et al. 2014; Van Dyk et al. 2014), +SN 2016gkg (Arcavi et al. 2017), SN 2017jgh (Armstrong +et al. 2021), and ZTF18aalrxas (Fremling et al. 2019), +among others. Most of these objects are nearby and had +follow-up observations scheduled hours after explosion, +which proved crucial to observing the rapidly-evolving +SCE. These studies have found that SNe IIb are con- +sistent with the explosions of stars with extended outer +envelopes, with the duration of the SCE dependent on +the extent of this envelope (Soderberg et al. 2012). +Numerical and analytical models of SCE can comple- +ment pre-explosion imaging in determining the progen- +itors of these objects. +Several models have been suc- +cessful in reproducing the observed early-time evolution +across all wavelengths. +Piro (2015, hereafter P15) is +one of the first to present a one-zone analytical descrip- +tion of the cooling of an extended low-mass envelope +shock-heated by the explosion of a compact massive +core. Piro et al. (2021, hereafter P21) extend this to +a two-zone model in order to better capture the emis- +sion from the outermost material in extended envelopes. +Sapir & Waxman (2017, hereafter SW17) calibrate ear- +lier models by Rabinak & Waxman (2011)—that depend +on the precise density structure of the outer material—to +numerical simulations for several days after explosion. +Comparing observed SCE to analytical and numeri- +cal models is one of the only ways of directly measuring +the radii and stellar structure of core-collapse progen- +itors from SN observations. This has been done for a +handful of SNe IIb as well as SNe of other subtypes, +including stripped-envelope Type Ib SNe (e.g., Modjaz +et al. 2009; Yao et al. 2020), short-plateau Type II SNe +(Hiramatsu et al. 2021), and exotic Ca-rich transients +(e.g., Jacobson-Gal´an et al. 2020, 2022). Analytical and +numerical modeling of double-peaked SNe IIb generally +yield large radii progenitors (≈ 100–500 R⊙) with low- +mass (≈ 10−2–10−1 M⊙) extended envelopes (Piro et +al. 2021, and references therein). These properties are +usually in agreement with those of SNe IIb progeni- +tors from pre-explosion Hubble Space Telescope images, +which have revealed them to be supergiants (Aldering +et al. 1994; Maund et al. 2011; Van Dyk et al. 2014). +In some cases, however, the progenitor radii estimated +from SCE modeling are in tension with those measured +from direct imaging (e.g., Arcavi et al. 2017; Tartaglia +et al. 2017, in the case of SN 2016gkg;). Potential bi- +nary companions to the progenitor, which have been +observed or inferred in a handful of cases (e.g., Maund +et al. 2004; Benvenuto et al. 2013) can further compli- +cate direct imaging estimates when the individual binary +members are unresolvable. +Here we present photometric and spectroscopic ob- +servations of SN 2020bio, an SN IIb showing remark- +ably strong SCE, obtained by Las Cumbres Observatory +(LCO) through the Global Supernova Project (GSP). +LCO extensively observed SN 2020bio from hours to +≈ 160 days after explosion, providing a detailed look +into the full evolution of a double-peaked SN IIb. In +this work, we analyze its light curve evolution, spectral +features, and fit analytic models to its full light-curve +evolution to estimate the radius, mass, and structure of +its progenitor star. We also compare its bolometric light +curve and spectra to numerical models in order to infer +its progenitor mass and the properties of its circumstel- +lar environment. + +The Double-peaked Type IIb SN 2020bio +3 +This paper is organized as follows. In Section 2 we +describe the discovery and follow-up observations of +SN 2020bio. +We present its full light curve and spec- +tral time series in Section 3 and compare observations +to analytical and numerical models in Section 4. +Fi- +nally, in Section 5 we discuss the potential progenitor +properties of SN 2020bio given the presented evidence. +2. DISCOVERY AND DATA DESCRIPTION +SN 2020bio was discovered by Koichi Itagaki on UT +2020 January 29.77 at the Itagaki Astronomical Obser- +vatory at an unfiltered Vega magnitude of 16.7. Analy- +sis of an image of the same field by the ATLAS survey +on the previous night yields a nondetection at c-band +magnitude 18.7. Soon after discovery rapid photometric +and spectroscopic follow-up observations were requested +by the GSP through the Las Cumbres global network of +telescopes. The GSP also triggered its Swift Key Project +(1518618: PI Howell) to obtain daily UV and optical +photometry. A classification spectrum obtained on the +2.0m Liverpool Telescope on 2020 January 31.19—ap- +proximately 1.5 days after the first detection—shows a +blue continuum superimposed with a narrow Hα emis- +sion feature and a broad possible He I λ 5876˚A feature, +consistent with a young core-collapse SN (Srivastav et +al. 2020). +SN 2020bio exploded at right ascension 13h55m37s.69 +and declination +40°28′39′′.1 in the spiral galaxy NGC +5371 at redshift z = 0.008533 (Springob et al. 2005). +The distance to NGC 5371 is uncertain due to its low +redshift. We adopt the mean of several distances mea- +sured using the method of Tully & Fisher (1977), which +gives d = 29.9 ± 5.1 Mpc (values from the NASA Ex- +tragalactic Database1). Using the Schlafly & Finkbeiner +(2011) dust map calibrations, we estimate a Galactic +line-of-sight extinction to SN 2020bio EMW (B − V ) = +0.008 mag. Given the location of SN 2020bio with re- +spect to its host galaxy, we also estimate host extinc- +tion using the Na I D equivalent widths measured in a +high-resolution spectrum of the SN. From the conver- +sions presented in Poznanski et al. (2012), we estimate +Ehost(B −V ) = 0.068 ± 0.038 mag for a total extinction +E(B − V ) = 0.076 ± 0.038 mag. The photometry of +SN 2020bio presented throughout this work is corrected +for this mean total extinction. +LCO photometric follow-up commenced less than a +day after discovery. UBgVri-band images were obtained +by the Sinistro and Spectral cameras mounted on LCO +1.0m and 2.0m telescopes, respectively, located at Mc- +1 https://ned.ipac.caltech.edu/ +Table 1. UV and Optical Photometry +JD +Filter +Magnitude +Uncertainty +Source +2458878.27 +Clear +16.77 +0.15 +Itagaki +2458878.33 +Clear +16.55 +0.15 +Itagaki +2458878.39 +Clear +16.51 +0.15 +Itagaki +2458879.27 +Clear +16.22 +0.15 +Itagaki +2458880.26 +Clear +16.49 +0.15 +Itagaki +2458881.25 +Clear +16.68 +0.15 +Itagaki +2458882.18 +Clear +16.82 +0.15 +Itagaki +2458883.26 +Clear +16.85 +0.15 +Itagaki +2458878.85 +UVW2 +13.56 +0.04 +Swift +2458879.89 +UVW2 +14.59 +0.05 +Swift +This table will be made available in its entirety in machine- +readable format. +Donald Observatory, Teide Observatory, and Haleakala +Observatory. +Data were reduced using lcogtsnpipe +(Valenti et al. 2016) which extracts point-spread func- +tion magnitudes after calculating zero-points and color +terms (Stetson 1987). UBV -band photometry was cali- +brated to Vega magnitudes using Landolt standard fields +(Landolt 1992) while gri-band photometry was cali- +brated to AB magnitudes (Smith et al. 2002) using Sloan +Digital Sky Survey (SDSS) catalogs. As SN 2020bio ex- +ploded coincident with its host galaxy, to remove host +galaxy light we performed template subtraction using +the HOTPANTS (Becker 2015) algorithm and template +images obtained after the SN had faded. Unfiltered im- +ages were obtained with the Itagaki Astronomical Ob- +servatory (Okayama and Kochi, Japan) 0.35 m tele- +scopes + KAF-1001E (CCD). Using our custom soft- +ware, the photometry was extracted after host subtrac- +tion and calibrated to the V-band magnitudes of +45 +field stars from the Fourth US Naval Observatory CCD +Astrograph Catalog (Zacharias et al. 2013). +UV and optical photometry were obtained with the +Ultraviolet and Optical Telescope (UVOT; Roming et +al. 2005) on the Neil Gehrels Swift observatory (Gehrels +et al. 2004). Swift data were reduced using a custom +adaptation of the Swift Optical/Ultraviolet Supernova +Archive (Brown et al. 2014) pipeline with the most re- +cent calibration files and the zeropoints of Breeveld et +al. (2011). Images from the final epoch, obtained after +the SN had sufficiently faded, were used as templates +to subtract the host galaxy light. All Swift photometry +is calibrated to Vega magnitudes. The entire UV and +optical data sets from LCO, Itagaki, and Swift UVOT +are given in Table 1. + +4 +Pellegrino et al. +58880 +58900 +58920 +58940 +58960 +58980 +59000 +59020 +59040 +MJD +10 +12 +14 +16 +18 +20 +22 +24 +Apparent Magnitude + Offset +58877.0 +58880.5 +58884.0 +10 +12 +14 +16 +18 +20 +UVW2 - 4 +UVM2 - 3 +UVW1 - 2 +U - 1 +B +g + 1 +V + 2 +r + 3 +i + 3.5 +Clear + 2 +0 +20 +40 +60 +80 +100 +120 +140 +160 +Days From Discovery +-22 +-20 +-18 +-16 +-14 +-12 +-10 +-8 +Absolute Magnitude + Offset +Figure 1. The full extinction-corrected light curves of SN 2020bio. Photometry in different filters have been offset for clarity. +Unfiltered photometry from the Itagaki Astronomical Observatory is included as clear points and calibrated to the V -band. +The inset focuses on the rapidly-evolving shock-cooling emission. +LCO spectra were obtained by the FLOYDS spectro- +graph on the 2.0m Faulkes Telescope North at Haleakala +Observatory. +Spectra cover a wavelength range of +3500–10,000 ˚A at a resolution R ≈ 300-600. +Data +were reduced using the floydsspec pipeline2, a custom +pipeline which performs cosmic ray removal, spectrum +extraction, and wavelength and flux calibration. +We +also present one spectrum obtained by the B&C spectro- +graph on the 2.3m Bok Telescope at Steward Observa- +tory, two spectra obtained by the Blue Channel Spectro- +graph on the 6.5m MMT at the Fred Lawrence Whipple +Observatory, and one spectrum obtained by the Optical +System for Imaging and low-Intermediate-Resolution In- +2 https://github.com/svalenti/FLOYDS pipeline/ +tegrated Spectroscopy spectrograph on the 10.4m Gran +Telescopio Canarias. Details of all these spectra are pre- +sented in Table 2. +3. PHOTOMETRIC AND SPECTRAL ANALYSIS +3.1. Light Curve and Color Evolution +In Figure 1 we show the full LCO and Swift extinction- +corrected light curve of SN 2020bio, from detection to +≈ 160 days after explosion. The discovery and subse- +quent follow-up photometry from Itagaki are included +as “Clear” data points. The inset shows in greater de- +tail the early-time evolution of the SCE, focusing on the +first week after discovery. The most distinctive feature +of the light curve is the luminous and rapidly-declining +SCE at early times. The peak SCE luminosity exceeds +that of the secondary peak ≈ 15 days later, but SCE +only dominates the light curve during the first several + +The Double-peaked Type IIb SN 2020bio +5 +-2 +0 +2 +UVW2 - B +-2 +0 +2 +UVW2 - V +-2 +0 +2 +UVM2 - B +-2 +0 +2 +UVM2 - V +0 +1 +2 +3 +4 +5 +6 +7 +8 +Days Since Discovery +-2 +0 +2 +UVW1 - B +0 +1 +2 +3 +4 +5 +6 +7 +8 +Days Since Discovery +-2 +0 +2 +UVW1 - V +SW17 Model +SN 2010jr +SN 2011dh +SN 2013df +SN 2016gkg +SN 2020bio +Figure 2. Swift colors of SN 2020bio compared with those of other SNe IIb with early-time Swift observations. We also include +the best fit SW17 model from Section 4 for comparison. SN 2020bio was bluer at earlier phases than the other SNe IIb. Data +for these comparison SNe were obtained from the following sources: Arcavi et al. (2011) (SN 2011dh); Morales-Garoffolo et al. +(2014) (SN 2013df); Arcavi et al. (2017) (SN 2016gkg); this work (SNe 2010jr and 2020bio). +days. Over this time the light curve falls by ≈ 4 mag +in the first week, making this phase difficult to observe +without rapid multi-wavelength follow-up. +After ≈ 4 days from discovery the slope of the light +curve decline changes as the luminosity from 56Ni de- +cay begins to dominate the light curve. +After about +a week the light curve re-brightens and reaches a sec- +ondary maximum ≈ 15 days after discovery. From this +point the emission settles onto the radioactive decay tail, +powered by 56Co decay, for the remainder of the obser- +vations. The secondary peak and overall late-time light +curve is relatively dim, peaking at M ≈ -14 mag in the +V -band, hinting at a small amount of 56Ni synthesized +in the explosion. +In Figure 2 we compare the early-time Swift UV- +optical colors of SN 2020bio to those of other SNe IIb +with observed SCE in the UV. All dates are given with +respect to the time of discovery and corrected for extinc- +tion according to the published values for each object. +SN 2020bio has both the earliest observations relative to +discovery and the bluest colors throughout its evolution +compared to the other objects. While objects such as +SN 2010jr and SN 2016gkg have more densely-sampled +light curves, their observations began later and their col- +ors evolved redward faster compared to SN 2020bio. +Of the 6 colors plotted, SN 2020bio is exceptionally +blue in the UVM2-B and UVM2-V colors, particularly +in the earliest epochs. +We plot a representative SCE +model color curve from Section 4.2 in each panel for +comparison. SN 2020bio is bluer than the model, which +more accurately reproduces the color evolution of the +other SNe IIb up to several days after the discovery. +This may be evidence for another luminosity contribu- +tion besides SCE, as we discuss in Section 5. +3.2. Spectral Comparison +Spectral coverage of SN 2020bio began fewer than 2 +days after the first detection—approximately 3 days +since the estimated explosion time (Section 4.2)—and +continued for 201 days. We plot the full spectral series + +6 +Pellegrino et al. +Table 2. Log of Spectroscopic Observations +Date of Observation +Days Since Discovery +Facility/Instrument +Exposure Time (s) +Wavelength Range (˚A) +2020-01-31 04:27:31 +1 +LT/SPRAT +1200 +4000–7925 +2020-02-03 14:32:18 +4 +LCO/FLOYDS-N +1800 +3500–10,000 +2020-02-05 12:19:05 +6 +LCO/FLOYDS-N +1800 +3500–10,000 +2020-02-15 09:35:59 +16 +Bok/B&C +600 +3850–7500 +2020-02-18 12:32:26 +19 +MMT/Blue Channel +300 +5700–7000 +2020-02-24 13:00:37 +25 +LCO/FLOYDS-N +1800 +3500–10,000 +2020-03-03 10:49:44 +33 +LCO/FLOYDS-N +2700 +3500–10,000 +2020-03-22 14:22:56 +52 +LCO/FLOYDS-N +3600 +3500–10,000 +2020-03-30 14:20:34 +60 +LCO/FLOYDS-N +3600 +3500–10,000 +2020-04-16 11:12:12 +77 +LCO/FLOYDS-N +3600 +3500–10,000 +2020-04-27 12:09:24 +88 +LCO/FLOYDS-N +3600 +3500–10,000 +2020-08-18 22:02:01 +201 +GTC/OSIRIS +1500 +3600–7808 +Note—All spectra will be made publicly-available on WiseRep (Yaron & Gal-Yam 2012). +in Figure 3. The earliest spectrum of SN 2020bio, re- +ported to the Transient Name Server (Srivastav et al. +2020), shows a hot blue continuum superimposed with +emission lines. +We identify narrow features of H and +Mg I as well as a potential weak, broad feature of He I +λ5876 ˚A. These lines are consistent with flash-ionized +features observed in other core-collapse SNe, which is +evidence of nearby CSM lost by the progenitor star. +After about a week post explosion, absorption fea- +tures begin to develop in the spectra. We identify lines +of He, O, and Ca. We also note persistent narrow H +emission features that last for several weeks. To deter- +mine if these features are produced by interaction with +CSM or by host galaxy emission, we fit the narrow Hα +emission line with a Gaussian function to estimate its +full-width at half-maximum (FWHM). The results are +shown in Figure 4. In our earliest spectrum we estimate +a FWHM of the Hα line of 1500 km s−1, greater than +the average widths of host galaxy emission lines, while +our spectrum obtained roughly two weeks after discov- +ery has a FWHM of ≈ 350 km s−1, more consistent with +host-galaxy emission at this resolution. The latter value +is also consistent with the FWHMs we measure for the +nearby host-dominated [N II] λ 6583 line throughout the +first several weeks. Therefore, we conclude that circum- +stellar interaction likely contributes to the H emission +during the first ≈ 2 weeks after explosion. +An absorption feature blueward of the rest-frame Hα +line matches He I λ 6678˚A absorption blueshifted by ≈ +7500 km s−1, which is commonly noted to cause “flat- +topped” Hα emission profiles in other SNe IIb (e.g., Fil- +ippenko et al. 1993). In general, the absorption features +in the SN 2020bio spectra are shallower than those of the +other SNe IIb, particularly SN 2011dh. Interaction with +CSM can produce absorption features that are weaker +and shallower than expected, which has been noted in +the spectra of SN 1993J and SN 2013df (Fremling et al. +2019). +To +further +investigate +the +differences +between +SN 2020bio and other SNe IIb, we plot comparison spec- +tra just after explosion (top), after two weeks (middle), +and three weeks (bottom) after explosion in Figure 5. +Among this sample, SN 2020bio is the only object to +show narrow features indicative of pre-existing CSM at +early times, despite similar phase coverage of the other +SNe IIb. This likely reflects differences in their circum- +stellar environments—if the narrow lines were formed +from the expanding outer envelopes of the progenitor, +they should be ubiquitous among SNe IIb at this phase. +Instead, the presence of narrow H and Mg lines in the +earliest spectrum of SN 2020bio more likely points to +confined CSM formed from material stripped from the +progenitor star. +Differences persist weeks after the estimated explo- +sion times. +While the other SNe IIb have developed +broad Hα and Hβ emission features, these same lines +are weaker in SN 2020bio. This could be partly caused +by He I λ 6678˚A absorption, which has an absorption +trough coincident with the Hα flux when blueshifted by +≈ 7500 km s−1. Another possibility is that the H emis- +sion from SN 2020bio is inherently weaker than in other +SNe IIb, which may be the case if the progenitor lost +more of its outer H envelope than the progenitors of the +other SNe IIb did. Weak H emission, combined with the + +The Double-peaked Type IIb SN 2020bio +7 +4000 +5000 +6000 +7000 +8000 +9000 +10000 +Rest-frame Wavelength (Å) +Normalized F + Constant +1d +4d +6d +16d +19d +25d +33d +52d +60d +77d +88d +201d +H +He I +Mg I +O III +Ca II +Figure 3. +The full spectral time series of SN 2020bio. +Phases with respect to the detection epoch are given above +each spectrum. Notable spectral features are identified with +dashed lines. +The first spectrum is the publicly-available +classification spectrum retrieved from the Transient Name +Server. +observed CSM features, point to a scenario in which the +progenitor of SN 2020bio underwent enhanced mass-loss, +shedding almost all of its outer H layer before explod- +ing. If this is the case, such a progenitor scenario to +SN 2020bio is unique among other well-studied SNe IIb. +4. LIGHT-CURVE MODELING AND +PROGENITOR INFERENCE +4.1. Shock-cooling Model Descriptions +A variety of analytical and numerical models of SCE +have been developed in recent years. Here we consider +6400 +6500 +6600 +6700 +Rest-frame Wavelength (Å) +Normalized F + Constant +1d +4d +6d +16d +Figure 4. Gaussian fits to the Hα emission line in the early- +time spectra of SN 2020bio. Phases relative to discovery are +given above each spectrum. The dashed line shows the rest- +frame Hα wavelength. The FWHMs decrease over time, evi- +dence that circumstellar interaction contributes to the emis- +sion profile. +3 analytical models that are commonly used to fit the +early-time emission of core-collapse SNe. The P15 model +extends the formalism of Nakar & Piro (2014) to repro- +duce the full shock-cooling peak. +It assumes a lower +mass extended envelope without assuming its specific +density structure. On the other hand, SW17 calibrates +to the numerical models of Rabinak & Waxman (2011) +and assumes specific polytropic indices for the extended +envelope. The methodology used to fit these models to +the data and derive resulting blackbody properties are +presented in Arcavi et al. (2017). +More recently, Piro et al. (2021) developed another +analytical model to better reproduce the early SCE ob- +served in a variety of transients (e.g., Arcavi et al. 2017; +Yao et al. 2020). They assume a two-zone extended en- +velope in homologous expansion and calculate the emis- +sion from this shocked material. This method begins by +assuming extended material in homologous expansion +separated into two regions—an outer density profile de- +scribed by ρ ∝ r−n, where n ≈ 10, and an inner region + +8 +Pellegrino et al. +Table 3. SCE Model Parameters +Model +Renv (R⊙) +Menv (10−2 M⊙) +va (104 km s−1) +t0 (days) +χ2 / d.o.f. +P15 +510+30 +−30 +1.14+0.02 +−0.02 +1.67+0.02 +−0.01 +0.67+0.02 +−0.02 +21.6 +P21 +1700+85 +−95 +1.60+0.03 +−0.02 +1.36+0.01 +−0.02 +0.98+0.01 +−0.01 +21.1 +SW17 (n=3/2) +160+12 +−10 +47.12+0.96 +−0.92 +1.69+0.04 +−0.04 +0.26+0.04 +−0.04 +8.7 +SW17 (n=3) +220+19 +−15 +322.60+6.10 +−6.20 +1.60+0.04 +−0.04 +0.25+0.04 +−0.04 +8.7 +aThe characteristic velocity for P15 and P21 and the shock velocity for SW17. +1.5d +2d +3d +2d +2d +17d +16d +17d +13d +17d +SN 2020bio +SN 1993J +SN 2013df +SN 2016gkg +SN 2011dh +4000 +5000 +6000 +7000 +8000 +9000 +Rest-frame Wavelength (Å) +26d +25d +25d +21d +25d +Normalized F + Constant +Figure 5. +Spectra of SN 2020bio compared with spectra +of other SNe IIb at similar phases. Phases with respect to +the estimated explosion time are given above each spectrum +and notable spectral features are identified with red (H) and +blue (He) vertical lines at their rest-frame wavelengths. The +spectra of SN 2016gkg are unpublished spectra obtained by +LCO while the other comparison spectra were retrieved from +WiseRep (Yaron & Gal-Yam 2012). +with ρ ∝ r−d, where δ ≈ 1.1. Assuming a transitional +velocity vt between the inner and outer regions of the +extended material, the time for the diffusion front to +reach this transition is given by +td = +� 3κKMe +(n − 1)vtc +�1/2 +(1) +where K = (n−3)(3−δ) +4π(n−δ) , κ is the optical opacity, and Me +is the mass of the extended material. The luminosity +from the cooling of the extended material is then defined +piecewise for times before and after this diffusion time: +L(t) ≈ π(n − 1) +3(n − 5) +cRev2 +t +κ +�td +t +�4/(n−2) +, t ≤ td +(2) +and +L(t) ≈ π(n − 1) +3(n − 5) +cRev2 +t +κ +exp +� +−1 +2 +�t2 +t2 +d +− 1 +�� +, t ≥ td +(3) +To fit the photometry in each band, we assume that +the material radiates as a blackbody at some photo- +spheric radius rph. The photosphere reaches the transi- +tion between the two regions at a time +tph = +� 3κKMe +2(n − 1)v2 +t +�1/2 +(4) +and the time evolution of the photospheric radius is +given relative to this characteristic time: +rph(t) = +�tph +t +�2/(n−1) +vtt, t ≤ tph +(5) +and +rph(t) = +� δ − 1 +n − 1 +� t2 +t2 +ph +− 1 +� ++ 1 +�−1/(δ−1) +vtt, t ≥ tph (6) +In addition, we attempt to fit the analytical models +of Shussman et al. (2016), which are calibrated to nu- +merical simulations from shock breakout to recombina- +tion. However, these model fits are unable to reproduce +the rapidly-declining shock-cooling emission in all fil- +ters during the week after explosion. It is possible this + +The Double-peaked Type IIb SN 2020bio +9 +0 +2 +4 +6 +8 +10 +Days from Discovery +10 +12 +14 +16 +18 +20 +22 +Apparent Magnitude + Offset +UVW2 - 3.5 +UVM2 - 3 +UVW1 - 2 +U - 1 +B +g + 1 +V + 2 +r + 3 +i + 3.5 +0 +2 +4 +6 +8 +10 +Days from Discovery +10 +12 +14 +16 +18 +20 +22 +SW17 +(n=3/2) +SW17 +(n=3) +Figure 6. Shock-cooling fits to the early-time photometry of SN 2020bio using the models of (left) P15 and P21; and (right) +SW17, assuming a constant optical opacity appropriate for solar-composition material. Photometry in each band has been offset +for clarity. Itagaki discovery photometry has been included in the V -band fits. +shortcoming is due to an unphysical application of the +model—which is calibrated to numerical simulations of +red supergiants—to the early light curve of SN 2020bio, +which likely had a different progenitor structure. De- +tailed comparisons between numerical models of SNe IIb +and the Shussman et al. (2016) models are beyond the +scope of this work. +4.2. Best-fit Analytic Models +We fit each model to the early-time photometry of +SN 2020bio. For the SW17 model we consider two poly- +tropic indices (n = 3/2 and n = 3), appropriate for con- +vective and radiative envelopes, respectively. Only data +taken up to 3.5 days after discovery are fit, as this is the +time when SCE dominates the luminosity over radioac- +tive decay (see Section 4.3 for a quantitative treatment +of the 56Ni light curve). Additionally, we ensure that the +phases we fit fall within the validity range of each model. +In each case we fit for the progenitor extended envelope +radius, Renv, the envelope mass, Menv, either the char- +acteristic velocity or the shock velocity v of the outer +material, and the offset time since explosion t0. We use +the emcee package (Foreman-Mackey et al. 2013) to per- +form Markov Chain Monte Carlo fitting of each model, +initializing 100 walkers with 1000 burn-in steps and run- +ning for an additional 1000 steps after burn-in. For each +step, the total luminosity is computed using the analyt- +ical model formalism, and the luminosity within each +filter is compared to the observed photometry assuming +a blackbody spectral energy distribution (SED). We fit +each model assuming an optical opacity κ = 0.34 cm2 +g−1, consistent with solar composition material. +The best-fit models to the multi-band SCE light +curves are shown in Figure 6, and best-fit parameters are +given in Table 3 with corner plots shown in Appendix +A. The Itagaki discovery data that capture the rise are +calibrated to the V -band. We find that all the mod- +els fit the early-time data well, reproducing the rapid +rise, luminous peak, and subsequent decline in all filters. +Quantitatively the SW17 model for convective envelopes +(n = 3/2) has the lowest reduced χ2 value, indicating +the model most closely matches the observations. On +the other hand, the best-fit envelope mass for the SW17 +model with a radiative (n = 3) envelope is larger than +the total ejecta mass, estimated in Section 4.3. There- +fore, we do not consider this model representative of the +progenitor of SN 2020bio. +Based on the unusual properties of SN 2020bio com- +pared to other SNe IIb, including its weak H spectral +features and faint secondary light-curve peak, we test +whether a lower-opacity envelope better reproduces the +observed SCE. This could be the case if the progeni- +tor star was almost completely stripped of its outer H +envelope. We perform the same fitting routine but fix +the opacity κ = 0.20 cm2 g−1 for H-poor material. We +find no differences in goodness of fits for each model +between the two chosen opacities—both the H-rich and + +10 +Pellegrino et al. +0 +20 +40 +60 +80 +100 +Days Since Discovery +1040 +1041 +1042 +Pseudo-Bolometric Luminosity (erg s−1) +E = 0.9 × 1051 erg +MNi = 0.015 M⊙ +MNi = 0.017 M⊙ +MNi = 0.019 M⊙ +MNi = 0.020 M⊙ +SN2020bio +Figure 7. Numerical MESA and STELLA model light curves of +SN 2020bio for varying MNi. Both the secondary light-curve +peak and late-time light-curve slope are best reproduced with +≈ 0.02 M⊙ of 56Ni synthesized in the explosion. +H-poor envelopes produce similarly good fits. However, +there are differences in the fitted parameters between +the best-fit models. +In the H-rich case, the envelope +radii and masses from the best-fit SW17 model are con- +sistent with those estimated for other SNe IIb (i.e. radii +of ≈ 1×1013 cm and masses of 10−3–10−2 M⊙). In the +H-poor case, however, the radii are smaller (≈ 3×1012 +cm) and the envelope masses are larger (≈ 10−1 M⊙). +These values are more consistent with those estimated +for Type Ib and Ca-rich transients with observed SCE +(e.g., Yao et al. 2020; Jacobson-Gal´an et al. 2022). +4.3. Bolometric Luminosities and Numerical Modeling +SCE dominates the total luminosity only for several +days after explosion. The rest of the light curve is pow- +ered by the radioactive decay of 56Ni and its children +isotopes. Using our multi-band coverage of SN 2020bio +for ≈ 160 days after explosion, we construct a pseudo- +bolometric light curve to fit for the amount of 56Ni pro- +duced in the explosion. For epochs with observations +in more than 3 filters, we extrapolate the SED out to +the blue and red edges of the U - and i-band filters, +respectively, using a univariate spline. We choose to ex- +trapolate the (extinction-corrected) photometry rather +than fit a blackbody SED because the spectra are not +representative of a blackbody throughout the object’s +evolution. +To infer the properties of the pre-explosion progeni- +tor as well as the explosion itself, we compare numerical +MESA (Paxton et al. 2011, 2013, 2015, 2018, 2019) and +STELLA (Blinnikov et al. 1998, 2000, 2006) model ex- +plosions to our pseudo-bolometric light curve. We begin +with a MESA progenitor with MZAMS = 15 M⊙ and evolve +it to a final mass of 4.8 M⊙. At explosion the progenitor +has a H-rich envelope radius of 280 R⊙ and mass of 0.10 +M⊙, in agreement with values we find from our best- +fit SCE models. The explosion energy and ejecta mass +are fixed at 0.9 × 1051 erg and 2.9 M⊙, respectively, +and the mass of 56Ni (MNi) is varied between 0.015 and +0.020 M⊙. These explosion models are then run through +STELLA in order to reproduce the bolometric luminosity +evolution. For more information, see Hiramatsu et al. +(2021). +The resulting model light curves are shown in Fig- +ure 7, compared with the pseudo-bolometric light curve +of SN 2020bio. We find decent qualitative agreement be- +tween the numerical models and the observed light-curve +evolution, particularly at later times. +The secondary +light-curve peak and late-time light-curve slope are well +reproduced by an explosion which synthesizes ≈ 0.02 +M⊙ of 56Ni. +The secondary light-curve peak may be +overproduced, but the exact peak luminosity and time +of peak are uncertain given the gap in our observational +coverage. +Interestingly, however, the peak luminosity of the SCE +is not reproduced by these models. It may be that the +treatment of the SN shock and the subsequent cool- +ing of the outer envelope is too complex to fully sim- +ulate within these models. +On the other hand, it is +possible that an additional powering mechanism con- +tributes to the early-time evolution. +To test this, we +explore how the addition of different mass-loss rates +and timescales to the models affects the early-time light +curve through short-lived circumstellar interaction. To +the best-fit MESA model we attach a wind density profile +ρCSM(r) = ˙Mwind/4πr2vwind, where vwind = 10 km s−1. +These CSM models are shown in Figure 8. We find that +the best-fit models have a confined CSM with masses of +1 × 10−3 – 1 × 10−2 M⊙ lost by the progenitor within +the last several months before explosion. This hints that +circumstellar interaction may contribute to the rapidly- +fading early-time emission of SN 2020bio and possibly +other SNe IIb. If this is the case, then the information +estimated through SCE model fits may not be truly rep- +resentative of the true nature of their progenitors. +The values inferred from this numerical modelling, +particularly the 56Ni mass, are on the low end of the +distribution of values estimated for other well-studied +SNe IIb. SNe IIb with double-peaked light curves typi- +cally display secondary radioactive decay-powered peaks +equally or more luminous than the peak of the SCE, +implying a greater amount of 56Ni synthesized. Stud- + +The Double-peaked Type IIb SN 2020bio +11 +0 +3 +6 +9 +12 +Days Since Discovery +1041 +1042 +Pseudo-Bolometric Luminosity (erg s−1) +MNi = 0.019 M⊙ +vwind = 10 km s−1 +˙Mwind = 0.1 M⊙ yr−1 +twind = 0.1 yr +˙Mwind = 0.1 M⊙ yr−1 +twind = 0.01 yr +˙Mwind = 0.1 M⊙ yr−1 +twind = 1.0 yr +˙Mwind = 0.01 M⊙ yr−1 +twind = 0.1 yr +˙Mwind = 1.0 M⊙ yr−1 +twind = 0.1 yr +˙Mwind = 1.0 M⊙ yr−1 +twind = 0.01 yr +SN2020bio +Figure +8. +Numerical MESA and STELLA circumstellar +interaction-powered model light curves of SN 2020bio at early +times. Different color curves correspond to models with vary- +ing mass-loss rates and timescales. The early-time emission +excess is best reproduced with 0.001-0.01 M⊙ of CSM. +ies using samples of these objects have found average +56Ni masses of ≈ 0.10 – 0.15 M⊙ and average ejecta +masses of ≈ 2.2 – 4.5 M⊙ (Lyman et al. 2016; Pren- +tice et al. 2016; Taddia et al. 2018), in better agreement +with ejecta parameters of other stripped-envelope and +H-rich core-collapse SNe. +However, rare cases of un- +derluminous SNe IIb with low inferred MNi have been +discovered (e.g., Nakaoka et al. 2019; Maeda et al. 2023). +These objects have light curves that appear transitional +between standard SNe II-P and SNe IIb, which differ +from the observed photometric evolution of SN 2020bio. +On the other hand, in the case of SN 2018ivc, both a +low 56Ni mass (MNi ≤ 0.015M⊙) and progenitor mass +(MZAMS ≲ 12M⊙) are inferred (Maeda et al. 2023). It is +possible that other SNe IIb with little synthesized 56Ni +may be undercounted due to their rapidly-fading or un- +derluminous light curves. Maeda et al. (2023) also con- +cluded that the light curve of SN 2018ivc was powered at +least in part by circumstellar interaction. Sustained cir- +cumstellar interaction has been inferred for other SNe +IIb, either through late-time spectral features (Maeda +et al. 2015; Fremling et al. 2019) or through X-ray and +radio observations (Fransson et al. 1996). +It may be +that the mechanism that produced the confined CSM +inferred from our numerical models of SN 2020bio, and +possibly that seen in the case of SN 2018ivc, points to +more extreme mass-loss than found in other SNe IIb. +4.4. Comparison to Nebula Spectra Models +A trend between an increasing amount of synthesized +O and increasing core-collapse SN progenitor mass has +been extensively studied (e.g., Woosley & Heger 2007). +Jerkstrand et al. (2015) use this relationship to calibrate +the [O I] λλ 6300,6364 luminosity, normalized by the +radioactive decay luminosity at the same phase, with +numerical models of SNe IIb progenitors (see Eq. +1 +of Jerkstrand et al. 2015). The authors consider mod- +els with zero-age main-sequence masses between 12 M⊙ +and 17 M⊙. Comparing the observed normalized [O I] +luminosity for a handful of SNe IIb, such as SN 1993J, +SN 2008ax, and SN 2011dh, to these models allows for a +direct estimate of their progenitor masses—all of which +fall in the range of masses modeled. +Here we reproduce this analysis using a nebular spec- +trum of SN 2020bio, obtained 201 days after the esti- +mated explosion, shown in Figure 3. We estimate the +luminosity from the [O I] λλ 6300,6364 emission doublet +in the same way as Jerkstrand et al. (2015)—assuming +the width of the feature to be 5000 km s−1, we estimate +the continuum by finding the minimum flux redward +and blueward of this width and calculate the luminosity +within the continuum-subtracted feature. We normal- +ize this luminosity using the luminosity of 56Ni decay, +assuming the best-fit MNi from Section 4.3. +The +normalized +luminosity +at +201 +days +is +Lnorm(t=201) = 9×10−4 ± 2×10−5. This value is lower +than any of the numerical models analyzed by Jerk- +strand et al. (2015), implying a progenitor mass ≤ 12 +M⊙. +A low progenitor mass for SN 2020bio can also +be inferred from the ratio of the [Ca II] λλ 7311, 7324 +to [O I] λλ 6300, 6364 fluxes. +A higher ratio implies +a lower-mass progenitor, with SNe IIb from literature +having values ≲ 1 throughout their nebular phases (e.g., +Terreran et al. 2019; Hiramatsu et al. 2021). Using the +same procedure as above, we estimate a [Ca II] to [O I] +ratio of 1.34 ± 0.03—again pointing to a low-mass +progenitor star. +Based on its low synthesized 56Ni mass and nebular +spectral features, we conclude that SN 2020bio was likely +the core-collapse of a star with a lower mass than the +progenitors of most other SNe IIb. +5. DISCUSSION AND CONCLUSIONS +We have presented rapid multi-band photometric and +spectroscopic observations of SN 2020bio, a Type IIb SN +with luminous and rapidly-evolving SCE, beginning ≤ 1 +day after explosion. Compared with other well-observed +SNe IIb, SN 2020bio has the bluest colors at early times +as well as unique spectral features with signatures of +pre-existing CSM. Fitting analytical models of SCE to + +12 +Pellegrino et al. +the early-time light curve gives progenitor radii on the +order of 100 R⊙ – 500 R⊙ and envelope masses of 0.01 +M⊙ – 0.5 M⊙ for our best-fit models, which are slightly +greater than values derived for other SNe IIb progenitors +using the same methods (e.g., SN 2016gkg; Arcavi et al. +2017). The weak secondary peak powered by radioac- +tive decay is evidence of relatively little 56Ni synthe- +sized, MNi ≈ 0.02 M⊙, which is in tension with average +MNi estimates from samples of other SNe IIb. Numer- +ical modeling of the progenitor explosion within con- +fined circumstellar material is consistent with the ob- +served light curve, showing that circumstellar interac- +tion is likely needed to reproduce the complete pseudo- +bolometric light curve. Finally, comparing the nebular +spectra to numerical models implies a progenitor mass +≤ 12 M⊙. +It is difficult to explain all these peculiar features of +SN 2020bio in one consistent model. The combination of +its blue colors, early-time spectral features, and numer- +ical modeling points to interaction with confined H-rich +CSM that was stripped from the progenitor’s outer enve- +lope during the months prior to explosion. The best-fit +progenitor parameters, particularly the large envelope +radius and low envelope mass, may suggest an inflated +progenitor undergoing enhanced mass-loss immediately +before exploding. However, the very low 56Ni and ejecta +masses inferred from the later-time light curve, as well as +the nebular spectroscopy, point to a lower-mass progeni- +tor. It is possible that SN 2020bio was the collapse of an +unusually low-mass core within a dense CSM produced +from its lost H layers. Such extensive mass-loss likely re- +quires interaction with a binary companion, as inferred +for other SNe IIb (e.g., Maund et al. 2004; Benvenuto +et al. 2013). Interaction between the SN ejecta and this +CSM explains the blue colors and narrow H spectral +features at early times while the small 56Ni mass and +nebular spectrum indicate a low zero-age main-sequence +mass. +This interaction can lead to an over-estimated +progenitor radius—if the CSM was near enough to the +progenitor, we may have observed the shock-cooling of +this extended CSM instead of the outer envelope of the +progenitor. +In the future, more detailed models and +multi-wavelength observations, particularly in the radio +and X-rays, will be needed to infer SNe IIb progenitor +mass-loss rates and CSM masses. +Given the weak H spectral features when compared to +spectra of other SNe IIb, SN 2020bio may be an inter- +mediary object between the Type IIb and Ib subclasses, +representing a progenitor that was recently stripped al- +most entirely of its H-rich envelope. +Transitional ob- +jects between SNe IIb and SNe Ib have been observed +(Prentice & Mazzali 2017) and can be explained by dif- +ferent amounts of H remaining in the outer envelope at +the time of explosion. More difficult to explain are the +small 56Ni and ejecta masses, which are lower than those +measured for both SNe IIb and SNe Ib (e.g., Taddia et +al. 2018). Some objects that exist in the literature with +both low ejecta and 56Ni masses and observed SCE are +peculiar SNe Ib as well as Ca-rich transients. However, +it is difficult to reconcile the photospheric-phase spec- +tra of SN 2020bio, which are most similar to those of +other SNe IIb, with the spectra of these objects, which +are often used to argue for a degenerate or ultra-stripped +progenitor (Yao et al. 2020; Jacobson-Gal´an et al. 2022). +Instead, it is more likely that SN 2020bio had a massive +star progenitor more similar to the progenitors of other +SNe IIb based on their similar spectral features. +This study contributes to the overall diversity in the +progenitors of SNe IIb. More systematic studies of SNe +with observed SCE will be needed to search for simi- +larities and differences in their progenitor systems. In +particular, this work shows the importance of rapid, +multi-wavelength follow-up of these objects. It is par- +ticularly important to better understand the number of +SNe IIb with weak secondary light-curve peaks, such as +SN 2020bio. +These objects may have later-time (≥ 5 +days) luminosity below the detection threshold of cur- +rent all-sky surveys as well as rapid early-time emis- +sion which evolves too quickly to be extensively followed. +Therefore we may be under-counting the rates of core- +collapse, stripped-envelope SNe with low 56Ni and ejecta +masses. A better understanding of their progenitors will +be important for exploring the low-mass end of core- +collapse SNe. +This work made use of data from the Las Cumbres Ob- +servatory network. +The LCO group is supported by +AST-1911151 and AST-1911225 and NASA Swift grant +80NSSC19k1639. I.A. is a CIFAR Azrieli Global Scholar +in the Gravity and the Extreme Universe Program and +acknowledges support from that program, from the Eu- +ropean Research Council (ERC) under the European +Union’s Horizon 2020 research and innovation program +(grant agreement No. 852097), from the Israel Science +Foundation (grant No. 2752/19), from the United States +- Israel Binational Science Foundation (BSF), and from +the Israeli Council for Higher Education Alon Fellow- +ship. +Software: +Astropy +(Astropy +Collaboration +et +al. +2018), +emcee +(Foreman-Mackey +et +al. +2013), +lcogtsnpipe (Valenti et al. 2016), Matplotlib (Hunter +2007), MESA (Paxton et al. 2011, 2013, 2015, 2018, 2019), + +The Double-peaked Type IIb SN 2020bio +13 +Numpy (Harris et al. 2020), STELLA (Blinnikov et al. +1998, 2000, 2006) +REFERENCES +Aldering, G., Humphreys, R. M., & Richmond, M. 1994, +AJ, 107, 662. doi:10.1086/116886 +Anderson, J. 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Corner plots showing the fitted parameter distributions for the P15 model. + +16 +Pellegrino et al. +1.50 +1.56 +1.62 +1.68 +1.74 +Menv (10−2 M ⊙ ) +1.300 +1.325 +1.350 +1.375 +1.400 +v (104 km s−1) +1400 +1600 +1800 +2000 +Renv (R ⊙ ) +0.9775 +0.9790 +0.9805 +0.9820 +t0 (days) +1.50 +1.56 +1.62 +1.68 +1.74 +Menv (10−2 M ⊙ ) +1.300 +1.325 +1.350 +1.375 +1.400 +v (104 km s−1) +0.9775 +0.9790 +0.9805 +0.9820 +t0 (days) +Figure A2. Same as Figure A1, but for the P21 model. + +The Double-peaked Type IIb SN 2020bio +17 +44 +46 +48 +50 +Menv (10−2 M ⊙ ) +1.5 +1.6 +1.7 +1.8 +1.9 +v (104 km s−1) +125 +150 +175 +200 +Renv (R ⊙ ) +0.08 +0.16 +0.24 +0.32 +0.40 +t0 (days) +44 +46 +48 +50 +Menv (10−2 M ⊙ ) +1.5 +1.6 +1.7 +1.8 +1.9 +v (104 km s−1) +0.08 +0.16 +0.24 +0.32 +0.40 +t0 (days) +Figure A3. Same as Figure A1, but for the SW17 (n=3/2) model. + +18 +Pellegrino et al. +300 +315 +330 +345 +Menv (10−2 M ⊙ ) +1.4 +1.5 +1.6 +1.7 +v (104 km s−1) +160 +200 +240 +280 +Renv (R ⊙ ) +0.1 +0.2 +0.3 +0.4 +t0 (days) +300 +315 +330 +345 +Menv (10−2 M ⊙ ) +1.4 +1.5 +1.6 +1.7 +v (104 km s−1) +0.1 +0.2 +0.3 +0.4 +t0 (days) +Figure A4. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Vicolo dell’Osservatorio 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' I-35122 Padova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Italy 11Institute of Space Sciences (ICE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' CSIC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Campus UAB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Carrer de Can Magrans s/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 08193 Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Spain 12Itagaki Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Yamagata,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Yamagata 990-2492,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Japan 13Kaneda Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Sapporo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Hokkaido 005-0862,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Japan 14Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' University of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Charlottesville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' VA 22904 15Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' New York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' NY 10003,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' USA Submitted to ApJ ABSTRACT We present photometric and spectroscopic observations of SN 2020bio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' a double-peaked Type IIb supernova (SN) discovered within a day of explosion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' primarily obtained by Las Cumbres Observatory and Swift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' SN 2020bio displays a rapid and long-lasting initial decline throughout the first week of its light curve, similar to other well-studied Type IIb SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This early-time emission is thought to originate from the cooling of the extended outer envelope of the progenitor star that is shock-heated by the SN explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We compare SN 2020bio to a sample of other double-peaked Type IIb SNe to investigate its progenitor properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Analytical model fits to the early-time emission give progenitor radius (≈ 100– 1500 R⊙) and H-rich envelope mass (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='01–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='5 M⊙) estimates that are consistent with other Type IIb SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' However, SN 2020bio displays several peculiarities, including: 1) weak H spectral features and narrow emission lines indicative of pre-existing circumstellar material;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2) an underluminous secondary light curve peak which implies a small amount of synthesized 56Ni (MNi ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='02 M⊙);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' and 3) low- luminosity nebular [O I] features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' These observations are more consistent with a lower-mass progenitor (MZAMS ≈ 12 M⊙) that was stripped of most of its H envelope before exploding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This study adds to the growing diversity in the observed properties of Type IIb SNe and their progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Keywords: Circumstellar matter(241) — Core-collapse supernovae(304) — Supernovae(1668) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' INTRODUCTION Corresponding author: Craig Pellegrino cpellegrino@lco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='global ∗ LSST Catalyst Fellow While the majority of stars with initial masses ≳ 8 M⊙ end their lives as H-rich core-collapse supernovae (SNe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Janka 2012), some massive stars lose their outer H and even He envelopes and explode as stripped-envelope SNe (SESNe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Filippenko 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Gal-Yam 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A small but growing number of SNe have been observed arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='04662v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='HE] 11 Jan 2023 2 Pellegrino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' with spectra that show similarities to both these classes (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Classified as Type IIb SNe (SNe IIb), their spectra have H features at early times that gradually give way to He features, indicating that their progenitors were partially stripped of their outer en- velopes before exploding (Woosley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' It is unclear what mechanisms are responsible for this mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Common hypotheses include stellar winds, binary interaction, or late-stage stellar instabilities (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Smith 2014, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Recent studies have shown that mass loss is common during the late stages of massive star evolution, as inferred from early-time observations of core-collapse SNe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Ofek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Bruch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Strotjohann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A signif- icant fraction of core-collapse SNe show signatures of pre-existing circumstellar material (CSM) in their early- time spectra, obtained days after their estimated explo- sion epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This CSM is the material shed by the pro- genitor star in the months to years before core-collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' As the SN shock breaks out of the expanding ejecta the resulting X-ray and ultraviolet (UV) flash may ionize the surrounding CSM, producing narrow spectral fea- tures as the CSM cools and recombines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Fassia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Yaron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Interaction between the SN ejecta and CSM can also influence the early-time light-curve evolution (Morozova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Some SNe IIb are observed to have double-peaked light curves, with rapidly-fading luminosities during the first several days after explosion before the radioactive decay of 56Ni synthesized during the explosion causes a re-brightening that lasts for several weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The early- time emission is thought to be the cooling of the ex- tended envelope of the progenitor star that is heated by the SN shock (Soderberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This shock- cooling emission (SCE) has only been extensively ob- served in a handful of cases, including SN 1993J (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Woosley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Richmond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1994), SN 2011dh (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Arcavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Ergon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2014), SN2013 df (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Morales-Garoffolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Van Dyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2014), SN 2016gkg (Arcavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2017), SN 2017jgh (Armstrong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2021), and ZTF18aalrxas (Fremling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2019), among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Most of these objects are nearby and had follow-up observations scheduled hours after explosion, which proved crucial to observing the rapidly-evolving SCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' These studies have found that SNe IIb are con- sistent with the explosions of stars with extended outer envelopes, with the duration of the SCE dependent on the extent of this envelope (Soderberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Numerical and analytical models of SCE can comple- ment pre-explosion imaging in determining the progen- itors of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Several models have been suc- cessful in reproducing the observed early-time evolution across all wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Piro (2015, hereafter P15) is one of the first to present a one-zone analytical descrip- tion of the cooling of an extended low-mass envelope shock-heated by the explosion of a compact massive core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Piro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2021, hereafter P21) extend this to a two-zone model in order to better capture the emis- sion from the outermost material in extended envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Sapir & Waxman (2017, hereafter SW17) calibrate ear- lier models by Rabinak & Waxman (2011)—that depend on the precise density structure of the outer material—to numerical simulations for several days after explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Comparing observed SCE to analytical and numeri- cal models is one of the only ways of directly measuring the radii and stellar structure of core-collapse progen- itors from SN observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This has been done for a handful of SNe IIb as well as SNe of other subtypes, including stripped-envelope Type Ib SNe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Modjaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2020), short-plateau Type II SNe (Hiramatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2021), and exotic Ca-rich transients (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Jacobson-Gal´an et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Analytical and numerical modeling of double-peaked SNe IIb generally yield large radii progenitors (≈ 100–500 R⊙) with low- mass (≈ 10−2–10−1 M⊙) extended envelopes (Piro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2021, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' These properties are usually in agreement with those of SNe IIb progeni- tors from pre-explosion Hubble Space Telescope images, which have revealed them to be supergiants (Aldering et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Maund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Van Dyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' In some cases, however, the progenitor radii estimated from SCE modeling are in tension with those measured from direct imaging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Arcavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Tartaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2017, in the case of SN 2016gkg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Potential bi- nary companions to the progenitor, which have been observed or inferred in a handful of cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Maund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Benvenuto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2013) can further compli- cate direct imaging estimates when the individual binary members are unresolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Here we present photometric and spectroscopic ob- servations of SN 2020bio, an SN IIb showing remark- ably strong SCE, obtained by Las Cumbres Observatory (LCO) through the Global Supernova Project (GSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' LCO extensively observed SN 2020bio from hours to ≈ 160 days after explosion, providing a detailed look into the full evolution of a double-peaked SN IIb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' In this work, we analyze its light curve evolution, spectral features, and fit analytic models to its full light-curve evolution to estimate the radius, mass, and structure of its progenitor star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We also compare its bolometric light curve and spectra to numerical models in order to infer its progenitor mass and the properties of its circumstel- lar environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The Double-peaked Type IIb SN 2020bio 3 This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' In Section 2 we describe the discovery and follow-up observations of SN 2020bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We present its full light curve and spec- tral time series in Section 3 and compare observations to analytical and numerical models in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Fi- nally, in Section 5 we discuss the potential progenitor properties of SN 2020bio given the presented evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' DISCOVERY AND DATA DESCRIPTION SN 2020bio was discovered by Koichi Itagaki on UT 2020 January 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='77 at the Itagaki Astronomical Obser- vatory at an unfiltered Vega magnitude of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Analy- sis of an image of the same field by the ATLAS survey on the previous night yields a nondetection at c-band magnitude 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Soon after discovery rapid photometric and spectroscopic follow-up observations were requested by the GSP through the Las Cumbres global network of telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The GSP also triggered its Swift Key Project (1518618: PI Howell) to obtain daily UV and optical photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A classification spectrum obtained on the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='0m Liverpool Telescope on 2020 January 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='19—ap- proximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='5 days after the first detection—shows a blue continuum superimposed with a narrow Hα emis- sion feature and a broad possible He I λ 5876˚A feature, consistent with a young core-collapse SN (Srivastav et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' SN 2020bio exploded at right ascension 13h55m37s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='69 and declination +40°28′39′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1 in the spiral galaxy NGC 5371 at redshift z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='008533 (Springob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The distance to NGC 5371 is uncertain due to its low redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We adopt the mean of several distances mea- sured using the method of Tully & Fisher (1977), which gives d = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='9 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1 Mpc (values from the NASA Ex- tragalactic Database1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Using the Schlafly & Finkbeiner (2011) dust map calibrations, we estimate a Galactic line-of-sight extinction to SN 2020bio EMW (B − V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='008 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Given the location of SN 2020bio with re- spect to its host galaxy, we also estimate host extinc- tion using the Na I D equivalent widths measured in a high-resolution spectrum of the SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' From the conver- sions presented in Poznanski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2012), we estimate Ehost(B −V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='068 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='038 mag for a total extinction E(B − V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='076 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='038 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The photometry of SN 2020bio presented throughout this work is corrected for this mean total extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' LCO photometric follow-up commenced less than a day after discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' UBgVri-band images were obtained by the Sinistro and Spectral cameras mounted on LCO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='0m and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='0m telescopes, respectively, located at Mc- 1 https://ned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='edu/ Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' UV and Optical Photometry JD Filter Magnitude Uncertainty Source 2458878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='27 Clear 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='15 Itagaki 2458878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='33 Clear 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='15 Itagaki 2458878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='39 Clear 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='15 Itagaki 2458879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='27 Clear 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='15 Itagaki 2458880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='26 Clear 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='15 Itagaki 2458881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='25 Clear 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='15 Itagaki 2458882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='18 Clear 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='15 Itagaki 2458883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='26 Clear 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='15 Itagaki 2458878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='85 UVW2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='04 Swift 2458879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='89 UVW2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='05 Swift This table will be made available in its entirety in machine- readable format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Donald Observatory, Teide Observatory, and Haleakala Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Data were reduced using lcogtsnpipe (Valenti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2016) which extracts point-spread func- tion magnitudes after calculating zero-points and color terms (Stetson 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' UBV -band photometry was cali- brated to Vega magnitudes using Landolt standard fields (Landolt 1992) while gri-band photometry was cali- brated to AB magnitudes (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2002) using Sloan Digital Sky Survey (SDSS) catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' As SN 2020bio ex- ploded coincident with its host galaxy, to remove host galaxy light we performed template subtraction using the HOTPANTS (Becker 2015) algorithm and template images obtained after the SN had faded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Unfiltered im- ages were obtained with the Itagaki Astronomical Ob- servatory (Okayama and Kochi, Japan) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='35 m tele- scopes + KAF-1001E (CCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Using our custom soft- ware, the photometry was extracted after host subtrac- tion and calibrated to the V-band magnitudes of 45 field stars from the Fourth US Naval Observatory CCD Astrograph Catalog (Zacharias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' UV and optical photometry were obtained with the Ultraviolet and Optical Telescope (UVOT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Roming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2005) on the Neil Gehrels Swift observatory (Gehrels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Swift data were reduced using a custom adaptation of the Swift Optical/Ultraviolet Supernova Archive (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2014) pipeline with the most re- cent calibration files and the zeropoints of Breeveld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Images from the final epoch, obtained after the SN had sufficiently faded, were used as templates to subtract the host galaxy light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' All Swift photometry is calibrated to Vega magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The entire UV and optical data sets from LCO, Itagaki, and Swift UVOT are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 4 Pellegrino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 58880 58900 58920 58940 58960 58980 59000 59020 59040 MJD 10 12 14 16 18 20 22 24 Apparent Magnitude + Offset 58877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='0 58880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='5 58884.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='0 10 12 14 16 18 20 UVW2 - 4 UVM2 - 3 UVW1 - 2 U - 1 B g + 1 V + 2 r + 3 i + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='5 Clear + 2 0 20 40 60 80 100 120 140 160 Days From Discovery 22 20 18 16 14 12 10 8 Absolute Magnitude + Offset Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The full extinction-corrected light curves of SN 2020bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Photometry in different filters have been offset for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Unfiltered photometry from the Itagaki Astronomical Observatory is included as clear points and calibrated to the V -band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The inset focuses on the rapidly-evolving shock-cooling emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' LCO spectra were obtained by the FLOYDS spectro- graph on the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='0m Faulkes Telescope North at Haleakala Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Spectra cover a wavelength range of 3500–10,000 ˚A at a resolution R ≈ 300-600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Data were reduced using the floydsspec pipeline2, a custom pipeline which performs cosmic ray removal, spectrum extraction, and wavelength and flux calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We also present one spectrum obtained by the B&C spectro- graph on the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3m Bok Telescope at Steward Observa- tory, two spectra obtained by the Blue Channel Spectro- graph on the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='5m MMT at the Fred Lawrence Whipple Observatory, and one spectrum obtained by the Optical System for Imaging and low-Intermediate-Resolution In- 2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='com/svalenti/FLOYDS pipeline/ tegrated Spectroscopy spectrograph on the 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='4m Gran Telescopio Canarias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Details of all these spectra are pre- sented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' PHOTOMETRIC AND SPECTRAL ANALYSIS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Light Curve and Color Evolution In Figure 1 we show the full LCO and Swift extinction- corrected light curve of SN 2020bio, from detection to ≈ 160 days after explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The discovery and subse- quent follow-up photometry from Itagaki are included as “Clear” data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The inset shows in greater de- tail the early-time evolution of the SCE, focusing on the first week after discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The most distinctive feature of the light curve is the luminous and rapidly-declining SCE at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The peak SCE luminosity exceeds that of the secondary peak ≈ 15 days later, but SCE only dominates the light curve during the first several The Double-peaked Type IIb SN 2020bio 5 2 0 2 UVW2 - B 2 0 2 UVW2 - V 2 0 2 UVM2 - B 2 0 2 UVM2 - V 0 1 2 3 4 5 6 7 8 Days Since Discovery 2 0 2 UVW1 - B 0 1 2 3 4 5 6 7 8 Days Since Discovery 2 0 2 UVW1 - V SW17 Model SN 2010jr SN 2011dh SN 2013df SN 2016gkg SN 2020bio Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Swift colors of SN 2020bio compared with those of other SNe IIb with early-time Swift observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We also include the best fit SW17 model from Section 4 for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' SN 2020bio was bluer at earlier phases than the other SNe IIb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Data for these comparison SNe were obtained from the following sources: Arcavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2011) (SN 2011dh);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Morales-Garoffolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2014) (SN 2013df);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Arcavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2017) (SN 2016gkg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' this work (SNe 2010jr and 2020bio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Over this time the light curve falls by ≈ 4 mag in the first week, making this phase difficult to observe without rapid multi-wavelength follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' After ≈ 4 days from discovery the slope of the light curve decline changes as the luminosity from 56Ni de- cay begins to dominate the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' After about a week the light curve re-brightens and reaches a sec- ondary maximum ≈ 15 days after discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' From this point the emission settles onto the radioactive decay tail, powered by 56Co decay, for the remainder of the obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The secondary peak and overall late-time light curve is relatively dim, peaking at M ≈ -14 mag in the V -band, hinting at a small amount of 56Ni synthesized in the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' In Figure 2 we compare the early-time Swift UV- optical colors of SN 2020bio to those of other SNe IIb with observed SCE in the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' All dates are given with respect to the time of discovery and corrected for extinc- tion according to the published values for each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' SN 2020bio has both the earliest observations relative to discovery and the bluest colors throughout its evolution compared to the other objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' While objects such as SN 2010jr and SN 2016gkg have more densely-sampled light curves, their observations began later and their col- ors evolved redward faster compared to SN 2020bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Of the 6 colors plotted, SN 2020bio is exceptionally blue in the UVM2-B and UVM2-V colors, particularly in the earliest epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We plot a representative SCE model color curve from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='2 in each panel for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' SN 2020bio is bluer than the model, which more accurately reproduces the color evolution of the other SNe IIb up to several days after the discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This may be evidence for another luminosity contribu- tion besides SCE, as we discuss in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Spectral Comparison Spectral coverage of SN 2020bio began fewer than 2 days after the first detection—approximately 3 days since the estimated explosion time (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='2)—and continued for 201 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We plot the full spectral series 6 Pellegrino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Log of Spectroscopic Observations Date of Observation Days Since Discovery Facility/Instrument Exposure Time (s) Wavelength Range (˚A) 2020-01-31 04:27:31 1 LT/SPRAT 1200 4000–7925 2020-02-03 14:32:18 4 LCO/FLOYDS-N 1800 3500–10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='000 2020-02-05 12:19:05 6 LCO/FLOYDS-N 1800 3500–10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='000 2020-02-15 09:35:59 16 Bok/B&C 600 3850–7500 2020-02-18 12:32:26 19 MMT/Blue Channel 300 5700–7000 2020-02-24 13:00:37 25 LCO/FLOYDS-N 1800 3500–10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='000 2020-03-03 10:49:44 33 LCO/FLOYDS-N 2700 3500–10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='000 2020-03-22 14:22:56 52 LCO/FLOYDS-N 3600 3500–10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='000 2020-03-30 14:20:34 60 LCO/FLOYDS-N 3600 3500–10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='000 2020-04-16 11:12:12 77 LCO/FLOYDS-N 3600 3500–10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='000 2020-04-27 12:09:24 88 LCO/FLOYDS-N 3600 3500–10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='000 2020-08-18 22:02:01 201 GTC/OSIRIS 1500 3600–7808 Note—All spectra will be made publicly-available on WiseRep (Yaron & Gal-Yam 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The earliest spectrum of SN 2020bio, re- ported to the Transient Name Server (Srivastav et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2020), shows a hot blue continuum superimposed with emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We identify narrow features of H and Mg I as well as a potential weak, broad feature of He I λ5876 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' These lines are consistent with flash-ionized features observed in other core-collapse SNe, which is evidence of nearby CSM lost by the progenitor star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' After about a week post explosion, absorption fea- tures begin to develop in the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We identify lines of He, O, and Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We also note persistent narrow H emission features that last for several weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' To deter- mine if these features are produced by interaction with CSM or by host galaxy emission, we fit the narrow Hα emission line with a Gaussian function to estimate its full-width at half-maximum (FWHM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The results are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' In our earliest spectrum we estimate a FWHM of the Hα line of 1500 km s−1, greater than the average widths of host galaxy emission lines, while our spectrum obtained roughly two weeks after discov- ery has a FWHM of ≈ 350 km s−1, more consistent with host-galaxy emission at this resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The latter value is also consistent with the FWHMs we measure for the nearby host-dominated [N II] λ 6583 line throughout the first several weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Therefore, we conclude that circum- stellar interaction likely contributes to the H emission during the first ≈ 2 weeks after explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' An absorption feature blueward of the rest-frame Hα line matches He I λ 6678˚A absorption blueshifted by ≈ 7500 km s−1, which is commonly noted to cause “flat- topped” Hα emission profiles in other SNe IIb (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Fil- ippenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' In general, the absorption features in the SN 2020bio spectra are shallower than those of the other SNe IIb, particularly SN 2011dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Interaction with CSM can produce absorption features that are weaker and shallower than expected, which has been noted in the spectra of SN 1993J and SN 2013df (Fremling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' To further investigate the differences between SN 2020bio and other SNe IIb, we plot comparison spec- tra just after explosion (top), after two weeks (middle), and three weeks (bottom) after explosion in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Among this sample, SN 2020bio is the only object to show narrow features indicative of pre-existing CSM at early times, despite similar phase coverage of the other SNe IIb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This likely reflects differences in their circum- stellar environments—if the narrow lines were formed from the expanding outer envelopes of the progenitor, they should be ubiquitous among SNe IIb at this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Instead, the presence of narrow H and Mg lines in the earliest spectrum of SN 2020bio more likely points to confined CSM formed from material stripped from the progenitor star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Differences persist weeks after the estimated explo- sion times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' While the other SNe IIb have developed broad Hα and Hβ emission features, these same lines are weaker in SN 2020bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This could be partly caused by He I λ 6678˚A absorption, which has an absorption trough coincident with the Hα flux when blueshifted by ≈ 7500 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Another possibility is that the H emis- sion from SN 2020bio is inherently weaker than in other SNe IIb, which may be the case if the progenitor lost more of its outer H envelope than the progenitors of the other SNe IIb did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Weak H emission, combined with the The Double-peaked Type IIb SN 2020bio 7 4000 5000 6000 7000 8000 9000 10000 Rest-frame Wavelength (Å) Normalized F + Constant 1d 4d 6d 16d 19d 25d 33d 52d 60d 77d 88d 201d H He I Mg I O III Ca II Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The full spectral time series of SN 2020bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Phases with respect to the detection epoch are given above each spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Notable spectral features are identified with dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The first spectrum is the publicly-available classification spectrum retrieved from the Transient Name Server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' observed CSM features, point to a scenario in which the progenitor of SN 2020bio underwent enhanced mass-loss, shedding almost all of its outer H layer before explod- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' If this is the case, such a progenitor scenario to SN 2020bio is unique among other well-studied SNe IIb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' LIGHT-CURVE MODELING AND PROGENITOR INFERENCE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Shock-cooling Model Descriptions A variety of analytical and numerical models of SCE have been developed in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Here we consider 6400 6500 6600 6700 Rest-frame Wavelength (Å) Normalized F + Constant 1d 4d 6d 16d Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Gaussian fits to the Hα emission line in the early- time spectra of SN 2020bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Phases relative to discovery are given above each spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The dashed line shows the rest- frame Hα wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The FWHMs decrease over time, evi- dence that circumstellar interaction contributes to the emis- sion profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 3 analytical models that are commonly used to fit the early-time emission of core-collapse SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The P15 model extends the formalism of Nakar & Piro (2014) to repro- duce the full shock-cooling peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' It assumes a lower mass extended envelope without assuming its specific density structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' On the other hand, SW17 calibrates to the numerical models of Rabinak & Waxman (2011) and assumes specific polytropic indices for the extended envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The methodology used to fit these models to the data and derive resulting blackbody properties are presented in Arcavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' More recently, Piro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2021) developed another analytical model to better reproduce the early SCE ob- served in a variety of transients (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Arcavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' They assume a two-zone extended en- velope in homologous expansion and calculate the emis- sion from this shocked material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This method begins by assuming extended material in homologous expansion separated into two regions—an outer density profile de- scribed by ρ ∝ r−n, where n ≈ 10, and an inner region 8 Pellegrino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' SCE Model Parameters Model Renv (R⊙) Menv (10−2 M⊙) va (104 km s−1) t0 (days) χ2 / d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' P15 510+30 −30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='14+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='02 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='6 P21 1700+85 −95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='60+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='01 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1 SW17 (n=3/2) 160+12 −10 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='69+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='7 SW17 (n=3) 220+19 −15 322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='60+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='10 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='60+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='25+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='7 aThe characteristic velocity for P15 and P21 and the shock velocity for SW17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='5d 2d 3d 2d 2d 17d 16d 17d 13d 17d SN 2020bio SN 1993J SN 2013df SN 2016gkg SN 2011dh 4000 5000 6000 7000 8000 9000 Rest-frame Wavelength (Å) 26d 25d 25d 21d 25d Normalized F + Constant Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Spectra of SN 2020bio compared with spectra of other SNe IIb at similar phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Phases with respect to the estimated explosion time are given above each spectrum and notable spectral features are identified with red (H) and blue (He) vertical lines at their rest-frame wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The spectra of SN 2016gkg are unpublished spectra obtained by LCO while the other comparison spectra were retrieved from WiseRep (Yaron & Gal-Yam 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' with ρ ∝ r−d, where δ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Assuming a transitional velocity vt between the inner and outer regions of the extended material, the time for the diffusion front to reach this transition is given by td = � 3κKMe (n − 1)vtc �1/2 (1) where K = (n−3)(3−δ) 4π(n−δ) , κ is the optical opacity, and Me is the mass of the extended material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The luminosity from the cooling of the extended material is then defined piecewise for times before and after this diffusion time: L(t) ≈ π(n − 1) 3(n − 5) cRev2 t κ �td t �4/(n−2) , t ≤ td (2) and L(t) ≈ π(n − 1) 3(n − 5) cRev2 t κ exp � −1 2 �t2 t2 d − 1 �� , t ≥ td (3) To fit the photometry in each band, we assume that the material radiates as a blackbody at some photo- spheric radius rph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The photosphere reaches the transi- tion between the two regions at a time tph = � 3κKMe 2(n − 1)v2 t �1/2 (4) and the time evolution of the photospheric radius is given relative to this characteristic time: rph(t) = �tph t �2/(n−1) vtt, t ≤ tph (5) and rph(t) = � δ − 1 n − 1 � t2 t2 ph − 1 � + 1 �−1/(δ−1) vtt, t ≥ tph (6) In addition, we attempt to fit the analytical models of Shussman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2016), which are calibrated to nu- merical simulations from shock breakout to recombina- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' However, these model fits are unable to reproduce the rapidly-declining shock-cooling emission in all fil- ters during the week after explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' It is possible this The Double-peaked Type IIb SN 2020bio 9 0 2 4 6 8 10 Days from Discovery 10 12 14 16 18 20 22 Apparent Magnitude + Offset UVW2 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='5 UVM2 - 3 UVW1 - 2 U - 1 B g + 1 V + 2 r + 3 i + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='5 0 2 4 6 8 10 Days from Discovery 10 12 14 16 18 20 22 SW17 (n=3/2) SW17 (n=3) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Shock-cooling fits to the early-time photometry of SN 2020bio using the models of (left) P15 and P21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' and (right) SW17, assuming a constant optical opacity appropriate for solar-composition material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Photometry in each band has been offset for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Itagaki discovery photometry has been included in the V -band fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' shortcoming is due to an unphysical application of the model—which is calibrated to numerical simulations of red supergiants—to the early light curve of SN 2020bio, which likely had a different progenitor structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' De- tailed comparisons between numerical models of SNe IIb and the Shussman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2016) models are beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Best-fit Analytic Models We fit each model to the early-time photometry of SN 2020bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' For the SW17 model we consider two poly- tropic indices (n = 3/2 and n = 3), appropriate for con- vective and radiative envelopes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Only data taken up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='5 days after discovery are fit, as this is the time when SCE dominates the luminosity over radioac- tive decay (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3 for a quantitative treatment of the 56Ni light curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Additionally, we ensure that the phases we fit fall within the validity range of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' In each case we fit for the progenitor extended envelope radius, Renv, the envelope mass, Menv, either the char- acteristic velocity or the shock velocity v of the outer material, and the offset time since explosion t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We use the emcee package (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2013) to per- form Markov Chain Monte Carlo fitting of each model, initializing 100 walkers with 1000 burn-in steps and run- ning for an additional 1000 steps after burn-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' For each step, the total luminosity is computed using the analyt- ical model formalism, and the luminosity within each filter is compared to the observed photometry assuming a blackbody spectral energy distribution (SED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We fit each model assuming an optical opacity κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='34 cm2 g−1, consistent with solar composition material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The best-fit models to the multi-band SCE light curves are shown in Figure 6, and best-fit parameters are given in Table 3 with corner plots shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The Itagaki discovery data that capture the rise are calibrated to the V -band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We find that all the mod- els fit the early-time data well, reproducing the rapid rise, luminous peak, and subsequent decline in all filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Quantitatively the SW17 model for convective envelopes (n = 3/2) has the lowest reduced χ2 value, indicating the model most closely matches the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' On the other hand, the best-fit envelope mass for the SW17 model with a radiative (n = 3) envelope is larger than the total ejecta mass, estimated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' There- fore, we do not consider this model representative of the progenitor of SN 2020bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Based on the unusual properties of SN 2020bio com- pared to other SNe IIb, including its weak H spectral features and faint secondary light-curve peak, we test whether a lower-opacity envelope better reproduces the observed SCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This could be the case if the progeni- tor star was almost completely stripped of its outer H envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We perform the same fitting routine but fix the opacity κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='20 cm2 g−1 for H-poor material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We find no differences in goodness of fits for each model between the two chosen opacities—both the H-rich and 10 Pellegrino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 0 20 40 60 80 100 Days Since Discovery 1040 1041 1042 Pseudo-Bolometric Luminosity (erg s−1) E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='9 × 1051 erg MNi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='015 M⊙ MNi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='017 M⊙ MNi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='019 M⊙ MNi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='020 M⊙ SN2020bio Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Numerical MESA and STELLA model light curves of SN 2020bio for varying MNi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Both the secondary light-curve peak and late-time light-curve slope are best reproduced with ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='02 M⊙ of 56Ni synthesized in the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' H-poor envelopes produce similarly good fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' However, there are differences in the fitted parameters between the best-fit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' In the H-rich case, the envelope radii and masses from the best-fit SW17 model are con- sistent with those estimated for other SNe IIb (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' radii of ≈ 1×1013 cm and masses of 10−3–10−2 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' In the H-poor case, however, the radii are smaller (≈ 3×1012 cm) and the envelope masses are larger (≈ 10−1 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' These values are more consistent with those estimated for Type Ib and Ca-rich transients with observed SCE (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Jacobson-Gal´an et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Bolometric Luminosities and Numerical Modeling SCE dominates the total luminosity only for several days after explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The rest of the light curve is pow- ered by the radioactive decay of 56Ni and its children isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Using our multi-band coverage of SN 2020bio for ≈ 160 days after explosion, we construct a pseudo- bolometric light curve to fit for the amount of 56Ni pro- duced in the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' For epochs with observations in more than 3 filters, we extrapolate the SED out to the blue and red edges of the U - and i-band filters, respectively, using a univariate spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We choose to ex- trapolate the (extinction-corrected) photometry rather than fit a blackbody SED because the spectra are not representative of a blackbody throughout the object’s evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' To infer the properties of the pre-explosion progeni- tor as well as the explosion itself, we compare numerical MESA (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2011, 2013, 2015, 2018, 2019) and STELLA (Blinnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1998, 2000, 2006) model ex- plosions to our pseudo-bolometric light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We begin with a MESA progenitor with MZAMS = 15 M⊙ and evolve it to a final mass of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='8 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' At explosion the progenitor has a H-rich envelope radius of 280 R⊙ and mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='10 M⊙, in agreement with values we find from our best- fit SCE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The explosion energy and ejecta mass are fixed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='9 × 1051 erg and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='9 M⊙, respectively, and the mass of 56Ni (MNi) is varied between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='015 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='020 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' These explosion models are then run through STELLA in order to reproduce the bolometric luminosity evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' For more information, see Hiramatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The resulting model light curves are shown in Fig- ure 7, compared with the pseudo-bolometric light curve of SN 2020bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We find decent qualitative agreement be- tween the numerical models and the observed light-curve evolution, particularly at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The secondary light-curve peak and late-time light-curve slope are well reproduced by an explosion which synthesizes ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='02 M⊙ of 56Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The secondary light-curve peak may be overproduced, but the exact peak luminosity and time of peak are uncertain given the gap in our observational coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Interestingly, however, the peak luminosity of the SCE is not reproduced by these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' It may be that the treatment of the SN shock and the subsequent cool- ing of the outer envelope is too complex to fully sim- ulate within these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' On the other hand, it is possible that an additional powering mechanism con- tributes to the early-time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' To test this, we explore how the addition of different mass-loss rates and timescales to the models affects the early-time light curve through short-lived circumstellar interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' To the best-fit MESA model we attach a wind density profile ρCSM(r) = ˙Mwind/4πr2vwind, where vwind = 10 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' These CSM models are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We find that the best-fit models have a confined CSM with masses of 1 × 10−3 – 1 × 10−2 M⊙ lost by the progenitor within the last several months before explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This hints that circumstellar interaction may contribute to the rapidly- fading early-time emission of SN 2020bio and possibly other SNe IIb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' If this is the case, then the information estimated through SCE model fits may not be truly rep- resentative of the true nature of their progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The values inferred from this numerical modelling, particularly the 56Ni mass, are on the low end of the distribution of values estimated for other well-studied SNe IIb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' SNe IIb with double-peaked light curves typi- cally display secondary radioactive decay-powered peaks equally or more luminous than the peak of the SCE, implying a greater amount of 56Ni synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Stud- The Double-peaked Type IIb SN 2020bio 11 0 3 6 9 12 Days Since Discovery 1041 1042 Pseudo-Bolometric Luminosity (erg s−1) MNi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='019 M⊙ vwind = 10 km s−1 ˙Mwind = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1 M⊙ yr−1 twind = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1 yr ˙Mwind = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1 M⊙ yr−1 twind = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='01 yr ˙Mwind = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1 M⊙ yr−1 twind = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='0 yr ˙Mwind = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='01 M⊙ yr−1 twind = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1 yr ˙Mwind = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='0 M⊙ yr−1 twind = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1 yr ˙Mwind = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='0 M⊙ yr−1 twind = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='01 yr SN2020bio Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Numerical MESA and STELLA circumstellar interaction-powered model light curves of SN 2020bio at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Different color curves correspond to models with vary- ing mass-loss rates and timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The early-time emission excess is best reproduced with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='001-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='01 M⊙ of CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' ies using samples of these objects have found average 56Ni masses of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='10 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='15 M⊙ and average ejecta masses of ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='2 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='5 M⊙ (Lyman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Pren- tice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Taddia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2018), in better agreement with ejecta parameters of other stripped-envelope and H-rich core-collapse SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' However, rare cases of un- derluminous SNe IIb with low inferred MNi have been discovered (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Nakaoka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Maeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' These objects have light curves that appear transitional between standard SNe II-P and SNe IIb, which differ from the observed photometric evolution of SN 2020bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' On the other hand, in the case of SN 2018ivc, both a low 56Ni mass (MNi ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='015M⊙) and progenitor mass (MZAMS ≲ 12M⊙) are inferred (Maeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' It is possible that other SNe IIb with little synthesized 56Ni may be undercounted due to their rapidly-fading or un- derluminous light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Maeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2023) also con- cluded that the light curve of SN 2018ivc was powered at least in part by circumstellar interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Sustained cir- cumstellar interaction has been inferred for other SNe IIb, either through late-time spectral features (Maeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Fremling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2019) or through X-ray and radio observations (Fransson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' It may be that the mechanism that produced the confined CSM inferred from our numerical models of SN 2020bio, and possibly that seen in the case of SN 2018ivc, points to more extreme mass-loss than found in other SNe IIb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Comparison to Nebula Spectra Models A trend between an increasing amount of synthesized O and increasing core-collapse SN progenitor mass has been extensively studied (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Woosley & Heger 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Jerkstrand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2015) use this relationship to calibrate the [O I] λλ 6300,6364 luminosity, normalized by the radioactive decay luminosity at the same phase, with numerical models of SNe IIb progenitors (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1 of Jerkstrand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The authors consider mod- els with zero-age main-sequence masses between 12 M⊙ and 17 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Comparing the observed normalized [O I] luminosity for a handful of SNe IIb, such as SN 1993J, SN 2008ax, and SN 2011dh, to these models allows for a direct estimate of their progenitor masses—all of which fall in the range of masses modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Here we reproduce this analysis using a nebular spec- trum of SN 2020bio, obtained 201 days after the esti- mated explosion, shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We estimate the luminosity from the [O I] λλ 6300,6364 emission doublet in the same way as Jerkstrand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2015)—assuming the width of the feature to be 5000 km s−1, we estimate the continuum by finding the minimum flux redward and blueward of this width and calculate the luminosity within the continuum-subtracted feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' We normal- ize this luminosity using the luminosity of 56Ni decay, assuming the best-fit MNi from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The normalized luminosity at 201 days is Lnorm(t=201) = 9×10−4 ± 2×10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This value is lower than any of the numerical models analyzed by Jerk- strand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' (2015), implying a progenitor mass ≤ 12 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A low progenitor mass for SN 2020bio can also be inferred from the ratio of the [Ca II] λλ 7311, 7324 to [O I] λλ 6300, 6364 fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A higher ratio implies a lower-mass progenitor, with SNe IIb from literature having values ≲ 1 throughout their nebular phases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Terreran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Hiramatsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Using the same procedure as above, we estimate a [Ca II] to [O I] ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='03—again pointing to a low-mass progenitor star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Based on its low synthesized 56Ni mass and nebular spectral features, we conclude that SN 2020bio was likely the core-collapse of a star with a lower mass than the progenitors of most other SNe IIb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS We have presented rapid multi-band photometric and spectroscopic observations of SN 2020bio, a Type IIb SN with luminous and rapidly-evolving SCE, beginning ≤ 1 day after explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Compared with other well-observed SNe IIb, SN 2020bio has the bluest colors at early times as well as unique spectral features with signatures of pre-existing CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Fitting analytical models of SCE to 12 Pellegrino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' the early-time light curve gives progenitor radii on the order of 100 R⊙ – 500 R⊙ and envelope masses of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='01 M⊙ – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='5 M⊙ for our best-fit models, which are slightly greater than values derived for other SNe IIb progenitors using the same methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', SN 2016gkg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Arcavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The weak secondary peak powered by radioac- tive decay is evidence of relatively little 56Ni synthe- sized, MNi ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='02 M⊙, which is in tension with average MNi estimates from samples of other SNe IIb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Numer- ical modeling of the progenitor explosion within con- fined circumstellar material is consistent with the ob- served light curve, showing that circumstellar interac- tion is likely needed to reproduce the complete pseudo- bolometric light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Finally, comparing the nebular spectra to numerical models implies a progenitor mass ≤ 12 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' It is difficult to explain all these peculiar features of SN 2020bio in one consistent model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The combination of its blue colors, early-time spectral features, and numer- ical modeling points to interaction with confined H-rich CSM that was stripped from the progenitor’s outer enve- lope during the months prior to explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The best-fit progenitor parameters, particularly the large envelope radius and low envelope mass, may suggest an inflated progenitor undergoing enhanced mass-loss immediately before exploding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' However, the very low 56Ni and ejecta masses inferred from the later-time light curve, as well as the nebular spectroscopy, point to a lower-mass progeni- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' It is possible that SN 2020bio was the collapse of an unusually low-mass core within a dense CSM produced from its lost H layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Such extensive mass-loss likely re- quires interaction with a binary companion, as inferred for other SNe IIb (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Maund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Benvenuto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Interaction between the SN ejecta and this CSM explains the blue colors and narrow H spectral features at early times while the small 56Ni mass and nebular spectrum indicate a low zero-age main-sequence mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This interaction can lead to an over-estimated progenitor radius—if the CSM was near enough to the progenitor, we may have observed the shock-cooling of this extended CSM instead of the outer envelope of the progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' In the future, more detailed models and multi-wavelength observations, particularly in the radio and X-rays, will be needed to infer SNe IIb progenitor mass-loss rates and CSM masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Given the weak H spectral features when compared to spectra of other SNe IIb, SN 2020bio may be an inter- mediary object between the Type IIb and Ib subclasses, representing a progenitor that was recently stripped al- most entirely of its H-rich envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Transitional ob- jects between SNe IIb and SNe Ib have been observed (Prentice & Mazzali 2017) and can be explained by dif- ferent amounts of H remaining in the outer envelope at the time of explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' More difficult to explain are the small 56Ni and ejecta masses, which are lower than those measured for both SNe IIb and SNe Ib (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Taddia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Some objects that exist in the literature with both low ejecta and 56Ni masses and observed SCE are peculiar SNe Ib as well as Ca-rich transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' However, it is difficult to reconcile the photospheric-phase spec- tra of SN 2020bio, which are most similar to those of other SNe IIb, with the spectra of these objects, which are often used to argue for a degenerate or ultra-stripped progenitor (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Jacobson-Gal´an et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Instead, it is more likely that SN 2020bio had a massive star progenitor more similar to the progenitors of other SNe IIb based on their similar spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This study contributes to the overall diversity in the progenitors of SNe IIb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' More systematic studies of SNe with observed SCE will be needed to search for simi- larities and differences in their progenitor systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' In particular, this work shows the importance of rapid, multi-wavelength follow-up of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' It is par- ticularly important to better understand the number of SNe IIb with weak secondary light-curve peaks, such as SN 2020bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' These objects may have later-time (≥ 5 days) luminosity below the detection threshold of cur- rent all-sky surveys as well as rapid early-time emis- sion which evolves too quickly to be extensively followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Therefore we may be under-counting the rates of core- collapse, stripped-envelope SNe with low 56Ni and ejecta masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A better understanding of their progenitors will be important for exploring the low-mass end of core- collapse SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' This work made use of data from the Las Cumbres Ob- servatory network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' The LCO group is supported by AST-1911151 and AST-1911225 and NASA Swift grant 80NSSC19k1639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' is a CIFAR Azrieli Global Scholar in the Gravity and the Extreme Universe Program and acknowledges support from that program, from the Eu- ropean Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 852097), from the Israel Science Foundation (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2752/19), from the United States Israel Binational Science Foundation (BSF), and from the Israeli Council for Higher Education Alon Fellow- ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Software: Astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2018), emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2013), lcogtsnpipe (Valenti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2016), Matplotlib (Hunter 2007), MESA (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2011, 2013, 2015, 2018, 2019), The 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3847/1538-3881/aabc4f Becker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2015, Astrophysics Source Code Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' ascl:1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='004 Benvenuto, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Bersten, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', & Nomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2013, ApJ, 762, 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/0004-637X/762/2/74 Blinnikov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Lundqvist, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Bartunov, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2000, ApJ, 532, 1132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1086/308588 Blinnikov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1051/0004-6361:20054594 Breeveld, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Landsman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Holland, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2011, Gamma Ray Bursts 2010, 1358, 373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3621807 Brown, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Breeveld, A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='309 Filippenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' V.' 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C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Lundqvist, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', & Chevalier, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1996, ApJ, 461, 993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1086/177119 Fremling, C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2020, Nature, 585, 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1038/s41586-020-2649-2 Hiramatsu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Howell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Moriya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2021, ApJ, 913, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3847/1538-4357/abf6d6 Hunter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2007, Computing in Science and Engineering, 9, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1109/MCSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='55 Jacobson-Gal´an, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' V.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3847/1538-4357/ac67dc Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2012, Annual Review of Nuclear and Particle Science, 62, 407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1146/annurev-nucl-102711-094901 Jerkstrand, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Ergon, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/2041-8205/739/2/L37 Modjaz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Butler, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2009, ApJ, 702, 226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/0004-637X/702/1/226 Morales-Garoffolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Elias-Rosa, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Benetti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2014, MNRAS, 445, 1647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1093/mnras/stu1837 Morozova, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Piro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', & Valenti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2018, ApJ, 858, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3847/1538-4357/aab9a6 Nakaoka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Moriya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Tanaka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2019, ApJ, 875, 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3847/1538-4357/ab0dfe Nakar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' & Piro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2014, ApJ, 788, 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/0004-637X/788/2/193 Ofek, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Sullivan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Shaviv, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2014, ApJ, 789, 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/0004-637X/789/2/104 Paxton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Bildsten, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Dotter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2011, ApJS, 192, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/0067-0049/192/1/3 Paxton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Cantiello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Arras, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2013, ApJS, 208, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/0067-0049/208/1/4 Paxton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Marchant, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Schwab, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2015, ApJS, 220, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/0067-0049/220/1/15 14 Pellegrino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Paxton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Schwab, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Bauer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2018, ApJS, 234, 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3847/1538-4365/aaa5a8 Paxton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Smolec, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Schwab, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2019, ApJS, 243, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3847/1538-4365/ab2241 Piro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2015, ApJL, 808, L51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/2041-8205/808/2/L51 Piro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Haynie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', & Yao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2021, ApJ, 909, 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3847/1538-4357/abe2b1 Poznanski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Prochaska, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', & Bloom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2012, MNRAS, 426, 1465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='21796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='x Prentice, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Mazzali, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Pian, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2016, MNRAS, 458, 2973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1093/mnras/stw299 Prentice, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' & Mazzali, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2017, MNRAS, 469, 2672.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1093/mnras/stx980 Rabinak, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' & Waxman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2011, ApJ, 728, 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/0004-637X/728/1/63 Richmond, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Treffers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Filippenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1994, AJ, 107, 1022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1086/116915 Roming, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Kennedy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Mason, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2005, SSRv, 120, 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1007/s11214-005-5095-4 Sapir, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' & Waxman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2017, ApJ, 838, 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3847/1538-4357/aa64df Schlafly, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' & Finkbeiner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2011, ApJ, 737, 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/0004-637X/737/2/103 Shussman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Waldman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', & Nakar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2016, arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='05323 Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Tucker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Kent, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2002, AJ, 123, 2121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1086/339311 Smith, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Filippenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2011, MNRAS, 412, 1522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='17229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='x Smith, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2014, ARA&A, 52, 487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1146/annurev-astro-081913-040025 Soderberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Margutti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Zauderer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1086/131977 Strotjohann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Ofek, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Gal-Yam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2021, ApJ, 907, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3847/1538-4357/abd032 Taddia, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Stritzinger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Bersten, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2018, A&A, 609, A136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1051/0004-6361/201730844 Tartaglia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Fraser, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Sand, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2019, ApJ, 883, 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3847/1538-4357/ab3e37 Tully, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' & Fisher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1977, A&A, 54, 661 Valenti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Howell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Stritzinger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2016, MNRAS, 459, 3939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1093/mnras/stw870 Van Dyk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Zheng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Fox, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2014, AJ, 147, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/0004-6256/147/2/37 Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Eastman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Weaver, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1994, ApJ, 429, 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1086/174319 Woosley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' & Heger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2007, PhR, 442, 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='physrep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='009 Yao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', De, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Kasliwal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2020, ApJ, 900, 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3847/1538-4357/abaa3d Yaron, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' & Gal-Yam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2012, PASP, 124, 668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1086/666656 Yaron, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Perley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Gal-Yam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2017, Nature Physics, 13, 510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1038/nphys4025 Zacharias, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Finch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', Girard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 2013, AJ, 145, 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1088/0004-6256/145/2/44 The Double-peaked Type IIb SN 2020bio 15 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' CORNER PLOTS In Figures A1, A2, A3, and A4 we present distributions of the fitted parameters of the models detailed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='4 t0 (days) 300 315 330 345 Menv (10−2 M ⊙ ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='7 v (104 km s−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content='4 t0 (days) Figure A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} +page_content=' Same as Figure A1, but for the SW17 (n=3) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E3T4oBgHgl3EQfrwpm/content/2301.04662v1.pdf'} diff --git a/-NAzT4oBgHgl3EQfSvt8/content/tmp_files/2301.01237v1.pdf.txt b/-NAzT4oBgHgl3EQfSvt8/content/tmp_files/2301.01237v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f00c9093b84fcd7f45bde50c3dd24fb016f2f205 --- /dev/null +++ b/-NAzT4oBgHgl3EQfSvt8/content/tmp_files/2301.01237v1.pdf.txt @@ -0,0 +1,1928 @@ +Safe Path following for Middle Ear Surgery +Bassem Dahroug1, Brahim Tamadazte2, and Nicolas Andreff1 +1Bassem Dahroug and Nicolas Andreff are with FEMTO-ST, AS2M, +Univ. Bourgogne Franche-Comt´e, +CNRS/ENSMM, 25000 Besan¸con, France +2Brahim Tamadazte is with Sorbonne Universit´e, +CNRS UMR 7222, INSERM U1150, ISIR, F-75005, Paris, France. +brahim.tamadazte@cnrs.fr +January 4, 2023 +Abstract +This article formulates a generic representation of a path-following +controller operating under contained motion, which was developed in +the context of surgical robotics. It reports two types of constrained +motion: i) Bilateral Constrained Motion, also called Remote Center +Motion (RCM), and ii) Unilaterally Constrained Motion (UCM). In +the first case, the incision hole has almost the same diameter as the +robotic tool, while in the second state, the diameter of the incision +orifice is larger than the tool diameter. The second case offers more +space where the surgical instrument moves freely without constraints +before touching the incision wall. +The proposed method aims to combine two tasks that must oper- +ate hierarchically: i) respect the RCM or UCM constraints formulated +by equality or inequality, respectively, and ii) perform a surgical as- +signment, e.g., scanning or ablation expressed as a 3D path-following +task. The proposed methods and materials were successfully tested +first on our simulator that mimics realistic conditions of middle ear +surgery, then on an experimental platform. Different validation sce- +narios were carried out experimentally to assess quantitatively and +qualitatively each developed approach. Although ultimate precision +was not the goal of this work, our concept is validated with enough +accuracy (≤ 100µm) for the ear surgery. +keywords: Medical Robotics, Constrained motion, Path Follow- +ing, Visual Servoing. +1 +arXiv:2301.01237v1 [cs.RO] 3 Jan 2023 + +1 +INTRODUCTION +Surgical robots are gaining more popularity due to their advantages for both +the patient and the physician [11,37,40]. It is particularly valid for so-called +Minimally-Invasive Surgery (MIS) approaches. For instance, a laparoscopy +or keyhole surgery [23] performs incision around 10mm. It is tiny compared +to the larger incisions needed in laparotomy (open surgery). Another sit- +uation where the surgical instruments could be inserted through a natural +orifice (e.g., mouth, nasal clefts, urethra, anus) to reach the targeted organ. +In both cases, the entry space (i.e., the incision hole or the natural orifice) +restricts the surgical tool motion, consequently the surgeon’s hands and the +robot carrying the instrument [7]. +This article mainly discusses two types of constrained motion that result +directly from MIS procedures: +1. Remote Center Motion (RCM), also known as fulcrum effect, implies +the incision hole has almost the same diameter as that of the surgical +tool [2]; +2. Unilaterally Center Motion (UCM) implies the incision diameter size +is bigger than that of the tool, offering more freedom for the tool +motion [12]. +The first type of motion was initially achieved by designing a particu- +lar robotic structure that imposes the constrained motion mechanically [2, +21, 29]. The RCM dictates that the center-line of the surgical tool is al- +ways coincident with the center point of the incision orifice (trocar point). +Consequently, the linear movement of the tool is prohibited along two axes. +The main advantage of RCM mechanisms is to reduce the risk of damag- +ing the trocar wall because their kinematic structures ensure the pivoting +motion. Their modest controller is also easy to implement. However, this +kind of mechanism is restricted to a unique configuration and cannot provide +enough flexibility for shifting the location of the trocar point. +An alternative solution proposes a software RCM for overcoming the +previous problem by guiding a general-purpose robot with the advantage of +being flexible enough for achieving complex tasks [6]. This solution is con- +venient for diverse medical applications (e.g., laparoscopic [32] and eye [28] +surgeries). However, we claim that the RCM approach is not the best choice +for other surgery types (e.g., ear, nose, mouth, knee arthroscopy). In latter +cases, the orifice diameter is generally bigger than the tool diameter. Con- +sequently, the RCM controller imposes too strong limitations on the tool +2 + +motion. Indeed, the RCM is a mathematical equality constraint (i.e., the +distance between the tool body and the center point of the incision orifice +must be equal to zero). As such, RCM motion can be named as a bilaterally +constrained motion. On the contrary, UCM is a weaker restriction since the +unilateral constraints are inequality equations (i.e., the latter distance could +be greater or less than zero) [19]. +In the literature, the term forbidden-region virtual fixtures [1] are used +for collaboration tasks where the user can either manipulate a robotic de- +vice [5] or telemanipulate a master device [33]. +These fixtures could be +defined as geometric forms [12,39] or vector field [26] around the tool. Then +a kinematic control [12] or dynamic one [26, 38, 39] is applied to guide the +robot during the desired task. +The theoretical contribution of this article lies in the improvement of the +generic formulation of constrained motion. It has the objective to achieve a +velocity controller that can maintain the RCM or UCM depending on the +configuration of the surgical procedure. Besides that, it reveals a new path- +following controller integrated with a task-hierarchy controller for imposing +a priority between the RCM/UCM and the path-following tasks. +Nevertheless, the technical contribution lies in the assessment of such +approaches. Therefore, we developed a simulator including surgical tools and +a numerical twin mimicking the middle ear cavity. Based on the auspicious +evaluation, we also carried out a pre-clinical setup that takes up the diverse +components of the simulator to assess the proposed methods experimentally. +Various scenarios are also implemented to accomplish these evaluations. The +obtained performances in terms of behavior and accuracy are promising. +The remainder of the article is organized as follows. Section 2 presents +the clinical needs and challenges. The methodology followed to design the +proposed controllers will be discussed in Section 3. After that, Section 4 +focuses on both the numerical and experimental validations of the proposed +approaches. Ultimately, Section 5 presents the conclusion and perspectives. +2 +MEDICAL MOTIVATIONS +2.1 +Treated Disease +The work discussed in this article represents a part of a long-term project. +It deals with the development of a robotic system that is dedicated to +cholesteatoma surgery. The system will aim to achieve an MIS within the +middle ear cavity by passing through the external ear canal or an incision +orifice made on the mastoid portion. +3 + +Cholesteatoma is a frequent disease that invades the middle ear. It in- +fects the middle ear by introducing abnormal skin (lesional tissue) in the +middle ear-cavity. The most common explanation [31] is due to the immi- +gration of the epidermal cells, which are the cells type in the external ear +canal, and cover up the mucosa of the middle ear cavity, as shown in Fig.1. +These cells gradually proliferate within the temporal bone and destroy the +adjacent bony structures. +Figure 1: Evolution of cholesteatoma disease within the middle ear, which +is located behind the tympanic membrane. +The evolution of cholesteatoma is life-threatening in the long run. The +complications can be classified as follows [3]: i) destruction of the ossicular +chain, ii) facial paralysis, iii) labyrinthitis, iv) extracranial complications, +and v) intracranial complications. It can notice the irreversible effects that +cholesteatoma can cause in a patient. Despite that, there is no drug therapy +for the treatment. The only solution is surgical intervention. +2.2 +Current Surgical Procedure +As claimed above, the only treatment for cholesteatoma is a surgical pro- +cedure. It aims to eradicate all cholesteatoma tissue and reconstruct the +anatomy of the middle ear [18]. +For reaching the middle-ear cavity, the surgeon often drills the temporal +bone behind the auricular, as shown in Fig. 2. This surgical procedure is +called mastoidectomy where the surgeon maintains the wall of the exter- +nal ear canal. This technique creates an incision that forms a triangular +(around 40 × 40 × 30mm) with a depth of about 30mm. The latter pro- +4 + +pdemni +Mucosa +Demis +Submucosa +Muscularis +Retraction and perforation of the tympanic +membrane +Normal tympanic +Cholestetoma +membraneFigure 2: Mastoidectomy procedure with canal-wall-up indicates that the +external ear canal is preserved. (a) side view of the mastoidectomy tunnel +and (b) top view of the mastoidectomy tunnel. +cedure can also become more invasive by sacrificing the posterior portion +of the external ear canal (i.e., canal-wall-down). Furthermore, even if the +surgical orifice is relatively large, the surgical procedure remains complex +and requires high expertise and dexterity from the surgeon. Also, even with +an experienced clinician in the cholesteatoma case, the clinical outcomes +remain unsatisfactory in terms of effectiveness. Indeed, there is a high risk +that the cholesteatoma could regrow a few months after the surgical inter- +vention. It occurs due to residual cholesteatoma cells. Consequently, 10 to +40% of patients perform more than one surgery to get definitively over this +disease [4]. +Due to the complexity of the temporal bone cavity, the surgeon mainly +faces numerous difficulties during the procedure (Fig.3): i) lack of ergonomy +of the tools; ii) limited field of view of the oto-microscope (the surgeon can- +not visualize the lateral regions hidden (blind spots) in the middle ear cavity) +and iii) access with the conventional rigid instruments requires considerable +expertise to handle. +Therefore, it is increasingly important to overcome the previous problems +and evolve this procedure towards less invasive. +It implies reducing the +incision orifice size, improving the cholesteatoma ablation efficiency, and +avoiding the current high surgical recurrence rate for this kind of surgery. +3 +METHODOLOGY +This section begins by presenting a brief summary of the new surgical pro- +tocol associated with the robotic system. After that, it discusses the hier- +archical controller for managing simultaneously the various tasks. It then +5 + +Wall of external +Wallof +earcanal +external +earcanal +Removed +bone +Cholesteatoma +Cholesteatoma +(a) +(b)Figure 3: Conceptual scheme to demonstrate the ”blind spot” during the +cholesteatoma surgery. +explains separately the path-following, the RCM, and the UCM controllers. +3.1 +New Surgical Protocol +In collaboration with surgeons experts in middle ear surgery, especially +cholesteatoma treatment, we have attempted to set up a new and more +efficient surgical protocol reported in [11]. +Firstly, the idea is to make +cholesteatoma surgery less invasive compared to the traditional one. Thus, a +macro-micro robotic system should pass through a millimetric incision made +behind the ear (in the mastoid portion) to access the middle ear cavity [35]. +Secondly, cholesteatoma surgery needs to be more efficient by eliminating +the residual cases. This second objective can be accomplished by removing +a large part of the cholesteatoma tissue using rigid miniature mechanical +resection tools. After that, a bendable actuated tool [16,36] could be used +to guide a laser fiber. This fiber carbonizes the residual cholesteatoma (re- +sulting from the mechanical resection phase) [22]. +Both mechanical resection and laser ablation should be performable ei- +ther in automatic or semi-automatic mode. While the mechanical resection +does not require high accuracy, the laser ablation requires higher precision +since the residual cholesteatoma cells can be a few tens of micrometers in +size. Therefore, the contributions of robotics and vision-based control are +essential to fundamental this kind of task. In this work, we investigated +6 + +Microscope +Wall of the external +Incision of +the mastoidectomy +ear canal +Outside +blind spot +Within +blind spot +Suction tool +Cholesteatoma +Knifethe use of path-following control schemes under constrained motion (due to +the incision orifice) to carry out the notions requested by the cholesteatoma +removal (i.e., mechanical resection and laser ablation). +3.2 +Task Hierarchical Controller +A surgical procedure can be considered as a set of sequential or overlap- +ping sub-tasks. The hierarchical methods ensure the execution of several +tasks simultaneously. Consequently, the required tasks do not enter into +conflict [13,34]. In the case of cholesteatoma surgery, various sub-tasks can +be involved during the procedure, such as constraint enforcement (RCM +or UCM) and ablation tools for the pathological tissues. Therefore, these +sub-tasks must be carried out according to a defined hierarchical scheme. +To express a controller that manages simultaneous sub-tasks, let us start +by assuming that a generic sub-task (˙ei ∈ Rmi) given by +˙ei = Li eve, +where i=1,2,...,j +(1) +where eve ∈ se(3) is the end-effector twist velocity to be computed in the +end-effector frame Fe, and Li ∈ Rmi×n is the interaction matrix which +relates the vector eve to the error ˙ei. +The inverse solution of the previous equation is not guaranteed since the +interaction matrix Li could be non-square, and the matrix rank is locally +deficient. Thanks to the least-square method, an approximate solution can +be found by minimizing ∥˙ei − Li eve∥ over eve, and using numerical proce- +dures (such as QR or SVD). The formal result of it can be simply written +as eve = L† +i ˙ei, where L† +i is the pseudo-inverse of Li. If Li does not have +full rank then it has at least one singular vector z1, located in its null-space +(Liz1 = 0). +The vector z1 is also described as the null space of ei, be- +cause any twist vector parallel to z1 will leave ei unchanged. Therefore, the +projection gradient general form [27] is given by +eve = L† +1˙e1 + (I − L† +1L1)z1 +(2) +In order to define z1, let us first consider a secondary sub-task ˙e2 = +L2 eve. Since the control vector must include the first sub-task, equation +(2) is injected in the latter expression, resulting in +˙e2 = L2 +� +L† +1˙e1 + (I − L† +1L1)z1 +� += L2L† +1˙e1 + L2(I − L† +1L1) +� +�� +� +˜L2 +z1 +(3) +7 + +From the previous equation, the vector z1 is deduced as +z1 = ˜L† +2(˙e2 − L2L† +1˙e1) + (I − ˜L† +2˜L2)z2 +(4) +with another criteria vector z2 which is projected in the null-space of the +secondary sub-task. By introducing (4) in (2), a recursive form of the pro- +jection gradient is obtained as +eve = L† +1 ˙e1 + (I − L† +1L1) +� +˜L† +2(˙e2 − L2L† +1 ˙e1) + (I − ˜L† +2˜L2)z2 +� += L† +1 ˙e1 + (I − L† +1L1)˜L† +2(˙e2 − L2L† +1 ˙e1) ++ (I − L† +1L1)(I − ˜L† +2˜L2)z2 +(5) +The right-hand side of the previous equation can further be simplified as [24] +eve = L† +1˙e1 + ˜L† +2(˙e2 − L2L† +1˙e1) +. +(6) +The latter equation finds a solution to satisfy both sub-tasks ˙e1 and ˙e2. +It also ensures a form of hierarchy/priority between them. The analytical +expression of each sub-task with its Li is presented in the coming sections. +3.3 +6D Approach Controller +This section is dedicated to mathematically describing how to control the +tool-tip for regulating its position and orientation with respect to a reference +frame, e.g., the orifice frame Fr. This task is applied when the tool locates +outside the incision orifice, and its pose must be adjusted with respect to +the orifice before it starts another task inside the orifice. +To do this, a traditional 3D position-based visual servo [8] is applied. +The feature vector s += +(rtt, θ rut) is defined as the pose vector which +describes the tool-tip frame Ft with respect to the orifice frame Fr. This +vector gathers the translation t of the tool-tip and its rotation θu in form +of angle/axis parameterization. The desired feature vector s∗ = (0, 0) is +set to a zero vector since it is required to make coincident the frame Ft with +Fr. Thus, the approach task error eapp is deduced as the difference between +the current features vector and the desired one, i.e., +eapp = s − s∗ +(7) +The time variation of the latter error is related to the spatial velocity of +the tool-tip tvt by the interaction matrix L3D ∈ R6×6 as +˙eapp = L3D tvt +(8) +8 + +where tvt = (tvt,t ω) gathers the instantaneous linear and angular velocities +of the tool-tip. Since the desired feature vector equals to 06×1, then the +interaction matrix L3D is determined by +L3D = +� −I3×3 +03×3 +03×3 +Lθu +� +(9) +where I3×3 is a 3 × 3 identity matrix, 03×3 is a 3 × 3 zero matrix, and Lθu +is given by [25] +Lθu = I3×3 − θ +2 [u]× + +� +1 − sinc θ +sinc2 θ +2 +� +[u]2 +× +(10) +in which sinc x is the sinus cardinal. +Finally, the spatial velocity tvt is determined for ensuring an exponential +decoupled reduction of the error (i.e., ˙e = −λe) as +tvt = −γL−1 +3Deapp +(11) +where γ is a gain coefficient, and L−1 +3D is the inverse of the interaction matrix +since it is square and has a closed-form inverse [25]. +The command velocity of the robot end-effector eve = eVt tvt is deduced +by the following twist matrix +eVt = +� +eRt +[ett]× +eRt +03×3 +eRt +� +(12) +since the tool body is rigid and the transformation between the end-effector +frame Fe and the tool-tip frame Ft is fixed. Finally, the controller stability +was demonstrated in [25] to be globally exponentially stable. +3.4 +3D Path-Following Controller +This section will focus on a generic modelling of a 3D path-following scheme. +The advantage of using such as controller is the separation between i) the +geometric curve (desired path Sp) which is planned by the surgeon based on +pre-operative images, and ii) the advance speed (vtis) of the tool-tip along +the desired path which is controlled by the surgeon during the operation. In +this manner, the collaboration surgeon/robot ensures that the robot guides +the tool along the path while the surgeon controls the robot progression +without planning the robot velocity direction. +9 + +Figure 4: Orthogonal projection of the tool-tip onto a geometric curve. +Fig. 4 depicts the surgical instrument and its reference frames with re- +spect to the desired path Sp. By projecting the tool-tip Ot onto the reference +path, the resultant orthogonal distance dpf is considered as the error (i.e., +lateral deviation) which must be controlled to zero. Therefore, the 3D vec- +tor distance between the tool-tip Ot and the projection point pp′ calculated +as +dpf = Ot − pp′. +(13) +In order to express the command velocity, the time-derivative of (13) +provides the tool-tip velocity vt as discussed in [10] +˙dpf = +� +�I3×3 − +kpk⊤ +p +1 − d⊤ +pf +� +Cp(sp) × kp +� +� +� vt +(14) +where Cp(sp) is the path curvature in function of the path curve length, kp +is the unit-vector of the instantaneous tangential vector (Fig. 4). +At this stage, it requires to choose the adequate velocity of the tool-tip +vt in the latter equation to ensure that the lateral error dpf is regulated +to zero while progressing along the path. An intuitive solution consists of +decomposing the control velocity into two orthogonal components (Fig. 5): i) +the advance velocity (vadv) along the path, and ii) the return velocity (vret) +for regulating the tool deviation from the reference path. +The previous +10 + +desired 3D path +mk-1 +tool body +mk +mk+1 +M +mk+2Figure 5: Representation of the different velocities involved in the path- +following controller. +concept is formulated as follows: +vt = αkp +���� +vadv ++ βdpf +� �� � +vret +. +(15) +The tuning coefficients of the controller α and β allow adjusting the +priority between the advance and return velocities, respectively. +Besides +that, the controller stability demonstrated in [10] shows that α should be +a positive scalar while β must be a negative scalar to ensure the system +stability. +The choice of these gain factors can be imposed by a function of a con- +stant velocity vtis > 0 that depends on the interaction between the surgical +tool and the lesional tissue. +This velocity could be tuned easily by the +surgeon before or during the intervention. Therefore, (15) yields +v2 +tis +���� +=∥vt∥2 += α2∥kp∥2 +� �� � +=1 ++ β2∥dpf∥2 +� +�� +� +=∥vret∥2 +. +(16) +The gain factor α is thus determined as +α = +� � +v2 +tis − ∥vret∥2 +∥vret∥2 < v2 +tis +0 +∥vret∥2 > v2 +tis +. +(17) +If the tool is not far from the reference path, the first condition in (17) is +selected. Otherwise, the priority is returning the tool-tip to the reference +path, and the advance velocity is null (i.e., second condition in (17)). +11 + +desired 3D path +tool body +mk +ret +adv +mk+1The latter strategy proposed in [10] applies a constant value for the gain +factor β. However, this section presents a new formulation of β to make +the controller sensitive to the path curvature. Thus, it is calculated by the +following equation +β = β′ +� +1 + sign +� +d⊤ +pf (Cp(sp) × kp) +� � +1 − eγc∥Cp(sp)∥� � +(18) +where β′ is a negative gain for returning to path, sign(•) is a sign function +to determine the direction along the reference path, and γc is a negative +gain for sensing the amount of path curvature. +The ratio between the gain factors (i.e., vtis and β′) forms an acceptable +error band around the reference path. For instance, if β′ is higher than vtis, +then the error band will be small. On the contrary, in the case where vtis +is bigger than β′, then the error band will be large since the priority is to +advance along the reference path. The effect of this ratio is presented in +section 4. +Furthermore, the control velocity of the tool-tip (15) could be repre- +sented with respect to any desired frame. +Note that if the end-effector +frame is selected, then the end-effector twist velocity eve is related to the +linear velocity of the tool-tip evt as +evt = [I3×3 +− [eet]×] +� +�� +� +Lpf∈R3×6 +� eve +eωe +� +� +�� +� +eve +(19) +whereby [eet]× is the skew-symmetric matrix associated to the vector eet, +and Lpf is the interaction matrix related to the path-following task. +Finally, the control velocity for the path-following task is deduced as +eve = L† +pf +evt +. +(20) +3.5 +Bilateral Constrained Motion Controller +As claimed above, the resection/ablation task is performed in a minimally +invasive procedure. Therefore, the robot should perform the surgical task +under the constraints of the incision point. This section begins with the +description of RCM (bilateral constraints), while the following section de- +scribes the UCM (unilateral constraints). The RCM imposes that the center- +line of tool body St should be coincident with the point Or. Simultaneously, +the tool-tip must follow the desired path inside the incision orifice. +12 + +Figure 6: Geometric scheme of the bilateral linear error drcm. +Fig. 6 shows a straight tool which is located far from the center-point +of incision orifice Or. The previous works [10,12] built the controller based +on the angular error between the vectors et′ and er while the proposed +controller in this section is based on the linear error drcm. This new choice +offers the controller to become independent of the tool shape. Let us imagine +that the tool-tip position in Fig. 6 is fixed in space, but its length can change. +In the case of angular error, when the tool length increases, the error reduces +its value. +However, the linear error stays constant when the tool length +changes. Therefore, the new choice grants better numerical computing. +The error drcm is deduced by the orthogonal projection of the point Or +onto the tool body St. The point pt′ is resultant from the latter projection +that is calculated as follows +ept′ = +euet eu⊤ +et +eer +(21) +whereby euet is the unit vector of et expressed in Fe, and eer represents the +vector between both points Oe and Or which is expressed in Fe. +In case the surgical tool is curved, the point pt′ is determined by dis- +cretizing the tool body. Then the closest point onto the tool body is located. +After that, the orthogonal projection is performed with respect to this point +and the previous one on the tool center-line. Thus, the error drcm is de- +duced as +drcm = +eOr − ept′ +. +(22) +13 + +center of the incision hole +er +et' +tool body +Pt +rcm +X +M +incision wallThe controller task is to find the spatial velocity of the robot end-effector +eve for eliminating the rate-of-change of the bilateral linear error drcm. +Thereby, the time-derivative of the latter equation results in +˙drcm = +evr − evt′ +(23) +where evt′ is the linear velocity of the projected point pt′ along the tool +body, and evr is the linear velocity of the trocar point described in Fe. +Indeed, the velocity of the projected point depends on the movement of the +tool body with respect to the trocar point. Hence, this velocity is computed +as [12] +evt′ = +ekt ekT +t +1 + dTrcm(Ct(st) × ekt) +evr +(24) +whereby Ct(st) is the tool curvature in the function of its arc length, and +ekt is the instantaneous tangential unit-vector onto the tool curve/shape. +Since the calculation is done in the perspective of the end-effector frame +Fe, it implies that this frame is fixed, and the other ones are dynamic with +respect to it. Consequently, the incision orifice virtually moves, and its linear +velocity evr is related to the spatial velocity of the robot end-effector thanks +to the following formula +evr = +� +I3×3 +− [eOr]× +� +� +�� +� +Lr∈R3×6 +eve +. +(25) +By injecting the latter equation in (24) then the resultant in (23), the +time-derivative of the error drcm equals to +˙drcm = +� +I3 − +ekt ekT +t +1 + dTrcm(Ct(st) × ekt) +� � +I3×3 +− [eOr]× +� +� +�� +� +Lrcm∈R3×6 +eve +(26) +where Lrcm is the interaction matrix which relates between the end-effector +velocity eve and the rate-of-change of the error drcm. +Furthermore, a linearized proportional controller is applied to reduce +the bilateral linear error in an exponential decay form. It defines the control +velocity of the end-effector as +eve = −λ L† +rcm drcm. +(27) +whereby λ is a positive gain which allows tuning the rate of exponential +decay, and L† +rcm is the pseudo-inverse of the interaction matrix Lrcm. +14 + +Finally, the RCM task can be combined as the highest priority with the +path-following task as the secondary criteria. The hierarchical controller +deduces the control velocity, by replacing the equations (27) and (20) in +equation (6), as +eve = −λL† +rcmdrcm + ˜L† +pf +� +evt + λLpfL† +rcmdrcm +� +, +with +˜Lpf = Lpf +� +I − L† +rcmLrcm +� +. +(28) +In the opposite case, the hierarchical controller sets the path-following task +(20) as the highest priority while the RCM task (27) as the secondary one. +The control velocity is deduced from equation (6) as +eve = L† +pf +evt − ˜L† +rcm +� +λdrcm + LrcmL† +pf +evt +� +, +(29) +with +˜Lrcm = Lrcm +� +I − L† +pfLpf +� +. +(30) +3.6 +Unilaterally Constrained Motion Controller +This section continues with the design of the path-following controller under +unilateral constraints. Notice that the UCM task assumes the incision orifice +is larger than the tool diameter. Consequently, it imposes on the tool-tip +to follow the incision/ablation path while the tool body is free to move +within the incision orifice as long as it does not damage the orifice wall. +Therefore, the formulation of the previous section needs to extend to satisfy +the unilateral constraints. +Fig. 7(left image 1) shows how the point pt′ is orthogonally projected +onto the orifice wall in order to determine the closest point ph′ on the orifice +wall Sh. The distance between the latter two points forms the vector error +ducm which can be defined as (left image 2 of Fig. 7) +ducm = +et′r +���� +=drcm +− eh′r +���� +=dwall +. +(31) +The question now is how to maintain the value of the error ducm greater +or equal to zero. For security issues, three regions are defined around the +projected point ph′, as shown in the left image of Fig. 7: +1. critical zone (dark red circle) which its border is defined by a minimal +distance dmin; +15 + +Figure 7: Geometric modelling of the unilateral linear error ducm. +2. dangerous zone (light green circle) which its border is defined by a +maximal distance dmax; and +3. safe zone which is the remain region outside the dangerous zone. +When the Euclidean norm ∥ducm∥ is larger than the ”dangerous” dis- +tance dmax, the tool can follow the reference path without any constraints +since its location is in the safe zone. +However, an admittance control is +activated, which is composed of a virtual damper µobs, when the tool body +passes the dangerous zone border. Indeed, the admittance control imposes +unilateral constraint towards the safe point ps by generating a compensation +velocity in the opposite direction to the orifice wall. +By differentiating equation (31) with respect to time for deducing the +velocity twist of the end-effector, it becomes equal to +˙ducm = ( evr − evt′) +� +�� +� +˙drcm +− ( evr − evh′) +� +�� +� +˙dwall += +evh′ − evt′ +(32) +The velocity of the projected point ph′ is deduced in the same way as +equation (24) +evh′ = +ekh ekT +h +1 + dTucm (Ch(sh) × ekh) +evt′ +. +(33) +16 + +Ph +tool body +(1) +centerofthe +dangerous zone +incision hole +et' +incision wall +ducm +d +rcm +d +11DM, +30 +critical zone +25mm +(2)where Ch(sh) is the orifice curvature in function of its arc length, and ekh is +the instantaneous tangential unit-vector onto the orifice curve. In another +perspective, the latter equation describes how the projection of the point +pt′ onto the geometric curve of the orifice wall Sh evolves with time. +The velocity evt′ is deduced by combining equations (24) and (25) +evt′ = +ekt ekT +t +1 + dTrcm (Ct(st) × ekt) +� +I3×3 +− [eOr]× +� +� +�� +� +Lvt′ ∈R3×6 +eve +. +(34) +Replacing equations (33) and (34) in (32) yields +˙ducm = +� +ekh ekT +h +1 + dTucm (Ch(sh) × ekh) − I3×3 +� +Lvt′ +� +�� +� +Lucm∈R3×6 +eve +(35) +whereas Lucm is the interaction matrix that relates the twist end-effector +with the rate of change of the error ducm. +Thereby, the control velocity of the UCM task is defined as +eve = −µobsλL† +ucmducm +. +(36) +The damping coefficient µobs changes following a sigmoid function that +depends on the vector ducm. It means that the gain µobs reaches its mini- +mal value when ducm is higher than the safe distance dmax, where the tool +location in the dangerous zone. However, µobs gradually increases until it +reaches its maximal value when ducm is smaller than the critical distance +dmin, where the tool location in the critical zone. This behaviour is modeled +as +µobs = +σmax +1 + e +� +σstep +� +∥ducm∥−σmin +�� +(37) +where σmax, σmin and σstep are tunable parameters for modifying the sigmoid +form. +Finally, the path-following task can be combined as the highest priority +with the UCM task as the secondary criteria. The hierarchical controller +deduces the control velocity, by replacing the equations (36) and (20) in +equation (6), as +eve = L† +pf +evt − ˜L† +ucm +� +µobsλducm + LucmL† +pf +evt +� +, +with +˜Lucm = Lucm +� +I − L† +pfLpf +� +. +(38) +17 + +4 +VALIDATION +This section discusses several scenarios to evaluate qualitatively and quan- +titatively the proposed methods and materials. The developed controllers +were first tested using our simulator framework and then in an experimental +set-up that takes up the various components of the simulator. +4.1 +Implementation Issues +This part begins by converting the patient’s ear to its numerical-twin and +then its 3D printed-twin. The first step to accomplish this job is the scan of +the patient’s ear during the preoperative phase for getting DICOM (Digital +Imaging and Communications in Medicine) images, as depicted in Fig. 8. +The DICOM images are handled by the software 3D Slicer which converts +these images to a 3D surface model after a segmentation process. +Prior +works were done in relation to this subject for achieving an automated seg- +mentation process (e.g., [15, 30]). However, the segmentation process that +we have done manually is not automated since this is not the focus of this +article. In the future, we believe that our segmentation process needs to be +done again in an automated manner for efficiency. +The 3D Slicer software exports the segmentation results as STL files for +each anatomical structure. Afterward, the software MeshLab treats the STL +files for smoothing the surface and reducing the number of vertices and faces +to cut down the final STL file size. This step produces the numerical-twin +of the patient’s ear. +The next step creates the 3D printed-twin for conducting the experimen- +tal validation. Indeed, a simplified version of the numerical-twin is imported +in Solidworks for i) adding some thickness to the middle ear cavity and ii) +creating the incision orifice through the mastoid. +After that, the planning stage of the desired path within the middle +ear cavity begins. The path planning step can be optimized (e.g., [14,20]). +However, this step was done manually on Solidworks to generate text files +that contain the geometry of the reference path and the orifice wall as a +sequence of 3D points. These files are inputs for the controller. This step +should be investigated in the future and add to the adequate functions in +the simulator. +Fig. 9 presents the proposed control architect with the TCP/IP com- +munication. This architect allows easy interchangeability between the real- +system (robot) and its numerical-twin (simulator). The latter figure (the +red block at the left-hand side) also shows that the implemented controller +18 + +Figure 8: The steps done to achieve a numerical and physical model of the +middle ear cavity. +is firstly initialized with the end-effector and the incision orifice poses, ⋆Te +and ⋆Tr respectively. These poses must be described in the same frame +(e.g., the world frame Fw or the camera Fc). Indeed, the tool geometry +19 + +DICOM images of the patient's ear +Preoperative scanning for the patient +Surface structure (numerical ear twin) of the patient's ear +Ossicles +Inner ear +Temporal bone +Middle ear +External ear +Chorda +cavity +canal +External ear canal +tympani nerve +3D printed ear twin version for +experimental validation +Preoperative planning phase +Red region +will be preserved +Mastoidectomy +Temporal bone +(Canal wall up) +Green region +External ear canal +will be removedFigure 9: Block diagram of the TCP/IP communication between the client +(proposed controller) and the server (simulator or robot) or vice-versa. +20 + +Robot +Contro +connection +unit +Simulator +Robot control (case 2) +Control Unit +Robot control (case 1) +Simulator controlSt is defined with respect to the end-effector frame Fe while the reference +path Sp and the orifice wall Sh are described in the incision orifice frame +Fr. Furthermore, the controllers should be initialized by the different gain +coefficients before the control-loop starts. +The hierarchy controller arranges throughout the control-loop the prior- +ity between the different tasks (i.e., the approach task, the path-following +task, and the RCM/UCM constraints). Indeed, the control-loop is mainly +divided into three phases: +1. the outside phase: the tool corrects its initial pose with respect to the +incision orifice. This stage applies the approach task for regulating: i) +the tool-tip position to the point located before the orifice center point, +and ii) the tool-tip rotation as the rotation of the orifice reference +frame. This manoeuvrer is performed to ensure some security for the +next phase; +2. the transition phase: the tool-tip passes the center point of the inci- +sion orifice. The RCM controller could oscillate when the trocar point +is close to the tool-tip. These oscillations are generated because the +controller computes large rotation displacement, due to the lever phe- +nomena, for compensating the rotation error. Thus, the trocar point +is virtually moved to the first point on the reference path. +Conse- +quently, the tool body can rotate about this new point. This virtual +trocar point moves towards the orifice frame while the tool-tip ad- +vances along the reference path; +3. the inside phase: the tool-tip follows the desired path while the tool +body is constrained by the orifice wall or the orifice center point. +Therefore, the output of this block is the spatial velocity of the end-effector +expressed in its frame (eve) while its inputs are the instantaneous poses of +the end-effector and the incision orifice (⋆Te and ⋆Tr). The question now +is: what is the observation frame? +In the simulator case (the blue block at the right-hand side of Fig. 9), it is +straightforward since the user initializes the poses with respect to the world +frame Fw of the virtual scene. Thus, the spatial velocity eve is transformed +to wve then it is integrated over the sample time Te to deduce the new pose +of the end-effector. Consequently, the tool pose is updated in the virtual +scene, and this new pose is sent back to the control unit block for computing +a new iteration. +There are two options for designing the control architect in the exper- +imental case. The first one consists of using an exteroceptive sensor (e.g., +21 + +camera) for estimating the required poses. This option is depicted in the +green block of Fig. 9 named Robot control (case 1). The input of this block +is the spatial velocity eve that is transformed to deduce the angular velocity +of each joint ˙q with the help of the inverse differential kinematic model to +move mechanical structure of the robot. This motion is observed from the +camera frame Fc in order to estimate the new pose of the end-effector and +that of the orifice. These poses are the output of this block which are sent +back to the control unit block for calculating a new iteration. +However, +this option is uneasy for implementation since it needs a particular setup to +accurately track both the end-effector and the orifice [17]. +The second option is more fundamental than the first one. It is also +presented in the green block of Fig. 9 named Robot control (case 2). It uses +the proprioceptive sensors of the robot and its forward geometric model to +estimate the end-effector pose. Despite that, this option requires performing +a registration process [9, 17] between the robot and the orifice before the +control-loop. After that, the robot works blindly, and the user assumes that +the orifice does not move during the control-loop. +The simulator is implemented in C++. It uses Eigen library for linear +algebra (e.g., vectors, matrices, numerical solvers) and PCL (Point Cloud +Library) for visualizing the STL parts and converting them to point clouds. +This conversion is done to initialize the collision detection that is accom- +plished by VCollide library. Finally, ViSP library is used for manipulating +the camera images throughout the experimental work. +4.2 +Numerical Validation +A numerical simulator was developed, as the first step, to validate the func- +tioning of the diverse methods before physical implementation. It simulates +the geometric motion of the surgical tool through the incision orifice and +the middle ear cavity. The software interchangeability of the simulator and +the physical set-up allowed us also to tune the controller parameters before +the experimental validation. Therefore, this part presents three scenarios +for the demonstration: +• scenario 1 performs the path-following task without any constraint +applied on the tool motion. +It demonstrates the effect of the gain +coefficients vtis and β in equations (16) and (18), respectively, on the +performance of the path-following controller; +• scenario 2 performs the path-following task with RCM constraints. It +simulates the drilling of a minimal invasive tunnel (i.e., conical tunnel) +22 + +through the mastoid portion to reach the middle ear cavity; +• scenario 3 assumes the surgeon performed a standard mastoidectomy. +It simulates an inspection/resection task performed under the UCM +constraints. +4.2.1 +Simulation of the path-following task without constraints +Throughout this first trial, the value of vtis = 4mm/second in equation (16) +remains constant during all tests. Besides that, the same reference path is +tested during this trial, and it is defined as a spiral curve. +Figure 10: The effect of the ratio between vdes and β′ on the path-following +error dpf with a zoom and magnification on the orange region. +The first group of tests keeps the value of γc in equation (18) constant +while decreasing the value of β′ which its value varies from −4 to −16. +Fig. 10 shows the influence of the gain coefficient β′ on the path-following +error dpf. Indeed, this error computed as in equation (13). The ripples +appearing in this figure represent the linear error between the projected +point pt′ and the closest point on the reference path pp′. An orange rectangle +appeared in this figure for zooming on one of these ripples. One can observe +that the error reduced as designed exponentially. +The latter figure also demonstrates that the best ratio between β′ and +vtis should be greater than −2 (the saddle-brown line with star markers), +and less than or equal −3 (the olive line with square markers). If the ratio is +less than or equal to −1, the controller response is relatively slow, and there +is a steady-state error (the maroon line with round markers in Fig. 10). On +23 + +0.8 +0.040 +Udes = 4.0, β = -4.0, %c = -1.0 +0.7 +0.035 +Udes = 4.0, β = -8.0, %c = -1.0 +Udes = 4.0, β =-12.0, %c = -1.0 +Udes = 4.0, β = -16.0, % = -1.0 +0.6 +0.030 +0.025 +0.5 +0.020 +(mm) +0.4 +ldpfll +0.015 +0.3 +0.010 +0.005 +0.2 +0.000 +1200 +1210 +1220 +1230 +1240 +1250 +1260 +1270 +1280 +1290 +0.1 +- +1000 +2000 +3000 +4000 +5000 +Iterationsthe opposite, if the ratio is higher than or equal to −4, the system begins +to oscillate (having over-shoots). However, the controller reduces the error +faster than the previous cases (the sea-green line with triangular markers in +Fig. 10). +The second group of tests chose a constant ratio −2 while decreasing +the value of γc from −2 to −16. This group shows that the best value of +γc is to be near from β′. If γc is higher than β′, the system begins to have +over-shoots, but it reduces faster the path-following error. +4.2.2 +Simulation of a robotic drilling task under RCM constraint +The surgeon perforates manually until now the mastoid portion in the tem- +poral bone for reaching the middle ear cavity. The resultant mastoidectomy +orifice is invasive. Thereby, a less invasive tunnel is proposed in this trial. +Besides that, the drilling procedure becomes automated so that the surgeon +can concentrate on other essential tasks. Indeed, this drilling procedure is +achieved by merging the approach task, the 3D path-following task, and the +RCM task. +(a) +(b) +Figure 11: Numerical validation of the 3D path-following under a RCM +constraint (see Extension 2). (a) The tool pose with respect to the desired +path. (b) Sequence of zoom images during the tool motion. +24 + +Tool body +Incision center +point +3D pathFig. 11 depicts the tool motion throughout the drilling procedure. The +subplot (a) draws the tool geometry and its poses at different instances (or- +ange straight-lines). It also shows the drilling path defined as a combination +of spiral and linear portions (sea-green dotted-line). One can view that the +tool body is always coincident with the orifice center point. The subplot +(b1) shows the path done by the tool-tip (dodger-blue line) to accomplish +the outside phase by i) approaching towards the point located before the +orifice center point, and ii) regulating the rotation of the tool-tip frame to +be as that of the orifice reference frame. The subplot (b2) depicts an in- +stantaneous zoom on the tool pose during the inside phase to visualize the +RCM effect. +Figure 12: The approach task error eapp, where the left column is the linear +error and the right column represents the angular error. +The approach task error eapp computed in equation (11) is visualized in +Fig. 12 which depicts the linear errors in the column and the angular errors +in the right one. Over this period, the error is reduced in an exponential +form as planned. +At the end of the latter period, the transition phase starts.The task- +hierarchical controller becomes active, and it arranges the path-following +task as the highest priority while the RCM task is the second one. The +errors of these tasks presented in the left columns of Fig. 14 and 13 which +are obtained from equations (13) and (22) for the path-following and RCM +errors, respectively. One can observe a peak appeared around 4 seconds in +the path-following figure due to the initial error when the controller becomes +activated. Then, it attenuates the error until it attains stability. Further- +more, one can visualize in the RCM figure that three peaks appeared at the +end of this phase. This behaviour happened due to the movement of the +virtual trocar point. +25 + +20 +12 +Eappr +Eappr +15 +Eappy +Eappy +10 +Eapp: +Eapp: +10 +8 +eapp +5 +(mm) +(deg) +6 +ddpa +Eappe +.5 +4 +.10 +2 +0 +-20 +0.5 +1.0 +1.52.02.5 +3.0 +3.5 +4.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Time (second) +Time (second)Figure 13: The RCM task error drcm, where the left column shows the error +evolution during the transition phase while the right column presents the +error during the inside phase. +Figure 14: The path-following task error dpf, where the left column shows +the error evolution during the transition phase while the right column +presents the error during the inside phase. +After the previous period, the inside phase starts where the hierarchical +controller modifies the priority by setting the RCM task as the highest one +while the path-following is the secondary one. The RCM task error drcm was +26 + +outside/transition phases +inside phase +0.20 +0.015 +drcms +0.15 +drcmy +0.010 +drcm +drcm: +0.10 +drcm +Idrcml +0.05 +(mm) +0.005 +0.00 +0.000 +d +-0.05 +-0.10 +-0.005 +-0.15 +-0.20 +-0.010, +1 +2 +3 +5 +6 +7 +8 +10 +20 +30 + 40 +50 +60 +Time (second) +Time (second)outside/transition phases +inside phase +0.25 +0.08 +drcms +0.06 +0.20 +drcm +0.04 +Idpf ll +0.15 +0.02 +(mm) +TAAAA +0.10 +0.00 +dpf +0.02 +0.05 +drcmr +-0.04 +drcmy +0.00 +-0.06 +II dp ll +-0.05 +-0.08 +1 +2 +3 +4 +5 +6 +7 +8 +10 +20 +30 +40 +50 +60 +Time (second) +Time (second)computed as 0.002 ± 0.002 mm (mean error ± STD (STandard Deviation) +error), as shown in the right column of Fig. 13, while the path-following error +dpf was 0.008±0.009 mm, as shown in the right column of Fig. 14. The gain +values used for this trail were equal to λ = 1, γ = 1, vtis = 4 mm/second, +β′ = −10, γc = −10 and Te = 0.008 second. +4.2.3 +Simulation of an ablation/excision surgical task under UCM +constraint +In this trial, the incision orifice size is larger than the instrument diameter. +The tool is consequently subject to the UCM for providing more freedom to +the tool movements inside the incision orifice. This behaviour is shown in +Fig. 15a where the orifice wall is represented by the red surface. The latter +figure also presents the curved tool employed during this trial which performs +an ablation or scanning process. The desired 3D path is thus composed of a +linear portion to reach the middle ear cavity and a spiral curve to simulate +the required surgical task. This selected path can reach some regions where +a straight tool cannot attain (see Extension 4 to visualize the collision of +the latter one with the orifice wall). +The subplot (b1) of Fig. 15b indicates the path done by the tool during +the outside phase. It also presents an instantaneous pose of the tool body +throughout the transition phase. +As explained in the previous trial, the +proposed controller executes the same tasks over these two phases. Subplot +(b2) presents the tool motion during the inside phase, where the dangerous +and critical zones are represented by the green and red circles, respectively. +The center point of these circles corresponds to the point ph′ obtained by +projecting pt′ onto the orifice wall Sh. +Throughout the inside phase, the hierarchical controller combines the +UCM task with the path-following task as described in (38). Fig. 16 shows +the UCM task error ducm which is deduced as in equation (31). +It also +presents the boundaries of the critical and dangerous zones. One can observe +that the error ducm begins with a considerable value, compared to the error +drcm, since the previous phase delivers the tool to the center point of the +incision orifice. Then, the error ducm reduced, while the error drcm increased +because the tool approached the incision wall to follow the reference path. +However, the error ducm did not exceed the dmin, which implies the tool +body did not enter the critical zone. +Fig. 17 presents the path-following error dpf during the inside phase. It +was measured was 0.005 ± 0.006 mm. The gain values used for this trail +were equal to λ += +0.8, γ += +0.8, vtis += +4 mm/second, β′ += +− 10, +27 + +(a) +(b) +Figure 15: Numerical validation of the 3D path-following under a UCM +constraint (see Extension 3). (a) The tool pose with respect to the desired +path. (b) Sequence of zoom images during the tool motion. +Figure 16: The UCM task error ducm during the inside phase along side the +error drcm. +28 + +reference curve +actual curve +tool body +orificewall5 +I drcm l +Ild ucmll +dmas +4 +dmin +Safe zone +3 +dangerous +zone +Critical +zone +0 +10 +15 +20 +25 +Time (second)Figure 17: The path-following task error dpf during the inside phase. +γc = +− 10 and Te = 0.008 second. +4.3 +Experimental Validation +This part is devoted to the physical implementation of the blocks Robot +control that is shown in Fig. 9. Its physical correspondence is presented in +Fig. 18. The robotic work-cell in the latter figure consists of: +• a serial robot from Universal Robot (UR3) with ±0.03 mm pose re- +peatability. It communicates with the proposed controller via TCP/IP +for receiving the command velocity of the end-effector. It also sends +the end-effector pose to the controller if the block Robot control (case +1) is required to be executed; +• a monocular camera from Guppy (with image size 640 × 420 pixels) +and an optical objective lens from Computar with distortion (model +MLM3X-MP) are used for the control purpose. This optical system +tracks and estimates the poses of the end-effector and the incision +orifice. +It then sends these poses to the proposed controller if the +block Robot control (case 2) is needed to be executed; +29 + +0.06 +urcm +0.04 +dpf +0.02 +(mm) +WAWWAAMWWM +dpf +0.00 +-0.02 +-0.04 +10 +15 +20 +25 +Time (second)• two visualization cameras provide other views for recording the mul- +timedia videos. +Figure 18: Configuration of the experimental setup. +The numerical twin of the ear model shown previously in Fig. 8 is mod- +ified for implementing its 3D printed twin. This modification holds up the +(a) +(b) +Figure 19: The printed ear model used during the different tests. (a) The +different parts of the ear model and the rigid tools. (b) After assembling +the different parts. +30 + +Robot +controller +End-effector +Tool body +Visualization +Control +cameras +Incision orifice- +cameramastoidectomy orifice with the middle ear cavity and a planar grid/marker. +Fig. 19 presented the fabricated parts before and after the assembly, along- +side the rigid tools used during the validation tests. +The trials of this part have the objective to evaluate the performance +of the path-following controller under constraints. Therefore, a curved tool +follows the same planned path, one time under the RCM constraint and the +second time under the UCM constraint. +4.3.1 +Path-Following under RCM Constraint +(a) +(b) +Figure 20: Experimental validation of the 3D path-following under a RCM +constraint (see Extension 5). (a) The tool pose with respect to the desired +path. (b) Sequence of zoom images during the tool motion. +Fig. 20 presents the desired path (sea-green dotted line), the resultant +motion of the curved tool (orange line), and the path done by tool-tip +(dodger-blue line). One can observe in Fig. 20b(1) that the tool approaches +to the incision orifice by executing the controller given in equation (11). The +approach task error eapp computed from equation (7). Fig. 21 presents the +latter error and it converges toward zero by the end of this phase. +Afterward, the transition phase starts so that the tool passes the center +point of the incision orifice, as explained previously. The hierarchical con- +troller (equation 29) arranges the path-following task as the highest priority +while the RCM task is the second one. This behaviour is demonstrated in +the left column of Fig. 22-23, where the hierarchical controller has been ac- +tivated around 4 second. One can visualize that the RCM task error drcm +31 + +reference curve +actualcurve +tool bodyFigure 21: The approach task error eapp, where the left column is the linear +error and the right column represents the angular error. +has some steps due to the movements of the virtual trocar point while the +path-following error dpf maintained its value around zero. +Figure 22: The RCM task error drcm, where the left column shows the +error evolution during the outside/transition phases while the right column +presents the error during the inside phase. +When the tool passes the center point of the incision orifice, the inside +phase begins. The hierarchical controller (equation 28) modifies its priorities +by setting the RCM task as the highest one and the path-following as the +32 + +25 +50 +linear +angular +20 +40 +(mm) +15 +10 +20 +5 +10 +0.5 +1.0 +1.52.0 +2.5 +3.0 +3.5 +4.0 +0.5 +1.0 +1.52.02.5 +3.0 +3.5 +4.0 +Time (second) +Time (second)outside/transition phases +inside phase +3.0 +0.7 +0.6 +2.5 +0.5 +2.0 +(u) +0.4 +1.5 +0.3 +1.0 +0.2 +0.5 +0.1 +0.0. +0.0 +0 +2 +4 +6 +8 +10 +12 +14 +16 +5 +20 +25 +30 +40 +45 +50 +Time (second) +Time (second)Figure 23: The path-following task error dpf, where the left column shows +the error evolution during the outside/transition phases while the right col- +umn presents the error during the inside phase. +second one. The system performances during the inside phase are shown +in the right columns of Fig. 22-23. During this phase, the RCM task error +drcm measured as 0.06 ± 0.05mm (mean error ± standard deviation (STD) +error) while the path-following error dpf was 0.05 ± 0.03mm. +A exteroceptive sensor used to close the control loop, as presented in +Fig. 9 by the block Robot control (case 2). Besides that, the gain values +used in this experiment were equal to λ = 1, γ = 1, vtis = 0.5 mm/second, +β′ = −1.25, γc = −10 and Te = 0.008 second. +Another trial was conducted for testing the block Robot control (case 1) +by using the proprioceptive sensor in the control loop. The system perfor- +mances are better than the exteroceptive test (see test 2 in Table 1). The +errors drcm and dpf are reduced to almost half. It implies that our vision +system needed amelioration in terms of accuracy. +From the surgeon’s perspective, it is required to target the residual cells +of cholesteatoma. It implies that the robot should detect/remove a human +cell whose size is around 0.1mm. The proposed controller reached the re- +quirements since the error dpf is smaller than the human cell size. Besides +that, the surgical tool does not damage the entry orifice (patient’s head). +By increasing the tool velocity vtis = 2 mm/second and maintain the +same ratio β′/vtis = −2, the system performances deteriorated as expected. +The errors drcm and dpf are almost increase by half (see tests 2 and 4 in +33 + +outside/transition phases +inside phase +0.30 +0.20 +0.25 +0.15 +0.20 +(mm) +0.15 +0.10 +0.10 +0.05 +0.05 +0.00 +0.00 +0 +2 +4 +6 +8 +10 +12 +14 +16 +15 +20 +30 +3540 +45 +50 +Time (second) +Time (second)Table 1). Therefore, the choice of the gain coefficients effects the system per- +formances. +4.3.2 +Path Following under UCM Constraint +This second trial assumes the same conditions as the previous one. It in- +volves the same curved tool and the desired path. However, this trial im- +posed a unilateral constraint on the tool motion. Consequently, the tool can +leave the center point of the incision orifice and move near the orifice wall. +This behaviour is demonstrated in Fig. 24. The sub-figure (b1) of the lat- +ter figure shows the path done by the tool-tip during the outside/transition +phases, while the sub-figure (b2) presents the tool-tip path during the inside +phase. The dangerous and critical regions are presented by the green and +red circles in the latter sub-figure. +(a) +(b) +Figure 24: Experimental validation of the 3D path-following under a UCM +constraint (see Extension 6). (a) The tool motion during the different phases. +(b) Sequence of zoom images during the tool motion. +Throughout the inside phase, the hierarchical controller arranges the +different tasks as explained in section 3.6. The highest priority is the path- +following task when the tool is located in the safe zone. However, the highest +priority changes to the UCM task when the tool body passes the danger +zone. The system performances are shown in Fig. 25-26. One can observe +from the UCM task error ducm (Fig. 25) that the tool body is maintained in +34 + +referencecurve +actualcurve +tool body +orificewallFigure 25: The UCM task error ducm during the inside phase. +Figure 26: The path-following task error dpf during the inside phase. +the dangerous zone since the error ducm changes its value between dmax and +dmin. Besides that, the path-following error dpf (Fig. 26) was 0.05±0.03mm +(mean error ± STD error) and its median error was 0.05mm. +A exteroceptive sensor used as the feedback sensor. Additionally, the +gain values used for this second trial were equal to λ = 1, γ = 1, vtis = +0.5 mm/second, β′ = −1.25, γc = −10 and Te = 0.008 second. +The error dpf of this trial remains almost the same as the previous trial. +It implies that the UCM constraint does not deteriorate the path-following +error. Indeed, it provides the surgical tool to move with more liberty in +order to take advantage of the large size of the entry orifice. +35 + +5 +Idrcm +I d ucm ll +dmax +4 +dmin +Safe zone +(mm) +3 +dangerous +zone +Critical +zone +0 +15 +20 +25 +30 +35 +40 +Time (second)inside phase +0.18 +0.16 +0.14 +0.12 +(mm) +0.10 +0.08 +d +0.06 +0.04 +0.02 +0.0Q +10 +15 +20 +25 +30 +35 +40 +45 +Time (second)N° +constraint +feedback +type of +error +mean (∥e∥) ± STD +1 +RCM +exteroceptive +drcm +dpf +0.06±0.05 +0.05±0.02 +2 +RCM +exteroceptive +drcm +dpf +0.15±0.06 +0.08±0.05 +3 +RCM +proprioceptive +drcm +dpf +0.02±0.05 +0.02±0.01 +4 +RCM +proprioceptive +drcm +dpf +0.03±0.08 +0.03±0.02 +5 +UCM +exteroceptive +drcm +dpf +3.30±0.93 +0.05±0.03 +6 +UCM +exteroceptive +drcm +dpf +3.30±0.93 +0.09±0.06 +7 +UCM +proprioceptive +drcm +dpf +2.74±0.77 +0.02±0.01 +8 +UCM +proprioceptive +drcm +dpf +2.69±0.67 +0.03±0.02 +Table 1: Summary of different trials achieved with the curved tool during +the experimental tests. +∥e∥ (in mm) is the absolute average of the linear error along x − y − z axes, +and STD is the related standard deviation (in mm). +Results obtained with the following parameters: λ = 1, vtis = 0, 5 mm/s, +and Te = 0, 008 second. The while trials applied β′ = −1.25, while the blue +ones applied β′ = −5. +5 +CONCLUSION AND FUTURE WORK +This article discussed the design of an original controller for guiding a rigid +instrument under constrained motions such as RCM or UCM. The proposed +methodology allows a generic formulation, in the same controller, two tasks: +i) the constrained motion (RCM or UCM), and ii) a revisited 3D path- +following scheme by increasing the sensitivity to the path complexity (e.g., +curvature radius) and then reducing the path-following error. To manage +the achievement of two or more tasks without conflicts, we also implemented +a task prioritizing paradigm. Consequently, the developed control scheme +can be integrated easily with various robotic systems without an accurate +knowledge of the robot inverse kinematics. +36 + +Experimental validation was also successfully conducted using a 6-DoF +robotic system. The obtained results are promising in terms of behavior +and precision. These performances, even if they meet the specifications of +the targeted middle ear surgery, may be considered improvements. +The +positioning error depends directly on the registration process that is not +treated optimally in this work. Furthermore, the pose estimation of the tool- +tip was done based on a geometric model of the instrument. Its estimation +could be another source of error. Thus, it would be interesting to find out +another method for estimating the tool shape and the pose of its tip. +The forthcoming work will implement the discussed methods in a clinical +context using a realistic phantom and a human cadaver. Besides that, a force +control could be added to increase the robot sensitivity to its environment +and increase the level of security. +ACKNOWLEDGMENTS +This work was supported by the Inserm ROBOT Project: ITMO Cancer +no 17CP068-00. +References +[1] Jake J Abbott, Panadda Marayong, and Allison M Okamura. 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IEEE T. on Biomed. +Eng., 65:797–808, 2018. +40 + diff --git a/-NAzT4oBgHgl3EQfSvt8/content/tmp_files/load_file.txt b/-NAzT4oBgHgl3EQfSvt8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d508358a35e273ca9e5959b7500cc3654e5465d --- /dev/null +++ b/-NAzT4oBgHgl3EQfSvt8/content/tmp_files/load_file.txt @@ -0,0 +1,974 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf,len=973 +page_content='Safe Path following for Middle Ear Surgery Bassem Dahroug1, Brahim Tamadazte2, and Nicolas Andreff1 1Bassem Dahroug and Nicolas Andreff are with FEMTO-ST, AS2M, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Bourgogne Franche-Comt´e, CNRS/ENSMM, 25000 Besan¸con, France 2Brahim Tamadazte is with Sorbonne Universit´e, CNRS UMR 7222, INSERM U1150, ISIR, F-75005, Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' brahim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='tamadazte@cnrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='fr January 4, 2023 Abstract This article formulates a generic representation of a path-following controller operating under contained motion, which was developed in the context of surgical robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It reports two types of constrained motion: i) Bilateral Constrained Motion, also called Remote Center Motion (RCM), and ii) Unilaterally Constrained Motion (UCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' In the first case, the incision hole has almost the same diameter as the robotic tool, while in the second state, the diameter of the incision orifice is larger than the tool diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The second case offers more space where the surgical instrument moves freely without constraints before touching the incision wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The proposed method aims to combine two tasks that must oper- ate hierarchically: i) respect the RCM or UCM constraints formulated by equality or inequality, respectively, and ii) perform a surgical as- signment, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', scanning or ablation expressed as a 3D path-following task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The proposed methods and materials were successfully tested first on our simulator that mimics realistic conditions of middle ear surgery, then on an experimental platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Different validation sce- narios were carried out experimentally to assess quantitatively and qualitatively each developed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Although ultimate precision was not the goal of this work, our concept is validated with enough accuracy (≤ 100µm) for the ear surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' keywords: Medical Robotics, Constrained motion, Path Follow- ing, Visual Servoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='01237v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='RO] 3 Jan 2023 1 INTRODUCTION Surgical robots are gaining more popularity due to their advantages for both the patient and the physician [11,37,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It is particularly valid for so-called Minimally-Invasive Surgery (MIS) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' For instance, a laparoscopy or keyhole surgery [23] performs incision around 10mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It is tiny compared to the larger incisions needed in laparotomy (open surgery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Another sit- uation where the surgical instruments could be inserted through a natural orifice (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', mouth, nasal clefts, urethra, anus) to reach the targeted organ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' In both cases, the entry space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', the incision hole or the natural orifice) restricts the surgical tool motion, consequently the surgeon’s hands and the robot carrying the instrument [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This article mainly discusses two types of constrained motion that result directly from MIS procedures: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Remote Center Motion (RCM), also known as fulcrum effect, implies the incision hole has almost the same diameter as that of the surgical tool [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Unilaterally Center Motion (UCM) implies the incision diameter size is bigger than that of the tool, offering more freedom for the tool motion [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The first type of motion was initially achieved by designing a particu- lar robotic structure that imposes the constrained motion mechanically [2, 21, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The RCM dictates that the center-line of the surgical tool is al- ways coincident with the center point of the incision orifice (trocar point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Consequently, the linear movement of the tool is prohibited along two axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The main advantage of RCM mechanisms is to reduce the risk of damag- ing the trocar wall because their kinematic structures ensure the pivoting motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Their modest controller is also easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, this kind of mechanism is restricted to a unique configuration and cannot provide enough flexibility for shifting the location of the trocar point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' An alternative solution proposes a software RCM for overcoming the previous problem by guiding a general-purpose robot with the advantage of being flexible enough for achieving complex tasks [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This solution is con- venient for diverse medical applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', laparoscopic [32] and eye [28] surgeries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, we claim that the RCM approach is not the best choice for other surgery types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', ear, nose, mouth, knee arthroscopy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' In latter cases, the orifice diameter is generally bigger than the tool diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Con- sequently, the RCM controller imposes too strong limitations on the tool 2 motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Indeed, the RCM is a mathematical equality constraint (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', the distance between the tool body and the center point of the incision orifice must be equal to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' As such, RCM motion can be named as a bilaterally constrained motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' On the contrary, UCM is a weaker restriction since the unilateral constraints are inequality equations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', the latter distance could be greater or less than zero) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' In the literature, the term forbidden-region virtual fixtures [1] are used for collaboration tasks where the user can either manipulate a robotic de- vice [5] or telemanipulate a master device [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' These fixtures could be defined as geometric forms [12,39] or vector field [26] around the tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Then a kinematic control [12] or dynamic one [26, 38, 39] is applied to guide the robot during the desired task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The theoretical contribution of this article lies in the improvement of the generic formulation of constrained motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It has the objective to achieve a velocity controller that can maintain the RCM or UCM depending on the configuration of the surgical procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Besides that, it reveals a new path- following controller integrated with a task-hierarchy controller for imposing a priority between the RCM/UCM and the path-following tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Nevertheless, the technical contribution lies in the assessment of such approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, we developed a simulator including surgical tools and a numerical twin mimicking the middle ear cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Based on the auspicious evaluation, we also carried out a pre-clinical setup that takes up the diverse components of the simulator to assess the proposed methods experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Various scenarios are also implemented to accomplish these evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The obtained performances in terms of behavior and accuracy are promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The remainder of the article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Section 2 presents the clinical needs and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The methodology followed to design the proposed controllers will be discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' After that, Section 4 focuses on both the numerical and experimental validations of the proposed approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Ultimately, Section 5 presents the conclusion and perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 2 MEDICAL MOTIVATIONS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='1 Treated Disease The work discussed in this article represents a part of a long-term project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It deals with the development of a robotic system that is dedicated to cholesteatoma surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The system will aim to achieve an MIS within the middle ear cavity by passing through the external ear canal or an incision orifice made on the mastoid portion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 3 Cholesteatoma is a frequent disease that invades the middle ear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It in- fects the middle ear by introducing abnormal skin (lesional tissue) in the middle ear-cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The most common explanation [31] is due to the immi- gration of the epidermal cells, which are the cells type in the external ear canal, and cover up the mucosa of the middle ear cavity, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' These cells gradually proliferate within the temporal bone and destroy the adjacent bony structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Figure 1: Evolution of cholesteatoma disease within the middle ear, which is located behind the tympanic membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The evolution of cholesteatoma is life-threatening in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The complications can be classified as follows [3]: i) destruction of the ossicular chain, ii) facial paralysis, iii) labyrinthitis, iv) extracranial complications, and v) intracranial complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It can notice the irreversible effects that cholesteatoma can cause in a patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Despite that, there is no drug therapy for the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The only solution is surgical intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='2 Current Surgical Procedure As claimed above, the only treatment for cholesteatoma is a surgical pro- cedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It aims to eradicate all cholesteatoma tissue and reconstruct the anatomy of the middle ear [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' For reaching the middle-ear cavity, the surgeon often drills the temporal bone behind the auricular, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This surgical procedure is called mastoidectomy where the surgeon maintains the wall of the exter- nal ear canal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This technique creates an incision that forms a triangular (around 40 × 40 × 30mm) with a depth of about 30mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The latter pro- 4 pdemni Mucosa Demis Submucosa Muscularis Retraction and perforation of the tympanic membrane Normal tympanic Cholestetoma membraneFigure 2: Mastoidectomy procedure with canal-wall-up indicates that the external ear canal is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (a) side view of the mastoidectomy tunnel and (b) top view of the mastoidectomy tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' cedure can also become more invasive by sacrificing the posterior portion of the external ear canal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', canal-wall-down).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Furthermore, even if the surgical orifice is relatively large, the surgical procedure remains complex and requires high expertise and dexterity from the surgeon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Also, even with an experienced clinician in the cholesteatoma case, the clinical outcomes remain unsatisfactory in terms of effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Indeed, there is a high risk that the cholesteatoma could regrow a few months after the surgical inter- vention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It occurs due to residual cholesteatoma cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Consequently, 10 to 40% of patients perform more than one surgery to get definitively over this disease [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Due to the complexity of the temporal bone cavity, the surgeon mainly faces numerous difficulties during the procedure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='3): i) lack of ergonomy of the tools;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' ii) limited field of view of the oto-microscope (the surgeon can- not visualize the lateral regions hidden (blind spots) in the middle ear cavity) and iii) access with the conventional rigid instruments requires considerable expertise to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, it is increasingly important to overcome the previous problems and evolve this procedure towards less invasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It implies reducing the incision orifice size, improving the cholesteatoma ablation efficiency, and avoiding the current high surgical recurrence rate for this kind of surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 3 METHODOLOGY This section begins by presenting a brief summary of the new surgical pro- tocol associated with the robotic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' After that, it discusses the hier- archical controller for managing simultaneously the various tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It then 5 Wall of external Wallof earcanal external earcanal Removed bone Cholesteatoma Cholesteatoma (a) (b)Figure 3: Conceptual scheme to demonstrate the ”blind spot” during the cholesteatoma surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' explains separately the path-following, the RCM, and the UCM controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='1 New Surgical Protocol In collaboration with surgeons experts in middle ear surgery, especially cholesteatoma treatment, we have attempted to set up a new and more efficient surgical protocol reported in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Firstly, the idea is to make cholesteatoma surgery less invasive compared to the traditional one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Thus, a macro-micro robotic system should pass through a millimetric incision made behind the ear (in the mastoid portion) to access the middle ear cavity [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Secondly, cholesteatoma surgery needs to be more efficient by eliminating the residual cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This second objective can be accomplished by removing a large part of the cholesteatoma tissue using rigid miniature mechanical resection tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' After that, a bendable actuated tool [16,36] could be used to guide a laser fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This fiber carbonizes the residual cholesteatoma (re- sulting from the mechanical resection phase) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Both mechanical resection and laser ablation should be performable ei- ther in automatic or semi-automatic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' While the mechanical resection does not require high accuracy, the laser ablation requires higher precision since the residual cholesteatoma cells can be a few tens of micrometers in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, the contributions of robotics and vision-based control are essential to fundamental this kind of task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' In this work, we investigated 6 Microscope Wall of the external Incision of the mastoidectomy ear canal Outside blind spot Within blind spot Suction tool Cholesteatoma Knifethe use of path-following control schemes under constrained motion (due to the incision orifice) to carry out the notions requested by the cholesteatoma removal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', mechanical resection and laser ablation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='2 Task Hierarchical Controller A surgical procedure can be considered as a set of sequential or overlap- ping sub-tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The hierarchical methods ensure the execution of several tasks simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Consequently, the required tasks do not enter into conflict [13,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' In the case of cholesteatoma surgery, various sub-tasks can be involved during the procedure, such as constraint enforcement (RCM or UCM) and ablation tools for the pathological tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, these sub-tasks must be carried out according to a defined hierarchical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' To express a controller that manages simultaneous sub-tasks, let us start by assuming that a generic sub-task (˙ei ∈ Rmi) given by ˙ei = Li eve, where i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=',j (1) where eve ∈ se(3) is the end-effector twist velocity to be computed in the end-effector frame Fe, and Li ∈ Rmi×n is the interaction matrix which relates the vector eve to the error ˙ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The inverse solution of the previous equation is not guaranteed since the interaction matrix Li could be non-square, and the matrix rank is locally deficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Thanks to the least-square method, an approximate solution can be found by minimizing ∥˙ei − Li eve∥ over eve, and using numerical proce- dures (such as QR or SVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The formal result of it can be simply written as eve = L† i ˙ei, where L† i is the pseudo-inverse of Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' If Li does not have full rank then it has at least one singular vector z1, located in its null-space (Liz1 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The vector z1 is also described as the null space of ei, be- cause any twist vector parallel to z1 will leave ei unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, the projection gradient general form [27] is given by eve = L† 1˙e1 + (I − L† 1L1)z1 (2) In order to define z1, let us first consider a secondary sub-task ˙e2 = L2 eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Since the control vector must include the first sub-task, equation (2) is injected in the latter expression, resulting in ˙e2 = L2 � L† 1˙e1 + (I − L† 1L1)z1 � = L2L† 1˙e1 + L2(I − L† 1L1) � �� � ˜L2 z1 (3) 7 From the previous equation, the vector z1 is deduced as z1 = ˜L† 2(˙e2 − L2L† 1˙e1) + (I − ˜L† 2˜L2)z2 (4) with another criteria vector z2 which is projected in the null-space of the secondary sub-task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' By introducing (4) in (2), a recursive form of the pro- jection gradient is obtained as eve = L† 1 ˙e1 + (I − L† 1L1) � ˜L† 2(˙e2 − L2L† 1 ˙e1) + (I − ˜L† 2˜L2)z2 � = L† 1 ˙e1 + (I − L† 1L1)˜L† 2(˙e2 − L2L† 1 ˙e1) + (I − L† 1L1)(I − ˜L† 2˜L2)z2 (5) The right-hand side of the previous equation can further be simplified as [24] eve = L† 1˙e1 + ˜L† 2(˙e2 − L2L† 1˙e1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (6) The latter equation finds a solution to satisfy both sub-tasks ˙e1 and ˙e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It also ensures a form of hierarchy/priority between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The analytical expression of each sub-task with its Li is presented in the coming sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='3 6D Approach Controller This section is dedicated to mathematically describing how to control the tool-tip for regulating its position and orientation with respect to a reference frame, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', the orifice frame Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This task is applied when the tool locates outside the incision orifice, and its pose must be adjusted with respect to the orifice before it starts another task inside the orifice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' To do this, a traditional 3D position-based visual servo [8] is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The feature vector s = (rtt, θ rut) is defined as the pose vector which describes the tool-tip frame Ft with respect to the orifice frame Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This vector gathers the translation t of the tool-tip and its rotation θu in form of angle/axis parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The desired feature vector s∗ = (0, 0) is set to a zero vector since it is required to make coincident the frame Ft with Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Thus, the approach task error eapp is deduced as the difference between the current features vector and the desired one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', eapp = s − s∗ (7) The time variation of the latter error is related to the spatial velocity of the tool-tip tvt by the interaction matrix L3D ∈ R6×6 as ˙eapp = L3D tvt (8) 8 where tvt = (tvt,t ω) gathers the instantaneous linear and angular velocities of the tool-tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Since the desired feature vector equals to 06×1, then the interaction matrix L3D is determined by L3D = � −I3×3 03×3 03×3 Lθu � (9) where I3×3 is a 3 × 3 identity matrix, 03×3 is a 3 × 3 zero matrix, and Lθu is given by [25] Lθu = I3×3 − θ 2 [u]× + � 1 − sinc θ sinc2 θ 2 � [u]2 × (10) in which sinc x is the sinus cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Finally, the spatial velocity tvt is determined for ensuring an exponential decoupled reduction of the error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', ˙e = −λe) as tvt = −γL−1 3Deapp (11) where γ is a gain coefficient, and L−1 3D is the inverse of the interaction matrix since it is square and has a closed-form inverse [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The command velocity of the robot end-effector eve = eVt tvt is deduced by the following twist matrix eVt = � eRt [ett]× eRt 03×3 eRt � (12) since the tool body is rigid and the transformation between the end-effector frame Fe and the tool-tip frame Ft is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Finally, the controller stability was demonstrated in [25] to be globally exponentially stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='4 3D Path-Following Controller This section will focus on a generic modelling of a 3D path-following scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The advantage of using such as controller is the separation between i) the geometric curve (desired path Sp) which is planned by the surgeon based on pre-operative images, and ii) the advance speed (vtis) of the tool-tip along the desired path which is controlled by the surgeon during the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' In this manner, the collaboration surgeon/robot ensures that the robot guides the tool along the path while the surgeon controls the robot progression without planning the robot velocity direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 9 Figure 4: Orthogonal projection of the tool-tip onto a geometric curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 4 depicts the surgical instrument and its reference frames with re- spect to the desired path Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' By projecting the tool-tip Ot onto the reference path, the resultant orthogonal distance dpf is considered as the error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', lateral deviation) which must be controlled to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, the 3D vec- tor distance between the tool-tip Ot and the projection point pp′ calculated as dpf = Ot − pp′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (13) In order to express the command velocity, the time-derivative of (13) provides the tool-tip velocity vt as discussed in [10] ˙dpf = � �I3×3 − kpk⊤ p 1 − d⊤ pf � Cp(sp) × kp � � � vt (14) where Cp(sp) is the path curvature in function of the path curve length, kp is the unit-vector of the instantaneous tangential vector (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' At this stage, it requires to choose the adequate velocity of the tool-tip vt in the latter equation to ensure that the lateral error dpf is regulated to zero while progressing along the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' An intuitive solution consists of decomposing the control velocity into two orthogonal components (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 5): i) the advance velocity (vadv) along the path, and ii) the return velocity (vret) for regulating the tool deviation from the reference path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The previous 10 desired 3D path mk-1 tool body mk mk+1 M mk+2Figure 5: Representation of the different velocities involved in the path- following controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' concept is formulated as follows: vt = αkp ���� vadv + βdpf � �� � vret .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (15) The tuning coefficients of the controller α and β allow adjusting the priority between the advance and return velocities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Besides that, the controller stability demonstrated in [10] shows that α should be a positive scalar while β must be a negative scalar to ensure the system stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The choice of these gain factors can be imposed by a function of a con- stant velocity vtis > 0 that depends on the interaction between the surgical tool and the lesional tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This velocity could be tuned easily by the surgeon before or during the intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, (15) yields v2 tis ���� =∥vt∥2 = α2∥kp∥2 � �� � =1 + β2∥dpf∥2 � �� � =∥vret∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (16) The gain factor α is thus determined as α = � � v2 tis − ∥vret∥2 ∥vret∥2 < v2 tis 0 ∥vret∥2 > v2 tis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (17) If the tool is not far from the reference path, the first condition in (17) is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Otherwise, the priority is returning the tool-tip to the reference path, and the advance velocity is null (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', second condition in (17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 11 desired 3D path tool body mk ret adv mk+1The latter strategy proposed in [10] applies a constant value for the gain factor β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, this section presents a new formulation of β to make the controller sensitive to the path curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Thus, it is calculated by the following equation β = β′ � 1 + sign � d⊤ pf (Cp(sp) × kp) � � 1 − eγc∥Cp(sp)∥� � (18) where β′ is a negative gain for returning to path, sign(•) is a sign function to determine the direction along the reference path, and γc is a negative gain for sensing the amount of path curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The ratio between the gain factors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', vtis and β′) forms an acceptable error band around the reference path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' For instance, if β′ is higher than vtis, then the error band will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' On the contrary, in the case where vtis is bigger than β′, then the error band will be large since the priority is to advance along the reference path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The effect of this ratio is presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Furthermore, the control velocity of the tool-tip (15) could be repre- sented with respect to any desired frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Note that if the end-effector frame is selected, then the end-effector twist velocity eve is related to the linear velocity of the tool-tip evt as evt = [I3×3 − [eet]×] � �� � Lpf∈R3×6 � eve eωe � � �� � eve (19) whereby [eet]× is the skew-symmetric matrix associated to the vector eet, and Lpf is the interaction matrix related to the path-following task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Finally, the control velocity for the path-following task is deduced as eve = L† pf evt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (20) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 Bilateral Constrained Motion Controller As claimed above, the resection/ablation task is performed in a minimally invasive procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, the robot should perform the surgical task under the constraints of the incision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This section begins with the description of RCM (bilateral constraints), while the following section de- scribes the UCM (unilateral constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The RCM imposes that the center- line of tool body St should be coincident with the point Or.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Simultaneously, the tool-tip must follow the desired path inside the incision orifice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 12 Figure 6: Geometric scheme of the bilateral linear error drcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 6 shows a straight tool which is located far from the center-point of incision orifice Or.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The previous works [10,12] built the controller based on the angular error between the vectors et′ and er while the proposed controller in this section is based on the linear error drcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This new choice offers the controller to become independent of the tool shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Let us imagine that the tool-tip position in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 6 is fixed in space, but its length can change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' In the case of angular error, when the tool length increases, the error reduces its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, the linear error stays constant when the tool length changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, the new choice grants better numerical computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The error drcm is deduced by the orthogonal projection of the point Or onto the tool body St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The point pt′ is resultant from the latter projection that is calculated as follows ept′ = euet eu⊤ et eer (21) whereby euet is the unit vector of et expressed in Fe, and eer represents the vector between both points Oe and Or which is expressed in Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' In case the surgical tool is curved, the point pt′ is determined by dis- cretizing the tool body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Then the closest point onto the tool body is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' After that, the orthogonal projection is performed with respect to this point and the previous one on the tool center-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Thus, the error drcm is de- duced as drcm = eOr − ept′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=" (22) 13 center of the incision hole er et' tool body Pt rcm X M incision wallThe controller task is to find the spatial velocity of the robot end-effector eve for eliminating the rate-of-change of the bilateral linear error drcm." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Thereby, the time-derivative of the latter equation results in ˙drcm = evr − evt′ (23) where evt′ is the linear velocity of the projected point pt′ along the tool body, and evr is the linear velocity of the trocar point described in Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Indeed, the velocity of the projected point depends on the movement of the tool body with respect to the trocar point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Hence, this velocity is computed as [12] evt′ = ekt ekT t 1 + dTrcm(Ct(st) × ekt) evr (24) whereby Ct(st) is the tool curvature in the function of its arc length, and ekt is the instantaneous tangential unit-vector onto the tool curve/shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Since the calculation is done in the perspective of the end-effector frame Fe, it implies that this frame is fixed, and the other ones are dynamic with respect to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Consequently, the incision orifice virtually moves, and its linear velocity evr is related to the spatial velocity of the robot end-effector thanks to the following formula evr = � I3×3 − [eOr]× � � �� � Lr∈R3×6 eve .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (25) By injecting the latter equation in (24) then the resultant in (23), the time-derivative of the error drcm equals to ˙drcm = � I3 − ekt ekT t 1 + dTrcm(Ct(st) × ekt) � � I3×3 − [eOr]× � � �� � Lrcm∈R3×6 eve (26) where Lrcm is the interaction matrix which relates between the end-effector velocity eve and the rate-of-change of the error drcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Furthermore, a linearized proportional controller is applied to reduce the bilateral linear error in an exponential decay form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It defines the control velocity of the end-effector as eve = −λ L† rcm drcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (27) whereby λ is a positive gain which allows tuning the rate of exponential decay, and L† rcm is the pseudo-inverse of the interaction matrix Lrcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 14 Finally, the RCM task can be combined as the highest priority with the path-following task as the secondary criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The hierarchical controller deduces the control velocity, by replacing the equations (27) and (20) in equation (6), as eve = −λL† rcmdrcm + ˜L† pf � evt + λLpfL† rcmdrcm � , with ˜Lpf = Lpf � I − L† rcmLrcm � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (28) In the opposite case, the hierarchical controller sets the path-following task (20) as the highest priority while the RCM task (27) as the secondary one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The control velocity is deduced from equation (6) as eve = L† pf evt − ˜L† rcm � λdrcm + LrcmL† pf evt � , (29) with ˜Lrcm = Lrcm � I − L† pfLpf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (30) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='6 Unilaterally Constrained Motion Controller This section continues with the design of the path-following controller under unilateral constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Notice that the UCM task assumes the incision orifice is larger than the tool diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Consequently, it imposes on the tool-tip to follow the incision/ablation path while the tool body is free to move within the incision orifice as long as it does not damage the orifice wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, the formulation of the previous section needs to extend to satisfy the unilateral constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 7(left image 1) shows how the point pt′ is orthogonally projected onto the orifice wall in order to determine the closest point ph′ on the orifice wall Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The distance between the latter two points forms the vector error ducm which can be defined as (left image 2 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 7) ducm = et′r ���� =drcm − eh′r ���� =dwall .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (31) The question now is how to maintain the value of the error ducm greater or equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' For security issues, three regions are defined around the projected point ph′, as shown in the left image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 7: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' critical zone (dark red circle) which its border is defined by a minimal distance dmin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 15 Figure 7: Geometric modelling of the unilateral linear error ducm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' dangerous zone (light green circle) which its border is defined by a maximal distance dmax;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' safe zone which is the remain region outside the dangerous zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' When the Euclidean norm ∥ducm∥ is larger than the ”dangerous” dis- tance dmax, the tool can follow the reference path without any constraints since its location is in the safe zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, an admittance control is activated, which is composed of a virtual damper µobs, when the tool body passes the dangerous zone border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Indeed, the admittance control imposes unilateral constraint towards the safe point ps by generating a compensation velocity in the opposite direction to the orifice wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' By differentiating equation (31) with respect to time for deducing the velocity twist of the end-effector, it becomes equal to ˙ducm = ( evr − evt′) � �� � ˙drcm − ( evr − evh′) � �� � ˙dwall = evh′ − evt′ (32) The velocity of the projected point ph′ is deduced in the same way as equation (24) evh′ = ekh ekT h 1 + dTucm (Ch(sh) × ekh) evt′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=" (33) 16 Ph tool body (1) centerofthe dangerous zone incision hole et' incision wall ducm d rcm d 11DM, 30 critical zone 25mm (2)where Ch(sh) is the orifice curvature in function of its arc length, and ekh is the instantaneous tangential unit-vector onto the orifice curve." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' In another perspective, the latter equation describes how the projection of the point pt′ onto the geometric curve of the orifice wall Sh evolves with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The velocity evt′ is deduced by combining equations (24) and (25) evt′ = ekt ekT t 1 + dTrcm (Ct(st) × ekt) � I3×3 − [eOr]× � � �� � Lvt′ ∈R3×6 eve .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (34) Replacing equations (33) and (34) in (32) yields ˙ducm = � ekh ekT h 1 + dTucm (Ch(sh) × ekh) − I3×3 � Lvt′ � �� � Lucm∈R3×6 eve (35) whereas Lucm is the interaction matrix that relates the twist end-effector with the rate of change of the error ducm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Thereby, the control velocity of the UCM task is defined as eve = −µobsλL† ucmducm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (36) The damping coefficient µobs changes following a sigmoid function that depends on the vector ducm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It means that the gain µobs reaches its mini- mal value when ducm is higher than the safe distance dmax, where the tool location in the dangerous zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, µobs gradually increases until it reaches its maximal value when ducm is smaller than the critical distance dmin, where the tool location in the critical zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This behaviour is modeled as µobs = σmax 1 + e � σstep � ∥ducm∥−σmin �� (37) where σmax, σmin and σstep are tunable parameters for modifying the sigmoid form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Finally, the path-following task can be combined as the highest priority with the UCM task as the secondary criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The hierarchical controller deduces the control velocity, by replacing the equations (36) and (20) in equation (6), as eve = L† pf evt − ˜L† ucm � µobsλducm + LucmL† pf evt � , with ˜Lucm = Lucm � I − L† pfLpf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (38) 17 4 VALIDATION This section discusses several scenarios to evaluate qualitatively and quan- titatively the proposed methods and materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The developed controllers were first tested using our simulator framework and then in an experimental set-up that takes up the various components of the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='1 Implementation Issues This part begins by converting the patient’s ear to its numerical-twin and then its 3D printed-twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The first step to accomplish this job is the scan of the patient’s ear during the preoperative phase for getting DICOM (Digital Imaging and Communications in Medicine) images, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The DICOM images are handled by the software 3D Slicer which converts these images to a 3D surface model after a segmentation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Prior works were done in relation to this subject for achieving an automated seg- mentation process (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', [15, 30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, the segmentation process that we have done manually is not automated since this is not the focus of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' In the future, we believe that our segmentation process needs to be done again in an automated manner for efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The 3D Slicer software exports the segmentation results as STL files for each anatomical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Afterward, the software MeshLab treats the STL files for smoothing the surface and reducing the number of vertices and faces to cut down the final STL file size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This step produces the numerical-twin of the patient’s ear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The next step creates the 3D printed-twin for conducting the experimen- tal validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Indeed, a simplified version of the numerical-twin is imported in Solidworks for i) adding some thickness to the middle ear cavity and ii) creating the incision orifice through the mastoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' After that, the planning stage of the desired path within the middle ear cavity begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The path planning step can be optimized (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', [14,20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, this step was done manually on Solidworks to generate text files that contain the geometry of the reference path and the orifice wall as a sequence of 3D points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' These files are inputs for the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This step should be investigated in the future and add to the adequate functions in the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 9 presents the proposed control architect with the TCP/IP com- munication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This architect allows easy interchangeability between the real- system (robot) and its numerical-twin (simulator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The latter figure (the red block at the left-hand side) also shows that the implemented controller 18 Figure 8: The steps done to achieve a numerical and physical model of the middle ear cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' is firstly initialized with the end-effector and the incision orifice poses, ⋆Te and ⋆Tr respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' These poses must be described in the same frame (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', the world frame Fw or the camera Fc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' the tool geometry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content="DICOM images of the patient's ear " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='Preoperative scanning for the patient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content="Surface structure (numerical ear twin) of the patient's ear " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='Ossicles ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='Inner ear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='Temporal bone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='Middle ear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='External ear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='Chorda ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='cavity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='canal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='External ear canal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='tympani nerve ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='3D printed ear twin version for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='experimental validation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='Preoperative planning phase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='Red region ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='will be preserved ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='Mastoidectomy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='Temporal bone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='(Canal wall up) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='Green region ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='External ear canal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='will be removedFigure 9: Block diagram of the TCP/IP communication between the client ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='(proposed controller) and the server (simulator or robot) or vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 20 Robot Contro connection unit Simulator Robot control (case 2) Control Unit Robot control (case 1) Simulator controlSt is defined with respect to the end-effector frame Fe while the reference path Sp and the orifice wall Sh are described in the incision orifice frame Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Furthermore, the controllers should be initialized by the different gain coefficients before the control-loop starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The hierarchy controller arranges throughout the control-loop the prior- ity between the different tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', the approach task, the path-following task, and the RCM/UCM constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Indeed, the control-loop is mainly divided into three phases: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' the outside phase: the tool corrects its initial pose with respect to the incision orifice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This stage applies the approach task for regulating: i) the tool-tip position to the point located before the orifice center point, and ii) the tool-tip rotation as the rotation of the orifice reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This manoeuvrer is performed to ensure some security for the next phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' the transition phase: the tool-tip passes the center point of the inci- sion orifice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The RCM controller could oscillate when the trocar point is close to the tool-tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' These oscillations are generated because the controller computes large rotation displacement, due to the lever phe- nomena, for compensating the rotation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Thus, the trocar point is virtually moved to the first point on the reference path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Conse- quently, the tool body can rotate about this new point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This virtual trocar point moves towards the orifice frame while the tool-tip ad- vances along the reference path;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' the inside phase: the tool-tip follows the desired path while the tool body is constrained by the orifice wall or the orifice center point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, the output of this block is the spatial velocity of the end-effector expressed in its frame (eve) while its inputs are the instantaneous poses of the end-effector and the incision orifice (⋆Te and ⋆Tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The question now is: what is the observation frame?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' In the simulator case (the blue block at the right-hand side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 9), it is straightforward since the user initializes the poses with respect to the world frame Fw of the virtual scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Thus, the spatial velocity eve is transformed to wve then it is integrated over the sample time Te to deduce the new pose of the end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Consequently, the tool pose is updated in the virtual scene, and this new pose is sent back to the control unit block for computing a new iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' There are two options for designing the control architect in the exper- imental case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The first one consists of using an exteroceptive sensor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', 21 camera) for estimating the required poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This option is depicted in the green block of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 9 named Robot control (case 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The input of this block is the spatial velocity eve that is transformed to deduce the angular velocity of each joint ˙q with the help of the inverse differential kinematic model to move mechanical structure of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This motion is observed from the camera frame Fc in order to estimate the new pose of the end-effector and that of the orifice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' These poses are the output of this block which are sent back to the control unit block for calculating a new iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, this option is uneasy for implementation since it needs a particular setup to accurately track both the end-effector and the orifice [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The second option is more fundamental than the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It is also presented in the green block of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 9 named Robot control (case 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It uses the proprioceptive sensors of the robot and its forward geometric model to estimate the end-effector pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Despite that, this option requires performing a registration process [9, 17] between the robot and the orifice before the control-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' After that, the robot works blindly, and the user assumes that the orifice does not move during the control-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The simulator is implemented in C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It uses Eigen library for linear algebra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', vectors, matrices, numerical solvers) and PCL (Point Cloud Library) for visualizing the STL parts and converting them to point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This conversion is done to initialize the collision detection that is accom- plished by VCollide library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Finally, ViSP library is used for manipulating the camera images throughout the experimental work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='2 Numerical Validation A numerical simulator was developed, as the first step, to validate the func- tioning of the diverse methods before physical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It simulates the geometric motion of the surgical tool through the incision orifice and the middle ear cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The software interchangeability of the simulator and the physical set-up allowed us also to tune the controller parameters before the experimental validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, this part presents three scenarios for the demonstration: scenario 1 performs the path-following task without any constraint applied on the tool motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It demonstrates the effect of the gain coefficients vtis and β in equations (16) and (18), respectively, on the performance of the path-following controller;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' scenario 2 performs the path-following task with RCM constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It simulates the drilling of a minimal invasive tunnel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', conical tunnel) 22 through the mastoid portion to reach the middle ear cavity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' scenario 3 assumes the surgeon performed a standard mastoidectomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It simulates an inspection/resection task performed under the UCM constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='1 Simulation of the path-following task without constraints Throughout this first trial, the value of vtis = 4mm/second in equation (16) remains constant during all tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Besides that, the same reference path is tested during this trial, and it is defined as a spiral curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Figure 10: The effect of the ratio between vdes and β′ on the path-following error dpf with a zoom and magnification on the orange region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The first group of tests keeps the value of γc in equation (18) constant while decreasing the value of β′ which its value varies from −4 to −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 10 shows the influence of the gain coefficient β′ on the path-following error dpf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Indeed, this error computed as in equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The ripples appearing in this figure represent the linear error between the projected point pt′ and the closest point on the reference path pp′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' An orange rectangle appeared in this figure for zooming on one of these ripples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' One can observe that the error reduced as designed exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The latter figure also demonstrates that the best ratio between β′ and vtis should be greater than −2 (the saddle-brown line with star markers), and less than or equal −3 (the olive line with square markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' If the ratio is less than or equal to −1, the controller response is relatively slow, and there is a steady-state error (the maroon line with round markers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' On 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='040 Udes = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0, β = -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0, %c = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='035 Udes = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0, β = -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0, %c = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 Udes = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0, β =-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0, %c = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 Udes = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0, β = -16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0, % = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='020 (mm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='4 ldpfll 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='000 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='1 1000 2000 3000 4000 5000 Iterationsthe opposite, if the ratio is higher than or equal to −4, the system begins to oscillate (having over-shoots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, the controller reduces the error faster than the previous cases (the sea-green line with triangular markers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The second group of tests chose a constant ratio −2 while decreasing the value of γc from −2 to −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This group shows that the best value of γc is to be near from β′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' If γc is higher than β′, the system begins to have over-shoots, but it reduces faster the path-following error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='2 Simulation of a robotic drilling task under RCM constraint The surgeon perforates manually until now the mastoid portion in the tem- poral bone for reaching the middle ear cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The resultant mastoidectomy orifice is invasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Thereby, a less invasive tunnel is proposed in this trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Besides that, the drilling procedure becomes automated so that the surgeon can concentrate on other essential tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Indeed, this drilling procedure is achieved by merging the approach task, the 3D path-following task, and the RCM task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (a) (b) Figure 11: Numerical validation of the 3D path-following under a RCM constraint (see Extension 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (a) The tool pose with respect to the desired path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (b) Sequence of zoom images during the tool motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 24 Tool body Incision center point 3D pathFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 11 depicts the tool motion throughout the drilling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The subplot (a) draws the tool geometry and its poses at different instances (or- ange straight-lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It also shows the drilling path defined as a combination of spiral and linear portions (sea-green dotted-line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' One can view that the tool body is always coincident with the orifice center point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The subplot (b1) shows the path done by the tool-tip (dodger-blue line) to accomplish the outside phase by i) approaching towards the point located before the orifice center point, and ii) regulating the rotation of the tool-tip frame to be as that of the orifice reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The subplot (b2) depicts an in- stantaneous zoom on the tool pose during the inside phase to visualize the RCM effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Figure 12: The approach task error eapp, where the left column is the linear error and the right column represents the angular error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The approach task error eapp computed in equation (11) is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 12 which depicts the linear errors in the column and the angular errors in the right one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Over this period, the error is reduced in an exponential form as planned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' At the end of the latter period, the transition phase starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='The task- hierarchical controller becomes active, and it arranges the path-following task as the highest priority while the RCM task is the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The errors of these tasks presented in the left columns of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 14 and 13 which are obtained from equations (13) and (22) for the path-following and RCM errors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' One can observe a peak appeared around 4 seconds in the path-following figure due to the initial error when the controller becomes activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Then, it attenuates the error until it attains stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Further- more, one can visualize in the RCM figure that three peaks appeared at the end of this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This behaviour happened due to the movement of the virtual trocar point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 25 20 12 Eappr Eappr 15 Eappy Eappy 10 Eapp: Eapp: 10 8 eapp 5 (mm) (deg) 6 ddpa Eappe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='10 2 0 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 Time (second) Time (second)Figure 13: The RCM task error drcm, where the left column shows the error evolution during the transition phase while the right column presents the error during the inside phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Figure 14: The path-following task error dpf, where the left column shows the error evolution during the transition phase while the right column presents the error during the inside phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' After the previous period, the inside phase starts where the hierarchical controller modifies the priority by setting the RCM task as the highest one while the path-following is the secondary one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The RCM task error drcm was 26 outside/transition phases inside phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='015 drcms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='15 drcmy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='010 drcm drcm: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='10 drcm Idrcml 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05 (mm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='000 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='010, 1 2 3 5 6 7 8 10 20 30 40 50 60 Time (second) Time (second)outside/transition phases inside phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='08 drcms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='20 drcm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='04 Idpf ll 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02 (mm) TAAAA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='00 dpf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05 drcmr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='04 drcmy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='06 II dp ll 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='08 1 2 3 4 5 6 7 8 10 20 30 40 50 60 Time (second) Time (second)computed as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='002 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='002 mm (mean error ± STD (STandard Deviation) error), as shown in the right column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 13, while the path-following error dpf was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='008±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='009 mm, as shown in the right column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The gain values used for this trail were equal to λ = 1, γ = 1, vtis = 4 mm/second, β′ = −10, γc = −10 and Te = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='008 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='3 Simulation of an ablation/excision surgical task under UCM constraint In this trial, the incision orifice size is larger than the instrument diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The tool is consequently subject to the UCM for providing more freedom to the tool movements inside the incision orifice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This behaviour is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 15a where the orifice wall is represented by the red surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The latter figure also presents the curved tool employed during this trial which performs an ablation or scanning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The desired 3D path is thus composed of a linear portion to reach the middle ear cavity and a spiral curve to simulate the required surgical task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This selected path can reach some regions where a straight tool cannot attain (see Extension 4 to visualize the collision of the latter one with the orifice wall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The subplot (b1) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 15b indicates the path done by the tool during the outside phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It also presents an instantaneous pose of the tool body throughout the transition phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' As explained in the previous trial, the proposed controller executes the same tasks over these two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Subplot (b2) presents the tool motion during the inside phase, where the dangerous and critical zones are represented by the green and red circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The center point of these circles corresponds to the point ph′ obtained by projecting pt′ onto the orifice wall Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Throughout the inside phase, the hierarchical controller combines the UCM task with the path-following task as described in (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 16 shows the UCM task error ducm which is deduced as in equation (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It also presents the boundaries of the critical and dangerous zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' One can observe that the error ducm begins with a considerable value, compared to the error drcm, since the previous phase delivers the tool to the center point of the incision orifice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Then, the error ducm reduced, while the error drcm increased because the tool approached the incision wall to follow the reference path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, the error ducm did not exceed the dmin, which implies the tool body did not enter the critical zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 17 presents the path-following error dpf during the inside phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It was measured was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='005 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='006 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The gain values used for this trail were equal to λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='8, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='8, vtis = 4 mm/second, β′ = − 10, 27 (a) (b) Figure 15: Numerical validation of the 3D path-following under a UCM constraint (see Extension 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (a) The tool pose with respect to the desired path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (b) Sequence of zoom images during the tool motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Figure 16: The UCM task error ducm during the inside phase along side the error drcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 28 reference curve actual curve tool body orificewall5 I drcm l Ild ucmll dmas 4 dmin Safe zone 3 dangerous zone Critical zone 0 10 15 20 25 Time (second)Figure 17: The path-following task error dpf during the inside phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' γc = − 10 and Te = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='008 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='3 Experimental Validation This part is devoted to the physical implementation of the blocks Robot control that is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Its physical correspondence is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The robotic work-cell in the latter figure consists of: a serial robot from Universal Robot (UR3) with ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='03 mm pose re- peatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It communicates with the proposed controller via TCP/IP for receiving the command velocity of the end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It also sends the end-effector pose to the controller if the block Robot control (case 1) is required to be executed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' a monocular camera from Guppy (with image size 640 × 420 pixels) and an optical objective lens from Computar with distortion (model MLM3X-MP) are used for the control purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This optical system tracks and estimates the poses of the end-effector and the incision orifice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It then sends these poses to the proposed controller if the block Robot control (case 2) is needed to be executed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='06 urcm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='04 dpf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02 (mm) WAWWAAMWWM dpf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='04 10 15 20 25 Time (second)• two visualization cameras provide other views for recording the mul- timedia videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Figure 18: Configuration of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The numerical twin of the ear model shown previously in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 8 is mod- ified for implementing its 3D printed twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This modification holds up the (a) (b) Figure 19: The printed ear model used during the different tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (a) The different parts of the ear model and the rigid tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (b) After assembling the different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 30 Robot controller End-effector Tool body Visualization Control cameras Incision orifice- cameramastoidectomy orifice with the middle ear cavity and a planar grid/marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 19 presented the fabricated parts before and after the assembly, along- side the rigid tools used during the validation tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The trials of this part have the objective to evaluate the performance of the path-following controller under constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, a curved tool follows the same planned path, one time under the RCM constraint and the second time under the UCM constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='1 Path-Following under RCM Constraint (a) (b) Figure 20: Experimental validation of the 3D path-following under a RCM constraint (see Extension 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (a) The tool pose with respect to the desired path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (b) Sequence of zoom images during the tool motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 20 presents the desired path (sea-green dotted line), the resultant motion of the curved tool (orange line), and the path done by tool-tip (dodger-blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' One can observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 20b(1) that the tool approaches to the incision orifice by executing the controller given in equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The approach task error eapp computed from equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 21 presents the latter error and it converges toward zero by the end of this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Afterward, the transition phase starts so that the tool passes the center point of the incision orifice, as explained previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The hierarchical con- troller (equation 29) arranges the path-following task as the highest priority while the RCM task is the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This behaviour is demonstrated in the left column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 22-23, where the hierarchical controller has been ac- tivated around 4 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' One can visualize that the RCM task error drcm 31 reference curve actualcurve tool bodyFigure 21: The approach task error eapp, where the left column is the linear error and the right column represents the angular error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' has some steps due to the movements of the virtual trocar point while the path-following error dpf maintained its value around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Figure 22: The RCM task error drcm, where the left column shows the error evolution during the outside/transition phases while the right column presents the error during the inside phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' When the tool passes the center point of the incision orifice, the inside phase begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The hierarchical controller (equation 28) modifies its priorities by setting the RCM task as the highest one and the path-following as the 32 25 50 linear angular 20 40 (mm) 15 10 20 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 Time (second) Time (second)outside/transition phases inside phase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 (u) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0 0 2 4 6 8 10 12 14 16 5 20 25 30 40 45 50 Time (second) Time (second)Figure 23: The path-following task error dpf, where the left column shows the error evolution during the outside/transition phases while the right col- umn presents the error during the inside phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The system performances during the inside phase are shown in the right columns of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 22-23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' During this phase, the RCM task error drcm measured as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05mm (mean error ± standard deviation (STD) error) while the path-following error dpf was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='03mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' A exteroceptive sensor used to close the control loop, as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 9 by the block Robot control (case 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Besides that, the gain values used in this experiment were equal to λ = 1, γ = 1, vtis = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 mm/second, β′ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='25, γc = −10 and Te = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='008 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Another trial was conducted for testing the block Robot control (case 1) by using the proprioceptive sensor in the control loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The system perfor- mances are better than the exteroceptive test (see test 2 in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The errors drcm and dpf are reduced to almost half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It implies that our vision system needed amelioration in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' From the surgeon’s perspective, it is required to target the residual cells of cholesteatoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It implies that the robot should detect/remove a human cell whose size is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='1mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The proposed controller reached the re- quirements since the error dpf is smaller than the human cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Besides that, the surgical tool does not damage the entry orifice (patient’s head).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' By increasing the tool velocity vtis = 2 mm/second and maintain the same ratio β′/vtis = −2, the system performances deteriorated as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The errors drcm and dpf are almost increase by half (see tests 2 and 4 in 33 outside/transition phases inside phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='20 (mm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='00 0 2 4 6 8 10 12 14 16 15 20 30 3540 45 50 Time (second) Time (second)Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Therefore, the choice of the gain coefficients effects the system per- formances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='2 Path Following under UCM Constraint This second trial assumes the same conditions as the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It in- volves the same curved tool and the desired path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, this trial im- posed a unilateral constraint on the tool motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Consequently, the tool can leave the center point of the incision orifice and move near the orifice wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' This behaviour is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The sub-figure (b1) of the lat- ter figure shows the path done by the tool-tip during the outside/transition phases, while the sub-figure (b2) presents the tool-tip path during the inside phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The dangerous and critical regions are presented by the green and red circles in the latter sub-figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (a) (b) Figure 24: Experimental validation of the 3D path-following under a UCM constraint (see Extension 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (a) The tool motion during the different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' (b) Sequence of zoom images during the tool motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Throughout the inside phase, the hierarchical controller arranges the different tasks as explained in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The highest priority is the path- following task when the tool is located in the safe zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' However, the highest priority changes to the UCM task when the tool body passes the danger zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The system performances are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 25-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' One can observe from the UCM task error ducm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 25) that the tool body is maintained in 34 referencecurve actualcurve tool body orificewallFigure 25: The UCM task error ducm during the inside phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Figure 26: The path-following task error dpf during the inside phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' the dangerous zone since the error ducm changes its value between dmax and dmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Besides that, the path-following error dpf (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 26) was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='03mm (mean error ± STD error) and its median error was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' A exteroceptive sensor used as the feedback sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Additionally, the gain values used for this second trial were equal to λ = 1, γ = 1, vtis = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='5 mm/second, β′ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='25, γc = −10 and Te = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='008 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The error dpf of this trial remains almost the same as the previous trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' It implies that the UCM constraint does not deteriorate the path-following error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Indeed, it provides the surgical tool to move with more liberty in order to take advantage of the large size of the entry orifice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 35 5 Idrcm I d ucm ll dmax 4 dmin Safe zone (mm) 3 dangerous zone Critical zone 0 15 20 25 30 35 40 Time (second)inside phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='12 (mm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='08 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='0Q 10 15 20 25 30 35 40 45 Time (second)N° constraint feedback type of error mean (∥e∥) ± STD 1 RCM exteroceptive drcm dpf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='06±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02 2 RCM exteroceptive drcm dpf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05 3 RCM proprioceptive drcm dpf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='01 4 RCM proprioceptive drcm dpf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02 5 UCM exteroceptive drcm dpf 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='03 6 UCM exteroceptive drcm dpf 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='06 7 UCM proprioceptive drcm dpf 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='01 8 UCM proprioceptive drcm dpf 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='69±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='02 Table 1: Summary of different trials achieved with the curved tool during the experimental tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' ∥e∥ (in mm) is the absolute average of the linear error along x − y − z axes, and STD is the related standard deviation (in mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Results obtained with the following parameters: λ = 1, vtis = 0, 5 mm/s, and Te = 0, 008 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The while trials applied β′ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='25, while the blue ones applied β′ = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 5 CONCLUSION AND FUTURE WORK This article discussed the design of an original controller for guiding a rigid instrument under constrained motions such as RCM or UCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The proposed methodology allows a generic formulation, in the same controller, two tasks: i) the constrained motion (RCM or UCM), and ii) a revisited 3D path- following scheme by increasing the sensitivity to the path complexity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=', curvature radius) and then reducing the path-following error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' To manage the achievement of two or more tasks without conflicts, we also implemented a task prioritizing paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Consequently, the developed control scheme can be integrated easily with various robotic systems without an accurate knowledge of the robot inverse kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' 36 Experimental validation was also successfully conducted using a 6-DoF robotic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The obtained results are promising in terms of behavior and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' These performances, even if they meet the specifications of the targeted middle ear surgery, may be considered improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The positioning error depends directly on the registration process that is not treated optimally in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Furthermore, the pose estimation of the tool- tip was done based on a geometric model of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Its estimation could be another source of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Thus, it would be interesting to find out another method for estimating the tool shape and the pose of its tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' The forthcoming work will implement the discussed methods in a clinical context using a realistic phantom and a human cadaver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Besides that, a force control could be added to increase the robot sensitivity to its environment and increase the level of security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the Inserm ROBOT Project: ITMO Cancer no 17CP068-00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' References [1] Jake J Abbott, Panadda Marayong, and Allison M Okamura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} +page_content=' Haptic virtual fixtures for robot-assisted manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf'} 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a/6dAyT4oBgHgl3EQfpfjS/content/tmp_files/2301.00528v1.pdf.txt b/6dAyT4oBgHgl3EQfpfjS/content/tmp_files/2301.00528v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..98ad84a12260b103f85a1bd9c8750f6d3536e1d4 --- /dev/null +++ b/6dAyT4oBgHgl3EQfpfjS/content/tmp_files/2301.00528v1.pdf.txt @@ -0,0 +1,505 @@ +Adaptive Quantum Amplitude Estimation +Xi Lu1 and Hongwei Lin1, ∗ +1School of Mathematical Science, Zhejiang University, Hangzhou, 310027, China +The maximum likelihood amplitude estimation (MLAE) algorithm is a practical solution to the +quantum amplitude estimation problem, which has a theoretically quadratic speedup over classical +Monte Carlo method. However, we find that MLAE is not unbiased, which is one of the major causes +of its inaccuracy. We propose an adaptive quantum amplitude estimation (AQAE) algorithm by +choosing MLAE parameters adaptively to avoid critical points. We also do numerical experiments +to show that our algorithm is approximately unbiased and more efficient than MLAE. +I. +INTRODUCTION +Quantum computing is an emerging subject that studies faster solutions on quantum computers over clas- +sical ones. Early quantum algorithms have achieved astonishing speedups over known classical algorithms, +such as the quadratic speedup of Grover’s search [1], and the exponential speedup of Shor’s integer factor- +ization [2]. Later algorithms like quantum approximate optimization algorithms (QAOA) [3–5], variational +quantum eigen solver (VQE) [6, 7] and quantum neural networks (QNN) [8, 9] also shows great potentials in +quantum computing. +The amplitude estimation problem [10] is one of the most fundamental problems in quantum computing, a +quantum variant of the classical Monte Carlo problem. Let A be any quantum algorithm that performs the +following unitary transformation, +A |00 · · · 0⟩ = +√ +1 − a |ψ0⟩ |0⟩ + √a |ψ1⟩ |1⟩ = cos φ |ψ0⟩ |0⟩ + sin φ |ψ1⟩ |1⟩ . +(1) +The goal of amplitude estimation problem is to estimate a. It is derived from the well-known phase estimation +problem, and has been widely applied in quantum chemistry [11–13] and machine learning [14, 15] in recent +studies. +The earliest solution [10] is a combination of quantum phase estimation and Grover’s search. There are some +later researches [16–19] that improve the robustness of phase estimation. The modified Grover’s operator [20] +is an approach that is designed to perform robustly under depolarizing noise. However, most of the recent +researches study amplitude estimation algorithms without the use of phase estimation, since it is believed that +the controlled amplification operations required by phase estimation can be different to implement on noise +intermediate-scale quantum devices. The maximum likelihood amplitude estimation (MLAE) [21] algorithm is +an approach without phase estimation, which is proved to have an error convergence O(N −1) asymptotically +when using an exponential incremental sequence (EIS), which is quadratically faster than O(N −1/2) for +classical Monte Carlo algorithm. The error convergence O(N −1) is also known as the Heisenberg limit [22]. +There is a variant of MLAE [23] that is built for noisy devices without estimating the noise parameters. +The iterative quantum amplitude estimation (IQPE) [24] is another approach without phase estimation, +which is proved rigorously to achieve a quadratic speedup up to a double-logarithmic factor compared to +classical Monte Carlo (MC) estimation. The variational amplitude estimation [25] is a variational quantum +algorithm based on constant-depth quantum circuits that also outperforms MC. There are also several other +approaches [26, 27]. +In this paper, we dive further into MLAE. In more precise experiments we find that the MLAE algorithm +is not unbiased, and the bias behaves periodically with respect to the ground truth a, as shown in Fig. 1. +Moreover, statistics theories show that the variance of any estimation ˜a follows the Cram´er-Rao inequality [28], +E[(˜a − a)2] ≥ [1 + b′(a)]2 +F(a) ++ b(a)2, +(2) +∗ hwlin@zju.edu.cn +arXiv:2301.00528v1 [quant-ph] 2 Jan 2023 + +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.002 +0.000 +0.002 +0.004 +0.006 +0.008 +Bias +RMSE +CRLB +FIG. 1: The bias and root of mean squared error (RMSE) of MLAE, for different a. The unbiased +Cram´er-Rao lower bound (CRLB) is the ideal distribution of RMSE, which is equal to Eq. (2) where +b(a) = 0. +where b(a) = E[˜a − a] is the bias, and the Fisher information F is defined as, +F(a) = E +��∂ ln L(a) +∂a +�2� +, +(3) +where L is the likelihood function of MLAE. An estimation is fully efficient [29] if it is unbiased and saturates +the Cram´er-Rao inequality. From Fig. 1, we can see that MLAE is approximately unbiased and close to the +unbiased Cram´er-Rao lower bound in most area, except some periodical small intervals. We improve MLAE +and propose the adaptive quantum amplitude estimation (AQAE) algorithm in this paper by introducing an +adaptive rule to avoid these small intervals, and show that our estimation algorithm is approximately efficient +with numerical experiments. +II. +PRELIMINARY +Most amplitude estimation algorithms are based on a general procedure called amplitude amplification [10], +which performs the transformation +QmA |00 · · · 0⟩ = cos[(2m + 1)φ] |ψ0⟩ |0⟩ + sin[(2m + 1)φ] |ψ1⟩ |1⟩ , +(4) +where +Q = A(2 |00 · · · 0⟩⟨00 · · · 0| − I)A−1(I ⊗ Z). +(5) +By measuring the last qubit with respect to the computational basis we obtain one with probability +sin2[(2m + 1)φ], and zero with probability cos2[(2m + 1)φ]. Such amplitude amplification process requires +(2m + 1) calls to the oracle A. +The MLAE algorithm requires parameters {mk, Rk}K +k=1. For each k the state QmkA |00 · · · 0⟩ is measured +for Rk times. Let hk be the number of ones in all Rk measurement results. The final estimation ˜a is obtained +by maximizing the likelihood function +L(a) := +K +� +k=1 +ℓk(φ), +(6) +where a ≡ sin2 φ, and +ℓk(φ) := +� +sin2(Mkφ) +�hk � +cos2(Mkφ) +�Rk−hk , +(7) + +3 +1( ) +2( ) +3( ) +4( ) +5( ) +FIG. 2: An illustration of how MLAE works. The curves illustrate the function ℓk(φ) for each k. Here +M1 = 1, Mk = 2k + 1(k = 2, 3, 4, 5). +where Mk ≡ 2mk + 1. +The Fig. 2 illustrates how MLAE works. Generally the function ℓk(φ) has Mk peaks. For M1 = 1, there is +a single smooth peak in the likelihood function ℓ1(φ). For bigger Mks, the peaks are sharper and thus have +better estimation ability, but there is more than one peak. So we cannot get more accurate estimation with +ℓk(φ) alone. The MLAE algorithm combines the information of ℓk(φ) for different Mks by multiplying all +those likelihood functions, thus obtaining a likelihood function L that has only one sharp peak. +By calculation the Fisher information of MLAE is [21], +F(a) = +1 +a(1 − a) +� +k +RkM 2 +k. +(8) +In most application problems the major complexity lies in the oracle A itself. Therefore, the time cost of +MLAE is, +N = +� +k +RkMk. +(9) +The original article about MLAE algorithm [21] presents two strategies of choosing parameters, +• Linear Incremental Sequence (LIS): mk = k − 1 and Rk = R for k = 1, 2, · · · , K, which has error +convergence ε ∼ N −3/4; +• Exponential Incremental Sequence (EIS): m1 = 0, mk = 2k−2(k = 2, 3, · · · , K) and Rk = R(k = +1, 2, · · · , K), which has error convergence ε ∼ N −1. +As MLAE is approximately unbiased and saturates the Cram´er-Rao inequality in most area, the RMSE +has the same error convergence as F−1/2. The MLAE algorithm with EIS fixes R1 = · · · = RK = R, and +chooses M1 = 1, Mk = 2k−1 + 1(k ≥ 2), then N = O(R · 2K) and F−1/2 = O(R−1/2 · 2−K) = O(N −1), which +is quadratically faster than MC and reaches the Heisenberg limit. But in reality, the existence of the bias +term in Eq. (2) has a significant impact and violates the quadratic speedup, as is shown by the numerical +experiments in the next section. +III. +THEORY AND ALGORITHM +In the beginning of this section, we set up a model for the bias of MLAE. We call, +� +sin2 +� j +m +π +2 +�����j = 1, 2, · · · , m − 1 +� +(10) + +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.002 +0.000 +0.002 +0.004 +0.006 +0.008 +Bias +RMSE +CRLB +FIG. 3: The bias and RMSE of MLAE with parameters K = 5, R = 32. The vertical black dashed lines are +the critical points of order M5 = 24 + 1 = 17. +the critical points of order m. In MLAE, consider two values on each side of some critical point sin2(jπ/2MK), +namely a± = sin2(φ±) = sin2(jπ/2MK ± ε). It is harder for the likelihood function Eq. (6) to tell apart +a± = sin2(φ±) = sin2(jπ/2MK ± ε) when ε is small, as ℓK(φ+) = ℓK(φ−), and thus they can only be told +apart by other terms {ℓk(φ)}K−1 +k=1 that is less sharp than ℓK(φ). As a result, MLAE has a positive bias when +a− is the ground truth, and has a negative bias when a+ is the ground truth. It should be mentioned that +other smaller Mks can also bring bias around their critical points, which is anyway not so obvious as MK. +Our theory concludes that MLAE has obvious bias in the intervals centered at each critical point of order +MK, as shown in Fig. 3. +By this observation, it is hard to flatten the bias curve on the whole interval [0, 1] with a fixed parameter +set {Mk, Rk}K +k=1. The intuition of our AQAE algorithm is that we can adaptively and randomly choose the +next Mk with the hope of staying away from the critical points of order Mk, namely {sin2(jπ/2Mk) : j = +1, 2, · · · , Mk − 1}. +Define the score function, +s(M; ˆa) = sin2(2M ˆφ), +(11) +where ˆa ≡ sin2(ˆφ), which is close to zero when ˆa is close to a critical point of order M. +Suppose we have already performed amplitude amplification procedures with parameters {Mk, Rk}K′ +k=1, +and have got the results {hk}K′ +k=1. The Bayes theory tells us the posterior probability density distributions +of a is, +ρ(ˆa) ∝ +K′ +� +k=1 +ℓk(ˆφ). +(12) +Combining the score function Eq. (11), we define the weight function about M as the expectation of the +score function, +w(M) = Eˆa[s(M; ˆa)] ∝ +� 1 +0 +s(M; ˆa) +� +� +K′ +� +k=1 +ℓk(ˆφ) +� +� d ˆa. +(13) +Similar to the score function, if ˆa is distributed mostly around some critical point of M, then w(M) is +small. The weight function is our guidance for the adaptive choice of the subsequent parameters {Mk, Rk}. +To avoid critical points, the key idea of AQAE is that the smaller w(Mk) is, the smaller Rk will be. +The Eq. (4) enables us to generate a 0-1 distribution random variable with p(1) = sin2[Mφ] for any odd +number M. An important thing for AQAE is that we should generalize it to the even case. From Eq. (1) we + +5 +have, +cos φA−1(|ψ0⟩ |0⟩) + sin φA−1(|ψ1⟩ |1⟩) = |00 · · · 0⟩ . +(14) +By the orthogonality of A−1 we know that, +|ψ′⟩ := sin φA−1(|ψ0⟩ |0⟩) − cos φA−1(|ψ1⟩ |1⟩), +(15) +is orthogonal to |00 · · · 0⟩. That is, if we measure all qubits of |ψ′⟩ under the computational basis, we will +certainly get results that contain one. Moreover, +A−1 |ψ0⟩ |0⟩ = cos φ |00 · · · 0⟩ + sin φ |ψ′⟩ , +(16) +A−1 |ψ1⟩ |1⟩ = sin φ |00 · · · 0⟩ − cos φ |ψ′⟩ . +(17) +Define, +Q′ = A−1(I ⊗ Z)A(2 |00 · · · 0⟩⟨00 · · · 0| − I). +(18) +Then, +Q′ |00 · · · 0⟩ =A−1(I ⊗ Z)A |00 · · · 0⟩ +=A−1(I ⊗ Z)(cos φ |ψ0⟩ |0⟩ + sin φ |ψ1⟩ |1⟩) +=A−1(cos φ |ψ0⟩ |0⟩ − sin φ |ψ1⟩ |1⟩) += cos φ(cos φ |00 · · · 0⟩ + sin φ |ψ′⟩) − sin φ(sin φ |00 · · · 0⟩ − cos φ |ψ′⟩) += cos(2φ) |00 · · · 0⟩ + sin(2φ) |ψ′⟩ , +(19) +and, +Q′ |ψ′⟩ =A−1(I ⊗ Z)A(− |ψ′⟩) +=A−1(I ⊗ Z)(− sin φ |ψ0⟩ |0⟩ + cos φ |ψ1⟩ |1⟩) +=A−1(− sin φ |ψ0⟩ |0⟩ − cos φ |ψ1⟩ |1⟩) += − sin φ(cos φ |00 · · · 0⟩ + sin φ |ψ′⟩) − cos φ(sin φ |00 · · · 0⟩ − cos φ |ψ′⟩) += − sin(2φ) |00 · · · 0⟩ + cos(2φ) |ψ′⟩ . +(20) +Therefore, Q′ is a rotation by angle 2φ in the plane spanned by |00 · · · 0⟩ and |ψ′⟩. We can deduce that, +Q′m |00 · · · 0⟩ = cos(2mφ) |00 · · · 0⟩ + sin(2mφ) |ψ′⟩ . +(21) +By measuring all qubits under the computational basis we obtain all zero with probability cos2(2mφ), and +results containing one with probability sin2(2mφ). The extended amplitude amplification process requires +2m calls to the oracle A. +In summary, no matter M is odd or even, we can obtain a random variable rM with 0-1 distribution where +p(1) = sin2(Mφ), with a cost of M oracle calls to the oracle A. When M is odd, we measure the last qubit +of the state Q(M−1)/2A |00 · · · 0⟩, and obtain one with probability sin2(Mφ). When M is even, we measure +all qubits of the state Q′M/2 |00 · · · 0⟩, and the probability that the results contain one is sin2(Mφ). For +convenience, we use the terminology measuring rM to mean that we use the procedure above to obtain a +random variable of 0-1 distribution with p(1) = sin2(Mφ). The extended amplitude amplification is crucial + +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0000 +0.0005 +0.0010 +0.0015 +0.0020 +0.0025 +0.0030 +0.0035 +Bias +RMSE +CRLB +FIG. 4: The bias, RMSE and CRLB for AQAE, with parameters K = 5 and R = 32. The CRLB is +calculated as the average value of F−1/2 = [ +1 +a(1−a) +� +k RkM 2 +k]−1/2, according to Eq. (8). +to our proposed algorithm. +Algorithm 1: Adaptive Quantum Amplitude Estimation (AQAE) +Input +: K: Number of iterations; R: Number of measurements in each iteration; +Output: ˜a: Estimation of a; +1 Set M1 = 1 and R1 = R; +2 Measure r1 for R times and let h1 be the number of ones; +3 for i = 2..K do +4 +Calculate the weights {w(m)}2i−1 +m=2i−1; +5 +Set Mm = m and Rm = 0 for m = 2i−1, · · · , 2i − 1; +6 +for j = 1..R do +7 +Draw a random sample mj from {2i−1, · · · , 2i − 1}, with probabilities w(mj)/ �2i−1 +m=2i−1 w(m); +8 +Increase Rmj by one; +9 +end +10 +Measure rk for Rk times and let hk be the number of ones; +11 end +12 Calculate ˜a using MLE. +Our algorithm is shown in Alg. 1. First we set M1 = 1 and R1 = R, and measure the state Eq. (1) +directly for R times to obtain h1. In the second iteration, we set M2 = 2 and M3 = 3, and compute w(2) +and w(3). We draw R samples from {2, 3} with probabilities +� +w(2) +w(2)+w(3), +w(3) +w(2)+w(3) +� +, and set R2, R3 to be +the number of 2 and 3 in the outcome, respectively. In the third iteration we run the same procedure for +M4 = 4, M5 = 5, M6 = 6, M7 = 7. After all K iterations, we apply the MLE to obtain the result ˜a. +We carry out several numerical experiments to show the efficiency of AQAE algorithm. All the quantum +outputs in the experiments are obtained by sampling the theoretic distribution functions. First, in comparison +to Fig. 1, the bias and RMSE curve for AQAE is shown in Fig. 4. We find that the bias intensity of AQAE +is much lower than MLAE. Besides, the RMSE curve is smooth and close to the average CRLB curve. +To illustrate how the parameters chosen by AQAE vary with different as, we make statistics for two typical +as, as shown in Fig. 5. In Fig. 5 (a) all even Mks are chosen less frequently then odd Mks, since a is a critical +point of order 2. In Fig. 5 (b) all Mks that are multiples of 3 are chosen less frequently since a is a critical +point of order 3. This set of experiments show that AQAE can effectively avoid critical points. +Finally, we compare different amplitude estimation algorithms and take the time cost into consideration. +In this experiment we uniformly randomly draw 216 samples in the interval [0, 1] as a, and compare the +error behavior with respect to the time cost. For Monte Carlo (MC) estimation, suppose the state Eq. (1) is +prepared for R times, and by measuring the last qubit the result 1 is obtained for h times, then the estimation +to a is given by ˆa = h/R. The time cost for MC is N = R, as each preparation of the state Eq. (1) requires + +7 +0 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +30 +(a) When a = sin2(π/4) = 0.5, a critical point of order 2. +0 +5 +10 +15 +20 +25 +30 +0 +5 +10 +15 +20 +25 +30 +(b) When a = sin2(π/6) = 0.25, a critical point of order 3. +FIG. 5: The average Rk (y-axis) for each Mk (x-axis) chosen by AQAE when K = 5 and R = 32. The Mks +that are multiples of 2 in (a) or multiples of 3 in (b) are labelled orange. +102 +103 +104 +Time Cost +10 +3 +10 +2 +RMSE +MC +QPE +UQPE +IQAE +MLAE +AQAE +FIG. 6: The error behavior (y-axis) with respect to the time cost N (x-axis). +one call to the oracle A. For MLAE and AQAE, both algorithms require two parameters K and R, which +is chosen by pre-calculation that has the minimal RMSE among several parameter pairs with approximate +time cost. The time cost of MLAE is N = � +k RkMk = R(2K + K − 2). Since the parameter set {Mk, Rk} +is not fixed in AQAE, we calculate the average value of � +k RkMk chosen in numerical experiments as its +time cost. The quantum phase estimation (QPE) based amplitude estimation requires a parameter t as the +number of controlled qubits [30], with time cost N = �t−1 +j=0 2j = 2t − 1. An efficient way to reduce the +RMSE of QPE is to repeat for R times and use MLE to give the final estimation. The unbiased quantum +phase estimation (UQPE) [19] is an unbiased variant of QPE. The time cost for both QPE and UQPE in +our experiments is N = R(2t − 1). In our experiments we fix R = 4 and let t vary. For IQPE [24], we use +Clopper-Pearson confidence interval method, fix α = 0.05, Nshots = 100 and let ϵ vary. The results are shown +in Fig. 6. The MC algorithm have an error convergence of O(N −1/2), while all other algorithms have an + +8 +asymptotic O(N −1) error convergence. The UQPE performs the best among those algorithms. If we limit +the comparison in algorithms without phase estimation, as they are more likely to be implemented widely in +recent years, then our AQAE algorithm outperforms other algorithms. +IV. +CONCLUSION +The maximum likelihood amplitude estimation (MLAE) algorithm is a practical solution to the quan- +tum amplitude estimation problem, which has a theoretically quadratic speedup over classical Monte Carlo +method. We find that MLAE behaves efficient, i.e. unbiased and saturates the Cram´er-Rao inequality in +most area except some periodical small intervals. We analyze how the bias occurs around the so-called crit- +ical points, and propose an adaptive quantum amplitude estimation (AQAE) algorithm by choosing MLAE +parameters adaptively to avoid critical points. In the end, we do numerical experiments among some ampli- +tude estimation algorithms, including Monte Carlo estimation, quantum phase estimation and its unbiased +variant, iterative quantum amplitude estimation, maximum likelihood amplitude estimation and our adaptive +amplitude estimation. We show that our algorithm outperforms the original MLAE obviously, and it behaves +the best among all algorithms without phase estimation. +[1] Lov K. Grover. Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett., 79:325, 7 +1997. +[2] Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum com- +puter. 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Iterative quantum amplitude estimation. +NPJ Quantum Inf., 7:1–6, 3 2021. +[25] Kirill Plekhanov, Matthias Rosenkranz, Mattia Fiorentini, and Michael Lubasch. Variational quantum amplitude +estimation. Quantum, 6:670, March 2022. +[26] Scott Aaronson and Patrick Rall. Quantum approximate counting, simplified. In Symposium on Simplicity in +Algorithms, pages 24–32. SIAM, 2020. +[27] Kouhei Nakaji. Faster amplitude estimation. arXiv:2003.02417, 2020. +[28] S. Kullback. Certain inequalities in information theory and the cramer-rao inequality. The Annals of Mathematical +Statistics, 25(4):745–751, 1954. +[29] Ronald A Fisher. On the mathematical foundations of theoretical statistics. Philosophical transactions of the +Royal Society of London. Series A, containing papers of a mathematical or physical character, 222(594-604):309– +368, 1922. +[30] Michael A Nielsen and Isaac L Chuang. Quantum computation and quantum information. Cambridge University +Press, 2010. + diff --git a/6dAyT4oBgHgl3EQfpfjS/content/tmp_files/load_file.txt b/6dAyT4oBgHgl3EQfpfjS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a8213b00e277b65ab522bb5a60bdccaed06141ef --- /dev/null +++ b/6dAyT4oBgHgl3EQfpfjS/content/tmp_files/load_file.txt @@ -0,0 +1,355 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf,len=354 +page_content='Adaptive Quantum Amplitude Estimation Xi Lu1 and Hongwei Lin1, ∗ 1School of Mathematical Science, Zhejiang University, Hangzhou, 310027, China The maximum likelihood amplitude estimation (MLAE) algorithm is a practical solution to the quantum amplitude estimation problem, which has a theoretically quadratic speedup over classical Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' However, we find that MLAE is not unbiased, which is one of the major causes of its inaccuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' We propose an adaptive quantum amplitude estimation (AQAE) algorithm by choosing MLAE parameters adaptively to avoid critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' We also do numerical experiments to show that our algorithm is approximately unbiased and more efficient than MLAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' INTRODUCTION Quantum computing is an emerging subject that studies faster solutions on quantum computers over clas- sical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Early quantum algorithms have achieved astonishing speedups over known classical algorithms, such as the quadratic speedup of Grover’s search [1], and the exponential speedup of Shor’s integer factor- ization [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Later algorithms like quantum approximate optimization algorithms (QAOA) [3–5], variational quantum eigen solver (VQE) [6, 7] and quantum neural networks (QNN) [8, 9] also shows great potentials in quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The amplitude estimation problem [10] is one of the most fundamental problems in quantum computing, a quantum variant of the classical Monte Carlo problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Let A be any quantum algorithm that performs the following unitary transformation, A |00 · · · 0⟩ = √ 1 − a |ψ0⟩ |0⟩ + √a |ψ1⟩ |1⟩ = cos φ |ψ0⟩ |0⟩ + sin φ |ψ1⟩ |1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (1) The goal of amplitude estimation problem is to estimate a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' It is derived from the well-known phase estimation problem, and has been widely applied in quantum chemistry [11–13] and machine learning [14, 15] in recent studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The earliest solution [10] is a combination of quantum phase estimation and Grover’s search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' There are some later researches [16–19] that improve the robustness of phase estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The modified Grover’s operator [20] is an approach that is designed to perform robustly under depolarizing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' However, most of the recent researches study amplitude estimation algorithms without the use of phase estimation, since it is believed that the controlled amplification operations required by phase estimation can be different to implement on noise intermediate-scale quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The maximum likelihood amplitude estimation (MLAE) [21] algorithm is an approach without phase estimation, which is proved to have an error convergence O(N −1) asymptotically when using an exponential incremental sequence (EIS), which is quadratically faster than O(N −1/2) for classical Monte Carlo algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The error convergence O(N −1) is also known as the Heisenberg limit [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' There is a variant of MLAE [23] that is built for noisy devices without estimating the noise parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The iterative quantum amplitude estimation (IQPE) [24] is another approach without phase estimation, which is proved rigorously to achieve a quadratic speedup up to a double-logarithmic factor compared to classical Monte Carlo (MC) estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The variational amplitude estimation [25] is a variational quantum algorithm based on constant-depth quantum circuits that also outperforms MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' There are also several other approaches [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' In this paper, we dive further into MLAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' In more precise experiments we find that the MLAE algorithm is not unbiased, and the bias behaves periodically with respect to the ground truth a, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Moreover, statistics theories show that the variance of any estimation ˜a follows the Cram´er-Rao inequality [28], E[(˜a − a)2] ≥ [1 + b′(a)]2 F(a) + b(a)2, (2) ∗ hwlin@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='00528v1 [quant-ph] 2 Jan 2023 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='008 Bias RMSE CRLB FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 1: The bias and root of mean squared error (RMSE) of MLAE, for different a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The unbiased Cram´er-Rao lower bound (CRLB) is the ideal distribution of RMSE, which is equal to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (2) where b(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' where b(a) = E[˜a − a] is the bias, and the Fisher information F is defined as, F(a) = E ��∂ ln L(a) ∂a �2� , (3) where L is the likelihood function of MLAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' An estimation is fully efficient [29] if it is unbiased and saturates the Cram´er-Rao inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 1, we can see that MLAE is approximately unbiased and close to the unbiased Cram´er-Rao lower bound in most area, except some periodical small intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' We improve MLAE and propose the adaptive quantum amplitude estimation (AQAE) algorithm in this paper by introducing an adaptive rule to avoid these small intervals, and show that our estimation algorithm is approximately efficient with numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' PRELIMINARY Most amplitude estimation algorithms are based on a general procedure called amplitude amplification [10], which performs the transformation QmA |00 · · · 0⟩ = cos[(2m + 1)φ] |ψ0⟩ |0⟩ + sin[(2m + 1)φ] |ψ1⟩ |1⟩ , (4) where Q = A(2 |00 · · · 0⟩⟨00 · · · 0| − I)A−1(I ⊗ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (5) By measuring the last qubit with respect to the computational basis we obtain one with probability sin2[(2m + 1)φ], and zero with probability cos2[(2m + 1)φ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Such amplitude amplification process requires (2m + 1) calls to the oracle A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The MLAE algorithm requires parameters {mk, Rk}K k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' For each k the state QmkA |00 · · · 0⟩ is measured for Rk times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Let hk be the number of ones in all Rk measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The final estimation ˜a is obtained by maximizing the likelihood function L(a) := K � k=1 ℓk(φ), (6) where a ≡ sin2 φ, and ℓk(φ) := � sin2(Mkφ) �hk � cos2(Mkφ) �Rk−hk , (7) 3 1( ) 2( ) 3( ) 4( ) 5( ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 2: An illustration of how MLAE works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The curves illustrate the function ℓk(φ) for each k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Here M1 = 1, Mk = 2k + 1(k = 2, 3, 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' where Mk ≡ 2mk + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 2 illustrates how MLAE works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Generally the function ℓk(φ) has Mk peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' For M1 = 1, there is a single smooth peak in the likelihood function ℓ1(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' For bigger Mks, the peaks are sharper and thus have better estimation ability, but there is more than one peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' So we cannot get more accurate estimation with ℓk(φ) alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The MLAE algorithm combines the information of ℓk(φ) for different Mks by multiplying all those likelihood functions, thus obtaining a likelihood function L that has only one sharp peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' By calculation the Fisher information of MLAE is [21], F(a) = 1 a(1 − a) � k RkM 2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (8) In most application problems the major complexity lies in the oracle A itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Therefore, the time cost of MLAE is, N = � k RkMk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (9) The original article about MLAE algorithm [21] presents two strategies of choosing parameters, Linear Incremental Sequence (LIS): mk = k − 1 and Rk = R for k = 1, 2, · · · , K, which has error convergence ε ∼ N −3/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Exponential Incremental Sequence (EIS): m1 = 0, mk = 2k−2(k = 2, 3, · · · , K) and Rk = R(k = 1, 2, · · · , K), which has error convergence ε ∼ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' As MLAE is approximately unbiased and saturates the Cram´er-Rao inequality in most area, the RMSE has the same error convergence as F−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The MLAE algorithm with EIS fixes R1 = · · · = RK = R, and chooses M1 = 1, Mk = 2k−1 + 1(k ≥ 2), then N = O(R · 2K) and F−1/2 = O(R−1/2 · 2−K) = O(N −1), which is quadratically faster than MC and reaches the Heisenberg limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' But in reality, the existence of the bias term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (2) has a significant impact and violates the quadratic speedup, as is shown by the numerical experiments in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' THEORY AND ALGORITHM In the beginning of this section, we set up a model for the bias of MLAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' We call, � sin2 � j m π 2 �����j = 1, 2, · · · , m − 1 � (10) 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='008 Bias RMSE CRLB FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 3: The bias and RMSE of MLAE with parameters K = 5, R = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The vertical black dashed lines are the critical points of order M5 = 24 + 1 = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' the critical points of order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' In MLAE, consider two values on each side of some critical point sin2(jπ/2MK), namely a± = sin2(φ±) = sin2(jπ/2MK ± ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' It is harder for the likelihood function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (6) to tell apart a± = sin2(φ±) = sin2(jπ/2MK ± ε) when ε is small, as ℓK(φ+) = ℓK(φ−), and thus they can only be told apart by other terms {ℓk(φ)}K−1 k=1 that is less sharp than ℓK(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' As a result, MLAE has a positive bias when a− is the ground truth, and has a negative bias when a+ is the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' It should be mentioned that other smaller Mks can also bring bias around their critical points, which is anyway not so obvious as MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Our theory concludes that MLAE has obvious bias in the intervals centered at each critical point of order MK, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' By this observation, it is hard to flatten the bias curve on the whole interval [0, 1] with a fixed parameter set {Mk, Rk}K k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The intuition of our AQAE algorithm is that we can adaptively and randomly choose the next Mk with the hope of staying away from the critical points of order Mk, namely {sin2(jπ/2Mk) : j = 1, 2, · · · , Mk − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Define the score function, s(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' ˆa) = sin2(2M ˆφ), (11) where ˆa ≡ sin2(ˆφ), which is close to zero when ˆa is close to a critical point of order M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Suppose we have already performed amplitude amplification procedures with parameters {Mk, Rk}K′ k=1, and have got the results {hk}K′ k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The Bayes theory tells us the posterior probability density distributions of a is, ρ(ˆa) ∝ K′ � k=1 ℓk(ˆφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (12) Combining the score function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (11), we define the weight function about M as the expectation of the score function, w(M) = Eˆa[s(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' ˆa)] ∝ � 1 0 s(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' ˆa) � � K′ � k=1 ℓk(ˆφ) � � d ˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (13) Similar to the score function, if ˆa is distributed mostly around some critical point of M, then w(M) is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The weight function is our guidance for the adaptive choice of the subsequent parameters {Mk, Rk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' To avoid critical points, the key idea of AQAE is that the smaller w(Mk) is, the smaller Rk will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (4) enables us to generate a 0-1 distribution random variable with p(1) = sin2[Mφ] for any odd number M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' An important thing for AQAE is that we should generalize it to the even case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (1) we 5 have, cos φA−1(|ψ0⟩ |0⟩) + sin φA−1(|ψ1⟩ |1⟩) = |00 · · · 0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (14) By the orthogonality of A−1 we know that, |ψ′⟩ := sin φA−1(|ψ0⟩ |0⟩) − cos φA−1(|ψ1⟩ |1⟩), (15) is orthogonal to |00 · · · 0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' That is, if we measure all qubits of |ψ′⟩ under the computational basis, we will certainly get results that contain one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Moreover, A−1 |ψ0⟩ |0⟩ = cos φ |00 · · · 0⟩ + sin φ |ψ′⟩ , (16) A−1 |ψ1⟩ |1⟩ = sin φ |00 · · · 0⟩ − cos φ |ψ′⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (17) Define, Q′ = A−1(I ⊗ Z)A(2 |00 · · · 0⟩⟨00 · · · 0| − I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (18) Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Q′ |00 · · · 0⟩ =A−1(I ⊗ Z)A |00 · · · 0⟩ =A−1(I ⊗ Z)(cos φ |ψ0⟩ |0⟩ + sin φ |ψ1⟩ |1⟩) =A−1(cos φ |ψ0⟩ |0⟩ − sin φ |ψ1⟩ |1⟩) = cos φ(cos φ |00 · · · 0⟩ + sin φ |ψ′⟩) − sin φ(sin φ |00 · · · 0⟩ − cos φ |ψ′⟩) = cos(2φ) |00 · · · 0⟩ + sin(2φ) |ψ′⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (19) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Q′ |ψ′⟩ =A−1(I ⊗ Z)A(− |ψ′⟩) =A−1(I ⊗ Z)(− sin φ |ψ0⟩ |0⟩ + cos φ |ψ1⟩ |1⟩) =A−1(− sin φ |ψ0⟩ |0⟩ − cos φ |ψ1⟩ |1⟩) = − sin φ(cos φ |00 · · · 0⟩ + sin φ |ψ′⟩) − cos φ(sin φ |00 · · · 0⟩ − cos φ |ψ′⟩) = − sin(2φ) |00 · · · 0⟩ + cos(2φ) |ψ′⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (20) Therefore, Q′ is a rotation by angle 2φ in the plane spanned by |00 · · · 0⟩ and |ψ′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' We can deduce that, Q′m |00 · · · 0⟩ = cos(2mφ) |00 · · · 0⟩ + sin(2mφ) |ψ′⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (21) By measuring all qubits under the computational basis we obtain all zero with probability cos2(2mφ), and results containing one with probability sin2(2mφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The extended amplitude amplification process requires 2m calls to the oracle A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' In summary, no matter M is odd or even, we can obtain a random variable rM with 0-1 distribution where p(1) = sin2(Mφ), with a cost of M oracle calls to the oracle A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' When M is odd, we measure the last qubit of the state Q(M−1)/2A |00 · · · 0⟩, and obtain one with probability sin2(Mφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' When M is even, we measure all qubits of the state Q′M/2 |00 · · · 0⟩, and the probability that the results contain one is sin2(Mφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' For convenience, we use the terminology measuring rM to mean that we use the procedure above to obtain a random variable of 0-1 distribution with p(1) = sin2(Mφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The extended amplitude amplification is crucial 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='0035 Bias RMSE CRLB FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 4: The bias, RMSE and CRLB for AQAE, with parameters K = 5 and R = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The CRLB is calculated as the average value of F−1/2 = [ 1 a(1−a) � k RkM 2 k]−1/2, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' to our proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Algorithm 1: Adaptive Quantum Amplitude Estimation (AQAE) Input : K: Number of iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' R: Number of measurements in each iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Output: ˜a: Estimation of a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 1 Set M1 = 1 and R1 = R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 2 Measure r1 for R times and let h1 be the number of ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 3 for i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='.K do 4 Calculate the weights {w(m)}2i−1 m=2i−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 5 Set Mm = m and Rm = 0 for m = 2i−1, · · · , 2i − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 6 for j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='.R do 7 Draw a random sample mj from {2i−1, · · · , 2i − 1}, with probabilities w(mj)/ �2i−1 m=2i−1 w(m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 8 Increase Rmj by one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 9 end 10 Measure rk for Rk times and let hk be the number of ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 11 end 12 Calculate ˜a using MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Our algorithm is shown in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' First we set M1 = 1 and R1 = R, and measure the state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (1) directly for R times to obtain h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' In the second iteration, we set M2 = 2 and M3 = 3, and compute w(2) and w(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' We draw R samples from {2, 3} with probabilities � w(2) w(2)+w(3), w(3) w(2)+w(3) � , and set R2, R3 to be the number of 2 and 3 in the outcome, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' In the third iteration we run the same procedure for M4 = 4, M5 = 5, M6 = 6, M7 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' After all K iterations, we apply the MLE to obtain the result ˜a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' We carry out several numerical experiments to show the efficiency of AQAE algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' All the quantum outputs in the experiments are obtained by sampling the theoretic distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' First, in comparison to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 1, the bias and RMSE curve for AQAE is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' We find that the bias intensity of AQAE is much lower than MLAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Besides, the RMSE curve is smooth and close to the average CRLB curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' To illustrate how the parameters chosen by AQAE vary with different as, we make statistics for two typical as, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 5 (a) all even Mks are chosen less frequently then odd Mks, since a is a critical point of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 5 (b) all Mks that are multiples of 3 are chosen less frequently since a is a critical point of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' This set of experiments show that AQAE can effectively avoid critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Finally, we compare different amplitude estimation algorithms and take the time cost into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' In this experiment we uniformly randomly draw 216 samples in the interval [0, 1] as a, and compare the error behavior with respect to the time cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' For Monte Carlo (MC) estimation, suppose the state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (1) is prepared for R times, and by measuring the last qubit the result 1 is obtained for h times, then the estimation to a is given by ˆa = h/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The time cost for MC is N = R, as each preparation of the state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' (1) requires 7 0 5 10 15 20 25 30 0 5 10 15 20 25 30 (a) When a = sin2(π/4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='5, a critical point of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 0 5 10 15 20 25 30 0 5 10 15 20 25 30 (b) When a = sin2(π/6) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='25, a critical point of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 5: The average Rk (y-axis) for each Mk (x-axis) chosen by AQAE when K = 5 and R = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The Mks that are multiples of 2 in (a) or multiples of 3 in (b) are labelled orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 102 103 104 Time Cost 10 3 10 2 RMSE MC QPE UQPE IQAE MLAE AQAE FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 6: The error behavior (y-axis) with respect to the time cost N (x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' one call to the oracle A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' For MLAE and AQAE, both algorithms require two parameters K and R, which is chosen by pre-calculation that has the minimal RMSE among several parameter pairs with approximate time cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The time cost of MLAE is N = � k RkMk = R(2K + K − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Since the parameter set {Mk, Rk} is not fixed in AQAE, we calculate the average value of � k RkMk chosen in numerical experiments as its time cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The quantum phase estimation (QPE) based amplitude estimation requires a parameter t as the number of controlled qubits [30], with time cost N = �t−1 j=0 2j = 2t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' An efficient way to reduce the RMSE of QPE is to repeat for R times and use MLE to give the final estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The unbiased quantum phase estimation (UQPE) [19] is an unbiased variant of QPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The time cost for both QPE and UQPE in our experiments is N = R(2t − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' In our experiments we fix R = 4 and let t vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' For IQPE [24], we use Clopper-Pearson confidence interval method, fix α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='05, Nshots = 100 and let ϵ vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The MC algorithm have an error convergence of O(N −1/2), while all other algorithms have an 8 asymptotic O(N −1) error convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' The UQPE performs the best among those algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' If we limit the comparison in algorithms without phase estimation, as they are more likely to be implemented widely in recent years, then our AQAE algorithm outperforms other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' CONCLUSION The maximum likelihood amplitude estimation (MLAE) algorithm is a practical solution to the quan- tum amplitude estimation problem, which has a theoretically quadratic speedup over classical Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' We find that MLAE behaves efficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' unbiased and saturates the Cram´er-Rao inequality in most area except some periodical small intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' We analyze how the bias occurs around the so-called crit- ical points, and propose an adaptive quantum amplitude estimation (AQAE) algorithm by choosing MLAE parameters adaptively to avoid critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' In the end, we do numerical experiments among some ampli- tude estimation algorithms, including Monte Carlo estimation, quantum phase estimation and its unbiased variant, iterative quantum amplitude estimation, maximum likelihood amplitude estimation and our adaptive amplitude estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' We show that our algorithm outperforms the original MLAE obviously, and it behaves the best among all algorithms without phase estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' [1] Lov K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Grover.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Quantum computation and quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} +page_content=' Cambridge University Press, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'} diff --git a/7dAzT4oBgHgl3EQfgPzo/content/tmp_files/2301.01467v1.pdf.txt b/7dAzT4oBgHgl3EQfgPzo/content/tmp_files/2301.01467v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..31eb586a8f8ff28462fae90a7b4735af72365157 --- /dev/null +++ b/7dAzT4oBgHgl3EQfgPzo/content/tmp_files/2301.01467v1.pdf.txt @@ -0,0 +1,1188 @@ +arXiv:2301.01467v1 [cond-mat.supr-con] 4 Jan 2023 +Nodeless superconductivity in noncentrosymmetric LaRhSn +Z. Y. Nie,1, 2 J. W. Shu,1, 2 A. Wang,1, 2 H. Su,1, 2 W. Y. Duan,1, 2 A. D. Hillier,3 D. T. +Adroja,3, 4 P. K. Biswas,3 T. Takabatake,1, 5 M. Smidman,1, 2, ∗ and H. Q. Yuan1, 2, 6, † +1Center for Correlated Matter and School of Physics, Zhejiang University, Hangzhou 310058, China +2Zhejiang Province Key Laboratory of Quantum Technology and Device, +Department of Physics, Zhejiang University, Hangzhou 310058, China +3ISIS Facility, STFC Rutherford Appleton Laboratory, +Harwell Science and Innovation Campus, Oxfordshire, OX11 0QX, United Kingdom +4Highly Correlated Matter Research Group, Physics Department, +University of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa +5Department of Quantum Matter, AdSE, Hiroshima University, Higashi-Hiroshima 739-8530, Japan +6State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310058, China +(Dated: January 5, 2023) +The superconducting order parameter of the noncentrosymmetric superconductor LaRhSn is +investigated by means of low temperature measurements of the specific heat, muon-spin relax- +ation/rotation (µSR) and the tunnel-diode oscillator (TDO) based method. +The specific heat +and magnetic penetration depth [λ(T )] show an exponentially activated temperature dependence, +demonstrating fully gapped superconductivity in LaRhSn. The temperature dependence of λ−2(T ) +deduced from the TDO based method and µSR show nearly identical behavior, which can be well +described by a single-gap s-wave model, with a zero temperature gap value of ∆(0) = 1.77(4)kBTc. +The zero-field µSR spectra do not show detectable changes upon cooling below Tc, and therefore +there is no evidence for time-reversal-symmetry breaking in the superconducting state. +PACS number(s): +I. +INTRODUCTION +Noncentrosymmetric superconductors (NCS) have at- +tracted considerable interest, since in the absence of +inversion symmetry, an antisymmetric potential gradi- +ent gives rise to an antisymmetric spin-orbit coupling +(ASOC). The ASOC lifts the two-fold spin degeneracy +of the electronic bands, potentially allowing for uncon- +ventional superconducting properties such as the admix- +ture of spin-singlet and spin-triplet pairing states [1, 2]. +In the noncentrosymmetric heavy fermion superconduc- +tor CePt3Si, measurements of the magnetic penetration +depth, thermal conductivity and specific heat showed +the presence of line nodes in the energy gap [3–5], and +nodal superconductivity was subsequently found in other +NCS, such as Li2Pt3B [6, 7], Y2C3 [8], K2Cr3As3 [9, 10], +and ThCoC2 [11]. However, many NCS are found to be +fully gapped superconductors, such as Mo3Al2C [12, 13], +RT Si3 (R = La, Sr, Ba, Ca; T = transition metal) [14– +18], BiPd [19, 20], Re6T [21–23], La7T3 [24, 25], BeAu +[26] and PbTaSe2 [27–29]. Even though some of these +systems have been found to have multiple superconduct- +ing gaps, many NCS show evidence for single gap s- +wave superconductivity, indicating negligible contribu- +tions from a spin-triplet pairing component. +The pre- +dominance of such s-wave superconductivity even in sys- +tems with strong ASOC has posed the question as to +what conditions are required to give rise to mixed parity +pairing. In addition, even in NCS exhibiting unconven- +tional properties, unambiguosly demonstrating the pres- +ence of singlet-triplet mixing remains challenging, and +obtaining direct evidence may require probing associated +topological superconducting phenomena such as gapless +edge modes and Majorana modes [30, 31]. +Time reversal symmetry breaking (TRSB) has been +observed in the superconducting states of some weakly +correlated NCS, such as LaNiC2 [32], La7T3 [24, 25], and +several Re-based superconductors [21, 33, 34]. TRSB has +primarily been revealed by muon-spin relaxation mea- +surements, which detect the spontaneous appearance of +small magnetic fields in the superconducting state, even +in the absence of external applied fields [35]. +In most +cases, such systems have been found to have nodeless +superconducting gaps, which has often been difficult to +reconcile with the unconventional nature of the pairing +state implied by TRSB. On the other hand, different be- +havior was recently found in the weakly correlated NCS +CaPtAs, where there is evidence for both nodal supercon- +ductivity and TRSB [36, 37]. Consequently, it is impor- +tant to survey a wide range of different classes of NCS, +so as to look for novel behaviors arising from ASOC, as +well as to reveal the origin of any time reversal symmetry +breaking and to understand its relationship to the broken +inversion symmetry. +LaRhSn crystallizes in the noncentrosymmetric hexag- +onal ZrNiAl-type structure (space group P¯62m) dis- +played in the inset of Fig. 1, where the rare-earth atoms +form a distorted kagome lattice. +Compounds in this +family with a magnetic rare-earth atom have been ex- +tensively studied due to the interplay of strong elec- +tronic correlations and frustrated magnetism [38–40], +while several other systems with nonmagnetic rare-earth +elements are superconductors. For example, Sc(Ir,Rh)P, +LaRhSn, LaPdIn are superconductors with relatively low +transition temperatures Tc [41–44], while (Zr,Hf)RuP, + +2 +μ +FIG. 1. (Color online) Temperature dependence of the electri- +cal resistivity ρ(T ) of LaRhSn from room temperature down +to 0.5 K. The insets show ρ(T ) near the superconducting tran- +sition, and the crystal structure of LaRhSn. +ZrRu(As,Si) and Mo(Ni,Ru)P have Tc’s over 10 K [45– +49], where the higher Tc values may be a consequence +of the phonon spectra and electron-phonon coupling +strengths [47, 50, 51]. In this article, we study the order +parameter of LaRhSn via measurements of the electronic +specific heat and magnetic penetration depth, where the +latter is probed using both the tunnel-diode oscillator +(TDO) based method and muon-spin rotation (µSR). +The experimental results obtained by various techniques +can be consistently described by a single-gap s-wave +model corresponding to weak electron-phonon coupling. +In addition, zero-field µSR measurements do not exhibit +detectable changes below Tc, and therefore there is no +evidence for TRSB in the superconducting state. +II. +EXPERIMENTAL DETAILS +Single crystals of LaRhSn were synthesized using the +Czochralski method, as described in Ref 52. The specific +heat was measured in a Quantum Design Physical Prop- +erty Measurement System (PPMS) with a 3He insert. +The resistivity ρ(T ) was measured in a 3He cryostat from +room temperature down to 0.5 K, using a standard four- +probe method. µSR measurements were performed using +the MuSR spectrometer at the ISIS pulsed muon source +of the Rutherford Appleton Laboratory, UK [53, 54]. The +µSR experiments were conducted in transverse-field (TF) +and zero-field (ZF) configurations, so as to probe the flux +line lattice (FLL) and the presence or absence of time- +reversal symmetry breaking, respectively. Powdered sin- +gle crystals of LaRhSn were mounted on a high-purity +silver sample holder, which was mounted on a dilution +refrigerator, with a temperature range from 0.05 K to +2.5 K. With an active compensation system, the stray +0.0 +0.5 +1.0 +1.5 +2.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 + + + s-wave + +(0) = 1.76 k +B +T +c +C +el +/ +n +T (K) +0 +2 +4 +6 +8 +0 +20 +40 +C/T (mJ mole +-1 + K +-2 +) +T (K) +C/T= +n ++ +T + 2 ++ +T + 4 +FIG. 2. (Color online) Temperature dependence of the elec- +tronic specific heat as Cel(T )/γnT of LaRhSn, where the solid +line represents fitting with a single-gap s-wave model. The in- +set displays the total specific heat C(T )/T , where the dashed +line represents the fitting to the normal state contribution. +magnetic field at the sample position can be canceled to +within 1 µT. TF-µSR experiments were carried out in +several fields up to 60 mT. +The shift of the magnetic penetration depth from the +zero-temperature value ∆λ(T ) = λ(T ) − λ(0) was mea- +sured down to 0.3 K in a 3He cryostat, using a tunnel- +diode oscillator (TDO) based method [55–57], with an +operating frequency of 7 MHz and a noise level of 0.1 Hz. +Samples with typical dimensions of 550 × 450 × 300 µm3, +were mounted on a sapphire rod. The generated ac field is +about 2 µT, which is much smaller than the lower critical +field Hc1, ensuring that the sample remains in the Meiss- +ner state. ∆λ(T ) is proportional to the frequency shift +from zero temperature ∆f(T ), i.e., ∆λ(T ) = G∆f(T ), +where G is the calibration factor determined from the +geometry of the coil and sample [56]. +III. +RESULTS +A. +Electrical resistivity and specific heat +The single crystals of LaRhSn were characterized by +measurements of the electrical resistivity and specific +heat. Figure 1 displays the electrical resistivity ρ(T ) from +room temperature down to 0.5 K, which exhibits metallic +behavior in the normal state. The inset shows ρ(T ) at +low temperatures, where there is a sharp superconduct- +ing transition at around 2.0 K. +The inset of Figure. 2 displays the total specific heat +C(T )/T of LaRhSn in zero field, where there is a clear +superconducting transition with a midpoint Tc = 1.9 K, +in line with the behavior of ρ(T ). In the normal state, the +specific heat data are fitted by C(T )/T = γn+βT 2+δT 4, + +BP3 +0.5 +1.0 +1.5 +2.0 +2.5 +0.0 +0.5 +1.0 +1.5 +2.0 + + + 0.25 T + 0.20 T + 0.17 T + 0.15 T + 0.13 T + 0.10 T + 0.08 T + 0.05 T + 0.02 T + 0.00 T +C +el +/ +n +T +T (K) +(a) +0.0 +0.5 +1.0 +1.5 +0.0 +0.1 +0.2 +B +c2 + (T) +T (K) +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +(b) + This work + (0.38K) + MgB +2 + LaNiC +2 + Re +24 +Nb +5 + + +0.38K +(B) / +n +B/B +c2 +(0) +FIG. 3. (Color online) (a) Temperature dependence of the +electronic specific heat as Cel/γnT of LaRhSn under vari- +ous applied fields. +The inset displays the temperature de- +pendence of the upper critical field Bc2(T ), derived from the +specific heat measurements, where the solid line represents +fitting with the WHH model where Bc2(0) = 0.219(2) T. (b) +Field dependence of the residual Sommerfeld coefficient plot- +ted as γ0.38K(B)/γn versus B/Bc2(0) for LaRhSn, Re24Nb5 +[34], MgB2 [58] and LaNiC2 [59]. The dashed and dashed- +dotted lines correspond to the expected behaviors of nodal +and single-gap s-wave superconductivity, respectively. +with γn = 11.15(4) mJ mole−1 K−2, β = 0.410(6) mJ +mole−1 K−4 and δ = 0.87(1) µJ mole−1 K−6. Here γn is +the normal state Sommerfeld coefficient, and the latter +two terms represent the phonon contribution. The De- +bye temperature θD is estimated to be 241(1) K using +θD = (12π4Rn/5β)1/3, where R = 8.31 J mole−1 K−1 +is the molar gas constant and n = 3 is the number of +atoms per formula unit. The electron-phonon coupling +constant λel-ph can be approximated via +λel-ph = +1.04 + µ∗ln( +θD +1.45Tc ) +(1 − 0.62µ∗)ln( +θD +1.45Tc ) − 1.04. +(1) +Using the typical values for µ∗ of 0.1 – 0.15, λel-ph = 0.47 +– 0.57 are obtained, close to the derived values for +isostructural LaPdIn [44], indicating weakly coupled su- +perconductivity in LaRhSn. In addition, the value of γn +is very similar to that of LaPdIn, but larger than the +values for LuPdIn and LaPtIn which are not supercon- +ducting down to at least 0.5 K [44]. This is consistent +with the magnitude of the density of states at the Fermi +level playing an important role in giving rise to super- +conductivity in this family of compounds. +The main panel of Fig. 2 shows the low tempera- +ture electronic specific heat Cel(T )/γnT , from which the +phonon contribution has been subtracted. In the super- +conducting state, the entropy S can be calculated by [60] +S = − 3γn +π3 +� 2π +0 +� ∞ +0 +[flnf + (1 − f)ln(1 − f)]dεdφ, (2) +where the f(E, T ) = [1+exp(E/kBT )]−1 is the Fermi- +Dirac distribution function. Here, E = +� +ε2 + ∆2 +k, where +∆k(T ) = ∆(T )gk is the superconducting gap function. +Therefore, the electronic specific heat of superconducting +state can be obtained by Cel = T dS/dT . In the case of +a single-gap s-wave model, there is no angle dependent +component (gk = 1), and ∆(T ) was approximated by [61] +∆(T ) = ∆(0)tanh +� +1.82 [1.018 (Tc/T − 1)]0.51� +, +(3) +where ∆(0) is the zero-temperature superconducting gap +magnitude. As shown by the solid line in Fig. 2, the zero +field Cel/γnT can be well described by this single-gap +s-wave model, with ∆(0) = 1.76(1)kBTc. +Upon applying a magnetic field, the bulk supercon- +ducting transition is shifted to lower temperatures and +is completely suppressed at about 0.25 T (see Fig. +3 (a)). +The inset displays the extracted upper crit- +ical field Bc2(T ) and the corresponding fitting using +the Werthamer-Helfand-Hohenberg (WHH) model [62], +with a zero temperature upper critical field Bc2(0) = +0.219(2) T. Using λ(0) = +� +Φ0Bc2(0)/√24γn∆(0) [63], +where the units of Bc2(0) and γn +are gauss and +ergs cm−3 K−2, respectively, a penetration depth at zero +temperature λ(0) = 244(1) nm is estimated using ∆(0) = +1.76(1)kBTc. Combined with a Ginzburg-Landau (GL) +coherence length of ξGL = +� +Φ/2πBc2(0) = 38.7(2) nm, +the GL parameter κ is estimated to be 6.30(4), indicating +that LaRhSn is a type-II superconductor. Using the val- +ues of λ(0)=244(1) nm, a residual normal state resistivity +ρ0 = 25 µΩ cm and γn = 11.15(4) mJ mole−1K−2, the +mean free path ℓ and BCS coherence length ξBCS are es- +timated to be ℓ=17.91(8) nm and ξBCS=43.8(2) nm [64]. +The mean free path ℓ is smaller than ξBCS, indicating +that the sample is in the dirty limit. + +4 +0 +2 +4 +6 +8 +10 +12 +0.0 +0.1 +0.2 + + + 2.5 K + 0.1K + 2.5 K fit + 0.1 K fit +Asymmetry +Time ( +s) +FIG. 4. (Color online) ZF-µSR spectra of LaRhSn at 2.5 K +(T > Tc) and 0.1 K (T < Tc). The solid lines show the results +from fitting using Eq. 4. +Figure 3 (b) displays the field dependence of the Som- +merfeld coefficient value at 0.38 K, normalized by its +value in the normal-state, i.e., γ0.38K(B)/γn. It can be +seen that γ0.38K(B)/γn shows a nearly linear field depen- +dence, being similar to the fully gapped superconductor +Re24Nb5 [34]. On the other hand, γ0.38K(B)/γn clearly +deviates from the square-root field dependence (dashed +line) expected for line nodal superconductors, as well as +the typical behaviors of the multiband superconductors +MgB2 [58] and LaNiC2 [59]. Note that γ0.38K(B)/γn of +LaRhSn are determined from the specific heat at the low- +est measured temperature, and therefore even in zero- +field the data have a finite value. +B. +µSR measurements +Figure 4 displays the zero-field (ZF) µSR spectra col- +lected at 2.5 K (T > Tc) and 0.1 K (T < Tc). These +are fitted with a damped Gaussian Kubo-Toyabe (KT) +function +GZF(t) = A +�1 +3 + 2 +3(1 − δ2t2)exp +� +−δ2t2 +2 +�� +exp(−Λt)+Abg, +(4) +where A is the initial asymmetry, and Abg corresponds +to the time independent background term from muons +stopping in the silver sample holder. +δ and Λ are +the Gaussian and Lorentzian relaxation rates, respec- +tively. +Upon fitting with Eq. +4, δ = 0.086(3) µs−1 +and Λ = 0.0134(11) µs−1 were obtained at 2.5 K, while +δ = 0.082(3) µs−1 and Λ = 0.0157(10) µs−1 at 0.1 K. +Therefore, we find no evidence for TRSB in the super- +conducting state of LaRhSn, and these results suggest +that any spontaneous internal fields should be no larger +than 6.6 µT, which is smaller than the corresponding +fields in other reported TRSB superconductors [35]. +Transverse-field µSR (TF-µSR) measurements were +carried out in the mixed state with applied fields in the +range 40 mT to 60 mT, where the data were collected +0 +2 +4 +6 +8 +-0.2 +0.0 +0.2 + + + Time ( +s) +Asymmetry +(b) 0.05 K +-0.2 +0.0 +0.2 + + + +(a) 2.5 K +FIG. 5. +(Color online) Transverse field µSR spectra of +LaRhSn at (a) 2.5 K (T > Tc) and (b) 0.05 K (T < Tc) +in an applied field of 40 mT. The solid lines show the results +of fitting with Eq. 5 +upon field-cooling in order to probe a well-ordered flux- +line lattice (FLL). The results at 2.5 K and 0.05 K in a +field of 40 mT are displayed in Fig. 5. The significant +increase of the depolarization rate corresponds to the in- +homogeneous field distribution in the sample, character- +istic of the formation of a FLL. The TF-µSR asymmetry +were fitted to the sum of oscillations damped by Gaussian +decaying functions +GTF(t) = +n +� +i=1 +Aicos(γµBit + φ)e−(σit)2/2 + ABG, (5) +where Ai is the amplitude of the oscillating compo- +nent, which precesses about a local field Bi with a com- +mon phase offset φ and a Gaussian decay rate σi, while +γµ/2π = 135.5 MHz/T and ABG are the muon gyromag- +netic ratio and background term, respectively. The asym- +metry can be well fitted with three oscillatory compo- +nents (n = 3), where σ3 was fixed to zero, corresponding +to muons stopping in the silver sample holder. Figure +6(a) displays the temperature dependence of σ(T ) ob- +tained following the multiple-Gaussian method described +in Ref 65. Here, the first and second moment of the field +distribution are calculated as +⟨B⟩ = +n−1 +� +i=1 +Ai Bi +A1 + · · ·An−1 +, +(6) +⟨B2⟩ = +n−1 +� +i=1 +Ai +A1 + · · ·An−1 +[(σi/γµ)2 +[Bi −⟨B⟩]2], (7) +and σ = γµ +� +⟨B2⟩. The relaxation rate in the normal +state is ascribed to a temperature independent contri- +bution arising from quasistatic nuclear moments, with a +nuclear dipolar relaxation rate σN += 0.0851(27) µs−1. + +5 +40 +50 +60 +70 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + + +sc + ( +s +-1 +) +Field (mT) + 0.1 K + 0.3 K + 0.5 K + 0.7 K + 0.9 K + 1.0 K + 1.1 K + 1.2 K + 1.3 K + 1.4 K + 1.5 K +(b) +0 +1 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + 40 mT + 45 mT + 50 mT + 60 mT + + + ( +s +-1 +) +T (K) +N + = 0.0851 +s +-1 +(a) +FIG. 6. (Color online) (a) Temperature dependence of the +Gaussian relaxation rate of the TF-µSR spectra in different +applied fields between 40 mT and 60 mT. (b) Field depen- +dence of the superconducting contribution to the TF-µSR re- +laxation rate σsc at various temperatures, where the solid lines +correspond to fitting using Eq. 8. +The superconducting component of the variance σsc is +calculated as σsc = +� +σ2 − σ2 +N, and its field dependence +is displayed in Fig. 6(b) for several temperatures. +For small applied fields and large κ, σsc is field inde- +pendent and proportional to λ−2, which is not applicable +for the current measurements of LaRhSn. On the other +hand, for κ ≥ 5 and 0.25/κ1.3 ≤ b ≤ 1, σsc may be ap- +proximated by [66] +σsc = 4.854 × 104 1 +λ2 (1 − b)[1 + 1.21(1 − +√ +b)3], +(8) +where b = B/Bc2 is the applied field normalized by the +upper critical field. Since the κ of LaRhSn was deter- +mined to be about 6.30(4), the measurements of LaRhSn +are within the applicability of Eq. 8. Therefore by fixing +Bc2(T ) to the bulk values derived from the specific heat +0.4 +0.6 +0.8 +1.0 +0 +10 +20 + + + s-wave + ~T + 4.4 + ~T + 2 + (nm) +T (K) +1 +2 +0 +2 +4 +6 +8 +f (kHz) +T (K) +FIG. 7. (Color online) The change of magnetic penetration +depth ∆λ(T ) of LaRhSn at low temperatures. The solid red, +dashed blue and dashed-dotted magenta lines represent fitting +to an s-wave model, and power-law dependences ∼ T 4.4 and +∼ T 2, respectively. +The inset displays the frequency shift +∆f(T ) from 2.5 K down to 0.3 K, where there is a sharp +superconducting transition at around Tc = 2 K. +in Fig. 3, the temperature dependence of λ−2(T ) can be +obtained from fitting with Eq. 8 [Fig. 6(b)], and the re- +sults are shown in Fig. 8, together with the TDO results +described in following section. +C. +TDO measurements and superfluid density +analysis +Figure 7 shows the penetration depth shift ∆λ(T ) of +LaRhSn at low temperatures, with a calibration factor +G = 14.2 ˚A/Hz. The inset displays the frequency shift +∆f(T ) from 2.5 K down to the base temperature of 0.3 K, +where a sharp superconducting transition is observed at +Tc = 2 K, in accordance with other measurements. Upon +further cooling, ∆λ(T ) flattens at the lowest measured +temperatures, indicating fully gapped superconductivity +in LaRhSn. For an s-wave superconductor, the temper- +ature dependence of ∆λ(T ) for T ≪ Tc can be approxi- +mated by +∆λ(T ) = λ(0) +� +π∆(0) +2kBT exp +� +−∆(0) +kBT +� +. +(9) +As shown by the solid line, the experimental data be- +low Tc/3 can be well described by the s-wave model with +∆(0) = 1.80(1)kBTc, where λ(0) = 227.9 nm was fixed +to the value derived from TF-µSR. The data were also +fitted by a power law dependence ∆λ(T ) ∝ T n, from +0.3 K up to 0.75 K. A large exponent of n = 4.4 is ob- +tained, which is much larger than two, excluding nodal +superconductivity in LaRhSn. + +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +5 +10 +15 +20 + + + TDO + +SR + s-wave (clean) + s-wave (dirty) + d-wave + p-wave +-2 + ( +m +-2 +) +T/T +c +FIG. 8. (Color online) Temperature dependence of λ−2(T ) as +a function of the normalized temperature T/Tc. The data are +derived from measurements using the TDO based method and +TF-µSR measurements, which correspond to the empty circle +and solid symbols, respectively. The lines show the results +from fitting with different models for the gap structure. +To further characterize the superconducting pairing +state of LaRhSn, the temperature dependence of λ−2(T ) +was analyzed, which is proportional to the superfluid den- +sity ρs(T ) as ρs(T ) = [λ(0)/λ(T )]2. Figure 8 displays +λ−2(T ) as a function of the reduced temperature T/Tc, +where the data are derived from both the TDO and TF- +µSR measurements, which show nearly identical behav- +ior. Since the previous analysis suggested that the sample +is in the dirty limit, the results from TF-µSR were fitted +with the following expression for a dirty s-wave model +[67] +ρs(T ) = ∆(T ) +∆(0) tanh +� ∆(T ) +2kBT +� +. +(10) +As shown by the dashed line in Fig. +8, the dirty s- +wave model can well describe the experimental data, with +λ(0) = 227.9(9) nm and ∆(0) = 1.77(4)kBTc. The data +were also analyzed using the clean limit expression +ρs(T ) = λ−2(T ) +λ−2(0) = 1 + 2 +�� ∞ +∆k +EdE +� +E2 − ∆2 +k +∂f +∂E +� +FS +, +(11) +where a clean single gap s-wave model can also fit +the data well, yielding a larger gap value of ∆(0) = +2.05(3)kBTc. Here the gap value obtained from the dirty +s-wave model is in very good agreement to those derived +from the analysis of specific heat and low temperature +∆λ(T ), while the clean limit value is considerably larger, +which is in-line with the previous dirty limit calculation. +We note that due to the samples being in the dirty limit, +TABLE I. Superconducting parameters of LaRhSn, where the +parentheses with C and µSR denote the results from the spe- +cific heat and µSR, respectively. +Property +Unit +Value +Tc +K +1.9 +Bc2(0) +T +0.219(2) +γn +mJ mole−1K−2 +11.15(4) +ΘD +K +241(1) +λel−ph +0.47-0.57 +ξGL +nm +38.7(2) +ℓ +nm +17.91(8) +ξBCS +nm +43.8(2) +λ0(C) +nm +244(1) +λ0(µSR)dirty +nm +227.9(9) +κ(C) +6.30(4) +κ(µSR)dirty +5.89(4) +∆(0)(C) +kBTc +1.76(1) +∆(0)(µSR)dirty +kBTc +1.77(4) +we cannot exclude an anisotropic superconducting gap +in LaRhSn, since impurity scattering can suppress any +gap anisotropy. +On the other hand, as also shown in +Fig. 8, a d-wave model with gk = cos 2φ and p-wave +model with gk = sin θ (φ= azimuthal angle, θ= polar +angle) cannot account for the data, further indicating a +lack of nodal superconductivity in LaRhSn. Meanwhile, +the value of λ(0) obtained from µSR experiments is very +close to that from specific heat results. Using this value of +λ(0) = 227.9(9)nm, κ = 5.89(4) is estimated, which cor- +responds well to the value from the specific heat analysis. +The obtained superconducting parameters of LaRhSn are +displayed in Table I. Therefore, the results of specific +heat, TDO-based measurements and µSR can all be con- +sistently described by a single-gap s-wave model with a +gap magnitude very close to that of weak-coupling BCS +theory, and there is no evidence for time-reversal sym- +metry breaking below Tc. +IV. +SUMMARY +In summary, we have studied the order parameter of +the noncentrosymmetric superconductor LaRhSn. Both +the specific heat and magnetic penetration depth show +exponentially activated behavior at low temperatures, +providing strong evidence for fully gapped superconduc- +tivity. λ−2(T ) derived from the TDO based method and +TF-µSR, as well as the specific heat can be consistently +well described by a single-gap s-wave model, with a gap +magnitude very close to that of weak coupling BCS the- +ory. Together with findings for LaPdIn [44] and ZrRuAs +[47], our results suggest that fully gapped s-wave super- +conductivity, together with a lack of evidence for time + +7 +reversal symmetry breaking, are consistent common fea- +tures of weakly correlated NCS with the ZrNiAl-type +structure and there is a lack of significant singlet-triplet +mixing. +ACKNOWLEDGMENTS +This work was supported by the National Key R&D +Program of China (Grant No. 2017YFA0303100), the +Key R&D Program of Zhejiang Province, China (Grant +No. 2021C01002), the National Natural Science Foun- +dation of China (Grant No. 11874320, No. 11974306 +and No. 12034017), and the Zhejiang Provincial Natu- +ral Science Foundation of China (R22A0410240). 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Tinkham, Introduction to Superconductivity (Courier +Corporation, North Chelmsford, MA, 2004). + diff --git a/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf b/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..52c589f048a4df0304a13bdd013c99e105c25781 --- /dev/null +++ b/7dE4T4oBgHgl3EQf2Q2c/content/2301.05297v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0cadfce4eb7cc1141eda21e2e2c5ab84220a616ffae7ff50568c41671f0d0173 +size 1544777 diff --git a/7dE4T4oBgHgl3EQf2Q2c/vector_store/index.faiss b/7dE4T4oBgHgl3EQf2Q2c/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..da4a036a6ca53acecfefc7442f6f9d4565d4d153 --- /dev/null +++ b/7dE4T4oBgHgl3EQf2Q2c/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5f0e796f813dff3147f30dcf7b1a2a215c861cf939a3ea5e8c3454d462feb851 +size 2949165 diff --git a/7tE2T4oBgHgl3EQf7whk/content/tmp_files/2301.04212v1.pdf.txt b/7tE2T4oBgHgl3EQf7whk/content/tmp_files/2301.04212v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4fbfd4115e255f2709e7bf58a25d646bf8a340a0 --- /dev/null +++ b/7tE2T4oBgHgl3EQf7whk/content/tmp_files/2301.04212v1.pdf.txt @@ -0,0 +1,934 @@ +DEEP LEARNING BASED MULTI-LABEL IMAGE CLASSIFICATION OF PROTEST +ACTIVITIES +Yingzhou Lu, Kosaku Sato, Jialu Wang +Electrical and Computer Engineering Department +Virginia Tech +Arlington,VA 22203, USA +George Washington University +Washington, DC 20052, USA +Email: lyz66@vt.edu, Ksato@vt.edu, jialu@gwu.edu +ABSTRACT +With the rise of internet technology amidst increasing ur- +banization rates, sharing information has never been easier, +thanks to globally-adopted platforms for digital communi- +cation. +The resulting output of massive amounts of user- +generated data can be used to enhance our understanding +of significant societal issues, particularly for urbanizing ar- +eas. In order to better analyze protest behavior, we enhanced +the GSR dataset and manually labeled all the images. We +used deep learning techniques to analyze social media data +to detect social unrest through image classification, which +performed well in predicting multi-attributes. Then, we used +map visualization to display protest behaviors across the +country. +Index Terms— Machine Learning, Deep Learning, Im- +age Classification, Multi-Label Classification, Social Media +1. INTRODUCTION +The study of protest activities plays a profound role in so- +ciologists and scholars’ studying citizens’ political behavior. +With the advancement of social media networks, people now +share an unprecedented amount of user-generated content in +the form of text, images, and videos on the web. Classifi- +cation of social media data not only helps in understanding +online behavior, but also elucidates significant priorities of +urban populations that carry real-life consequences. Using +social media data, we focuses on social unrest in the form of +public protest images, specifically for Latin American coun- +tries. +The traditional approach to the study of social media +dataset focused on using natural language processing to mon- +itor how hashtags and links are used by the user and the +propagation of those items to other users. However, these ap- +proaches may not effectively capture some important features +or details of protest activities. For instance, we may be in- +terested in knowing details such as whether there was a large +crowd involved in the protest, if polices were present, or what +the demographics (young or adults) of protesters carrying a +sign. Our approach uses image processing to capture those +features of the protest activitiesFu et al. (2021). +We took several approaches in image classification of +social media data: our initial approach is to utilize traditional +machine learning methods such as Support Vector Machine +(SVM)Weston et al. (1999) and a deep learning method like +Convolutional Neural Networks (CNNs)Krizhevsky et al. +(2012) which have shown some advantages in large-scale im- +age and video analysis. Traditional machine learning method, +such as SVM, can be used to classify images with good accu- +racy; however, as the volume of data and number of classes +for recognition increases, the deep learning approaches be- +comes the more advanced approach for object recognition. +2. LITERATURE REVIEW +As our objective of our model is to detect protest activi- +ties using the image, the preliminary work relevant to our +study is the EMBERS system by Naren, Patrick and et +elRamakrishnan et al. (2014). The EMBERS system con- +tinuously monitor the social media dataset such as Twitter, +Facebook, news pages, and use data mining to process the +trend to predict the protest activities in South America re- +gions. Their planned protest model based on custom multi- +lingual lexicon matching predicted the protest activities with +precision and recall rate of 0.69 and 0.82 respectively. How- +ever, their approach does not capture the additional features +of the protests such as demography. +Moreover, image classification is classical topic in com- +puter vision area which aims to predict and assign each given +image a specific label from several categories. +However, +background clutter, occlusion and variation in image scale +make the computer vision tasks more challenging. The tra- +ditional approaches to perform image classification includes +k-nearest neighbor and SVM algorithmsYi et al. (2018). +k-nearest neighbor is one of the simplest classification al- +gorithm that aims at labeling an image based on the best fit +result, but the model is usually not robust to noise or im- +balanced class datasetZhang et al. (2006). Similarly, SVM, +which is originally proposed as a binary classification by +Cortes and VapnikCortes and Vapnik (1995) is another clas- +sical approach to perform classification and it has shown a +better performance than the k-nearest neighbor in some ap- +arXiv:2301.04212v1 [cs.CV] 10 Jan 2023 + +Fig. 1. Labeled GSR images in GSR dataset +plicationsBoiman et al. (2008). Furthermore, our approach +adopts a deep learning based on CNNs based on the study +of the visual cortex of the human brain which have shown +a great success recently in many computer vision applica- +tionsAghdam and Heravi. +3. OSI DATASET +The OSI (Open Source Indicators) dataset, provided by com- +puter science department from Virginia Tech, is MITRE’s +gold standard report (GSR) of protests organized by survey- +ing newspapers for civil unrest reports. +The dataset also +contains large samples of non-protest images that were col- +lected in the process. There are 48,713 images in GSR and +40,647 non-protest images. +High-confidence images indi- +cate a top image among multiple images embedded in the +articles. +High-confidence images within datasets indicate +a top image among multiple images embedded in the arti- +clesJayachandra et al. (2020). High-confidence GSR images +relevant to GSR articles based on social protest total 7,884, +and low-confidence GSR images total 40,829. +3.1. Details of the Image Labels +Table 1 shows the visual attributes that characterize the +protests which we used to label each image from the GSR +dataset. Out of 40,647, a total of 9,504 images were hand- +picked to train and test our prediction models by excluding +bad data points that are obviously irrelevant to the social ac- +tivities that we are interested in detectingYang et al. (2020). +The Annotated images consist of 327 fire, 1,943 flag, 7,347 +large crowd, 248 other, 2,159 police, 4,462 sign, and 1,233 +student images. Fig. 1 contains sample images with their +class labels. Each image has a label with vector of length +7 that has ”0”s and ”1”s corresponding to the index number +of visual attributes. The ”Other” label was inferred from the +absence of positive class labels across categories. +3.2. Challenges of the Dataset +There are a few inherent challenges in our training dataset. +First, some attributes of protest are commonly shared with +Fig. 2. Sample protest images from training set +Fig. 3. Sample non-protest images from training set +Table 1. Visual attributes of protest images +Class label +Description +Sample size +0 +Fire +Presence of active fire +327 +1 +Flag +Presence of flag +1943 +2 +Large Crowd +Presence of roughly more than 20 people +7347 +3 +Other +None of the above or the below +248 +4 +Police +Presence of police +2159 +5 +Sign +Presence of a protest sign +4462 +6 +Student +Presence of young students +1233 +non-protest images. For instance, Figure 2 shows a sample +of protest images used to train the machine learning and deep +learning models. Images with a protest class label often de- +picted fire, police, handwritten signs, and large crowds. How- +ever, Figure 3 shows a sample of non-protest images in which +large crowds are also frequently seen attribute while it also +comprised of a variety of other objects (such as animals or +soccer players). Second, we have imbalanced dataset where +it does not have exactly equal number of instances in each +class. This issue is mainly defined by the specific subject or +attribute we set up in our problem. The training of the classi- +fiers in imbalanced dataset can cause the trained model clas- +sifying images as images of majority class most of the times +and under-represent the minority class. +3.3. Image Augmentation for Imbalanced Dataset +One way to balance the imbalanced classes is to use image +augmentation. Data augmentation is a method to artificially +increase sample size of the training images through various +pre-processing or combinations of multiple pre-processing of +the image such as adding noise, flipping and re-scaling. Ta- +ble 2 summarizes the different techniques we adopted to per- +form image augmentation for minority classes. Image aug- +mentation has been considered as a promising method to im- +prove the performance of prediction model. +For instance, +adding noise to our observation can help make the prediction +Table 2. Increased sample size after image augmentation +Class label +Image transformation used +New sample size +Fire +Flipping, Scaling, Translation, Noise, +Affine Transform, Perspective Transform, +Intensity, Contrast, Filters, Crop, Shear +4,578 +Flag +Flipping, Noise, Affine Transform +5,829 +Large Crowd +– +7347 +Other +Flipping, Scaling, Translation, Noise, +Affine Transform, Perspective Transform, +Intensity, Contrast, Filters, Crop, Shear +3,472 +Police +Affine Transform, Noise +6,477 +Sign +– +4462 +Student +Flipping, Noise, Affine Transform, Crop +6,165 + +GSR Image with Labels +['0", "1', '1', "0", '0', "1', "0' +["0", "1', '1', '0', 0', '1',"1'] +["0",'0', "1', "0", "','0", "0] +["0", "0', "0', "1', 0', '0", "0"Fig. 4. Example images from the dataset after transformation +model more robust in the face of social media dataset and pre- +vents it from overfitting Perez and Wang (2017). Moreover, +Python’s scikit-image library has the full list of the avail- +able image transformation but we have adopted the thirteen +of them. Those are horizontal and vertical flipping, affine +transfomration, perspective transformation, rescaling, crop- +ping, blurring, changing contrast and intensity, gaussian and +exposure filter, translation, and shearing. Fig. 4 shows the +example of such transformations of the images. As a result +of implementing image augmentation, we were able to signif- +icantly improve the sample size of the minority classes; for +instance, the sample size of ’fire’ class increased from 327 to +4,578 as seen in Table 2. Also, as an alternative approach, +we considered oversampling using Synthetic Minority Over- +sampling Technique (SMOTE) Chawla et al. (2002). How- +ever, we believe that image augmentation can create more +variation in training images to prevent over fitting, and hence +we did not utilize SMOTE in this paper. +4. MULTI-LABEL IMAGE CLASSIFICATION +Multi-label learning is a form of supervised learning where +the classification algorithm learns from a set of images in +which an image can belong to one or more classes. The goal +of multi-label image classification is to predict a set of class +labels for the input image. +A more generalized approach +is multi-class learning where each image is limited to one +correct class label. Multi-label classification and prediction +is more practical since the many real world problems involve +multiple objects belonging to different categories. +Multi- +label classification is also applicable various domains such +as text, video, and scene classification. For a typical multi- +label image, objects of different categories in each image are +located at varying positions with differing scale, zoom, size, +and poseWon et al. (2017). For example, two images labeled +as ’police’ and ’fire’ may have different spatial arrangements +of identified objects. Although factors such as differing ar- +rangements or occlusion can contribute to the inaccuracy of +multi-label classification, we expect reasonable results with a +sufficiently large dataset. Details of implementation of multi- +label classification for SVM and CNNs will be discussed in +the following sections. +5. APPROACH OF IMAGE CLASSIFICATION +In our approach, we utilized multi-label SVM and CNNs to +detect protest attributes in image classification. First, multi- +label SVM will be explained in details. In our proposal, SVM +is a baseline model to evaluate the performance of the pre- +diction model using CNNs. As the problem requires a large +dataset for training and good accuracy of classifying many +protest attributes, we believe that CNNs will perform better +than SVM. +5.1. Support Vector Machine +SVM is originally proposed as a binary classification by +Cortes and VapnikCortes and Vapnik (1995), but the model +has been extended to apply to multi-label classification prob- +lemsWeston and Watkins (1998). +One-vs.-All: One-vs.-All is a classical approach to solve +k-class pattern recognition problem. It involves training a sin- +gle binary classifier per class, with the samples of one class +as positive samples while other samples are set as negative. +More specifically, using this method, n-th classifier finds a +hyperplane between class n and the rest of the classesWeston +and Watkins (1998). A point where the distance from the +margin is maximal is assigned to the class. We aim at de- +tecting seven classes so that this strategy requires the training +of seven different SVMs. During testing the models, all clas- +sifiers would vote ’true’ by predicting that a testing sample +belongs to their class. In the end of testing, a sample is classi- +fied by the ensemble as the class that has the highest number +of votes. One-vs.-All is widely used in multi-label classifica- +tion. +Weighting Hyper-Parameter for SVM: Imbalanced +dataset causes misclassification of images that belonging to +the minority class impacted more heavily than that of the ma- +jority class because the frequency of the minority class is rare +compared to that of the majority class. In order to mitigate +over fitting of training classifier resulted from the imbalanced +data, we propose the modification of hyperparameter C in +SVM’s objective function which determines the penalty for +misclassifying the objects. Instead of defaulting C to be one, +Ck belonging to class k will have different values as shown +in (4). +Ck = C · +n +knj +(1) +As you can see, the updated Ck value will be inversely propor- +tional to instances of j class in order to increase Ck value for +the minority class in order to mitigate under-representation +issue. k is the number of class and j is the sample size be- +longing to the class. +5.2. Convolutional Neural Networks +5.2.1. Architecture +CNNs consist of input, convolution, activation function, pool- +ing, deep layers, and output layers. Throughout the training +of the network, the parameters are updated except for the ones +between convolution and pooling. There are some important +properties of the convolution layer. Some patterns are smaller + +Original Image +Contrast Adjusted +Noise Added +ELPAQUETE +ELPAOUETE +Cropped +Horizontal Flip +Vertical Flip +ESEKKON +PAQUETE +ETBHOREIE +CIEN +NOAENERKOthan the entire image so that the image can be subsampled to +reduce the image size; this is to train fewer parameters in the +neural network. The same patterns can appear in different re- +gions so that the same set of parameters can be used to reduce +computation. +Convolutional Layers: Convolution of a filter on an input +image is a point-wise multiplication operation. The activation +function, which in our case is Rectified Linear Unit (ReLU) +activation, is applied on each image separately in an element- +wise fashion to create activation maps based on outputs of the +convolution Aghdam and Heravi. +Activation Function: Nair and Hinton introduced the +non-saturating nonlinearity f(x) = max(0, x), also known as +the ReLU, which has gained popularity in the deep-learning +community because of its fast computing time Krizhevsky +et al. (2012).Hence, our model applied the ReLU nonlinear- +ity function to the output of every convolutional and fully- +connectedKrizhevsky et al. (2012). +Pooling Layer: In our model, we applied a 2x2 filter size +with a 2-length stride after each layer with the option of max- +pooling. Max-pooling applies the filters and the stride to the +input and returns the maximum value, dropping the non-max +values in each sub-region that convolution is applied. +Fully Connected Layers: The fully connected layer uses +those inputs to produce N-dimensional vectors, where N is +the number of classes needed for prediction. +Loss Function: In our model, we used sigmoid cross- +entropy for multi-label classification. Cross-entropy is used +to define the loss function in training the network in which +the model is penalized if it estimates a low probability for the +target class Nielsen (2015). +J(θ) = − 1 +m +m +� +i=1 +K +� +k=1 +[yi +k log(ˆpi +k)] +(2) +For the loss function, we used Adaptive Moment Estima- +tion (ADAM) Kingma and Ba (2014). ADAM keeps track of +a learning rate for each network weight and computes indi- +vidual adaptive learning rates for different parameters based +on estimates of first and second moments of the gradients +Kingma and Ba (2014). Since ADAM is an adaptive rate +learning algorithm, it requires less tuning. The default learn- +ing rate 0.001 is often used to support usability of the algo- +rithm. +Table 3 shows our CNNs architecture. There are three +convolutional layers with ReLU, and max-pooling is used to +down-sample the image after each convolutional layer. The +filter size for max-pooling is 2 × 2 so that output image of the +max-pooling is half the size of the input. The convolutional +layers uses the filter size of 3 × 3 while length of stride equal +to 2. The first fully connected layer (FC1) is a vector with a +length of 1024, and the second fully connected layer (FC2) +is a vector of length 7 which is the number of class labels to +predict visual attributes of protest images. +5.2.2. Evaluation Method of Multi-label Classification +Evaluation of multi-label classification has a notion of being +partially correct. One way to evaluate the classification is +label-set based accuracy or exact match that considers par- +tially correct as incorrect. On the other hand, evaluation of +label-based accuracy is carried out on a per label basisChen +Table 3. Architecture of Multi-Label Classification CNN +Layer +Feature Map +Feature Size +Filter Size +Stride +Pad +Activation +FC2 +- +7 +- +- +- +Sigmoid +FC1 +- +1024 +- +- +- +ReLU +Max-Pooling +128 +4x4 +2x2 +2 +Same +- +Conv3 +128 +7x7 +3x3 +2 +Same +ReLU +Max-Pooling +64 +14x14 +2x2 +2 +Same +- +Conv2 +64 +28x28 +3x3 +2 +Same +ReLU +Max-Pooling +32 +56x56 +2x2 +2 +Same +- +Conv1 +32 +112x112 +3x3 +2 +Same +ReLU +Input +3(RGB) +224x224 +- +- +- +- +et al. (2021). The calculation method of label-set accuracy, +where a predicted set of labels ˆy must exactly match the +ground truth y, is shown in equation (3) Read et al. (2011); +0/1 loss dictates that any label vector not predicted perfectly +will be given a zero score. +0/1 +loss = 1 − 1 +N +N +� +i=1 +1yi= ˆ +yi +(3) +Label-based accuracy is more lenient approach to evaluate the +performance since it does not consider multi-label problem as +a whole. When each label has a separate binary evaluation, we +have hamming loss which is shown in the following equation: +Hamming +Loss = 1 − +1 +NL +N +� +i=l +L +� +j=l +1yi= ˆ +yi +(4) +We adapted both approaches to evaluate the performance of +our multi-label classifier. +5.2.3. Threshold Selection +The fully connected layer represents a vector containing prob- +ability for each class. The threshold function can be used to +obtain a multi-label prediction ˆy. Specifically, we used the +Matthews Correlation Coefficient (MCC) which is an evalu- +ation metric of binary classification. The MCC is a correla- +tion coefficient for ground truth versus predictions and varies +between -1 and 1, where 1 represents a perfect prediction +Gorodkin (2004). The MCC is given by the following equa- +tion (5). +MCC = +Tp × Tn − Fp × Fn +� +(Tp + Fp)(Tp + Fn)(Tn + Fp)(Tn + Fn) +(5) +For multi-label classification, the MCC is defined in terms +of a confusion Matrix C for K classes in the equation (6) . +MCC = +c × s − �K +k pkxtk +� +(s2 − �K +k p2 +k)(s2 − �K +k t2 +k) +(6) +The values tk = �K +i Cik is the number of times class k +truly happened. pk = �K +i Cki is the number of times class +k was predictedTian et al. (2021). c = �K +i Ckk is the total +number of samples correctly predicted. s = �K +i +�K +j Cij is +the total number of samples. + +5.3. Advantages and Disadvantages of SVM and CNN +In image classification, there are advantages and disadvan- +tages for both SVM and CNNs. Theoretically, SVM is very +good at finding the margin and hyperplane for classification, +and it is very robust for high dimensional dataZhang et al. +(2006). However, SVM model is sensitive to noise, for exam- +ple, if there is a noise in background or a visible object in one +image is occluded or partially blocked by scenes in other, it +will have a negative impact on the performance of the clas- +sification model Cortes and Vapnik (1995). Moreover, since +one-vs-all involves training a binary classifier for all classes, +computation time can be very expensive. +On the other hand, the main advantage of CNNs is that +it can be used to extract important image features with a suf- +ficiently large datasetLeCun et al. (2015). The performance +of CNNs classifier largely depend on the size of the dataset. +The bottle necks in training the CNNs model are computa- +tional time and memory used to retain activation from for- +ward pass and error gradients computation when dataset is +very largeSun et al. (2008). However, an efficient parallel +computation with a help of GPU or training a model in mini +batches can be used to mitigate those issues to some degrees. +Also, there are many parameters for CNNs that need to be +set by the users in order to train a robust and good prediction +model. +6. EXPERIMENT +We implemented K-class SVM and CNNs using scikit-learn +and TensorFlow libraries in Python 3.5. In the experiment, +we used our personal server desktop which runs on Windows +with 2 Intel Xeon E5-2630 V3 CPU 2.4 GHz and 8 small +cores with RAM size of 64GB. First, we split the sample im- +ages via image augmentation from Table 2 into training and +testing (the ration of 80% and 20% respectively) and then +we re-sized each image to 224x224 with 3 color channels. +However, the implementation of SVM in scikit-learn does not +adopt online learning so that we had to down-sample the sam- +ple images from 31,472 to 12,000 to avoid memory limit er- +ror. On the other hand, with a help of mini-batches, we used +all of the data points without any down-sampling for CNNs +model. +We trained a baseline SVM using One-vs.-All method. +Our final setting of SVM consisted of max iterations to be +ran as 4000 to ensure it converges. Also, we experimented +weighting of hyperparameter in equation (1) but it did not im- +prove the result so our final setting of the weight parameters +for each classes are set to 1. For CNNs, we used the learning +rate of 0.001 which is a standard rate and mini batch size of +202 images with numbers of batches as 125. +7. RESULTS AND DISCUSSION +7.1. Evaluation +Evaluation was conducted on each class. Since we have the +manual label as ground truth, we calculated accuracy, the pre- +cision rate, recall rate, and F1 score for each class respec- +tively. Specifically, the following is the equations we used to +calculate each evaluation criteria: Precision rate= +T P +T P +F P , +Recall rate= +T P +T P +F N , Accuracy= +T P +T N +T P +T N+F P +F N . +F1 +score is the harmonic mean of precision rate and recall rate. +Table 4. Evaluation of SVM Model Prediction +Fire +Flag +Large Crowd +Other +Police +Sign +Student +Accuracy (%) +88 +74 +60 +89 +77 +63 +79 +Precision (%) +53 +53 +63 +53 +52 +55 +50 +Recall (%) +35 +22 +60 +24 +30 +39 +19 +F1 Score (%) +42 +31 +61 +33 +38 +45 +27 +Fig. 5. SVM vs CNN Precision Rate +Fig. 6. SVM vs CNN Recall Rate +7.2. Result +Training SVM model with one-vs.-all method took longer +than 12 hours and consistently consumed 70-90% of avail- +able memory on our machine whereas the CNNs model only +took less than half of the training time with much less con- +sumption of memory with a help of mini-batch. Therefore, +we were able to obtain the results by CNNs easier and faster +than the SVM. Table 4 and 5 shows the accuracy, precison, +recall, and F1 score of each predicted lables for SVM and +CNNs respectively. We also plotted the precision and recall +of the two models side by side in histogram in Fig. 5 and 6 +to compare the performance of the two models. As you can +see, their overall performance is comparable to each other but +recall using CNNs is slightly better than that of SVM. +For the evaluation of CNNs prediction, We calculated +the best threshold using MCC to transform the probability of +each label from the fully connected layer into the 7 predicted +class labels: the calculated thresholds are 0.2, 0.4, 0.7, 0.6, +0.5, 0.4, 0.5 for ’fire’, ’flag’, ’large crowd’, ’other’, ’police’, +’sign’, and ’student’ respectively. +Prediction accuracy per +Table 5. Evaluation of CNNs Model Prediction +Fire +Flag +Large Crowd +Other +Police +Sign +Student +Accuracy (%) +91 +72 +71 +89 +76 +61 +76 +Precision (%) +76 +48 +74 +52 +45 +51 +32 +Recall (%) +67 +36 +73 +37 +32 +46 +23 +F1 Score (%) +71 +41 +73 +43 +38 +49 +27 + +SVM +■CNNs +76 +74 +63 +53 +55 +53 +53 +52 +52 +48 +5051 +45 +32 +FIRE +FLAG +LARGE +OTHER +POLICE +SIGN +STUDENT +CROWDSVMCNNs +73 +67 +60 +46 +37 +39 +35 +36 +30 32 +22 +24 +23 +19 +FIRE +FLAG +LARGE +OTHER +POLICE +SIGN +STUDENT +CROWDFig. 7. Image of a burning vehicle with police in background +(top); image of bikers and a flag (bottom) +label reached almost 77% on average. Fig. 7 shows sample +test images of ’fire’ and ’police’ on the left and ’large crowd’ +and ’police’ on the right. Our CNNs model predicted them +correctly but when we evaluated our classifier model with a +large dataset, we found that our label-set accuracy was very +low around 20% due to the challenges of exact matching on a +multi-label classifier. +7.3. Future work +From the experiment, we learnt that we were able to get rea- +sonable performance using both SVM and CNNs model to +predict each class label separately but the bottom line perfor- +mance of our prediction model is still not desirable: our goal +is to increase the accuracy of exact matching. Therefore, Fu- +ture work can be done in following aspects. First, we can +apply state-of-the-art algorithm like Generative Adversarial +Network to generate more training samples, which would be +helpful to prevent over fitting. Second, we modify the equa- +tion for SVM to enhance the classifier, and improve the deep +learning modelYan et al. (2018). Moreover, The main limi- +tation of our image classification approach is that it does not +consider the credibility of the source in decision making, and +hence requires assessment of the social media source or of +each image posted on the web. Also, there are privacy protec- +tion concern in using both social media and image dataChai +and Nayak (2018). In other further research, we will merge +image and text data like article headlines and descriptions +associated with each image which should help improve the +performance of prediction model. We will conduct privacy +protection procedure such as Randomized Response Chai and +Nayak (2019) to the data. Then, we can evaluate our model +using the OSI database as well as social media such as Twit- +ter to determine the level of generalization our model may be +able to achieve. +8. CONCLUSION +Our paper demonstrates a rapid means of image augmenta- +tion and identifying key aspects of protest activity from pub- +licly available image streams, using open source software. +Although there are additional work that need to be done to +improve our classifier models, our approach creates greater +opportunities for the collection of such data to enable work +for public good. While traditional efforts to monitor violence +and protests may largely be hampered by linguistic barriers +and reporting delays, images streams from social media pro- +vide a language-agnostic means of assessing such threats. By +demonstrating that we were able to get reasonable prediction +accuracy of key aspects of protest images using SVM and +CNNs, we hope to enable its application to improve moni- +toring of social unrest activities within unstable regionsSun +et al. 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Neural Computing and Ap- +plications, 17(1):59–64, 2008. +Jingwei Yan, Wenming Zheng, Zhen Cui, Chuangao Tang, +Tong Zhang, and Yuan Zong. Multi-cue fusion for emo- +tion recognition in the wild. Neurocomputing, 309:27–35, +2018. +Jichong Chai and Tapan K. Nayak. A criterion for privacy +protection in data collection and its attainment via random- +ized response procedures. Electronic Journal of Statistics, +12(2):4264–4287, 2018. +Jichong Chai and Tapan K. Nayak. +Minimax randomized +response methods for providing local differential privacy. +Statistics, 04, 2019. +Ning Sun, Ling Leng, Jixin Liu, and Guang Han. +Multi- +stream slowfast graph convolutional networks for skeleton- +based action recognition. +Image and Vision Computing, +109:104141, 2021. + diff --git a/7tE2T4oBgHgl3EQf7whk/content/tmp_files/load_file.txt b/7tE2T4oBgHgl3EQf7whk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..496015754e9dfb66f36b735ccaf756b61e0eae56 --- /dev/null +++ b/7tE2T4oBgHgl3EQf7whk/content/tmp_files/load_file.txt @@ -0,0 +1,502 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf,len=501 +page_content='DEEP LEARNING BASED MULTI-LABEL IMAGE CLASSIFICATION OF PROTEST ACTIVITIES Yingzhou Lu, Kosaku Sato, Jialu Wang Electrical and Computer Engineering Department Virginia Tech Arlington,VA 22203, USA George Washington University Washington, DC 20052, USA Email: lyz66@vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='edu, Ksato@vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='edu, jialu@gwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='edu ABSTRACT With the rise of internet technology amidst increasing ur- banization rates, sharing information has never been easier, thanks to globally-adopted platforms for digital communi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The resulting output of massive amounts of user- generated data can be used to enhance our understanding of significant societal issues, particularly for urbanizing ar- eas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' In order to better analyze protest behavior, we enhanced the GSR dataset and manually labeled all the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' We used deep learning techniques to analyze social media data to detect social unrest through image classification, which performed well in predicting multi-attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Then, we used map visualization to display protest behaviors across the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Index Terms— Machine Learning, Deep Learning, Im- age Classification, Multi-Label Classification, Social Media 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' INTRODUCTION The study of protest activities plays a profound role in so- ciologists and scholars’ studying citizens’ political behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' With the advancement of social media networks, people now share an unprecedented amount of user-generated content in the form of text, images, and videos on the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Classifi- cation of social media data not only helps in understanding online behavior, but also elucidates significant priorities of urban populations that carry real-life consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Using social media data, we focuses on social unrest in the form of public protest images, specifically for Latin American coun- tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The traditional approach to the study of social media dataset focused on using natural language processing to mon- itor how hashtags and links are used by the user and the propagation of those items to other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' However, these ap- proaches may not effectively capture some important features or details of protest activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' For instance, we may be in- terested in knowing details such as whether there was a large crowd involved in the protest, if polices were present, or what the demographics (young or adults) of protesters carrying a sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Our approach uses image processing to capture those features of the protest activitiesFu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' We took several approaches in image classification of social media data: our initial approach is to utilize traditional machine learning methods such as Support Vector Machine (SVM)Weston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (1999) and a deep learning method like Convolutional Neural Networks (CNNs)Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2012) which have shown some advantages in large-scale im- age and video analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Traditional machine learning method, such as SVM, can be used to classify images with good accu- racy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' however, as the volume of data and number of classes for recognition increases, the deep learning approaches be- comes the more advanced approach for object recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' LITERATURE REVIEW As our objective of our model is to detect protest activi- ties using the image, the preliminary work relevant to our study is the EMBERS system by Naren, Patrick and et elRamakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The EMBERS system con- tinuously monitor the social media dataset such as Twitter, Facebook, news pages, and use data mining to process the trend to predict the protest activities in South America re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Their planned protest model based on custom multi- lingual lexicon matching predicted the protest activities with precision and recall rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='69 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='82 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' How- ever, their approach does not capture the additional features of the protests such as demography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Moreover, image classification is classical topic in com- puter vision area which aims to predict and assign each given image a specific label from several categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' However, background clutter, occlusion and variation in image scale make the computer vision tasks more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The tra- ditional approaches to perform image classification includes k-nearest neighbor and SVM algorithmsYi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' k-nearest neighbor is one of the simplest classification al- gorithm that aims at labeling an image based on the best fit result, but the model is usually not robust to noise or im- balanced class datasetZhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Similarly, SVM, which is originally proposed as a binary classification by Cortes and VapnikCortes and Vapnik (1995) is another clas- sical approach to perform classification and it has shown a better performance than the k-nearest neighbor in some ap- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='04212v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='CV] 10 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Labeled GSR images in GSR dataset plicationsBoiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Furthermore, our approach adopts a deep learning based on CNNs based on the study of the visual cortex of the human brain which have shown a great success recently in many computer vision applica- tionsAghdam and Heravi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' OSI DATASET The OSI (Open Source Indicators) dataset, provided by com- puter science department from Virginia Tech, is MITRE’s gold standard report (GSR) of protests organized by survey- ing newspapers for civil unrest reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The dataset also contains large samples of non-protest images that were col- lected in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' There are 48,713 images in GSR and 40,647 non-protest images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' High-confidence images indi- cate a top image among multiple images embedded in the articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' High-confidence images within datasets indicate a top image among multiple images embedded in the arti- clesJayachandra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' High-confidence GSR images relevant to GSR articles based on social protest total 7,884, and low-confidence GSR images total 40,829.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Details of the Image Labels Table 1 shows the visual attributes that characterize the protests which we used to label each image from the GSR dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Out of 40,647, a total of 9,504 images were hand- picked to train and test our prediction models by excluding bad data points that are obviously irrelevant to the social ac- tivities that we are interested in detectingYang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The Annotated images consist of 327 fire, 1,943 flag, 7,347 large crowd, 248 other, 2,159 police, 4,462 sign, and 1,233 student images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 1 contains sample images with their class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Each image has a label with vector of length 7 that has ”0”s and ”1”s corresponding to the index number of visual attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The ”Other” label was inferred from the absence of positive class labels across categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Challenges of the Dataset There are a few inherent challenges in our training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' First, some attributes of protest are commonly shared with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Sample protest images from training set Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Sample non-protest images from training set Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Visual attributes of protest images Class label Description Sample size 0 Fire Presence of active fire 327 1 Flag Presence of flag 1943 2 Large Crowd Presence of roughly more than 20 people 7347 3 Other None of the above or the below 248 4 Police Presence of police 2159 5 Sign Presence of a protest sign 4462 6 Student Presence of young students 1233 non-protest images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' For instance, Figure 2 shows a sample of protest images used to train the machine learning and deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Images with a protest class label often de- picted fire, police, handwritten signs, and large crowds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' How- ever, Figure 3 shows a sample of non-protest images in which large crowds are also frequently seen attribute while it also comprised of a variety of other objects (such as animals or soccer players).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Second, we have imbalanced dataset where it does not have exactly equal number of instances in each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' This issue is mainly defined by the specific subject or attribute we set up in our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The training of the classi- fiers in imbalanced dataset can cause the trained model clas- sifying images as images of majority class most of the times and under-represent the minority class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Image Augmentation for Imbalanced Dataset One way to balance the imbalanced classes is to use image augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Data augmentation is a method to artificially increase sample size of the training images through various pre-processing or combinations of multiple pre-processing of the image such as adding noise, flipping and re-scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Ta- ble 2 summarizes the different techniques we adopted to per- form image augmentation for minority classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Image aug- mentation has been considered as a promising method to im- prove the performance of prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' For instance, adding noise to our observation can help make the prediction Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Increased sample size after image augmentation Class label Image transformation used New sample size Fire Flipping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Scaling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Translation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Affine Transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Perspective Transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Intensity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Contrast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Filters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Crop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Shear 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='578 Flag Flipping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Affine Transform 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='829 Large Crowd – 7347 Other Flipping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Scaling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Translation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Affine Transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Perspective Transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Intensity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Contrast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Filters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Crop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Shear 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='472 Police Affine Transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Noise 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='477 Sign – 4462 Student Flipping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Affine Transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Crop 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='165 GSR Image with Labels [\'0",' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=" 0'," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' \'0",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' "0"Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Example images from the dataset after transformation model more robust in the face of social media dataset and pre- vents it from overfitting Perez and Wang (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Moreover, Python’s scikit-image library has the full list of the avail- able image transformation but we have adopted the thirteen of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Those are horizontal and vertical flipping, affine transfomration, perspective transformation, rescaling, crop- ping, blurring, changing contrast and intensity, gaussian and exposure filter, translation, and shearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 4 shows the example of such transformations of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' As a result of implementing image augmentation, we were able to signif- icantly improve the sample size of the minority classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' for instance, the sample size of ’fire’ class increased from 327 to 4,578 as seen in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Also, as an alternative approach, we considered oversampling using Synthetic Minority Over- sampling Technique (SMOTE) Chawla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' How- ever, we believe that image augmentation can create more variation in training images to prevent over fitting, and hence we did not utilize SMOTE in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' MULTI-LABEL IMAGE CLASSIFICATION Multi-label learning is a form of supervised learning where the classification algorithm learns from a set of images in which an image can belong to one or more classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The goal of multi-label image classification is to predict a set of class labels for the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' A more generalized approach is multi-class learning where each image is limited to one correct class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Multi-label classification and prediction is more practical since the many real world problems involve multiple objects belonging to different categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Multi- label classification is also applicable various domains such as text, video, and scene classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' For a typical multi- label image, objects of different categories in each image are located at varying positions with differing scale, zoom, size, and poseWon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' For example, two images labeled as ’police’ and ’fire’ may have different spatial arrangements of identified objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Although factors such as differing ar- rangements or occlusion can contribute to the inaccuracy of multi-label classification, we expect reasonable results with a sufficiently large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Details of implementation of multi- label classification for SVM and CNNs will be discussed in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' APPROACH OF IMAGE CLASSIFICATION In our approach, we utilized multi-label SVM and CNNs to detect protest attributes in image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' First, multi- label SVM will be explained in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' In our proposal, SVM is a baseline model to evaluate the performance of the pre- diction model using CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' As the problem requires a large dataset for training and good accuracy of classifying many protest attributes, we believe that CNNs will perform better than SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Support Vector Machine SVM is originally proposed as a binary classification by Cortes and VapnikCortes and Vapnik (1995), but the model has been extended to apply to multi-label classification prob- lemsWeston and Watkins (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' One-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='-All: One-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='-All is a classical approach to solve k-class pattern recognition problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' It involves training a sin- gle binary classifier per class, with the samples of one class as positive samples while other samples are set as negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' More specifically, using this method, n-th classifier finds a hyperplane between class n and the rest of the classesWeston and Watkins (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' A point where the distance from the margin is maximal is assigned to the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' We aim at de- tecting seven classes so that this strategy requires the training of seven different SVMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' During testing the models, all clas- sifiers would vote ’true’ by predicting that a testing sample belongs to their class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' In the end of testing, a sample is classi- fied by the ensemble as the class that has the highest number of votes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' One-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='-All is widely used in multi-label classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Weighting Hyper-Parameter for SVM: Imbalanced dataset causes misclassification of images that belonging to the minority class impacted more heavily than that of the ma- jority class because the frequency of the minority class is rare compared to that of the majority class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' In order to mitigate over fitting of training classifier resulted from the imbalanced data, we propose the modification of hyperparameter C in SVM’s objective function which determines the penalty for misclassifying the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Instead of defaulting C to be one, Ck belonging to class k will have different values as shown in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Ck = C · n knj (1) As you can see, the updated Ck value will be inversely propor- tional to instances of j class in order to increase Ck value for the minority class in order to mitigate under-representation issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' k is the number of class and j is the sample size be- longing to the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Convolutional Neural Networks 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Architecture CNNs consist of input, convolution, activation function, pool- ing, deep layers, and output layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Throughout the training of the network, the parameters are updated except for the ones between convolution and pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' There are some important properties of the convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Some patterns are smaller Original Image Contrast Adjusted Noise Added ELPAQUETE ELPAOUETE Cropped Horizontal Flip Vertical Flip ESEKKON PAQUETE ETBHOREIE CIEN NOAENERKOthan the entire image so that the image can be subsampled to reduce the image size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' this is to train fewer parameters in the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The same patterns can appear in different re- gions so that the same set of parameters can be used to reduce computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Convolutional Layers: Convolution of a filter on an input image is a point-wise multiplication operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The activation function, which in our case is Rectified Linear Unit (ReLU) activation, is applied on each image separately in an element- wise fashion to create activation maps based on outputs of the convolution Aghdam and Heravi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Activation Function: Nair and Hinton introduced the non-saturating nonlinearity f(x) = max(0, x), also known as the ReLU, which has gained popularity in the deep-learning community because of its fast computing time Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='Hence, our model applied the ReLU nonlinear- ity function to the output of every convolutional and fully- connectedKrizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Pooling Layer: In our model, we applied a 2x2 filter size with a 2-length stride after each layer with the option of max- pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Max-pooling applies the filters and the stride to the input and returns the maximum value, dropping the non-max values in each sub-region that convolution is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Fully Connected Layers: The fully connected layer uses those inputs to produce N-dimensional vectors, where N is the number of classes needed for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Loss Function: In our model, we used sigmoid cross- entropy for multi-label classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Cross-entropy is used to define the loss function in training the network in which the model is penalized if it estimates a low probability for the target class Nielsen (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' J(θ) = − 1 m m � i=1 K � k=1 [yi k log(ˆpi k)] (2) For the loss function, we used Adaptive Moment Estima- tion (ADAM) Kingma and Ba (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' ADAM keeps track of a learning rate for each network weight and computes indi- vidual adaptive learning rates for different parameters based on estimates of first and second moments of the gradients Kingma and Ba (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Since ADAM is an adaptive rate learning algorithm, it requires less tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The default learn- ing rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='001 is often used to support usability of the algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Table 3 shows our CNNs architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' There are three convolutional layers with ReLU, and max-pooling is used to down-sample the image after each convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The filter size for max-pooling is 2 × 2 so that output image of the max-pooling is half the size of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The convolutional layers uses the filter size of 3 × 3 while length of stride equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The first fully connected layer (FC1) is a vector with a length of 1024, and the second fully connected layer (FC2) is a vector of length 7 which is the number of class labels to predict visual attributes of protest images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Evaluation Method of Multi-label Classification Evaluation of multi-label classification has a notion of being partially correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' One way to evaluate the classification is label-set based accuracy or exact match that considers par- tially correct as incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' On the other hand, evaluation of label-based accuracy is carried out on a per label basisChen Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Architecture of Multi-Label Classification CNN Layer Feature Map Feature Size Filter Size Stride Pad Activation FC2 7 Sigmoid FC1 1024 ReLU Max-Pooling 128 4x4 2x2 2 Same Conv3 128 7x7 3x3 2 Same ReLU Max-Pooling 64 14x14 2x2 2 Same Conv2 64 28x28 3x3 2 Same ReLU Max-Pooling 32 56x56 2x2 2 Same Conv1 32 112x112 3x3 2 Same ReLU Input 3(RGB) 224x224 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The calculation method of label-set accuracy, where a predicted set of labels ˆy must exactly match the ground truth y, is shown in equation (3) Read et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 0/1 loss dictates that any label vector not predicted perfectly will be given a zero score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 0/1 loss = 1 − 1 N N � i=1 1yi= ˆ yi (3) Label-based accuracy is more lenient approach to evaluate the performance since it does not consider multi-label problem as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' When each label has a separate binary evaluation, we have hamming loss which is shown in the following equation: Hamming Loss = 1 − 1 NL N � i=l L � j=l 1yi= ˆ yi (4) We adapted both approaches to evaluate the performance of our multi-label classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Threshold Selection The fully connected layer represents a vector containing prob- ability for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The threshold function can be used to obtain a multi-label prediction ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Specifically, we used the Matthews Correlation Coefficient (MCC) which is an evalu- ation metric of binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The MCC is a correla- tion coefficient for ground truth versus predictions and varies between -1 and 1, where 1 represents a perfect prediction Gorodkin (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The MCC is given by the following equa- tion (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' MCC = Tp × Tn − Fp × Fn � (Tp + Fp)(Tp + Fn)(Tn + Fp)(Tn + Fn) (5) For multi-label classification, the MCC is defined in terms of a confusion Matrix C for K classes in the equation (6) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' MCC = c × s − �K k pkxtk � (s2 − �K k p2 k)(s2 − �K k t2 k) (6) The values tk = �K i Cik is the number of times class k truly happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' pk = �K i Cki is the number of times class k was predictedTian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' c = �K i Ckk is the total number of samples correctly predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' s = �K i �K j Cij is the total number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Advantages and Disadvantages of SVM and CNN In image classification, there are advantages and disadvan- tages for both SVM and CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Theoretically, SVM is very good at finding the margin and hyperplane for classification, and it is very robust for high dimensional dataZhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' However, SVM model is sensitive to noise, for exam- ple, if there is a noise in background or a visible object in one image is occluded or partially blocked by scenes in other, it will have a negative impact on the performance of the clas- sification model Cortes and Vapnik (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Moreover, since one-vs-all involves training a binary classifier for all classes, computation time can be very expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' On the other hand, the main advantage of CNNs is that it can be used to extract important image features with a suf- ficiently large datasetLeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The performance of CNNs classifier largely depend on the size of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' The bottle necks in training the CNNs model are computa- tional time and memory used to retain activation from for- ward pass and error gradients computation when dataset is very largeSun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' However, an efficient parallel computation with a help of GPU or training a model in mini batches can be used to mitigate those issues to some degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Also, there are many parameters for CNNs that need to be set by the users in order to train a robust and good prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' EXPERIMENT We implemented K-class SVM and CNNs using scikit-learn and TensorFlow libraries in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' In the experiment, we used our personal server desktop which runs on Windows with 2 Intel Xeon E5-2630 V3 CPU 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='4 GHz and 8 small cores with RAM size of 64GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' First, we split the sample im- ages via image augmentation from Table 2 into training and testing (the ration of 80% and 20% respectively) and then we re-sized each image to 224x224 with 3 color channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' However, the implementation of SVM in scikit-learn does not adopt online learning so that we had to down-sample the sam- ple images from 31,472 to 12,000 to avoid memory limit er- ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' On the other hand, with a help of mini-batches, we used all of the data points without any down-sampling for CNNs model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' We trained a baseline SVM using One-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='-All method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Our final setting of SVM consisted of max iterations to be ran as 4000 to ensure it converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Also, we experimented weighting of hyperparameter in equation (1) but it did not im- prove the result so our final setting of the weight parameters for each classes are set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' For CNNs, we used the learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='001 which is a standard rate and mini batch size of 202 images with numbers of batches as 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' RESULTS AND DISCUSSION 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Evaluation Evaluation was conducted on each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Since we have the manual label as ground truth, we calculated accuracy, the pre- cision rate, recall rate, and F1 score for each class respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Specifically, the following is the equations we used to calculate each evaluation criteria: Precision rate= T P T P +F P , Recall rate= T P T P +F N , Accuracy= T P +T N T P +T N+F P +F N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' F1 score is the harmonic mean of precision rate and recall rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Evaluation of SVM Model Prediction Fire Flag Large Crowd Other Police Sign Student Accuracy (%) 88 74 60 89 77 63 79 Precision (%) 53 53 63 53 52 55 50 Recall (%) 35 22 60 24 30 39 19 F1 Score (%) 42 31 61 33 38 45 27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' SVM vs CNN Precision Rate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' SVM vs CNN Recall Rate 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Result Training SVM model with one-vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='-all method took longer than 12 hours and consistently consumed 70-90% of avail- able memory on our machine whereas the CNNs model only took less than half of the training time with much less con- sumption of memory with a help of mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Therefore, we were able to obtain the results by CNNs easier and faster than the SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Table 4 and 5 shows the accuracy, precison, recall, and F1 score of each predicted lables for SVM and CNNs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' We also plotted the precision and recall of the two models side by side in histogram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 5 and 6 to compare the performance of the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' As you can see, their overall performance is comparable to each other but recall using CNNs is slightly better than that of SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' For the evaluation of CNNs prediction, We calculated the best threshold using MCC to transform the probability of each label from the fully connected layer into the 7 predicted class labels: the calculated thresholds are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='5 for ’fire’, ’flag’, ’large crowd’, ’other’, ’police’, ’sign’, and ’student’ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Prediction accuracy per Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Evaluation of CNNs Model Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='Fire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='Flag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='Large Crowd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='Other ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='Police ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='Sign ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='Student ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='Accuracy (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='91 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='71 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='89 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='61 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='Precision (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='74 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='Recall (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='73 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='F1 Score (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='71 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='73 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='SVM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='■CNNs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='74 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='52 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='5051 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='FIRE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='FLAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='LARGE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='OTHER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='POLICE ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='30 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='FIRE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='FLAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='LARGE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='OTHER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='POLICE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='SIGN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='STUDENT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='CROWDFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Image of a burning vehicle with police in background (top);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' image of bikers and a flag (bottom) label reached almost 77% on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 7 shows sample test images of ’fire’ and ’police’ on the left and ’large crowd’ and ’police’ on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Our CNNs model predicted them correctly but when we evaluated our classifier model with a large dataset, we found that our label-set accuracy was very low around 20% due to the challenges of exact matching on a multi-label classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Future work From the experiment, we learnt that we were able to get rea- sonable performance using both SVM and CNNs model to predict each class label separately but the bottom line perfor- mance of our prediction model is still not desirable: our goal is to increase the accuracy of exact matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Therefore, Fu- ture work can be done in following aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' First, we can apply state-of-the-art algorithm like Generative Adversarial Network to generate more training samples, which would be helpful to prevent over fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Second, we modify the equa- tion for SVM to enhance the classifier, and improve the deep learning modelYan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Moreover, The main limi- tation of our image classification approach is that it does not consider the credibility of the source in decision making, and hence requires assessment of the social media source or of each image posted on the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Also, there are privacy protec- tion concern in using both social media and image dataChai and Nayak (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' In other further research, we will merge image and text data like article headlines and descriptions associated with each image which should help improve the performance of prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' We will conduct privacy protection procedure such as Randomized Response Chai and Nayak (2019) to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Then, we can evaluate our model using the OSI database as well as social media such as Twit- ter to determine the level of generalization our model may be able to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' CONCLUSION Our paper demonstrates a rapid means of image augmenta- tion and identifying key aspects of protest activity from pub- licly available image streams, using open source software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Although there are additional work that need to be done to improve our classifier models, our approach creates greater opportunities for the collection of such data to enable work for public good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' While traditional efforts to monitor violence and protests may largely be hampered by linguistic barriers and reporting delays, images streams from social media pro- vide a language-agnostic means of assessing such threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' By demonstrating that we were able to get reasonable prediction accuracy of key aspects of protest images using SVM and CNNs, we hope to enable its application to improve moni- toring of social unrest activities within unstable regionsSun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' ACKNOWLEDGMENT We thank Virginia Tech CS department for providing us with the OSI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' References Tianfan Fu, Cao Xiao, Cheng Qian, Lucas M Glass, and Ji- meng Sun.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Ning Sun, Ling Leng, Jixin Liu, and Guang Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Multi- stream slowfast graph convolutional networks for skeleton- based action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} +page_content=' Image and Vision Computing, 109:104141, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE2T4oBgHgl3EQf7whk/content/2301.04212v1.pdf'} diff --git a/89FST4oBgHgl3EQfajh_/vector_store/index.pkl b/89FST4oBgHgl3EQfajh_/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..8cd64ec77b345928e2789bd20b973234d512d150 --- /dev/null +++ b/89FST4oBgHgl3EQfajh_/vector_store/index.pkl @@ -0,0 +1,3 @@ +version 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Nicolas-Carlock5, Ana I. Bento6, +Benjamin M. Althouse7,8, Bernardo Gutierrez9,10,11, +Marina Escalera-Zamudio9,11, Arturo Reyes-Sandoval12,13, +Oliver G. Pybus9,14,18, Alessandro Vespignani1,2, +Jose Alberto Diaz-Quiñonez*†15, Samuel V. Scarpino*‡1,16,17, and +Moritz U.G. Kraemer*§9,18 +1Network Science Institute, Northeastern University, Boston, Massachusetts, USA +2Laboratory for the Modeling of Biological & Socio-technical Systems, +Northeastern University, Boston, Massachusetts, USA +3Massachusetts General Hospital, Boston, Massachusetts, USA +4Instituto Mexicano del Seguro Social, Ciudad de México, México +5Instituto de Investigaciones Jurídicas, Universidad Nacional Autónoma de México, +Ciudad de México, México +6Department of Epidemiology and Biostatistics, School of Public Health, +Indiana University, Bloomington, Indiana, USA +7Information School, University of Washington, Seattle, Washington, USA +8Department of Biology, New Mexico State University, Las Cruces, New Mexico, USA +9Department of Biology, University of Oxford, Oxford, UK +10School of Biological & Environmental Sciences, +Universidad San Francisco de Quito, Quito, Ecuador +11Consorcio Mexicano de Vigilancia Genómica +12The Jenner Institute, University of Oxford, Oxford, UK +13Instituto Politécnico Nacional, IPN, Ciudad de México, México +14Department of Pathobiology and Population Science, Royal Veterinary College, London, UK +15Instituto de Ciencias de la Salud, Universidad Autónoma del Estado de Hidalgo, +Pachuca, Hidalgo, México +16Institute for Experiential AI, Northeastern University, Boston, Massachusetts, USA +17Santa Fe Institute, Santa Fe, New Mexico, USA +18Pandemic Sciences Institute, University of Oxford, UK +February 1, 2023 +∗b.klein@northeastern.edu +†alberto_diaz@uaeh.edu.mx +‡s.scarpino@northeastern.edu +§moritz.kraemer@biology.ox.ac.uk +1 +arXiv:2301.13256v1 [physics.soc-ph] 30 Jan 2023 + +Abstract +During outbreaks of emerging infectious diseases, internationally connected cities +often experience large and early outbreaks, while rural regions follow after some delay +[1–6]. This hierarchical structure of disease spread is influenced primarily by the mul- +tiscale structure of human mobility [7–9]. However, during the COVID-19 epidemic, +public health responses typically did not take into consideration the explicit spatial +structure of human mobility when designing non-pharmaceutical interventions (NPIs). +NPIs were applied primarily at national or regional scales [10]. Here we use weekly +anonymized and aggregated human mobility data and spatially highly resolved data +on COVID-19 cases, deaths and hospitalizations at the municipality level in Mexico +to investigate how behavioural changes in response to the pandemic have altered the +spatial scales of transmission and interventions during its first wave (March - June +2020). We find that the epidemic dynamics in Mexico were initially driven by SARS- +CoV-2 exports from Mexico State and Mexico City, where early outbreaks occurred. +The mobility network shifted after the implementation of interventions in late March +2020, and the mobility network communities became more disjointed while epidemics +in these communities became increasingly synchronised. Our results provide actionable +and dynamic insights into how to use network science and epidemiological modelling +to inform the spatial scale at which interventions are most impactful in mitigating the +spread of COVID-19 and infectious diseases in general. +Table 1: Policy summary +Background +The establishment, persistence and growth rates of COVID-19 mainly depend +on human mobility and mixing. However, current approaches attempting to +limit transmission have been primarily based on administrative boundaries +instead of the natural scales of human mobility. +Main findings +& limitations +Using aggregated and anonymized human mobility and detailed COVID-19 +case data, we find that the scales of human mixing shift during the pandemic +and that transmission is highly clustered amongst mobility communities. +Policy +implications +Structuring interventions based on spatial mobility may be more effective com- +pared to interventions based on administrative boundaries. Future pandemic +control interventions should consider empirical human mobility networks when +designing interventions. +2 + +1 +Introduction +The transmission of infectious diseases is highly heterogeneous. Differences in population +structure, the landscape of prior immunity, and environmental factors, result in differences +in the timing of outbreaks, their magnitude, and duration [2, 3, 9, 11–20]. +For infec- +tious diseases, one principal component determining the spatial structure of outbreaks is +the frequency of interactions between susceptible and infectious individuals within and be- +tween regions. In most geographies, public health decision-making authority follows political +boundaries. However, from an epidemiological perspective, the relevant spatial units may +not strictly follow political boundaries but rather human mixing [8, 14, 21]. Evaluating +the spatial structure of COVID-19 transmission remains important in determining optimal +interventions (non-pharmaceutical and/or vaccination) to reduce transmission and limit the +risk of resurgence of cases [22–25]. +During the first half of 2020, Mexico experienced one of the largest SARS-CoV-2 epi- +demics worldwide, with more than 600,000 cases and 65,000 confirmed deaths reported be- +tween February and September 2020 [26] (Fig. 1a). The epidemic wave peaked in May in +the largest metropolitan areas of Mexico City and the State of Mexico and later ignited +epidemics in all other states [27], peaking between June and July 2020 (Fig. 1b). Here we +combine municipality level epidemiological data with weekly anonymized aggregated human +mobility data at the same scale, to characterise the spatial scales of the Mexican COVID-19 +pandemic and their implications for the implementation of spatially targeted interventions. +2 +Results +2.1 +Spatial expansion of COVID-19 in Mexico +In Mexico, the spatial range of transmission expanded rapidly after reports of the earliest +cases in March 2020, with over 700 municipalities reporting transmission by July 2020 (out of +2,448, Fig. 1c). During April and May the risk of positive RTq-PCR confirmed cases amongst +men aged 30-69 was 1.4 times higher than between July 1 and September 1 (Fig. 1d,e), +indicating that the epidemic spread initially within and through these age groups (Extended +Data Figure A.1). +This dynamic trend in the demographics of cases is similar to that +observed in other countries during the early stages of the pandemic [28, 29]. +States that experienced early transmission were the state of Mexico and Mexico City +(Fig. 1b) [27]. Due to the centrality of Mexico City connecting people from abroad (in- +ternational arrivals) and within Mexico we hypothesise that human mobility from these +states was a key driver of the spread of COVID-19 in Mexico. Using anonymized, opt-in +and aggregated human movement data from mobile phones (Materials and Methods) we +find that case growth rates across Mexican states were well predicted by a lagged model +of human movements from the State of Mexico and Mexico City between March and May +2020 (Fig. 2c, conditional R2 = 0.62; see Materials & Methods). Further, we observe that +the share of overall relative human mobility to and from Mexico and Mexico City increased +3 + +Apr. +May +Jun. +Jul. +Aug. +Sep. +0 +2 +4 +6 +8 +10 +12 +14 +Total reported cases as of Sept. 1, 2020: +(b) Daily new cases per 100,000, state level (7-day rolling avg.) +Apr. +May +Jun. +Jul. +Aug. +Sep. +0 +200 +400 +600 +800 +(c) +Municipalities reporting cases (2,448 total) +Apr. +May +Jun. +Jul. +Aug. +Sep. +0% +10% +20% +30% +40% +(d) +"early" +"late" +Percent of new cases (7-day rolling avg.) +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +1.4 +Early: April 1 - May 1 +Late: June 30 - Aug. 30 +F: under 30 +F: under 30 +F: 30-49 +F: 30-49 +F: 50-69 +F: 50-69 +F: over 70 +F: over 70 +M: under 30 +M: under 30 +M: 30-49 +M: 30-49 +M: 50-69 +M: 50-69 +M: over 70 +M: over 70 +(e) +Relative risk ("early" vs. "late" periods) +10 +100 +1000 +10000 +Cases per 100,000 +(as of September 1) +Figure 1: Epidemiological situation of COVID-19 in Mexico. (a) Map of cumulative +cases per 100,000 people, as of September 1, 2020. (b) Timeline of new cases per 100,000 +population at the state level (7-day rolling average), highlighting the 15 states with the most +severe cumulative outbreaks. (c) Number of municipalities that reported confirmed cases of +COVID-19 through time. (d) Age and sex distributions of confirmed COVID-19 cases across +Mexico, highlighting “early” and “late” periods during which the relative risk of infections +were calculated. (e) Age and sex relative risk ratios of infection, comparing the early vs. +late periods from panel (d). +markedly during that period (Fig. 2b) when overall human mobility between states declined +(Fig. 2b, Extended Data Figure A.2 showing state level data on change in human mobility). +This points towards a change in the network structure of human mobility in Mexico, as +documented in some other countries [30, 31]. Overall transmission, and the importance of +Mexico City driving the epidemic, declined after the implementation of NPIs through May +2020. However, after the lifting of physical distancing measures on June 1st (see table of +documented changes in NPIs, Table A.1), case growth rates in the country increased again as +4 + +a function of mobility from Mexico City, in line with models predicting that lifting lockdowns +can lead to reseeding of transmission chains from larger to smaller cities where epidemics +were successfully controlled (Fig. 2b, Table A.1, [7]). +Variation in weekly new cases within each state in Mexico are generally well predicted +by cases in Mexico City weighted by human mobility except for Baja California, More- +los, Chihuahua, Oaxaca, and Chiapas (Extended Data Figure A.3). We hypothesise that +epidemics there were possibly seeded from other countries (USA and Guatemala); further +SARS-CoV-2 genomic analyses of unbiased collections of samples will be needed to confirm +the SARS-CoV-2 lineage dynamics in these states [27, 32–36]. Human mobility data showing +cross border (US to Mexico) movements indicate higher overall mobility to bordering states +in Mexico and growth rates in US-Mexico border states appear higher in the period between +24 May - 28 June 2020 (Extended Data Figures A.4, A.5, A.6). The high mobility during +that phase resulted in larger case numbers in states bordering the US when compared to +other states in Mexico (Extended Data Figure A.5). +2.2 +The scales of COVID-19 transmission +It is well known that reductions in mobility (a proxy for reductions in population mixing) +have reduced the transmission of COVID-19 within a location [38]. However, it remains un- +clear how structural changes to the mobility network (shifts in the frequency and intensity +of mobility within and among regions) have impacted COVID-19 dynamics empirically [30, +31, 39–41]. Our underlying hypothesis is that more tightly connected communities exhibit +more synchronised epidemic dynamics and, conversely, that more disjointed individual com- +munities have less synchronised epidemics and their epidemics are more likely to fade out +[4–6] (here, communities are equivalent to municipalities and synchrony is defined as the +similarity among communities in weekly case growth rates [42]). Both processes have critical +implications for disease mitigation and eliminations locally, and at a country level [7, 43–47]. +The Mexican government announced stringent physical distancing policies on March 30th, +2020 which resulted in marked changes in the mobility network (Fig. 2a, Table A.1). +To quantify the degree to which mobility patterns are structured by geopolitical bound- +aries, we use a community detection algorithm that groups municipalities based on their +movement patterns [48]. Specifically, we aim to identify groups of municipalities such that +movements between municipalities within the same group, i.e., community, are more fre- +quent than movements to other municipalities in other communities. Community detection +is often accomplished via modularity maximization [49]; however, these approaches neglect +information about the flow of mobility through the network. Instead, we leverage the map +equation via an algorithm called InfoMap [48]. The InfoMap algorithm utilises an informa- +tion theoretic approach to derive expected connectivity patterns if the observed flows were +entirely determined by a random walk process. For this study, InfoMap is ideal because +it is conceptually related to infectious disease transmission models, which often also utilise +stochastic processes [50]. +The aim is to identify municipalities where frequent interactions between individuals +occur, such that the detected communities approximate the spatial scales of disease trans- +5 + +03-01 +04-12 +05-24 +07-05 +08-16 +20% +40% +60% +80% +100% +(b) +Percent of typical mobility +(total across all of Mexico) +03-01 +04-12 +05-24 +07-05 +08-16 +0.004 +0.002 +0.000 +0.002 +0.004 +0.006 +(c) +Coefficients of case growth rate +and mobility from Mexico City +03-01 +04-12 +05-24 +07-05 +08-16 +18% +19% +20% +21% +22% +23% +24% +(d) +Dynamics of states' outgoing +mobility to Mexico City +Community size distribution +(n = 16, using Infomap) +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +Figure 2: Human mobility and transmission of COVID-19 in Mexico. (a) Pre- +pandemic average of the inter-municipality mobility network, coloured by network commu- +nity (detected using the Infomap algorithm). Mobility flow data is based on the aggregated +Google Mobility Research dataset (see Materials & Methods). (b) Percent of typical weekly +mobility nationwide (typical refers to mobility between January 12 and February 29, 2020). +(c) Evolution of the coefficients of mobility flow from Mexico City in (lagged) correlations +with state-level case rates across the country, highlighting the key role that mobility from +Mexico City played in the early stage of the epidemic. (d) Average fraction of total outgoing +mobility from each state that is to Mexico City (black) and the median entropy of states’ +distributions of outgoing mobility. Error bands correspond to 95% confidence intervals. +mission (i.e., communities in which it is assumed that infection spreads via contacts within +a relatively homogeneously mixing population [51]). Accounting for spatial heterogeneity is +6 + +Network of average mobility flow: +(a) +2020-01-12 t0 2020-02-23 +Administrative +(state) boundary + Example: +Chiapas +Network community03-01 +03-29 +04-26 +05-24 +06-21 +07-19 +08-16 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +(c) +Standard deviation of municipality growth rates +within grouping (lower values: higher synchrony) +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +Growth rate (reported cases) +0.11 +Nayarit +Morelos +Michoacán de Ocampo +México +Jalisco +Hidalgo +Guerrero +Guanajuato +Durango +Distrito Federal +Chihuahua +Chiapas +Colima +Coahuila de Zaragoza +Campeche +Baja California Sur +Baja California +Aguascalientes +0.36 +0.26 +0.69 +0.43 +0.29 +0.30 +0.44 +0.31 +0.83 +0.42 +0.57 +0.47 +0.44 +0.37 +0.39 +0.48 +0.47 +... +... +... +... +0.465 +(mean) + Std. dev. of growth rates +(d) +Example: Variance in growth rates (2020-04-19) +municipalities grouped by administrative boundaries +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +Growth rate (reported cases) +0.38 +Comm. 01 +0.54 +Comm. 02 +0.35 +Comm. 03 +0.45 +Comm. 04 +0.50 +Comm. 05 +0.43 +Comm. 06 +0.30 +Comm. 07 +0.49 +Comm. 08 +0.37 +Comm. 09 +0.31 +Comm. 10 +0.59 +Comm. 11 +0.00 +Comm. 12 +0.00 +Comm. 13 +0.00 +Comm. 14 +0.00 +Comm. 15 +0.00 +Comm. 16 +0.00 +Comm. 17 +0.00 +Comm. 18 +... +... +... +... +0.223 +(mean) + Std. dev. of growth rates +(e) +Example: Variance in growth rates (2020-04-19) +municipalities grouped by network communities +Figure 3: Network structure determines the synchrony of epidemics. (a) Grouping +of municipalities based on the state administrative boundaries. Shaded municipalities are re- +moved from downstream analyses as they could not be assigned a movement community (see +Materials & Methods). (b) Example grouping of municipalities based on human movement +data and a community detection algorithm [37] (Materials and Methods). Colours indicate +movement communities. Grey municipalities have limited recorded movements and could not +be assigned to a community and were consequently excluded from analysis. (c) Synchrony of +weekly growth rates of epidemics across municipalities as measured by the pairwise standard +error between growth rates. The lower the error, the more synchronised epidemics are. Blue +line shows grouping by network communities, and orange shows groupings by state admin- +istrative boundaries. The green dashed line shows the nationwide trend in reported cases +during this period. For a visual intuition of the differences in within-community standard +deviations of growth rates, see Extended Data Figure A.9. +known to be important for assessing strategies for interventions [2], especially in areas that +have marked differences in urban and rural areas [52]. Using this algorithm, we identify 16 +communities before the first cases of COVID-19 were detected in Mexico (Fig. 3b). Com- +munity size and organisation changed following the announcement of the lockdown (March +23 and 30, 2020) in Mexico and communities generally became smaller (fewer municipalities +within each community (Extended Data Figures A.7 and A.8 show the communities for each +week during the study period). At the peak of the lockdown, we identified approximately 60 +movement communities (a 4-fold increase from the baseline period). +More specifically, there are two notable shifts in the network following the introduction of +NPIs. First, more communities are identified but importantly the size of these communities +shrinks disproportionately so that one community expands (Mexico City) and many very +7 + +b +Example network grouping +(infomap)a) +Administrative grouping +(states)small ones emerge (Fig. 2d). Further, as a result of the lockdown human movements across +municipalities decline more rapidly than movements within a community with one important +exception: Mexico City. There we observe that the ratio of within municipality movements +declines at a similar rate than movements across municipalities (Extended Data Figure A.2) +further proving its central importance in the mobility network in Mexico. +We then compared the weekly infection incidence growth rates within each community +and contrasted them to growth rates under a scenario in which municipalities are grouped +based on state boundaries (black lines, Fig. 3a,b). As expected, we find that epidemics in +municipalities that are grouped by human mobility were more synchronised compared to +those grouped by state (Fig. 3c; see Extended Data Figure A.9 for an illustration of the +variance in municipality epidemic growth rates for several example groups of municipalities +defined by administrative or network boundaries). +The synchrony among municipalities +within each community were maximised in April and May 2020, a period when cases were +rapidly rising across the country. After June, epidemics that are grouped by movement are +still more synchronised, but the differences with groupings by state appear to be smaller +(Fig. 3c). This later period (June to October 2020) is a time when Mexico City appears +to also lose importance in seeding the epidemic across the country, and local factors (e.g., +population size) become more important in determining the epidemic trajectory [53]. These +results are expected as local factors become more influential in determining disease dynamics +(population size, local mixing) and that the importance of continued virus re-importations +wanes through time [33]. +3 +Discussion & Limitations +We present a generalisable approach for understanding the spatial structure of transmission +of COVID-19 and other emerging infectious diseases by accounting for the variations of +the human mobility network. We aimed to differentiate the transmission dynamics at a +level defined by administrative boundaries from that defined by simple community detection +algorithms that are applied to aggregated anonymized weekly human mobility data. We +find that as human mobility network structures change, so does to spatial transmission. +Incorporating these findings into real-world public health decision-making may result in +more effective strategies to control an epidemic [54–57]. +The European Commission for +example published a report on Mobility Functional Areas (MFAs) which were informed by +mobile phone data but the adoption of these recommendations remained sparse [55]. +Our model and results are only as accurate as the data that go into them. The Mexican +COVID-19 database may suffer from underreporting due to testing shortages, changing case +definitions and spatial heterogeneity in reporting [58, 59]. For example, relatively few cases +were reported from Oaxaca (Fig. 1a) which may be due to barriers to access to testing [60]. +Future extensions of the model and as the pandemic continues will need to take into account +high-resolution SARS-CoV-2 cross-immunity. Further, our model is based on higher level +descriptions of the population (raw case data and population level human movement data) +and these do not capture the high contact heterogeneity within each municipality (e.g., de- +8 + +mographic heterogeneity and assortative mixing) shown to be important in the transmission +of COVID-19 [61]. Contact patterns may differ significantly by age group, employment sta- +tus and other factors not accounted for in this work. We did however observe heterogeneity +in the demographic makeup of cases during the earlier phases of the Mexican COVID-19 +pandemic. +Further, results should be interpreted in light of important limitations related to the +human mobility data. First, the Google mobility data is limited to smartphone users who +have opted into Google’s Location History feature, which is off by default. These data may +not be representative of the population as whole, and furthermore their representativeness +may vary by municipality. +Importantly, these limited data are only viewed through the +lens of differential privacy algorithms, specifically designed to protect user anonymity and +obscure fine detail. +Mexico is composed of 31 free and sovereign states and Mexico City, united under a +federation. +This means that each administrative region or state is governed by its own +constitution, although they are not completely independent of the federal jurisdiction. Fur- +thermore, each state is divided into municipalities, the nation’s basic administrative unit, +which possesses limited autonomy (discretionary power on how best to respond to, or apply +a public policy). Under a serious nationwide health threat or emergency, such as a pandemic, +the federal Ministry of Health (MoH) acquires full authority over the health policies to be +implemented nationwide. Nevertheless, Mexican law establishes that the General Health +Council (GHC), a collegial body that reports to the president of the republic has the char- +acter of health authority, and can emit obligatory norms to be abided by the MoH. The +GHC is presided by the Minister of Health, and is conformed by federal institutions (e.g.h, +Economy, Communication & Transport) as well as academic institutions, representatives +from pharmaceutical industry, and other health system actors [62]. Given its mandate and +position in the Mexican health system, the GHC constitutes a promising agent to drive pub- +lic policy outside of the margins or across geo-administrative units. Furthermore, there are +examples of inter-state and inter-municipality coordination to resolve problems that extend +beyond their borders such as waste management, tax, policing, and perhaps most relevant, +health provision. It is in these contexts where evidence-based interventions on innovative +approaches, such as the ones presented here become not only an option but a possibility, +with greater impact in reducing transmission as compared to approaches where interven- +tions are based on administrative boundaries. However, theory often differs from practice +and reality brings along additional and expected factors into play (e.g., economic [63] and +political interests) many of which are not accounted for in this work. Some state governors +for example refused to comply with federal health policies in the early relaxation phase in +May 2020 [64]. +Mexico has suffered a large and devastating epidemic, and we hope that our findings +contribute to a more rational implementation of interventions in the future that can account +for the substantial and changing spatial heterogeneity in transmission. Such analyses can +be updated and translated to any other country in the world for which aggregated human +mobility data is available. Future work should also focus on validating the inferred spatial +9 + +scales with genomic data [32, 33, 65] or other coarse-graining techniques [66, 67]. Developing +interventions using patterns observed in empirical mobility networks must be added to the +list of priorities for pandemic response and preparedness in the 21st century. +4 +Materials & Methods +Epidemiological data: +Epidemiological data include individual level information on pa- +tients with confirmed RTq-PCR COVID-19 infection between March - September 30th, 2020. +Data were downloaded from http://datosabiertos.salud.gob.mx/gobmx/salud/datos_ +abiertos/datos_abiertos_covid19.zip (last accessed October 24, 2020). Data include +information about patients demographics (age and sex) and municipality of residence. In all +analyses we used the date of onset of symptoms. +Population and travel data: +Human mobility and population data were extracted at the +municipality level based on the 2016 boundaries (INEGI 2016: https://www.inegi.org. +mx/app/mapa/espacioydatos/default.aspx). Population data were downloaded from the +COVID-19 indicator dataset, which was provided by INEGI (https://www.inegi.org.mx/ +investigacion/covid/). +Aggregated and anonymised human mobility data: +We used the Google COVID- +19 Aggregated Mobility Research Dataset described in detail in [68, 69], which contains +anonymized relative mobility flows aggregated over users who have turned on the Location +History setting, which is turned off by default. This is similar to the data used to show +how busy certain types of places are in Google Maps—helping identify when a local business +tends to be the most crowded. The mobility flux is aggregated per week, between pairs of +approximately 5km2 cells worldwide, and for the purpose of this study further aggregated +for municipalities in Mexico. +To produce this dataset, machine learning is applied to log data to automatically segment +it into semantic trips. To provide strong privacy guarantees [70], all trips were anonymized +and aggregated using a differentially private mechanism to aggregate flows over time (see +https://policies.google.com/technologies/anonymization). This research is done on +the resulting heavily aggregated and differentially private data. No individual user data was +ever manually inspected, only heavily aggregated flows of large populations were handled. All +anonymized trips are processed in aggregate to extract their origin and destination location +and time. For example, if n users travelled from location a to location b within time interval +t, the corresponding cell (a, b, t) in the tensor would be n±err, where err is Laplacian noise. +The automated Laplace mechanism adds random noise drawn from a zero mean Laplacian +distribution and yields (ϵ, δ)-differential privacy guarantee of ϵ = 0.66 and δ = 2.1 × 1029 +per metric. Specifically, for each week W and each location pair (A, B), we compute the +number of unique users who took a trip from location A to location B during week W. To +each of these metrics, we add Laplace noise from a zero-mean distribution of scale 1/0.66. +We then remove all metrics for which the noisy number of users is lower than 100, following +10 + +the process described in [70], and publish the rest. This yields that each metric we publish +satisfies (ϵ, δ)-differential privacy with values defined above. The parameter ϵ controls the +noise intensity in terms of its variance, while δ represents the deviation from pure ϵ-privacy. +The closer they are to zero, the stronger the privacy guarantees. +These results should be interpreted in light of several important limitations. First, the +Google mobility data is limited to smartphone users who have opted into Google’s Loca- +tion History feature, which is off by default. These data may not be representative of the +population as whole, and furthermore their representativeness may vary by location. Impor- +tantly, these limited data are only viewed through the lens of differential privacy algorithms, +specifically designed to protect user anonymity and obscure fine detail. Moreover, compar- +isons across rather than within locations are only descriptive since these regions can differ +in substantial ways. +Timeline of interventions: +The Mexican government has outlined four principle objec- +tives for the control of COVID-19: a) Reduce risk of acquiring infection, b) Reduce risk of +severe morbidity and mortality, c) Reduce risk and impact on society and d) Reduce risk +of transmission between infectious and susceptible individuals. We collated a full list of +interventions between February and September 2020 and details are provided in Table A.1, +including references. +Relative risk model: +Following Goldstein and Lipsitch [71] we used age stratified epi- +demiological data to assess the temporal shifts in the share of a given age group among all +cases of infection. To do so we use the relative risk (RR) [72, 73] statistic that estimates the +ratio of the proportion of a given age group among all detected cases of COVID-19 for a later +time period vs. an early time period. We selected the early time period to be the month of +April (the period right after the implementation of the lockdown) and the late period to be +June to September. We adopted the code and model from Goldstein and Lipsitch described +in detail [71]. +Community detection algorithm: +Human mobility networks, based on data from mo- +bile devices, can be used to capture important population-level trends. Microscopic descrip- +tions often remain too complex to extract meaningful information to describe the transmis- +sion process accurately [61]. We here use a community detection algorithm following [48] to +identify human movement communities (basins) where within-community mobility among +municipalities is higher than across-community mobility. We chose this community detec- +tion algorithm as it is conceptually related to infectious disease transmission models—both +utilising random walks. +Municipality level case growth rates: +To estimate the daily epidemic growth rates in +each municipality, we fit a mixed effects GLM of log new daily case counts in sliding 7-day +windows (fixed effect; approximately the generation time of COVID-19 in the earliest wave) +11 + +and a random effect for each municipality on the slope and intercept, using the R package +lme4 v.1.1-21 [74]. Daily case counts were determined using the date of symptom onset. +Relationship between case growth rates and mobility: +To test for an effect of mo- +bility from Mexico City on municipality growth rates, we fit a mixed effect GLM with log +mobility as a fixed effect, a random effect on the intercept for each municipality and a random +effect on the slope and intercept for log mobility each week. The conditional and marginal +coefficient of determination, i.e., R2, were calculated using the R package MuMIn v1.471. +[75] which implements the method developed by Nakagawa et al. 2017 [76]. Model selection +was performed using analysis of variance for mixed effects models as implemented in the R +package lmerTest v.3.1-3 [77]. +Additional information +Acknowledgments: +We thank all health care workers and those involved in the collection, +processing and publishing COVID-19 epidemiological data from Mexico. +Funding: +M.U.G.K., O.G.P., B.G. acknowledge funding from the Oxford Martin School +Pandemic Genomics programme. M.U.G.K. acknowledges funding from the European Hori- +zon 2020 programme MOOD (grant no. #874850), the Wellcome Trust, a Branco Weiss +Fellowship, The Rockefeller Foundation and Google.org. The contents of this publication +are the sole responsibility of the authors and do not necessarily reflect the views of the Euro- +pean Commission or the other funders. B.K., H.H., S.V.S., & A.V. acknowledge the support +of a grant from the John Templeton Foundation (61780). The opinions expressed in this +publication are those of the author(s) and do not necessarily reflect the views of the John +Templeton Foundation. +Author contributions: +S.V.S., M.U.G.K. and B.K. developed the idea, planned the re- +search and conducted analyses. A.C.Z. and D.B.S.C. collected government intervention data. +S.V.S., M.U.G.K. and B.K. wrote the first draft of the manuscript. All authors interpreted +the data, contributed to writing and approved the manuscript. +Competing interests: +We declare no conflicts of interest. +Data and materials availability: +Code, spatial, and epidemiological data are available +upon publication. The Google COVID-19 Aggregated Mobility Research Dataset used for +this study is available with permission from Google LLC. 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In: Journal of Statistical Software 82.13 +(2017), pp. 1–26. doi: 10.18637/jss.v082.i13. +20 + +A +Extended Data Figures +Figure A.1: Number of new cases per state and sex (7-day average). +21 + +Veracruz de Ignacio +Ciudad de México +Nuevo Leon +México +Tabasco +de la Llave +Puebla +Guanajuato +Sonora +Female +Female + Female +Female +emale +Female +400 +Male + Male +Male + Male +9300 +100 +0 +Coahuila de Zaragoza +Michoacan de Ocampo +Baja California +sedineel +Sinaloa +Jalisco +Guerrero +Oaxaca + Female + Female +— Female +— Female +— Female + Female + Female +400 +Male +Male +Male +Male +300 +100- +Yucatan +Quintana Roo +San Luis Potosi +Hidalgo +Chiapas +Chihuahua +Tlaxcala +Morelos +emale +Female +Female +Female + Female +emale +Female +400 - +Male +Male + Male +Male +Male +Male +Male +00m + 200 +100 - +- 0 +Campeche +Durango +Zacatecas +Aguascalientes +Baja California Sur +Querétaro +Nayarit +Colima +Female +Female +Female + Female +Female +Female +Female + Female +400 +Male + Male + Male + Male +Male + Male +100 +1 0 +04-01 05-01 06-01 07-01 +04-01 05-01 06-01 07-01 +04-01 05-01 06-01 07-01 +04-01 05-01 06-01 07-01 +04-01 05-01 06-01 07-01 +04-01 05-01 06-01 07-01 +04-01 05-01 06-01 07-01 +04-01 05-01 06-01 07-0150% +100% +150% +Percent typical +mobility +Distrito Federal +within-state +outgoing movement +incoming movement +México +within-state +outgoing movement +incoming movement +50% +100% +150% +Percent typical +mobility +Guanajuato +within-state +outgoing movement +incoming movement +Nuevo León +within-state +outgoing movement +incoming movement +Veracruz +within-state +outgoing movement +incoming movement +Tabasco +within-state +outgoing movement +incoming movement +Puebla +within-state +outgoing movement +incoming movement +Tamaulipas +within-state +outgoing movement +incoming movement +50% +100% +150% +Percent typical +mobility +Coahuila +within-state +outgoing movement +incoming movement +Sonora +within-state +outgoing movement +incoming movement +Jalisco +within-state +outgoing movement +incoming movement +San Luis Potosí +within-state +outgoing movement +incoming movement +Baja California +within-state +outgoing movement +incoming movement +Michoacán +within-state +outgoing movement +incoming movement +50% +100% +150% +Percent typical +mobility +Sinaloa +within-state +outgoing movement +incoming movement +Guerrero +within-state +outgoing movement +incoming movement +Yucatán +within-state +outgoing movement +incoming movement +Oaxaca +within-state +outgoing movement +incoming movement +Hidalgo +within-state +outgoing movement +incoming movement +Quintana Roo +within-state +outgoing movement +incoming movement +50% +100% +150% +Percent typical +mobility +Chihuahua +within-state +outgoing movement +incoming movement +Baja California Sur +within-state +outgoing movement +incoming movement +Querétaro de Arteaga +within-state +outgoing movement +incoming movement +Durango +within-state +outgoing movement +incoming movement +Tlaxcala +within-state +outgoing movement +incoming movement +Chiapas +within-state +outgoing movement +incoming movement +12-01 +01-26 +03-22 +05-17 +07-12 +50% +100% +150% +Percent typical +mobility +Aguascalientes +within-state +outgoing movement +incoming movement +12-01 +01-26 +03-22 +05-17 +07-12 +Zacatecas +within-state +outgoing movement +incoming movement +12-01 +01-26 +03-22 +05-17 +07-12 +Campeche +within-state +outgoing movement +incoming movement +12-01 +01-26 +03-22 +05-17 +07-12 +Morelos +within-state +outgoing movement +incoming movement +12-01 +01-26 +03-22 +05-17 +07-12 +Nayarit +within-state +outgoing movement +incoming movement +12-01 +01-26 +03-22 +05-17 +07-12 +Colima +within-state +outgoing movement +incoming movement +Figure A.2: Weekly relative change in human mobility within each state and between +states (incoming and outgoing) as compared to baseline. +22 + +Figure A.3: State-specific correlations of new reported cases (weekly) vs. mobility from +Mexico City times new reported cases in Mexico City (weekly). States with low mobility +and case count data coverage are included but not plotted in this figure. +23 + +Distrito Federal +México +Between 2020-03-29 and 2020-07-19 +6000 +(yellow dots = later) +8 +O +. +4000 - +· +2000 +. +50 +100 +5 +10 +15 +Tabasco +Veracruz de Ignacio +Guanajuato +Pueblal +Nuevo Leon +Sonora +4000 - +de la Llavel 4000 - +3000 - +3000 +3000 +. +8 +3000 +●2000 +2000 +2000 - +·· +8 +2000 - +2000 +. +. +. +. +: +1000 +· +1000 +1000 - +1000 - +8 +1000 +New +· +o +0.001 +0.002 +0.005 +0.010 +0.002 +0.004 +0.01 +0.02 +0.03 +0.005 +0.010 +0.001 +0.002 +0.003 +1400 +sed!inee +Baja California +Jalisco +Coahuila de Zaragoza +Sinaloa +Guerrero +2000 - +1200 +1500 +1500 +. +2000 +·1000 +9 +1000 +. +. +1000 +. +1000 - +·！ +800 +. +1000 - +. +500 - +. +500 +600 : +ob +0.0006 +0.0007 +0.005 +0.010 +0.005 0.010 0.015 +0.0005 +0.0010 +0.001 0.002 +0.003 +0.002 +0.004 +0.006 +San Luis Potosi +Oaxaca| 1500 +Michoacan de Ocampo +Yucatan +Quintana Roo +Hidalgo +2000 : +1000 +1500 +: +·1000- +· +1500 - +750 +1000 +.o. +1000 +oo +0 +0 +1000 - +500 +.0. +8 +500 - +500 - +500 - +500 - +New +. +250 +0 +0.0005 0.0010 0.0015 +0.000 +0.002 +0.004 +0.002 +0.004 +0.0025 0.0050 0.0075 +0.00 +0.01 +0.02 +0.02 +0.04 +Chihuahua +Chiapas +Tlaxcala +Campeche +Baja California Sur +Durango +009 +1000 +1000 +600 - +575 - +00.05 +750 +750 - +. +8 +400 - +550 - +500 +. +500 - +0.00 - +200 +525 +250 +250 +.o +500 +-0.05 +0.0000.002 0.004 0.006 +0.002 +0.004 +0.002 +0.004 +0.006 +0.00050 +0.00055 +0.00060 +0.000 +0.002 +0.004 +0.006 +0.05 +0.00 +0.05 +500 +Morelos +Querétaro de Arteaga +Nayarit +Zacatecas +Aguascalientes +Colimal +600 +400 - +0.05 +0.05 - +0.05 +· +400 - +300 - +8 +200 +0.00 +0.00 +0.00 +200 +New +100 +-0.05 +. +←0.05 +0.05 - +oh +0.0005 +0.0010 +0.02 +0.04 +0.005 +0.010 +-0.05 +0.00 +0.05 +-0.05 +0.00 +-0.05 +0.00 +0.05 +0.06 +0.05 +Movement from Mexico D.F. +Movement from Mexico D.F. + Movement from Mexico D.F. +Movement from Mexico D.F. +Movement from Mexico D.F. +Movement from Mexico D.F. +times Mexico D.F. new cases +times Mexico D.F. new cases +times Mexico D.F. new cases +times Mexico D.F. new cases +times Mexico D.F. new cases +times Mexico D.F. new casesFigure A.4: Weekly relative human mobility where the origin is the USA and the destina- +tion are states in Mexico divided into states that share a land border, Mexico and Mexico +City and all other states. +24 + +USA +States with +140% +USA border +States without +from +120% +USA border +Mexico and +Mexico City +mobility +100% +80% +typical +60% +40% +Percent of i +20% +0% +12-01 +02-09 +04-19 +06-28Figure A.5: Weekly new cases per 100,000 divided into cases in Mexico City and the state +of Mexico, states that share a land border with the USA, and all other states. +25 + + rate per 100,000 +100,000 +40 +65 +States with +USA border +States without +4 +USA border +30 +new cases per +Mexico and +Mexico City +20 +Weekly growth I +Z +10 +Weekly +L +0 +04-19 +06-28 +03-15 +03-15 +05-24 +04-19 +05-24 +06-28Figure A.6: Weekly number of cases among municipalities in Mexico coloured by their +geographic position to the USA (bordering vs. not bordering) and the sum of in-municipality +mobility × weekly new cases among origin nodes (both on the log scale). +26 + +8 +Municipalities without +U.S. connections +Sum of weekly new cases among +(log-scaled) +6 +Municipalities with +U.S. connections +4 +destination nodes ( +2 +0 +-4 +-2 +0 +2 +4 +6 +8 +10 +12 +Sum of in-municipality mobility x sum of +weekly new cases among origin nodes (log-scaled)Community size distribution +Community size distribution +Community size distribution +Community size distribution +Community size distribution +Community size distribution +Community size distribution +Community size distribution +Community size distribution +Figure A.7: Four-week snapshots of mobility in Mexico. Weekly human mobility in +Mexico at the municipality level. Thickness of lines represents intensity of relative mobility +flow. Colours represent the membership to movement communities as estimated using the +map equation (Materials & Methods). +27 + +Fr0m 2019-11-24 +(four-week total) +15 communitiesFrom 2019-12-29 +(four-week total) +15 communitiesFr0m 2020-01-26 +(four-week total) +18 communitiesFr0m 2020-02-23 +(four-week total) +18 communitiesFrom 2020-03-22 +(four-week total) +33 communitiesFr0m 2020-04-19 +(four-week total) +26 communitiesFr0m 2020-05-17 +(four-week total) +22 communitiesFr0m 2020-06-14 +(four-week total) +33 communitiesFr0m 2020-07-12 +(four-week total) +32 communities03-01 +03-29 +04-26 +05-24 +06-21 +07-19 +08-16 +20 +30 +40 +50 +60 +Number of communities detected over time +Figure A.8: +Number of communities detected each week during the first wave of the +COVID-19 epidemic in Mexico. +28 + +03-0103-29 +04-26 +05-24 +06-21 +07-19 +08-16 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +(c) +Standard deviation of municipality growth rates +within grouping (lower values: higher synchrony) +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +Growth rate (reported cases) +0.11 +Nayarit +Morelos +Michoacán de Ocampo +México +Jalisco +Hidalgo +Guerrero +Guanajuato +Durango +Distrito Federal +Chihuahua +Chiapas +Colima +Coahuila de Zaragoza +Campeche +Baja California Sur +Baja California +Aguascalientes +0.36 +0.26 +0.69 +0.43 +0.29 +0.30 +0.44 +0.31 +0.83 +0.42 +0.57 +0.47 +0.44 +0.37 +0.39 +0.48 +0.47 +... +... +... +... +0.465 +(mean) + Std. dev. of growth rates +Example: Variance in growth rates (2020-04-19) +municipalities grouped by administrative boundaries +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +Growth rate (reported cases) +0.38 +Comm. 01 +0.54 +Comm. 02 +0.35 +Comm. 03 +0.45 +Comm. 04 +0.50 +Comm. 05 +0.43 +Comm. 06 +0.30 +Comm. 07 +0.49 +Comm. 08 +0.37 +Comm. 09 +0.31 +Comm. 10 +0.59 +Comm. 11 +0.00 +Comm. 12 +0.00 +Comm. 13 +0.00 +Comm. 14 +0.00 +Comm. 15 +0.00 +Comm. 16 +0.00 +Comm. 17 +0.00 +Comm. 18 +... +... +... +... +0.223 +(mean) + Std. dev. of growth rates +Example: Variance in growth rates (2020-04-19) +municipalities grouped by network communities +o +. +(e) +(d) +o +. +Figure A.9: For one example week (April 19, 2020), comparison of the mean standard +deviations in municipality growth rates within either (left) administrative (states) boundaries +or (right) network communities. In each panel, the standard deviations of municipalities’ +infection growth rates within each grouping (state vs. network community) is shown on the +right. Figure 3c shows the average of these values over time. +29 + +b +Example network grouping +(infomap)a) +Administrative grouping +(states)Date +Intervention +March 16, 2020 +Mexican Secretariat of Public Education (SEP) suspend classes in +schools of preschool, primary, secondary education, as well as those +of the upper middle and higher types dependent on the SEP [1]. +March 17, 2020 +Universities begin to suspend classes and social events [2]. +March 20, 2020 +Mexican Secretariat of Public Education (SEP) cancels all civic and +sports events [3]. +March 21, 2020 +United States - Mexico border was closed to non-essential travel but +remained open for commerce. Closure extended until November 21, +2020 [4]. +March 23/24, 2020 +National period of social distancing begins. Schools closed and all +non-essential operations were closed including gatherings of 100+ +people [5]. +March 30, 2020 +National health emergency declared. Policies included: (1.) Non- +essential services suspended. (2.) Private sector is asked to require +employees to work from home. (3.) Sectors that kept operating nor- +mally: government, health (public and private), public safety, social +programs, critical infrastructure, and essential services. A full list +of essential services can be viewed here: https://www.dof.gob. +mx/nota_detalle.php?codigo=5590914&fecha=31/03/2020. (4.) +People over 60 years old are urged to stay home. (5.) Public gath- +erings of over 50 people are banned. (6.) There was no enforced +curfew. Expiration date: April 30, 2020. +April 5, 2020 +Hospital reconversion strategy guidelines published in order to con- +tain nosocomial transmission [6] +April 16, 2020 +The Federal Government announces the extension of the health +emergency and emphasizes the need to restrict movement to and +from areas of high transmissibility until May 30th [7]. +May 14, 2020 +Ministry of Health announces epidemiologic color-coded system to +re-open social, educational, economic activities at state level [8]. +May 18, 2020 +First phase of the “new normality” 324 municipalities with no +recorded COVID-19 cases are given green light to reopen businesses +and schools [9]. Car factories were meant to reopen on June 1st, +but began reopening on May 18th under US pressure (Factories +remained closed from March 23rd, to May 18th). +30 + +June 1, 2020 +Mexico’s national period of social distancing concludes. +A new +color-coded system was enacted across the country to assess how +quickly states can reopen their economies and schools: red, orange, +yellow, and green [10]. +July 20, 2020 +Daycare centers run by the country’s social security system re- +opened in coordination with local authority and based on color- +coded indicators. Currently, all but 4 states have daycare centers +open. [11] +October 18, 2020 +The Health Ministry announced that 17 states—Mexico City +included—were at alert level orange and 14 were at yellow. Only +one state, Campeche, was at green [12]. In states at the orange +level, businesses such as hotels and restaurants can reopen while +following health protocols such as enforcing limited capacity. Yel- +low allows for most economic activities to return to normal with +some occupancy limits. +October 22, 2020 +Some local governments have chosen to enact more stringent re- +strictions than the federal government guidelines, e.g., Jalisco and +Chihuahua [13]. +Table A.1: Timeline of government interventions in Mexico. +31 + +A.1 +Citation diversity statement +Recent work has quantified bias in citation practices across various scientific fields; namely, +women and other minority scientists are often cited at a rate that is not proportional to +their contributions to the field [14–21]. In this work, we aim to be proactive about the +research we reference in a way that corresponds to the diversity of scholarship in this field. +To evaluate gender bias in the references used here, we obtained the gender of the first/last +authors of the papers cited here through either 1) the gender pronouns used to refer to them +in articles or biographies or 2) if none were available, we used a database of common name- +gender combinations across a variety of languages and ethnicities. By this measure (excluding +citations to datasets/organizations, citations included in this section, and self-citations to the +first/last authors of this manuscript), our references contain 3% woman(first)-woman(last), +22% woman-man, 20% man-woman, 47% man-man, 0% nonbinary, 8% man solo-author, and +0% woman solo-author. This method is limited in that an author’s pronouns may not be +consistent across time or environment, and no database of common name-gender pairings is +complete or fully accurate. +Supplemental References +[1] +DOF - Diario Oficial de la Federación. url: https://www.dof.gob.mx/nota_ +detalle.php?codigo=5589479&fecha=16/03/2020. +[2] +Coronavirus en México: universidades suspenden clases y se intensifican las acciones +preventivas. 2020. url: https : / / www . infobae . com / america / mexico / 2020 / +03 / 13 / coronavirus - en - mexico - universidades - suspenden - clases - y - se - +intensifican-las-acciones-preventivas/. +[3] +Gobierno de México suspenderá todas las actividades escolares por coronavirus. 2020. +url: https : / / www . latimes . com / espanol / mexico / articulo / 2020 - 03 - 14 / +gobierno- de- mexico- suspendera- todas- las- actividades- escolares- por- +coronavirus. +[4] +U.S. Embassy & Consulates in Mexico. Mexico, U.S.M. to. Travel restrictions - Fact +sheet. 2021. url: https://mx.usembassy.gov/travel-restrictions-fact-sheet/. +[5] +Inicia fase 2 por coronavirus COVID-19 – Coronavirus. url: https://coronavirus. +gob.mx/2020/03/24/inicia-fase-2-por-coronavirus-covid-19/. +[6] +Gobierno de México and Secretaría de Salud COVID-19. Lineamiento de Reconversión +Hospitalaria. url: https://coronavirus.gob.mx/wp-content/uploads/2020/04/ +Documentos-Lineamientos-Reconversion-Hospitalaria.pdf. +[7] +Coronavirus en México: guía para entender las cuatro nuevas medidas de control y +prevención del COVID-19 cercanas a la Fase 3. url: https://www.infobae.com/ +america/mexico/2020/04/16/coronavirus-en-mexico-guia-para-entender- +las - cuatro - nuevas - medidas - de - control - y - prevencion - del - covid - 19 - +cercanas-a-la-fase-3/. +32 + +[8] +DOF - Diario Oficial de la Federación. url: https://dof.gob.mx/nota_detalle. +php?codigo=5593313&fecha=14/05/2020#gsc.tab=0. +[9] +Conferencia 16 de mayo – Coronavirus. url: https://coronavirus.gob.mx/2020/ +05/16/conferencia-16-de-mayo-2/. +[10] +Subsecretaría de Prevención y Promoción de la Salud. Semáforo de riesgo epidemi- +ológico: COVID-19: indicadores y metodología. url: https://coronavirus.gob.mx/ +wp-content/uploads/2020/06/Lineamiento_Semaforo_COVID_05Jun2020_1600. +pdf. +[11] +IMSS nurseries open on July 20, after supervision of health protocols. url: http: +//www.imss.gob.mx/prensa/archivo/202007/463. +[12] +COVID-19 MÉXICO Comunicado Técnico Diario - 18 Octubre 2022. url: http : +/ / saludsinaloa . gob . mx / wp - content / uploads / 2020 / reportescovid / +Covid19ReporteDiario18Noviembre2020.pdf. +[13] +State Government publishes new agreement on Red Light measures. url: https:// +chihuahua.gob.mx/contenidos/publica-gobierno-del-estado-nuevo-acuerdo- +de-medidas-del-semaforo-rojo. +[14] +Perry Zurn, Danielle S. Bassett, and Nicole C. Rust. “The citation diversity statement: +A practice of transparency, a way of life”. In: Trends in Cognitive Sciences 24.9 (2020), +pp. 669–672. doi: 10.1016/j.tics.2020.06.009. +[15] +Jordan D. Dworkin, Kristin A. Linn, Erin G. Teich, Perry Zurn, Russell T. Shinohara, +and Danielle S. Bassett. “The extent and drivers of gender imbalance in neuroscience +reference lists”. In: Nature Neuroscience 23.8 (2020), pp. 918–926. doi: 10.1038/ +s41593-020-0658-y. +[16] +Paula Chakravartty, Rachel Kuo, Victoria Grubbs, and Charlton McIlwain. “#Com- +municationSoWhite”. In: Journal of Communication 68.2 (2018), pp. 254–266. doi: +10.1093/joc/jqy003. +[17] +Daniel Maliniak, Ryan Powers, and Barbara F. Walter. The gender citation gap in in- +ternational relations. Vol. 67. 4. 2013, pp. 889–922. doi: 10.1017/S0020818313000209. +[18] +Michelle L. Dion, Jane Lawrence Sumner, and Sara Mc Laughlin Mitchell. “Gendered +citation patterns across political science and social science methodology fields”. In: +Political Analysis 26.3 (2018), pp. 312–327. doi: 10.1017/pan.2018.12. +[19] +Neven Caplar, Sandro Tacchella, and Simon Birrer. “Quantitative evaluation of gender +bias in astronomical publications from citation counts”. In: Nature Astronomy 1 (2017). +doi: 10.1038/s41550-017-0141. +[20] +Pierre Azoulay and Freda Lynn. “Self-citation, cumulative advantage, and gender in- +equality in science”. In: Sociological Science 7 (2020). doi: 10.15195/v7.a7. +[21] +Gita Ghiasi, Philippe Mongeon, Cassidy R. Sugimoto, and Vincent Larivière. “Gender +homophily in citations”. In: 23rd International Conference on Science and Technology +Indicators (2018), pp. 1519–1525. url: https://hdl.handle.net/1887/65291. +33 + diff --git a/9tFQT4oBgHgl3EQfJjU4/content/tmp_files/load_file.txt b/9tFQT4oBgHgl3EQfJjU4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b89a70f1f5054f9f4e5c2896d3d72221b3104406 --- /dev/null +++ b/9tFQT4oBgHgl3EQfJjU4/content/tmp_files/load_file.txt @@ -0,0 +1,2082 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf,len=2081 +page_content='Spatial scales of COVID-19 transmission in Mexico Brennan Klein∗1,2, Harrison Hartle1, Munik Shrestha1, Ana Cecilia Zenteno3, David Barros Sierra Cordera4, José R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Nicolas-Carlock5, Ana I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Bento6, Benjamin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Althouse7,8, Bernardo Gutierrez9,10,11, Marina Escalera-Zamudio9,11, Arturo Reyes-Sandoval12,13, Oliver G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Pybus9,14,18, Alessandro Vespignani1,2, Jose Alberto Diaz-Quiñonez*†15, Samuel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Scarpino*‡1,16,17, and Moritz U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Kraemer*§9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='18 1Network Science Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Northeastern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Massachusetts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' USA 2Laboratory for the Modeling of Biological & Socio-technical Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Northeastern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Massachusetts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' USA 3Massachusetts General Hospital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Massachusetts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' USA 4Instituto Mexicano del Seguro Social,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Ciudad de México,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' México 5Instituto de Investigaciones Jurídicas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Universidad Nacional Autónoma de México,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Ciudad de México,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' México 6Department of Epidemiology and Biostatistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' School of Public Health,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Indiana University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Bloomington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Indiana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' USA 7Information School,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Seattle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' USA 8Department of Biology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' New Mexico State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Las Cruces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' New Mexico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' USA 9Department of Biology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' UK 10School of Biological & Environmental Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Universidad San Francisco de Quito,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Quito,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Ecuador 11Consorcio Mexicano de Vigilancia Genómica 12The Jenner Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' UK 13Instituto Politécnico Nacional,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' IPN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Ciudad de México,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' México 14Department of Pathobiology and Population Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Royal Veterinary College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' UK 15Instituto de Ciencias de la Salud,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Universidad Autónoma del Estado de Hidalgo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Pachuca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Hidalgo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' México 16Institute for Experiential AI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Northeastern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Massachusetts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' USA 17Santa Fe Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Santa Fe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' New Mexico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' USA 18Pandemic Sciences Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' UK February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2023 ∗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='klein@northeastern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='edu †alberto_diaz@uaeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='mx ‡s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='scarpino@northeastern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='edu §moritz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='kraemer@biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='uk 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='13256v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='soc-ph] 30 Jan 2023 Abstract During outbreaks of emerging infectious diseases, internationally connected cities often experience large and early outbreaks, while rural regions follow after some delay [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' This hierarchical structure of disease spread is influenced primarily by the mul- tiscale structure of human mobility [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' However, during the COVID-19 epidemic, public health responses typically did not take into consideration the explicit spatial structure of human mobility when designing non-pharmaceutical interventions (NPIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' NPIs were applied primarily at national or regional scales [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Here we use weekly anonymized and aggregated human mobility data and spatially highly resolved data on COVID-19 cases, deaths and hospitalizations at the municipality level in Mexico to investigate how behavioural changes in response to the pandemic have altered the spatial scales of transmission and interventions during its first wave (March - June 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' We find that the epidemic dynamics in Mexico were initially driven by SARS- CoV-2 exports from Mexico State and Mexico City, where early outbreaks occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The mobility network shifted after the implementation of interventions in late March 2020, and the mobility network communities became more disjointed while epidemics in these communities became increasingly synchronised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Our results provide actionable and dynamic insights into how to use network science and epidemiological modelling to inform the spatial scale at which interventions are most impactful in mitigating the spread of COVID-19 and infectious diseases in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Table 1: Policy summary Background The establishment, persistence and growth rates of COVID-19 mainly depend on human mobility and mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' However, current approaches attempting to limit transmission have been primarily based on administrative boundaries instead of the natural scales of human mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Main findings & limitations Using aggregated and anonymized human mobility and detailed COVID-19 case data, we find that the scales of human mixing shift during the pandemic and that transmission is highly clustered amongst mobility communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Policy implications Structuring interventions based on spatial mobility may be more effective com- pared to interventions based on administrative boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Future pandemic control interventions should consider empirical human mobility networks when designing interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2 1 Introduction The transmission of infectious diseases is highly heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Differences in population structure, the landscape of prior immunity, and environmental factors, result in differences in the timing of outbreaks, their magnitude, and duration [2, 3, 9, 11–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' For infec- tious diseases, one principal component determining the spatial structure of outbreaks is the frequency of interactions between susceptible and infectious individuals within and be- tween regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' In most geographies, public health decision-making authority follows political boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' However, from an epidemiological perspective, the relevant spatial units may not strictly follow political boundaries but rather human mixing [8, 14, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Evaluating the spatial structure of COVID-19 transmission remains important in determining optimal interventions (non-pharmaceutical and/or vaccination) to reduce transmission and limit the risk of resurgence of cases [22–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' During the first half of 2020, Mexico experienced one of the largest SARS-CoV-2 epi- demics worldwide, with more than 600,000 cases and 65,000 confirmed deaths reported be- tween February and September 2020 [26] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The epidemic wave peaked in May in the largest metropolitan areas of Mexico City and the State of Mexico and later ignited epidemics in all other states [27], peaking between June and July 2020 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Here we combine municipality level epidemiological data with weekly anonymized aggregated human mobility data at the same scale, to characterise the spatial scales of the Mexican COVID-19 pandemic and their implications for the implementation of spatially targeted interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2 Results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1 Spatial expansion of COVID-19 in Mexico In Mexico, the spatial range of transmission expanded rapidly after reports of the earliest cases in March 2020, with over 700 municipalities reporting transmission by July 2020 (out of 2,448, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' During April and May the risk of positive RTq-PCR confirmed cases amongst men aged 30-69 was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='4 times higher than between July 1 and September 1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 1d,e), indicating that the epidemic spread initially within and through these age groups (Extended Data Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' This dynamic trend in the demographics of cases is similar to that observed in other countries during the early stages of the pandemic [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' States that experienced early transmission were the state of Mexico and Mexico City (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 1b) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Due to the centrality of Mexico City connecting people from abroad (in- ternational arrivals) and within Mexico we hypothesise that human mobility from these states was a key driver of the spread of COVID-19 in Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Using anonymized, opt-in and aggregated human movement data from mobile phones (Materials and Methods) we find that case growth rates across Mexican states were well predicted by a lagged model of human movements from the State of Mexico and Mexico City between March and May 2020 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2c, conditional R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='62;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' see Materials & Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Further, we observe that the share of overall relative human mobility to and from Mexico and Mexico City increased 3 Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' May Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 0 2 4 6 8 10 12 14 Total reported cases as of Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 1, 2020: (b) Daily new cases per 100,000, state level (7-day rolling avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=') Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' May Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 0 200 400 600 800 (c) Municipalities reporting cases (2,448 total) Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' May Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 0% 10% 20% 30% 40% (d) "early" "late" Percent of new cases (7-day rolling avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='4 Early: April 1 - May 1 Late: June 30 - Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 30 F: under 30 F: under 30 F: 30-49 F: 30-49 F: 50-69 F: 50-69 F: over 70 F: over 70 M: under 30 M: under 30 M: 30-49 M: 30-49 M: 50-69 M: 50-69 M: over 70 M: over 70 (e) Relative risk ("early" vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' "late" periods) 10 100 1000 10000 Cases per 100,000 (as of September 1) Figure 1: Epidemiological situation of COVID-19 in Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (a) Map of cumulative cases per 100,000 people, as of September 1, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (b) Timeline of new cases per 100,000 population at the state level (7-day rolling average), highlighting the 15 states with the most severe cumulative outbreaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (c) Number of municipalities that reported confirmed cases of COVID-19 through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (d) Age and sex distributions of confirmed COVID-19 cases across Mexico, highlighting “early” and “late” periods during which the relative risk of infections were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (e) Age and sex relative risk ratios of infection, comparing the early vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' late periods from panel (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' markedly during that period (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2b) when overall human mobility between states declined (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2b, Extended Data Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='2 showing state level data on change in human mobility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' This points towards a change in the network structure of human mobility in Mexico, as documented in some other countries [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Overall transmission, and the importance of Mexico City driving the epidemic, declined after the implementation of NPIs through May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' However, after the lifting of physical distancing measures on June 1st (see table of documented changes in NPIs, Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1), case growth rates in the country increased again as 4 a function of mobility from Mexico City, in line with models predicting that lifting lockdowns can lead to reseeding of transmission chains from larger to smaller cities where epidemics were successfully controlled (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2b, Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1, [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Variation in weekly new cases within each state in Mexico are generally well predicted by cases in Mexico City weighted by human mobility except for Baja California, More- los, Chihuahua, Oaxaca, and Chiapas (Extended Data Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' We hypothesise that epidemics there were possibly seeded from other countries (USA and Guatemala);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' further SARS-CoV-2 genomic analyses of unbiased collections of samples will be needed to confirm the SARS-CoV-2 lineage dynamics in these states [27, 32–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Human mobility data showing cross border (US to Mexico) movements indicate higher overall mobility to bordering states in Mexico and growth rates in US-Mexico border states appear higher in the period between 24 May - 28 June 2020 (Extended Data Figures A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='4, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The high mobility during that phase resulted in larger case numbers in states bordering the US when compared to other states in Mexico (Extended Data Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='2 The scales of COVID-19 transmission It is well known that reductions in mobility (a proxy for reductions in population mixing) have reduced the transmission of COVID-19 within a location [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' However, it remains un- clear how structural changes to the mobility network (shifts in the frequency and intensity of mobility within and among regions) have impacted COVID-19 dynamics empirically [30, 31, 39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Our underlying hypothesis is that more tightly connected communities exhibit more synchronised epidemic dynamics and, conversely, that more disjointed individual com- munities have less synchronised epidemics and their epidemics are more likely to fade out [4–6] (here, communities are equivalent to municipalities and synchrony is defined as the similarity among communities in weekly case growth rates [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Both processes have critical implications for disease mitigation and eliminations locally, and at a country level [7, 43–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The Mexican government announced stringent physical distancing policies on March 30th, 2020 which resulted in marked changes in the mobility network (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2a, Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' To quantify the degree to which mobility patterns are structured by geopolitical bound- aries, we use a community detection algorithm that groups municipalities based on their movement patterns [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Specifically, we aim to identify groups of municipalities such that movements between municipalities within the same group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', community, are more fre- quent than movements to other municipalities in other communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Community detection is often accomplished via modularity maximization [49];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' however, these approaches neglect information about the flow of mobility through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Instead, we leverage the map equation via an algorithm called InfoMap [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The InfoMap algorithm utilises an informa- tion theoretic approach to derive expected connectivity patterns if the observed flows were entirely determined by a random walk process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' For this study, InfoMap is ideal because it is conceptually related to infectious disease transmission models, which often also utilise stochastic processes [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The aim is to identify municipalities where frequent interactions between individuals occur, such that the detected communities approximate the spatial scales of disease trans- 5 03-01 04-12 05-24 07-05 08-16 20% 40% 60% 80% 100% (b) Percent of typical mobility (total across all of Mexico) 03-01 04-12 05-24 07-05 08-16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content="006 (c) Coefficients of case growth rate and mobility from Mexico City 03-01 04-12 05-24 07-05 08-16 18% 19% 20% 21% 22% 23% 24% (d) Dynamics of states' outgoing mobility to Mexico City Community size distribution (n = 16, using Infomap) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='3 Figure 2: Human mobility and transmission of COVID-19 in Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (a) Pre- pandemic average of the inter-municipality mobility network, coloured by network commu- nity (detected using the Infomap algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Mobility flow data is based on the aggregated Google Mobility Research dataset (see Materials & Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (b) Percent of typical weekly mobility nationwide (typical refers to mobility between January 12 and February 29, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (c) Evolution of the coefficients of mobility flow from Mexico City in (lagged) correlations with state-level case rates across the country, highlighting the key role that mobility from Mexico City played in the early stage of the epidemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (d) Average fraction of total outgoing mobility from each state that is to Mexico City (black) and the median entropy of states’ distributions of outgoing mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Error bands correspond to 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' mission (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', communities in which it is assumed that infection spreads via contacts within a relatively homogeneously mixing population [51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Accounting for spatial heterogeneity is 6 Network of average mobility flow: (a) 2020-01-12 t0 2020-02-23 Administrative (state) boundary Example: Chiapas Network community03-01 03-29 04-26 05-24 06-21 07-19 08-16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='8 (c) Standard deviation of municipality growth rates within grouping (lower values: higher synchrony) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='465 (mean) Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' of growth rates (d) Example: Variance in growth rates (2020-04-19) municipalities grouped by administrative boundaries 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='0 Growth rate (reported cases) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='38 Comm.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='223 (mean) Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' of growth rates (e) Example: Variance in growth rates (2020-04-19) municipalities grouped by network communities Figure 3: Network structure determines the synchrony of epidemics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (a) Grouping of municipalities based on the state administrative boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Shaded municipalities are re- moved from downstream analyses as they could not be assigned a movement community (see Materials & Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (b) Example grouping of municipalities based on human movement data and a community detection algorithm [37] (Materials and Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Colours indicate movement communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Grey municipalities have limited recorded movements and could not be assigned to a community and were consequently excluded from analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (c) Synchrony of weekly growth rates of epidemics across municipalities as measured by the pairwise standard error between growth rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The lower the error, the more synchronised epidemics are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Blue line shows grouping by network communities, and orange shows groupings by state admin- istrative boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The green dashed line shows the nationwide trend in reported cases during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' For a visual intuition of the differences in within-community standard deviations of growth rates, see Extended Data Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' known to be important for assessing strategies for interventions [2], especially in areas that have marked differences in urban and rural areas [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Using this algorithm, we identify 16 communities before the first cases of COVID-19 were detected in Mexico (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Com- munity size and organisation changed following the announcement of the lockdown (March 23 and 30, 2020) in Mexico and communities generally became smaller (fewer municipalities within each community (Extended Data Figures A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='7 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='8 show the communities for each week during the study period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' At the peak of the lockdown, we identified approximately 60 movement communities (a 4-fold increase from the baseline period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' More specifically, there are two notable shifts in the network following the introduction of NPIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' First, more communities are identified but importantly the size of these communities shrinks disproportionately so that one community expands (Mexico City) and many very 7 b Example network grouping (infomap)a) Administrative grouping (states)small ones emerge (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Further, as a result of the lockdown human movements across municipalities decline more rapidly than movements within a community with one important exception: Mexico City.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' There we observe that the ratio of within municipality movements declines at a similar rate than movements across municipalities (Extended Data Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='2) further proving its central importance in the mobility network in Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' We then compared the weekly infection incidence growth rates within each community and contrasted them to growth rates under a scenario in which municipalities are grouped based on state boundaries (black lines, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 3a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' As expected, we find that epidemics in municipalities that are grouped by human mobility were more synchronised compared to those grouped by state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 3c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' see Extended Data Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='9 for an illustration of the variance in municipality epidemic growth rates for several example groups of municipalities defined by administrative or network boundaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The synchrony among municipalities within each community were maximised in April and May 2020, a period when cases were rapidly rising across the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' After June, epidemics that are grouped by movement are still more synchronised, but the differences with groupings by state appear to be smaller (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' This later period (June to October 2020) is a time when Mexico City appears to also lose importance in seeding the epidemic across the country, and local factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', population size) become more important in determining the epidemic trajectory [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' These results are expected as local factors become more influential in determining disease dynamics (population size, local mixing) and that the importance of continued virus re-importations wanes through time [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 3 Discussion & Limitations We present a generalisable approach for understanding the spatial structure of transmission of COVID-19 and other emerging infectious diseases by accounting for the variations of the human mobility network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' We aimed to differentiate the transmission dynamics at a level defined by administrative boundaries from that defined by simple community detection algorithms that are applied to aggregated anonymized weekly human mobility data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' We find that as human mobility network structures change, so does to spatial transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Incorporating these findings into real-world public health decision-making may result in more effective strategies to control an epidemic [54–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The European Commission for example published a report on Mobility Functional Areas (MFAs) which were informed by mobile phone data but the adoption of these recommendations remained sparse [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Our model and results are only as accurate as the data that go into them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The Mexican COVID-19 database may suffer from underreporting due to testing shortages, changing case definitions and spatial heterogeneity in reporting [58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' For example, relatively few cases were reported from Oaxaca (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 1a) which may be due to barriers to access to testing [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Future extensions of the model and as the pandemic continues will need to take into account high-resolution SARS-CoV-2 cross-immunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Further, our model is based on higher level descriptions of the population (raw case data and population level human movement data) and these do not capture the high contact heterogeneity within each municipality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', de- 8 mographic heterogeneity and assortative mixing) shown to be important in the transmission of COVID-19 [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Contact patterns may differ significantly by age group, employment sta- tus and other factors not accounted for in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' We did however observe heterogeneity in the demographic makeup of cases during the earlier phases of the Mexican COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Further, results should be interpreted in light of important limitations related to the human mobility data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' First, the Google mobility data is limited to smartphone users who have opted into Google’s Location History feature, which is off by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' These data may not be representative of the population as whole, and furthermore their representativeness may vary by municipality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Importantly, these limited data are only viewed through the lens of differential privacy algorithms, specifically designed to protect user anonymity and obscure fine detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Mexico is composed of 31 free and sovereign states and Mexico City, united under a federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' This means that each administrative region or state is governed by its own constitution, although they are not completely independent of the federal jurisdiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Fur- thermore, each state is divided into municipalities, the nation’s basic administrative unit, which possesses limited autonomy (discretionary power on how best to respond to, or apply a public policy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Under a serious nationwide health threat or emergency, such as a pandemic, the federal Ministry of Health (MoH) acquires full authority over the health policies to be implemented nationwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Nevertheless, Mexican law establishes that the General Health Council (GHC), a collegial body that reports to the president of the republic has the char- acter of health authority, and can emit obligatory norms to be abided by the MoH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The GHC is presided by the Minister of Health, and is conformed by federal institutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='h, Economy, Communication & Transport) as well as academic institutions, representatives from pharmaceutical industry, and other health system actors [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Given its mandate and position in the Mexican health system, the GHC constitutes a promising agent to drive pub- lic policy outside of the margins or across geo-administrative units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Furthermore, there are examples of inter-state and inter-municipality coordination to resolve problems that extend beyond their borders such as waste management, tax, policing, and perhaps most relevant, health provision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' It is in these contexts where evidence-based interventions on innovative approaches, such as the ones presented here become not only an option but a possibility, with greater impact in reducing transmission as compared to approaches where interven- tions are based on administrative boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' However, theory often differs from practice and reality brings along additional and expected factors into play (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', economic [63] and political interests) many of which are not accounted for in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Some state governors for example refused to comply with federal health policies in the early relaxation phase in May 2020 [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Mexico has suffered a large and devastating epidemic, and we hope that our findings contribute to a more rational implementation of interventions in the future that can account for the substantial and changing spatial heterogeneity in transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Such analyses can be updated and translated to any other country in the world for which aggregated human mobility data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Future work should also focus on validating the inferred spatial 9 scales with genomic data [32, 33, 65] or other coarse-graining techniques [66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Developing interventions using patterns observed in empirical mobility networks must be added to the list of priorities for pandemic response and preparedness in the 21st century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 4 Materials & Methods Epidemiological data: Epidemiological data include individual level information on pa- tients with confirmed RTq-PCR COVID-19 infection between March - September 30th, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Data were downloaded from http://datosabiertos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='salud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='gob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='mx/gobmx/salud/datos_ abiertos/datos_abiertos_covid19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='zip (last accessed October 24, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Data include information about patients demographics (age and sex) and municipality of residence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' In all analyses we used the date of onset of symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Population and travel data: Human mobility and population data were extracted at the municipality level based on the 2016 boundaries (INEGI 2016: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='inegi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' mx/app/mapa/espacioydatos/default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='aspx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Population data were downloaded from the COVID-19 indicator dataset, which was provided by INEGI (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='inegi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='mx/ investigacion/covid/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Aggregated and anonymised human mobility data: We used the Google COVID- 19 Aggregated Mobility Research Dataset described in detail in [68, 69], which contains anonymized relative mobility flows aggregated over users who have turned on the Location History setting, which is turned off by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' This is similar to the data used to show how busy certain types of places are in Google Maps—helping identify when a local business tends to be the most crowded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The mobility flux is aggregated per week, between pairs of approximately 5km2 cells worldwide, and for the purpose of this study further aggregated for municipalities in Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' To produce this dataset, machine learning is applied to log data to automatically segment it into semantic trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' To provide strong privacy guarantees [70], all trips were anonymized and aggregated using a differentially private mechanism to aggregate flows over time (see https://policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='com/technologies/anonymization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' This research is done on the resulting heavily aggregated and differentially private data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' No individual user data was ever manually inspected, only heavily aggregated flows of large populations were handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' All anonymized trips are processed in aggregate to extract their origin and destination location and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' For example, if n users travelled from location a to location b within time interval t, the corresponding cell (a, b, t) in the tensor would be n±err, where err is Laplacian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The automated Laplace mechanism adds random noise drawn from a zero mean Laplacian distribution and yields (ϵ, δ)-differential privacy guarantee of ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='66 and δ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1 × 1029 per metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Specifically, for each week W and each location pair (A, B), we compute the number of unique users who took a trip from location A to location B during week W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' To each of these metrics, we add Laplace noise from a zero-mean distribution of scale 1/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' We then remove all metrics for which the noisy number of users is lower than 100, following 10 the process described in [70], and publish the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' This yields that each metric we publish satisfies (ϵ, δ)-differential privacy with values defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The parameter ϵ controls the noise intensity in terms of its variance, while δ represents the deviation from pure ϵ-privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The closer they are to zero, the stronger the privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' These results should be interpreted in light of several important limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' First, the Google mobility data is limited to smartphone users who have opted into Google’s Loca- tion History feature, which is off by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' These data may not be representative of the population as whole, and furthermore their representativeness may vary by location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Impor- tantly, these limited data are only viewed through the lens of differential privacy algorithms, specifically designed to protect user anonymity and obscure fine detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Moreover, compar- isons across rather than within locations are only descriptive since these regions can differ in substantial ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Timeline of interventions: The Mexican government has outlined four principle objec- tives for the control of COVID-19: a) Reduce risk of acquiring infection, b) Reduce risk of severe morbidity and mortality, c) Reduce risk and impact on society and d) Reduce risk of transmission between infectious and susceptible individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' We collated a full list of interventions between February and September 2020 and details are provided in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1, including references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Relative risk model: Following Goldstein and Lipsitch [71] we used age stratified epi- demiological data to assess the temporal shifts in the share of a given age group among all cases of infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' To do so we use the relative risk (RR) [72, 73] statistic that estimates the ratio of the proportion of a given age group among all detected cases of COVID-19 for a later time period vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' an early time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' We selected the early time period to be the month of April (the period right after the implementation of the lockdown) and the late period to be June to September.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' We adopted the code and model from Goldstein and Lipsitch described in detail [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Community detection algorithm: Human mobility networks, based on data from mo- bile devices, can be used to capture important population-level trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Microscopic descrip- tions often remain too complex to extract meaningful information to describe the transmis- sion process accurately [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' We here use a community detection algorithm following [48] to identify human movement communities (basins) where within-community mobility among municipalities is higher than across-community mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' We chose this community detec- tion algorithm as it is conceptually related to infectious disease transmission models—both utilising random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Municipality level case growth rates: To estimate the daily epidemic growth rates in each municipality, we fit a mixed effects GLM of log new daily case counts in sliding 7-day windows (fixed effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' approximately the generation time of COVID-19 in the earliest wave) 11 and a random effect for each municipality on the slope and intercept, using the R package lme4 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1-21 [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Daily case counts were determined using the date of symptom onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Relationship between case growth rates and mobility: To test for an effect of mo- bility from Mexico City on municipality growth rates, we fit a mixed effect GLM with log mobility as a fixed effect, a random effect on the intercept for each municipality and a random effect on the slope and intercept for log mobility each week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The conditional and marginal coefficient of determination, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', R2, were calculated using the R package MuMIn v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [75] which implements the method developed by Nakagawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2017 [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Model selection was performed using analysis of variance for mixed effects models as implemented in the R package lmerTest v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1-3 [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Additional information Acknowledgments: We thank all health care workers and those involved in the collection, processing and publishing COVID-19 epidemiological data from Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Funding: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' acknowledge funding from the Oxford Martin School Pandemic Genomics programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' acknowledges funding from the European Hori- zon 2020 programme MOOD (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' #874850), the Wellcome Trust, a Branco Weiss Fellowship, The Rockefeller Foundation and Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The contents of this publication are the sole responsibility of the authors and do not necessarily reflect the views of the Euro- pean Commission or the other funders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' acknowledge the support of a grant from the John Templeton Foundation (61780).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of the John Templeton Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Author contributions: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' developed the idea, planned the re- search and conducted analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' collected government intervention data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' wrote the first draft of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' All authors interpreted the data, contributed to writing and approved the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Competing interests: We declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Data and materials availability: Code, spatial, and epidemiological data are available upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' The Google COVID-19 Aggregated Mobility Research Dataset used for this study is available with permission from Google LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Correspondence and requests for materials should be addressed to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='D-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', or M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 12 References [1] Dirk Brockmann and Dirk Helbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' “The Hidden Geometry of Complex, Network- Driven Contagion Phenomena”.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' In: PLOS Computational Biology 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='12 (2013), e1003327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='pcbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 1003327.' metadata={'source': 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Emmanuel Afolabi, Simon I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Hay, Robert C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Reiner, Samson Kiware, and David L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' “Spatial Dynamics of Malaria Transmission”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' In: medRxiv (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' doi: 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='2000596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [72] Edward Goldstein, Hieu H Nguyen, Patrick Liu, Cecile Viboud, Claudia A Steiner, Colin J Worby, and Marc Lipsitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' “On the Relative Role of Different Age Groups During Epidemics Associated With Respiratory Syncytial Virus”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' In: The Journal of Infectious Diseases 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='2 (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 238–244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1093/infdis/jix575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [73] Edward Goldstein, Virginia E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Pitzer, Justin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' O’Hagan, and Marc Lipsitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' “Tem- porally Varying Relative Risks for Infectious Diseases: Implications for Infectious Disease Control”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' In: Epidemiology 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 136–144.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' url: https://CRAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='R-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='org/package=MASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [75] Kamil Bartoń.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' MuMIn: Multi-Model Inference.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 19 [76] Shinichi Nakagawa, Paul C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Johnson, and Holger Schielzeth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' “The coefficient of deter- mination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' In: Journal of the Royal Society Interface 14.' metadata={'source': 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B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Brockhoff, and Rune H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Christensen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' “lmerTest Pack- age: Tests in Linear Mixed Effects Models”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' In: Journal of Statistical Software 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='13 (2017), pp.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='03-22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='05-17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='07-12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='Colima ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='within-state ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='outgoing movement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='incoming movement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='2: Weekly relative change in human mobility within each state and between states (incoming and outgoing) as compared to baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 22 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='3: State-specific correlations of new reported cases (weekly) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' mobility from Mexico City times new reported cases in Mexico City (weekly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' States with low mobility and case count data coverage are included but not plotted in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 23 Distrito Federal México Between 2020-03-29 and 2020-07-19 6000 (yellow dots = later) 8 O .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 4000 - 2000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 50 100 5 10 15 Tabasco Veracruz de Ignacio Guanajuato Pueblal Nuevo Leon Sonora 4000 - de la Llavel 4000 - 3000 - 3000 3000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 8 3000 2000 2000 2000 - ·· 8 2000 - 2000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' : 1000 1000 1000 - 1000 - 8 1000 New o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='002 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='05 Movement from Mexico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Movement from Mexico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Movement from Mexico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Movement from Mexico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Movement from Mexico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Movement from Mexico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' times Mexico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' new cases times Mexico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' new cases times Mexico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' new cases times Mexico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' new cases times Mexico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' new cases times Mexico D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' new casesFigure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='4: Weekly relative human mobility where the origin is the USA and the destina- tion are states in Mexico divided into states that share a land border, Mexico and Mexico City and all other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 24 USA States with 140% USA border States without from 120% USA border Mexico and Mexico City mobility 100% 80% typical 60% 40% Percent of i 20% 0% 12-01 02-09 04-19 06-28Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5: Weekly new cases per 100,000 divided into cases in Mexico City and the state of Mexico, states that share a land border with the USA, and all other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 25 rate per 100,000 100,000 40 65 States with USA border States without 4 USA border 30 new cases per Mexico and Mexico City 20 Weekly growth I Z 10 Weekly L 0 04-19 06-28 03-15 03-15 05-24 04-19 05-24 06-28Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='6: Weekly number of cases among municipalities in Mexico coloured by their geographic position to the USA (bordering vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' not bordering) and the sum of in-municipality mobility × weekly new cases among origin nodes (both on the log scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 26 8 Municipalities without U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' connections Sum of weekly new cases among (log-scaled) 6 Municipalities with U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' connections 4 destination nodes ( 2 0 4 2 0 2 4 6 8 10 12 Sum of in-municipality mobility x sum of weekly new cases among origin nodes (log-scaled)Community size distribution Community size distribution Community size distribution Community size distribution Community size distribution Community size distribution Community size distribution Community size distribution Community size distribution Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='7: Four-week snapshots of mobility in Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Weekly human mobility in Mexico at the municipality level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Thickness of lines represents intensity of relative mobility flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Colours represent the membership to movement communities as estimated using the map equation (Materials & Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='Fr0m 2019-11-24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='(four-week total) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='15 communitiesFrom 2019-12-29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='(four-week total) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='15 communitiesFr0m 2020-01-26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='(four-week total) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='18 communitiesFr0m 2020-02-23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='(four-week total) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='18 communitiesFrom 2020-03-22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='(four-week total) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='33 communitiesFr0m 2020-04-19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='(four-week total) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='26 communitiesFr0m 2020-05-17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='(four-week total) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='22 communitiesFr0m 2020-06-14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='(four-week total) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='33 communitiesFr0m 2020-07-12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='(four-week total) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='32 communities03-01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='03-29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='04-26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='05-24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='06-21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='07-19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='08-16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='Number of communities detected over time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='8: Number of communities detected each week during the first wave of the COVID-19 epidemic in Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 28 03-0103-29 04-26 05-24 06-21 07-19 08-16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='8 (c) Standard deviation of municipality growth rates within grouping (lower values: higher synchrony) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='0 Growth rate (reported cases) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='11 Nayarit Morelos Michoacán de Ocampo México Jalisco Hidalgo Guerrero Guanajuato Durango Distrito Federal Chihuahua Chiapas Colima Coahuila de Zaragoza Campeche Baja California Sur Baja California Aguascalientes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='36 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='465 (mean) Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' of growth rates Example: Variance in growth rates (2020-04-19) municipalities grouped by administrative boundaries 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='54 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='35 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='45 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='50 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='43 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='30 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='49 Comm.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='00 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='00 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='00 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='00 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='00 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='00 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='00 Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='223 (mean) Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' of growth rates Example: Variance in growth rates (2020-04-19) municipalities grouped by network communities o .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (e) (d) o .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='9: For one example week (April 19, 2020), comparison of the mean standard deviations in municipality growth rates within either (left) administrative (states) boundaries or (right) network communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' In each panel, the standard deviations of municipalities’ infection growth rates within each grouping (state vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' network community) is shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Figure 3c shows the average of these values over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 29 b Example network grouping (infomap)a) Administrative grouping (states)Date Intervention March 16, 2020 Mexican Secretariat of Public Education (SEP) suspend classes in schools of preschool, primary, secondary education, as well as those of the upper middle and higher types dependent on the SEP [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' March 17, 2020 Universities begin to suspend classes and social events [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' March 20, 2020 Mexican Secretariat of Public Education (SEP) cancels all civic and sports events [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' March 21, 2020 United States - Mexico border was closed to non-essential travel but remained open for commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Closure extended until November 21, 2020 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' March 23/24, 2020 National period of social distancing begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Schools closed and all non-essential operations were closed including gatherings of 100+ people [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' March 30, 2020 National health emergency declared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Policies included: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=') Non- essential services suspended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=') Private sector is asked to require employees to work from home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=') Sectors that kept operating nor- mally: government, health (public and private), public safety, social programs, critical infrastructure, and essential services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' A full list of essential services can be viewed here: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='dof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='gob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' mx/nota_detalle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='codigo=5590914&fecha=31/03/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=') People over 60 years old are urged to stay home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=') Public gath- erings of over 50 people are banned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=') There was no enforced curfew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Expiration date: April 30, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' April 5, 2020 Hospital reconversion strategy guidelines published in order to con- tain nosocomial transmission [6] April 16, 2020 The Federal Government announces the extension of the health emergency and emphasizes the need to restrict movement to and from areas of high transmissibility until May 30th [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' May 14, 2020 Ministry of Health announces epidemiologic color-coded system to re-open social, educational, economic activities at state level [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' May 18, 2020 First phase of the “new normality” 324 municipalities with no recorded COVID-19 cases are given green light to reopen businesses and schools [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Car factories were meant to reopen on June 1st, but began reopening on May 18th under US pressure (Factories remained closed from March 23rd, to May 18th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 30 June 1, 2020 Mexico’s national period of social distancing concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' A new color-coded system was enacted across the country to assess how quickly states can reopen their economies and schools: red, orange, yellow, and green [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' July 20, 2020 Daycare centers run by the country’s social security system re- opened in coordination with local authority and based on color- coded indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Currently, all but 4 states have daycare centers open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [11] October 18, 2020 The Health Ministry announced that 17 states—Mexico City included—were at alert level orange and 14 were at yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Only one state, Campeche, was at green [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' In states at the orange level, businesses such as hotels and restaurants can reopen while following health protocols such as enforcing limited capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Yel- low allows for most economic activities to return to normal with some occupancy limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' October 22, 2020 Some local governments have chosen to enact more stringent re- strictions than the federal government guidelines, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=', Jalisco and Chihuahua [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1: Timeline of government interventions in Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 31 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='1 Citation diversity statement Recent work has quantified bias in citation practices across various scientific fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' namely, women and other minority scientists are often cited at a rate that is not proportional to their contributions to the field [14–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' In this work, we aim to be proactive about the research we reference in a way that corresponds to the diversity of scholarship in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' To evaluate gender bias in the references used here, we obtained the gender of the first/last authors of the papers cited here through either 1) the gender pronouns used to refer to them in articles or biographies or 2) if none were available, we used a database of common name- gender combinations across a variety of languages and ethnicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' By this measure (excluding citations to datasets/organizations, citations included in this section, and self-citations to the first/last authors of this manuscript), our references contain 3% woman(first)-woman(last), 22% woman-man, 20% man-woman, 47% man-man, 0% nonbinary, 8% man solo-author, and 0% woman solo-author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' This method is limited in that an author’s pronouns may not be consistent across time or environment, and no database of common name-gender pairings is complete or fully accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Supplemental References [1] DOF - Diario Oficial de la Federación.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='dof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='gob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='mx/nota_ detalle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='codigo=5589479&fecha=16/03/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [2] Coronavirus en México: universidades suspenden clases y se intensifican las acciones preventivas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' url: https : / / www .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' infobae .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' com / america / mexico / 2020 / 03 / 13 / coronavirus - en - mexico - universidades - suspenden - clases - y - se - intensifican-las-acciones-preventivas/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [3] Gobierno de México suspenderá todas las actividades escolares por coronavirus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' url: https : / / www .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' latimes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' com / espanol / mexico / articulo / 2020 - 03 - 14 / gobierno- de- mexico- suspendera- todas- las- actividades- escolares- por- coronavirus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [4] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Embassy & Consulates in Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Mexico, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Travel restrictions - Fact sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' url: https://mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='usembassy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='gov/travel-restrictions-fact-sheet/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [5] Inicia fase 2 por coronavirus COVID-19 – Coronavirus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' url: https://coronavirus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' gob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='mx/2020/03/24/inicia-fase-2-por-coronavirus-covid-19/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [6] Gobierno de México and Secretaría de Salud COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Lineamiento de Reconversión Hospitalaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' url: https://coronavirus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='gob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='mx/wp-content/uploads/2020/04/ Documentos-Lineamientos-Reconversion-Hospitalaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [7] Coronavirus en México: guía para entender las cuatro nuevas medidas de control y prevención del COVID-19 cercanas a la Fase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='infobae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='com/ america/mexico/2020/04/16/coronavirus-en-mexico-guia-para-entender- las - cuatro - nuevas - medidas - de - control - y - prevencion - del - covid - 19 - cercanas-a-la-fase-3/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 32 [8] DOF - Diario Oficial de la Federación.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' url: https://dof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='gob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='mx/nota_detalle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='codigo=5593313&fecha=14/05/2020#gsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='tab=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [9] Conferencia 16 de mayo – Coronavirus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' url: https://coronavirus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='gob.' metadata={'source': 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+page_content='mx/prensa/archivo/202007/463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' [12] COVID-19 MÉXICO Comunicado Técnico Diario - 18 Octubre 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' url: http : / / saludsinaloa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' gob .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' mx / wp - content / uploads / 2020 / reportescovid / Covid19ReporteDiario18Noviembre2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='pdf.' metadata={'source': 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Mongeon, Cassidy R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' Sugimoto, and Vincent Larivière.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' “Gender homophily in citations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' In: 23rd International Conference on Science and Technology Indicators (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 1519–1525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' url: https://hdl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content='net/1887/65291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} +page_content=' 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFQT4oBgHgl3EQfJjU4/content/2301.13256v1.pdf'} diff --git a/AtAyT4oBgHgl3EQfRvdA/content/tmp_files/2301.00071v1.pdf.txt b/AtAyT4oBgHgl3EQfRvdA/content/tmp_files/2301.00071v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d81abb560cfc4350c24bcc6e39008a9065fc26ed --- /dev/null +++ b/AtAyT4oBgHgl3EQfRvdA/content/tmp_files/2301.00071v1.pdf.txt @@ -0,0 +1,1455 @@ +arXiv:2301.00071v1 [math.CO] 30 Dec 2022 +SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE +PRINCIPLE +M.M. SKRIGANOV +Abstract. Stolarsky’s invariance principle, known for point distributions on +the Euclidean spheres Sd [25], has been extended to the real RP n, complex +CP n, and quaternionic HP n projective spaces and the octonionic OP 2 pro- +jective plane in our previous paper [23]. Geometric features of such spaces as +well as their models in terms of Jordan algebras have been used very essen- +tially in the proof. In the present paper, we give a new pure analytic proof of +Stolarsky’s invariance principle relying on the theory of spherical functions on +compact symmetric Riemannian manifolds of rank one. +1. Introduction and main results +1.1 Introduction. In 1973 Kenneth B. Stolarsky [25] established the following +remarkable formula for point distributions on the Euclidean spheres. Let Sd = +{x ∈ Rd+1 : ∥x∥ = 1} be the standard d-dimensional unit sphere in Rd+1 with +the geodesic (great circle) metric θ and the Lebesgue measure µ normalized by +µ(Sd) = 1. We write C(y, t) = {x ∈ Sd : (x, y) > t} for the spherical cap of height +t ∈ [−1, 1] centered at y ∈ Sd. Here we write (·, ·) and ∥ · ∥ for the inner product +and the Euclidean norm in Rd+1. +For an N-point subset DN ⊂ Sd, the spherical cap quadratic discrepancy is +defined by +λcap[DN] = +� 1 +−1 +� +Sd ( #{|C(y, t) ∩ DN} − Nµ(C(y, t)) )2 dµ(y) dt. +(1.1) +We introduce the sum of pairwise Euclidean distances between points of DN +τ[DN] = 1 +2 +� +x1,x2∈DN ∥x1 − x2∥ = +� +x1,x2∈DN sin 1 +2θ(x1, x2), +(1.2) +and write ⟨τ⟩ for the average value of the Euclidean distance on Sd, +⟨τ⟩ = 1 +2 +�� +Sd×Sd ∥y1 − y2∥ dµ(y1) dµ(y2). +(1.3) +The study of the quantities (1.1) and (1.2) falls within the subjects of discrepancy +theory and geometry of distances, see [1,6] and references therein. It turns out that +the quantities (1.1) and (1.2) are not independent and are intimately related by the +following remarkable identity +γ(Sd)λcap[DN] + τ[DN] = ⟨τ⟩N 2, +(1.4) +2010 Mathematics Subject Classification. 11K38, 22F30, 52C99. +Key words and phrases. Geometry of distances, discrepancies, spherical functions, projective +spaces, Jacobi polynomials. + +2 +M.M. SKRIGANOV +for an arbitrary N-point subset DN ⊂ Sd. Here γ(Sd) is a positive constant inde- +pendent of DN, +γ(Sd) = d √π Γ(d/2) +2 Γ((d + 1)/2) . +(1.5) +The identity (1.4) is known in the literature as Stolarsky’s invariance principle. +Its original proof given in [25] has been simplified in [7,10]. simplified in [7,10]. +In our previous paper [23] Stolarsky’s invariance principle (1.4) has been ex- +tended to the real RP n, the complex CP n, the quaternionic HP n projective spaces, +and the octonionic OP 2 projective plane. Geometric features of such spaces as well +as their models in terms of Jordan algebras have been used very essentially in the +proof. The aim of the present paper is to give an alternative pure analytic proof +relying on the theory of spherical functions. +1.2 Discrepancies and metrics. L1-invariance principles. Let us consider Sto- +larsky’s invariance principle in a broader context. Let M be a compact metric +measure space with a fixed metric θ and a finite Borel measure µ, normalized, for +convenience, by +diam(M, θ) = π, +µ(M) = 1, +(1.6) +where diam(E, ρ) = sup{ρ(x1, x2) : x1, x2 ∈ E} denotes the diameter of a subset +E ⊆ M with respect to a metric ρ. +We write B(y, r) = {x ∈ M : θ(x, y) < r} for the ball of radius r ∈ I centered at +y ∈ M and of volume v(y, r) = µ(B(y, r)). Here I = {r = θ(x1, x2) : x1, x2 ∈ M} +denotes the set of all possible radii. If the space M is connected, we have I = [ 0, π ]. +We consider distance-invariant metric spaces. Recall that a metric space M is +called distance-invariant, if the volume of any ball v(r) = v(y, r) is independent of +y ∈ M. The typical examples of distance-invariant spaces are homogeneous spaces +M = G/H with G-invariant metrics θ and measures µ. +For an N-point subset DN ⊂ M, the ball quadratic discrepancy is defined by +λ[ξ, DN] = +� +I +� +M +( #{B(y, r) ∩ DN} − Nv(r)) )2 dµ(y) dξ(r), +(1.7) +where ξ is a finite measure on the set of radii I. +Notice that for Sd spherical caps and balls are related by C(y, t) = B(y, r), +t = cos r, and the discrepancies (1.1) and (1.7) are related by λcap[DN] = λ[ξ♮, DN], +where dξ♮(r) = sin r dr, r ∈ I = [0, π]. +The ball quadratic discrepancy (1.7) can be written in the form +λ[ξ, DN] = +� +x1,x2∈DN λ(ξ, x1, x2) +(1.8) +with the kernel +λ(ξ, x1, x2) = +� +I +� +M +Λ(B(y, r), x1) Λ(B(y, r), x2) dµ(y) dξ(r) , +(1.9) +where +Λ(B(y, r), x) = χ(B(y, r), x) − v(r), +(1.10) +and χ(E, ·) denotes the characteristic function of a subset E ⊆ M. +For an arbitrary metric ρ on M we introduce the sum of pairwise distances +ρ[DN] = +� +x1,x2∈DN ρ(x1, x2). +(1.11) + +SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE +3 +and the average value +⟨ρ⟩ = +� +M×M +ρ(y1, y2) dµ(y1) dµ(y2). +(1.12) +We introduce the following symmetric difference metrics on the space M +θ∆(ξ, y1, y2) = 1 +2 +� +I +µ(B(y1, r)∆B(y2, r)) dξ(r) += 1 +2 +� +I +� +M +χ(B(y1, r)∆B(y2, r), y) dµ(y) dξ(r), +(1.13) +where +B(y1, r)∆B(y2, r) = B(y1, r) ∪ B(y2, r) \ B(y1, r) ∩ B(y2, r) +is the symmetric difference of the balls B(y1, r) and B(y2, r). +In line with the definitions (1.11) and (1.12), we put +θ∆[ξ, DN] = +� +x1,x2∈DN θ∆(ξ, x1, x2). +and +⟨θ∆(ξ)⟩ = +� +M×M +θ∆(ξ, y1, y2) dµ(y1) dµ(y2) . +A direct calculation leads to the following result. +Proposition 1.1. Let a compact metric measure space M be distance-invariant, +then we have +λ(ξ, y1, y2) + θ∆(ξ, y1, y2) = ⟨θ∆(ξ)⟩. +(1.14) +In particular, we have the following invariance principle +λ[ ξ, DN ] + θ∆[ ξ, DN ] = ⟨θ∆(ξ)⟩ N 2 +(1.15) +for an arbitrary N-point subset DN ⊂ M. +Proof. In view of the symmetry of the metric θ, we have +χ(B(x, r), y) = χ(B(y, r), x) = χ0(r − θ(y, x)) , +(1.16) +where χ0(·) is the characteristic function of the half-axis (0, ∞). Therefore +χ(B(y1, r)∆B(y2, r), y) = χ(B(y1, r), y) + χ(B(y2, r), y) +−2χ(B(y1, r) ∩ B(y2, r), y) , +and +� +M χ(B(x, r), y)dµ(x) = +� +M χ(B(x, r), y)dµ(y) = v(r). +Using these relations, we obtain +λ(ξ, x1, x2) = +� +I +� +µ(B(x1, r) ∩ B(x2, r)) − v(r)2� +dξ(r) , +θ∆(ξ, y1, y2) = +� +I +� +v(r) − µ(B(y1, r) ∩ B(y2, r)) +� +dξ(r) , +⟨θ∆(ξ)⟩ = +� +I +� +v(r) − v(r)2� +dξ(r) . + + + + + + + + + + + + + + + +(1.17) +These relations imply (1.14). +□ + +4 +M.M. SKRIGANOV +In the case of spheres Sd, relations of the type (1.14) and (1.15) were given +in [25]. Their extensions to more general metric measure spaces were given in [21, +Theorem 2.1], [22, Eq. (1.30)] and [23, Proposition 1.1]. +Notice that +χ(B(y1, r)∆B(y2, r), y) = |χ(B(y1, r), y) − χ(B(y2, r), y)| , +(1.18) +and hence +θ∆(ξ, y1, y2) = 1 +2 +� +I +� +M +|χ(B(y1, r), y) − χ(B(y2, r), y)| dµ(y) dξ(r) +(1.19) +is an L1-metric. +Recall that a metric space M with a metric ρ is called isometrically Lq-embeddable +(q = 1 or 2), if there exists a mapping ϕ : M ∋ x → ϕ(x) ∈ Lq, such that +ρ(x1, x2) = ∥ϕ(x1)−ϕ(x2)∥Lq for all x1, x2 ∈ M. Notice that the L2-embeddability +is stronger and implies the L1-embeddability, see [13, Sec. 6.3]. +It follows from (1.19) that the space M with the symmetric difference metrics +θ∆(ξ) is isometrically L1-embeddable by the formula +M ∋ x → χ(B(x, r), y) ∈ L1(M × I) , +(1.20) +The identity (1.15) can be called the L1-invariance principle, while Stolarsky’s +invariance principle (1.4) should be called the L2-invariance principle, because it +involves the Euclidean metric. The identities of such a type including correspond- +ingly L1 and L2 metrics could be also called weak and strong invariance principles. +1.3 L2-invariance principles. Recall the definition and necessary facts on two- +point homogeneous spaces. Let G = G(M) be the group of isometries of a metric +space M with a metric θ, i.e. θ(gx1, gx2) = θ(x1, x2) for all x1, x2 ∈ M and g ∈ G. +The space M is called two-point homogeneous, if for any two pairs of points x1, +x2 and y1, y2 with θ(x1, x2) = θ(y1, y2) there exists an isometry g ∈ G, such that +y1 = gx1, y2 = gx2. In this case, the group G is obviously transitive on M and +M = G/H is a homogeneous space, where the subgroup K ⊂ G is the stabilizer of +a point x0 ∈ M. Furthermore, the homogeneous space M is symmetric, i.e. for +any two points y1, y2 ∈ M there exists an isometry g ∈ G, such that gy1 = y2, +gy2 = y1. +There is a very large number of two-point homogeneous spaces. For example, all +Hamming spaces, known in the coding theory, are two-point homogeneous. We will +consider compact connected two-point homogeneous spaces. The assumption that +the space is connected turns out to be a strong restriction. All compact connected +two-point homogeneous spaces Q = G/H are known, and by Wang’s classifications +they are the following, see [16,17,20,29,30]: +(i) The d-dimensional Euclidean spheres Sd = SO(d + 1)/SO(d) × {1}, d ⩾ 2, +and S1 = O(2)/O(1) × {1}. +(ii) The real projective spaces RP n = O(n + 1)/O(n) × O(1). +(iii) The complex projective spaces CP n = U(n + 1)/U(n) × U(1). +(iv) The quaternionic projective spaces HP n = Sp(n + 1)/Sp(n) × Sp(1), +(v) The octonionic projective plane OP 2 = F4/ Spin(9). +Here we use the standard notation from the theory of Lie groups; in particular, +F4 is one of the exceptional Lie groups in Cartan’s classification. +All these spaces are Riemannian symmetric manifolds of rank one. +Geomet- +rically, this means that all geodesic sub-manifolds in Q are one-dimensional and + +SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE +5 +coincide with geodesics. From the spectral stand point, this also means that all +operators on Q commuting with the action of the group G are functions of the +Laplace–Beltrami operator on Q, see [16,17,29,30] for more details. +The spaces FP n as Riemannian manifolds have dimensions d, +d = dimR FP n = nd0, +d0 = dimR F, +(1.21) +where d0 = 1, 2, 4, 8 for F = R, C, H, O, correspondingly. +For the spheres Sd we put d0 = d by definition. Projective spaces of dimension +d0 (n = 1) are homeomorphic to the spheres Sd0: RP 1 ≈ S1, CP 1 ≈ S2, HP 1 ≈ +S4, OP 1 ≈ S8. We can conveniently agree that d > d0 (n ⩾ 2) for projective spaces, +while the equality d = d0 holds only for spheres. Under this convention, the dimen- +sions d = nd0 and d0 define uniquely (up to homeomorphism) the corresponding +homogeneous space which we denote by Q = Q(d, d0). +We consider Q(d, d0) as a metric measure space with the metric θ and measure +µ proportional to the invariant Riemannian distance and measure on Q(d, d0). The +coefficients of proportionality are defined to satisfy (1.6). In what follows we always +assume that n = 2 if F = O, since projective spaces OP n do not exist for n > 2. +Any space Q(d, d0) is distance-invariant and the volume of balls in the space is +given by +v(r) = κ +� r +0 +(sin 1 +2u)d−1(cos 1 +2u)d0−1 du +r ∈ [ 0, π ] += κ 21−d/2−d0/2 +� 1 +cos r +(1 − z) +d +2 −1 (1 + z) +d0 +2 −1 dz, +(1.22) +where κ = κ(d, d0) = B(d/2, d0/2)−1; B(a, b) = Γ(a)Γ(b)/Γ(a + b) and Γ(a) are +the beta and gamma functions. Equivalent forms of (1.22) can be found in the +literature, see, for example, [15, pp. 177–178], [17, pp. 165–168], [18, pp. 508–510]. +For even d0, the integrals (1.22) can be calculated explicitly that gives convenient +expressions for v(r) in the case of CP n, HP n and OP 2, see, for example, [20]. +The chordal metric on the spaces Q(d, d0) is defined by +τ(x1, x2) = sin 1 +2θ(x1, x2) = +� +1 − cos(x1, x2) +2 +, +x1, x2 ∈ Q(d, d0). +(1.23) +The formula (1.23) defines a metric because the function ϕ(θ) = sin θ/2, 0 ⩽ θ ⩽ π, +is concave, increasing, and ϕ(0) = 0, that implies the triangle inequality. For the +sphere Sd we have +τ(x1, x2) = sin 1 +2θ(x1, x2) = 1 +2 ∥x1 − x2∥, +x1, x2 ∈ Sd. +(1.24) +Lemma 1.1. The space Q(d, d0), d = nd0, can be embedded into the unit sphere +Π : Q(d, d0) ∋ x → Π(x) ∈ Sm−1 ⊂ Rm, +m = 1 +2(n + 1)(d + 2) − 1, +(1.25) +such that +τ(x1, x2) = +� +d +2(d + d0) +�1/2 +∥Π(x1) − Π(x2)∥, +x1, x2 ∈ Q(d, d0), +(1.26) +where ∥ · ∥ is the Euclidean norm in Rm. + +6 +M.M. SKRIGANOV +Hence, the metric τ(x1, x2) is proportional to the Euclidean length of a segment +joining the corresponding points Π(x1) and Π(x2) on the unit sphere. The chordal +metric τ on the complex projective space CP n is known as the Fubini–Study metric. +Lemma 1.1 will be proved in Section 2, and the embedding (1.25) will be de- +scribed explicitly in terms of spherical functions on the space Q(d, d0). Note that +the embedding (1.25) can be described in different ways, see, for example, [23,27]. +The following general result has been established in [23, Theorems 1.1 and 1.2]. +Theorem 1.1. For each space Q = Q(d, d0), we have the equality +τ(x1, x2) = γ(Q) θ∆(ξ♮, x1, x2). +(1.27) +where dξ♮(r) = sin r dr, r ∈ [0, π] and +γ(Q) = +√π +4 (d + d0) +Γ(d0/2) +Γ((d0 + 1)/2) = d + d0 +2d0 +γ(Sd0) , +(1.28) +where γ(Sd0) is defined by (1.5). +Comparing Theorem 1.1 with Proposition 1.1, we arrive to the following. +Corollary 1.1. We have the following L2-invariance principle +γ(Q) λ[ξ♮, DN] + τ[DN] = ⟨τ⟩N 2 +(1.29) +for an arbitrary N-point subset DN ⊂ Q. +The constant γ(Q) has the following geometric interpretation +γ(Q) = +⟨τ⟩ +⟨θ∆(ξ♮)⟩ = +diam(Q, τ) +diam(Q, θ∆(ξ♮)) . +(1.30) +Indeed, it suffices to calculate the average values (1.12) of both metrics in (1.27) to +obtain the first equality in (1.30). Similarly, writing (1.27) for any pair of antipodal +points x1, x2, θ(x1, x2) = π, we obtain the second equality in (1.30). The average +value ⟨τ⟩ of the chordal metric τ can be easily calculated with the help of the +formulas (1.12) and (1.22): +⟨τ⟩ = B(d/2, d0/2)−1 B((d + 1)/2, d0/2) . +(1.31) +In the case of spheres Sd, the identity (1.29) coincides with (1.4). The identity +(1.29) can be thought of as an extension of Stolarsky’s invariance principle to all +projective spaces. +Applications of L1- and L2-invariance principles and similar identities to the +discrepancy theory, geometry of distances, and information theory have been given +in many papers, see, for example, [1,3–10,21–23,25]. +It is worth noting that the equality (1.27) is of interest by itself. +Since the +integrand in (1.19) takes the values 0 and 1 only, we can write +θ∆(ξ, y1, y2) = +� +θ∆ +p (ξ, y1, y2) +�p +, p > 0, +(1.32) +where +θ∆ +p (ξ, y1, y2) = +�1 +2 +� π +0 +� +Q +|χ(B(y1, r), y) − χ(B(y2, r), y)|p) dµ(y) dξ(r) +�1/p +, (1.33) +is an Lp-metric for p ⩾ 1. + +SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE +7 +Comparing (1.27) and (1.32), we see that the chordal metric τ is proportional +to the p-th power of the metric θ∆ +p (ξ♮) for all p ⩾ 1. This is a nontrivial fact. For +example, we have for p = 2 +τ(x1, x2) = γ(Q) +2 +� π +0 +� +Q +|χ(B(x1, r), y) − χ(B(x2, r), y)|2 dµ(y) dξ♮(r), +(1.34) +and the equality (1.34) implies the existence of Gaussian random fields on the +spaces Q(d, d0), see [12,15]. However, a detailed considerations of these questions +is beyond the scope of the present paper. +In the context of our discussion, the following open problems are of interest: +- Do there exist measures ξ on the set of radii for spaces Q(d, d0) (for spheres Sd, +say) other than the measure ξ♮ such that the corresponding symmetric difference +metrics θ∆(ξ) are the L2-metrics? +- Do there exist compact measure metric spaces other than spheres Sd and pro- +jective spaces FP n for which the L2-invariance principle is also true? +1.4 Proof of Theorem 1.2. In the present paper we use the theory of spherical +functions to prove the following result. +Theorem 1.2. The equality (1.27) is equivalent to the following series of formulas +for Jacobi polynomials +� 1 +−1 +� +P (d/2,d0/2) +l−1 +(t) +�2 +(1 − t)d (1 + t)d0 dt += 2d+d0+1 (1/2)l−1 +((l − 1)!)2 +B(d + 1, d0 + 1) Tl−1(d/2, d0/2) +(1.35) +for all l ⩾ 1, where +Tl−1(d/2, d0/2) == +Γ(d/2 + l) Γ(d0/2 + l) Γ(d/2 + d0/2 + 3/2)) +Γ(d/2 + 1) Γ(d0/2 + 1) Γ(d/2 + d0/2 + 1/2 + l) . +(1.36) +Here P (α,β) +n +(t), t ∈ [−1, 1], α > −1, β > −1, are the standard Jacobi polynomials +of degree n normalized by +P (α,β) +n +(1) = +�α + n +n +� += +Γ(α + n + 1) +Γ(n + 1)Γ(α + 1) , +(1.37) +and P (α,β) +n +can be given by Rodrigues’ formula +P (α,β) +n +(t) = (−1)n +2nn! (1 − t)−α(1 + t)−β dn +dtn +� +(1 − t)n+α(1 + t)n+β� +. +(1.38) +Notice that |P (α,β) +n +(t)| ⩽ P (α,β) +n +(1) for t ∈ [−1, 1]. Recall also that P (α,β) +n +are +orthogonal polynomials with the following orthogonality relations +π +� +0 +P (α,β) +l +(cos u)P (α,β) +l′ +(cos u)(sin 1 +2u)2α+1(cos 1 +2u)2β+1 du += 2−α−β−1 +1 +� +−1 +P (α,β) +l +(z)P (α,β) +l′ +(z)(1 − z)α(1 + z)β dz = M −1 +l +δll′, +(1.39) + +8 +M.M. SKRIGANOV +where M0 = B(α + 1, β + 1)−1 and +Ml = Ml(α, β) = (2l + α + β + 1)Γ(l + 1)Γ(l + α + β + 1) +Γ(l + α + 1)Γ(l + β + 1), +l ⩾ 1. +(1.40) +All necessary facts about Jacobi polynomials can be found in [2, 26]. We also +use the notation +(a)0 = 1, +(a)k = a(a + 1) . . . (a + k − 1) = Γ(a + k) +Γ(a) +(1.41) +for the rising factorial powers (Pochhammer’s symbol). +Theorem 1.2 reduces the proof of Theorem 1.1 to the proof of the formulas (1.35). +Perhaps such formulas are known but I could not find them in the literature. For +spheres Jacobi polynomials P (d/2,d/2) +n +with equal parameters coincide (up to con- +stant factors) with Gegenbauer polynomials, and in this case very general formulas +for weighted L2-norms of Gegenbauer polynomials are given in the paper [11]. +In the present paper we will prove the following statement. +Lemma 1.2. For all n ⩾ 0, Re α > −1/2 and Re β > −1/2, we have +� 1 +−1 +� +P (α,β) +n +(t) +�2 +(1 − t)2α(1 + t)2β dt += 22α+2β+1 (1/2)n +(n!)2 +B(2α + 1, 2β + 1) Tn(α, β), +(1.42) +where +Tn(α, β) = (α + 1)n (β + 1)n +(α + β + 3/2)n += Γ(α + n + 1) Γ(β + n + 1) Γ(α + β + 3/2)) +Γ(α + 1) Γ(β + 1) Γ(α + β + 3/2 + n) +(1.43) +is a rational function of α and β. +The integral (1.42) converges for Re α > −1/2 and Re β > −1/2, and represents +in this region a holomorphic function of two complex variables. The equality (1.42) +defines an analytic continuation of the integral (1.42) to α ∈ C and β ∈ C. +For α = d/2, β = d0/2 and n = l − 1 the equality (1.42) coincides with (1.35). +This proves Theorem 1.1. +Lemma 1.2 will be proved in Section 3. The crucial point in the proof is Watson’s +theorem on the value of hypergeometric series 3F2(1). +2. Spherical functions. Proofs of Lemma 1.1 and Theorem 1.2 +2.1. Invariant kernels and spherical functions. The general theory of spherical +functions on homogeneous spaces can be found in [16,17,28,30]. The homogeneous +spaces Q(d, d0) of interest to us belong to the class of so-called commutative spaces +or symmetric Gelfand pairs. In this case the theory becomes significantly simpler. +For Euclidean spheres Sd this theory is well known, see, for example, [14, 19]. +However, the theory of spherical functions on general spaces Q(d, d0) is probably +not commonly known. In this section we describe the basic facts about spherical +functions on spaces Q(d, d0) in a form convenient for us. + +SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE +9 +Let us consider the quasi-regular representation U(g)f(x) = f(g−1x), f ∈ L2(Q), +x ∈ Q, g ∈ G, and its decomposition into the orthogonal sum +U(g) = � +� +l⩾0 Ul(g), +L2(Q) = � +� +l⩾0 Vl , +(2.1) +of irreducible representations Ul(g) in mutually orthogonal subspaces Vl of dimen- +sions ml < ∞. +Let A denote the algebra of Hilbert–Schmidt operators in L2(Q) commuting +with the representation U. Each K ∈ A is an integral operator +Kf(x) = +� +Q +K(x, y) f(y) dµ(y), +with the invariant kernel: +K(gx1, gx2) = K(x1, x2), x1, x2 ∈ Q, g ∈ G, +(2.2) +which satisfies the condition +||K||2 +HS = Tr KK∗ += +� +Q×Q +|K(x, y)|2 dµ(x)dµ(y) = +� +Q +|K(x, y)|2 dµ(x) < ∞, +(2.3) +where Tr denotes the trace of an operator, and the second integral is independent +of y in view of (2.2). +Since the space Q is two-point homogeneous, the condition (2.2) implies that the +kernel K(x1, x2) depends only on the distance θ(x1, x2), and can be written as +K(x1, x2) = K(θ(x1, x2)) = k(cos θ(x1, x2)), x1, x2 ∈ Q, +(2.4) +with function K(z), z ∈ [0, π] and k(z), z ∈ [−1, 1]. The cosine is presented here for +convenience in further calculations. The formula (2.4) can be also written as +K(x1, x2) = K(θ(x, x0)) = k(cos θ(x, x0)), +(2.5) +where x1 = g1x0, x2 = g2x0, x = g−1 +2 g1x0, g1, g2 ∈ G and x0 ∈ Q is the fixed point +of the subgroup H. Moreover, K(hx, x0) = K(x, x0), h ∈ H. Therefore, invariant +kernels can be thought of as functions on the double co-sets H \ G/H. +In terms of the function K(·) and k(·), the Hilbert-Schmidt norm (2.3) takes the +form +||K||2 +HS = +� π +0 +|K(u)|2 dv(u) +=κ +� π +0 +|k(cos u)|2(sin 1 +2u)d−1(cos 1 +2u)d0−1 du +=κ 21−d/2−d0/2 +� 1 +−1 +|k(z)|2 (1 − z) +d +2 −1 (1 + z) +d0 +2 −1 dz, +(2.6) +where v(·) is the volume function (1.22). +We conclude from (2.2) and (2.4) that for K ∈ A its kernel K(x1, x2) = +K(x2, x1), the value K(x, x) = k(1) is independent of x ∈ Q, and if an opera- +tor K is self-adjoint, then its kernel is real-valued. + +10 +M.M. SKRIGANOV +It follows from (2.2) and (2.4) that the algebra A is commutative. Indeed, +(K1K2)(x1, x2) = +� +Q +K1(x1, x)K2(x, x2)dµ(x) += +� +Q +K2(x2, x)K1(x, x1)dµ(x) = (K2K1)(x2, x1) = (K2K1)(x1, x2). +Therefore, the decomposition (2.1) is multiplicity-free, that is any two representa- +tions Ul and Ul′, l ̸= l′, are non-equivalent, because otherwise the algebras A could +not be commutative. +Let Pl denote orthogonal projectors in L2(Q) onto the subspaces Vl in (2.1), +P ∗ +l = Pl , +Pl Pl′ = δl,l′ Pl , +� +l⩾0 Pl = I , +(2.7) +where δl,l′ is Kronecker’s symbol and I is the identity operator in L2(Q). By Schur’s +lemma, we have for K ∈ A +Pl K Pl′ = δl,l′ cl(K) Pl, , +(2.8) +where cl(K) is a constant. Calculating the trace of both sides of the equality (2.8), +we find cl(K) = m−1 +l +Tr KPl. Therefore, we have the expansions +K = +� +l,l′⩾0 Pl K Pl′ = +� +l⩾0 cl(K) Pl, +(2.9) +with Parseval’s identity +||K||2 +HS = +� +l⩾0 ml |cl(K)|2 , +(2.10) +and for K1, K2 ∈ A, we have +K1 K2 = +� +l⩾0 cl(K1) cl(K2) Pl, +(2.11) +The equality (2.10) implies that the series (2.11) converges in the Hilbert-Schmidt +norm (2.3), while the series (2.11) converges in the norm (2.3) for the subclass of +nuclear operators. +Since Vl are invariant subspaces, Pl ∈ A, their kernels Pl(·, ·) are symmetric and +real-valued, and can be written as follows +Pl(x1, x2) = pl(cos θ(x1, x2)) = +�ml +1 +ψl,j(x1) ψl,j(x2), +(2.12) +where {ψl,j(·)}ml +1 +is an orthonormal and real-valued basis in Vl. Hence, subspace +Vl and irreducible representations Ul in (2.1) can be thought of as defined over the +field of reals, this means that all representations Ul in (2.1) are of the real type. +Using (2.12), we obtain the formulas +||Pl||2 +HS = ml, +Tr Pl = pl(1) = ml > 0. +(2.13) +Furthermore, +Pl(x, x) = pl(1) = +�ml +1 +ψl,j(x)2. +(2.14) +is independent of x ∈ Q. Applying Cauchy-Schwartz inequality to (2.12) and taking +(2.14) into account, we obtain the bound +|Pl(x1, x2)| = |pl(cos θ(x1, x2))| ⩽ pl(1). +(2.15) +It follows from (2.14) and (2.13) that the mapping +Πl : Q ∋ x → (m−1/2 +l +ψl,1(x) . . . m−1/2 +l +ψl,ml(x)) ∈ Sml−1 ⊂ Rml +(2.16) + +SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE +11 +defines an embedding of the space Q into the unite sphere in Rml. +By definition the (zonal) spherical function are kernels of the operators Φl = +m−1 +l +Pl: +Φl(x1, x2) = φl(cos θ(x1, x2)) = pl(cos θ(x1, x2)) +pl(1) +. +(2.17) +From (2.14) and (2.17) we conclude that |φl(cos θ(x1, x2))| ⩽ φl(1) = 1. Comparing +(2.13), (2.14) and (2.17), we find the formulas for dimensions +ml = ||Φl||−2 +HS = +� +κ +� π +0 +|φl(cos u)|2(sin 1 +2u)d−1(cos 1 +2u)d0−1 du +�−1 += +� +κ 21−d/2−d0/2 +� 1 +−1 +|φl(z)|2 (1 − z) +d +2 −1 (1 + z) +d0 +2 −1 dz +�−1 +. +(2.18) +In terms of spherical functions the formulas (2.9) and (2.11) take the form +k(cos θ(x1, x2)) = +� +l⩾0 cl(K) ml φl(cos θ(x1, x2)), +(2.19) +where +cl(K) = Tr KΦl = +� +Q +K(x1, x2) Φ(x1, x2) dµ(x1)dµ(x2) += κ +� π +0 +k(cos u) φl(cos u) (sin 1 +2u)d−1(cos 1 +2u)d0−1 du += κ 21−d/2−d0/2 +� 1 +−1 +k(z) φl(z) (1 − z) +d +2 −1 (1 + z) +d0 +2 −1 dz. +(2.20) +and +� +Q +k1(cos θ(x1, y)) k2( cos θ(y, x2)) dµ(y) += +� +l⩾0 cl(K1) cl(K2) ml φl(cos θ(x1, x2)), +(2.21) +for K1, K2 ∈ A. It follows from (2.11) with K1 = K and K2 = K∗ that +||K||2 +HS = +� +l⩾0 ml |cl(K)|2 , +(2.22) +The above facts are valid for all compact two-point homogeneous spaces. Since +spaces Q are also symmetric Riemannian manifolds of rank one, the invariant ker- +nels pl(cos θ(x, x0)) are eigenfunctions of the radial part of the Laplace–Beltrami +operator on Q (in the spherical coordinates centered at x0). +This leads to the +following explicit formula for spherical functions +Φ(x1, x2) = φl(cos θ(x1, x2)) = P +( d +2 −1, d0 +2 −1) +l +(cos θ(x1, x2)) +P +( d +2 −1, d0 +2 −1) +l +(1) +, +l ⩾ 0. +(2.23) +where P (α,β) +n +(t), t ∈ [−1, 1], are Jacobi polynomials (1.38). We refer to [15, p. 178], +[17, Chap. V, Theorem 4.5], [18, pp. 514–512, 543–544], [28, Chapters 2 and 17]: [30, +Theorem 11.4.21] for more details. +From (1.37) and (1.40) we obtain +P +( d +2 −1, d0 +2 −1) +n +(1) = +Γ(n + d/2) +Γ(n + 1)Γ(1 + d/2) , +(2.24) + +12 +M.M. SKRIGANOV +and Ml = Ml(d/2 − 1, d0/2 − 1), where M0 = B(d/2, d0/2)−1 and +Ml = (2l − 1 + (d + d0)/2)Γ(l + 1)Γ(l − 1 + (d + d0)/2) +Γ(l + d/2)Γ(l + d0/2) +, l ⩾ 1, +(2.25) +Substituting (2.23), into (2.18) and using (2.24) and (2.25), we obtain the following +explicit formulas for dimensions of irreducible representations (2.1) : m0 = 1 and +ml =Ml B(d/2, d0/2) +� +P ( d +2 −1, d0 +2 −1)(1) +�2 +=(2l − 1 + (d + d0)/2) Γ(l − 1 + (d + d0)/2)Γ(l + d/2)Γ(d0/2) +Γ((d + d0)/2)Γ(l + d0/2)Γ(d/2)Γ(l + 1) +l ⩾ 1. (2.26) +The formulas (2.19) for invariant kernels coincide with Fourier-Jacobi expansions. +Suppose that a function k(t), t ∈ [−1, 1], has the expansion +k(cos r) = +� +l⩾0 +Ml Cl(F) P +( d +2 −1, d0 +2 −1) +l +(cos r), +(2.27) +with Fourier-Jacobi coefficients +Cl(k) = +� π +0 +k(cos u) P +( d +2 −1, d0 +2 −1) +l +(cos u) (sin 1 +2u)d−1 (cos 1 +2u)d0−1 du, +(2.28) +then the corresponding invariant kernel k(cos θ(x1, x2)) has the expansion (2.19) +with coefficients +cl(k) = Cl(k) +κ(d, d0) +P +( d +2 −1, d0 +2 −1) +l +(1) +, +l ⩾ 0. +(2.29) +Lemma 2.1. (i) For the chordal metric (1.23), we have +τ(x1, x2) = 1 +2 +� +l⩾1 Ml Cl [ 1 − φl(x1, x2) ] , +(2.30) +where +Cl = B((d + 1)/2, l + d0/2) Γ(l + 1)−1 (1/2)l−1 P +(( d +2 −1, d0 +2 −1)) +l +(1) . +(2.31) +(ii) For the symmetric difference metrics (1.13), we have +θ∆(ξ, x1, x2) = κ(d, d0) +� +l⩾1 l−2MlAl(ξ) [ 1 − φl(x1, x2) ] , +(2.32) +where +Al(ξ) = +� π +0 +� +P +( d +2 , d0 +2 ) +l−1 +(cos r) +�2 +(sin 1 +2r)2d(cos 1 +2r)2d0 dξ(r). +(2.33) +The series (2.30) and (2.32) converge absolutely and uniformly. +The expansions (2.30) and (2.32) have been established in [23, Lemma 4.1] and +[22, Theorema 4.1(ii)], correspondingly. +2.3 Proof of Lemma 1.1. Let us consider the embedding (2.16) for l = 1. From the +formula (2.26) we find +m1 = d(d + d0 + 2) +2d0 += (n + 1)(d + 2) +2 +− 1, +d = nd0, +(2.34) +and for x1, x2 ∈ Q, we have +∥Π1(x1) − Π1(x2)∥2 = 2 − 2(Π1(x1), Π1(x2)) = 2(1 − φ1(cos θ(x1, x2)), +(2.35) +where ∥ · ∥ and (·, ·) are the Euclidean norm and inner product in Rm1. + +SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE +13 +On the other hand, from Rodrigues’ formula (1.38) we obtain +P +( d +2 −1, d0 +2 −1) +1 +(t) = ((d + d0)t + d − d0)/4, +P +( d +2 −1, d0 +2 −1) +1 +(1) = d/2, and +1 − t +2 += +d +d + d0 + +1 − P +( d +2 −1, d0 +2 −1) +1 +(t) +P +( d +2 −1, d0 +2 −1) +1 +(1) + + . +Therefore, +1 − cos θ(x1, x2) +2 += +d +d + d0 +� +1 − φ1(cos θ(x1, x2)) +� +. +(2.36) +Comparing (1.23), (2.35) and (2.36), we complete the proof. +□ +2.3 Proof of Theorem 1.2. Since zonal spherical functions are mutually orthogonal, +we conclude from the expansions (2.30) and (2.32) that the equality (1.27) is equiv- +alent to the formulas +γ(Q) l−2 B(d/2, d0/2)−1 Al(ξ♮) = Cl/2 , +l ⩾ 1 . +(2.37) +The integral (2.33) with the special measure dξ♮(r) = sin r dr takes the form +Al(ξ♮) = +� π +0 +� +P +( d +2 , d0 +2 ) +l−1 +(cos r) +�2 +(sin 1 +2r)2d(cos 1 +2r)2d0 sin r dr += 2−d−d0 +� 1 +−1 +� +P (d/2,d0/2) +l−1 +(t) +�2 +(1 − t)d (1 + t)d0 dt . +(2.38) +Hence, the formulas (2.37) can be written as follows +� 1 +−1 +� +P (d/2,d0/2) +l−1 +(t) +�2 +(1 − t)d (1 + t)d0 dt += 2d+d0+1 (1/2)l−1 +((l − 1)!)2 +B(d + 1, d0 + 1) T ∗, +(2.39) +where +T ∗ = +(l!)2 B(d/2, d0/2) Cl +4 (1/2)l−1 B(d + 1, d0 + 1) γ(Q) . +(2.40) +On the other hand, using (1.37) and (??), we find +Cl = (l!)−1 (1/2)l−1 +Γ(d/2 + 1/2) Γ(l + d/2) Γ(l + d0/2) +Γ(l + 1/2 + d/2 + d0/2) Γ(d/2) +. +(2.41) +Substituting (2.41) and (1.28) into (2.40), we obtain +T ∗ =π−1/2 (d + d0)−1 +Γ(d + d0 + 2) +Γ(d + 1) Γ(d0 + 1) × +× Γ(d/2 + 1/2) Γ(l + d/2) Γ(d0/2 + 1/2) Γ(l + d0/2) +Γ(d/2 + d0/2) Γ(l + d/2 + d0/2 + 1/2) +. +(2.42) +Applying the duplication formula for gamma function +Γ(2z) = π−1/2 22z−1 Γ(z) Γ(z + 1/2) +(2.43) + +14 +M.M. SKRIGANOV +to the first co-factor in (2.42), we find +π−1/2 (d + d0)−1 +Γ(d + d0 + 2) +Γ(d + 1) Γ(d0 + 1) += +Γ(d/2 + d0/2) Γ(d/2 + d0/2 + 3/2) +Γ(d/2 + 1/2) Γ(d0/2 + 1) Γ(d0/2 + 1/2) Γ(d0/2 + 1) , +(2.44) +where the relation Γ(z + 1) = zΓ(z) with z = d/2 + d0/2 has been also used. +Substituting (2.44) into (2.42), we find that T ∗ = Tl−1(d/2, d0/2). +□ +3. Proof of Lemma 1.2 +Lemma 1.2 follows from Lemma 3.1 and Lemma 3.2 given below. +Lemma 3.1. For all n ⩾ 0, Re α > −1/2 and Re β > −1/2, we have +� 1 +−1 +� +P (α,β) +n +(t) +�2 +(1 − t)2α(1 + t)2β dt += 22α+2β+1 +(n!)2 +B(2α + 1, 2β + 1) +Wn(α, β) +(2α + 2β + 2)2n +, +(3.1) +where +Wn(α, β) += +�2n +k=0 +(−1)n+k +k! +⟨2n⟩k ⟨α + n⟩k ⟨β + n⟩2n−k (2α + 1)2n−k (2β + 1)k (3.2) +is a polynomial of α and β. +Proof. Using Rodrigues’ formula (1.38), we can write +� 1 +−1 +� +P (α,β) +n +(t) +�2 +(1 − t)2α(1 + t)2β dt = +� +1 +2n n! +�2 +In(α, β) . +(3.3) +where +In(α, β) = +� 1 +−1 +� dn +dtn +� +(1 − t)n+α(1 + t)n+β� �2 +dt . +(3.4) +Integrating in (3.4) n times by part, we obtain +In(α, β) += (−1)n +� 1 +−1 +� +(1 − t)n+α(1 + t)n+β� d2n +dt2n +� +(1 − t)n+α(1 + t)n+β� +dt , +(3.5) +since all terms outside the integral vanish. By Leibniz’s rule, +d2n +dt2n +� +(1 − t)n+α (1 + t)n+β� += +�2n +k=0 +�2n +k +� dk +dtk (1 − t)n+α d2n−k +dt2n−k (1 + t)n+β , + +SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE +15 +where +�2n +k +� += ⟨2n⟩k/k! and +dk +dtk (1 − t)n+α = (−1)k ⟨α + n⟩k (1 − t)n−k+α , +d2n−k +dt2n−k (1 + t)n+β = ⟨β + n⟩2n−k (1 + t)−n+k+β . +Substituting these formulas into (3.5), we obtain +In(α, β) += 22α+2β+2n+1 �2n +k=0 +(−1)n+k +k! +⟨2n⟩k ⟨α + n⟩k ⟨β + n⟩2n−k I(k) +n (α, β) , (3.6) +where +I(k) +n (α, β) = B(2α + 2n − k + 1, 2β + k + 1). +(3.7) +Here we have used the following Euler’s integral +21−a−b +� 1 +−1 +(1 − t)a−1 (1 + t)b−1 dt = B(a, b) = Γ(a)Γ(b) +Γ(a + b) +(3.8) +with Re a > 0, Re b > 0. +The formula (3.7) can be written as follows +I(k) +n (α, β) = Γ(2α + 2n − k + 1) Γ(2β + k + 1) +Γ(2α + 2β + 2n + 2) +=Γ(2α + 2n − k + 1) +Γ(2α + 1) +Γ(2β + k + 1) +Γ(2β + 1) +Γ(2α + 1) Γ(2β + 1) +Γ(2α + 2β + 2) +Γ(2α + 2β + 2) +Γ(2α + 2β + 2n + 2) +=(2α + 1)2n−k (2β + 1)k +(2α + 2β + 2)2n +B(2α + 1, 2β + 1) . +(3.9) +Combining the formulas (3.9), (3.6) and (3.3), we obtain (3.1). +□ +The next Lemma 3.2 is more specific, it relies on Watson’s theorem for general- +ized hypergeometric series, see [2,24]. We consider the series of the form +3F2(a, b, c; d, e; z) = +� +k⩾0 +(a)k (b)k (c)k +(d)k (e)k k! z , +(3.10) +where neither d nor e are negative integers. The series absolutely converges for +|z| ⩽ 1, if Re(d + e) > Re(a + b + c). The series (3.10) terminates, if one of the +numbers a, b, c is a negative integer. +Watson’s theorem.We have +3F2(a,b, c; (a + b + 1)/2, 2c; 1) += +Γ(1/2) Γ(c + 1/2) Γ((a + b + 1)/2) Γ(c − (a + b − 1)/2) +Γ((a + 1)/2) Γ((b + 1)/2) Γ(c − (a − 1)/2) Γ(c − (b − 1)/2) . +(3.11) +provided that +Re (2c − a − b + 1) > 0. +(3.12) +The condition (3.12) ensures the convergence of hypergeometric series in (3.11). +Furthermore, this condition is necessary for the truth of equality (3.11) even in +the case of terminated series. The proof of Watson’s theorem can be found in [2, +Therem 3.5.5], [24, p.54, Eq.(2.3.3.13)]. + +16 +M.M. SKRIGANOV +Lemma 3.2. For all n ⩾ 0, α ∈ C and β ∈ C, the polynomial (3.2) is equal to +Wn(α, β) =22n (α + 1)n (β + 1)n (α + β + 1)n +=22n Γ(α + 1 + n) Γ(β + 1 + n) Γ(α + β + 1 + n) +Γ(α + 1) Γ(β + 1) Γ(α + β + 1) +. +(3.13) +In particular, +Wn(α, β) +(2α + 2β + 2)2n += (α + 1)n (β + 1)n +(α + β + 3/2)n +. +(3.14) +Proof. Since Wn(α, β) is a polynomial, it suffers to check the equality (3.13) for α +and β in an open subset in C2. As such a subset we shall take the following region +O = { α, β : Re α < 0, Re β < 0, Im α > 0, Im β > 0 }. +(3.15) +For α and β in O, the co-factors in terms in (3.2) may be rearranged as follows: +⟨2n⟩k = (−1)k (−2n)k , +⟨α + n⟩k = (−1)k (−α − n)k , +⟨β + n⟩2n−k = (−1)k (−β − n)2n−k = (−β − n)2n +(β + 1 − n)k +, +(2α + 1)2n−k = (−1)k(2α + 1)2n +(−2α − 2n)k +, + + + + + + + + + + + + + +(3.16) +Here we have used the following elementary relation for the rising factorial powers +(a)m−k = +(−1)k (a)m +(1 − a − m)k +, +m ⩾ 0 , 0 ⩽ k ⩽ m . +(3.17) +Substituting (3.16) into (3.2), we find that +Wn(α, β) = (−1)n (2α + 1)2n (−β − n)2n Fn(α, β) += (−1)n Γ(2α + 1 + 2n) Γ(−β + n) +Γ(2α + 1) Γ(−β − n) +Fn(α, β) , +(3.18) +where +Fn(α, β) = +�2n +k=0 +(−2n)k (2β + 1)k (−α − n)k +(β + 1 − n)k (−2α − 2n)k k! +(3.19) +In view of the definition (3.10), we have +Fn(α, β) = 3F2 (−2n, 2β + 1, −α − 1; β + 1 − n, −2α − 2n; 1) . +(3.20) +The parameters in hypergeometric series (3.20) are identical with those in (3.11) +for a = −2n, b = 2β + 1, c = −α − n, and in this case, (a + b + 1)/2 = 2β + 1 + n, +2c = −2α − 2n. The condition (3.12) also holds for α and β in the region O, since +Re (2c − a − b + 1) = Re (−2α − 2β) > 0. Therefore, Watson’s theorem (3.11) can +be applied to obtain +Fn(α, β) = +Γ(1/2) Γ(−α − n − 1/2) Γ(β + 1 − n) Γ(−α − β) +Γ(−n + 1/2) Γ(β + 1) Γ(−α + 1/2) Γ(−α − β − n) . +(3.21) +Substituting the expression (3.21) into (3.18) , we may write +Wn(α, β) = c0 c1(α) c2(β) c3(α + β) , +(3.22) + +SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE +17 +where +c0 = (−1)n Γ(1/2) +Γ(−n + 1/2) , +c1(α) = Γ(2α + 2n + 1) Γ(−α − n + 1/2) +Γ(2α + 1) Γ(−α + 1/2) +, +c2(β) = Γ(β + 1 − n) Γ(−β + n) +Γ(β + 1) Γ(−β − n) +, +c3(α + β) = +Γ(−α − β) +Γ(−α − β − n) . + + + + + + + + + + + + + + + + + + + + + + + + + +(3.23) +Using the duplication formula (2.43) and reflection formulas, see [2, Sec. 1.2], +Γ(1 − z)Γ(z) = +π +sin πz , +Γ(1/2 − z)Γ(1/2 + z) = +π +cos πz , +(3.24) +we may rearrange the expressions in (3.23) as follows. For c0, we have +c0 = +(−1)n Γ(1/2)2 +Γ(−n + 1/2) Γ(n + 1/2) +Γ(n + 1/2) +Γ(1/2) += (1/2)n , +since Γ(1/2) = √π. For c1(α) and c2(β), we have +c1(α) =22n Γ(α + n + 1) Γ(α + n + 1/2) Γ(−α − n + 1/2) +Γ(α + 1) Γ(α + 1/2) Γ(−α + 1/2) +=22n cos πα Γ(α + n + 1) +cos π(α + n) Γ(α + 1) = 22n (−1)n (α + 1)n +and +c2(β) = Γ(β + 1 − n) Γ(−β + n) +Γ(β + 1) Γ(−β − n) += sin π(β + n) Γ(β + 1 + n) +sin π(β − n) Γ(β + 1) += (β + 1)n . +Finally, +c3(α + β) = sin π(α + β) Γ(α + β + 1 + n) +sin π(α + β + n) Γ(α + β + 1) = (−1)n (α + β + 1)n . +Substituting these expressions into (3.22), we obtain (3.13). +It follows from (2.31) and the duplication formula (2.43) that +(2α + 2β + 2)2n = 22n (α + β + 1)n (α + β + 3/2)n . +(3.25) +Using (3.13) together with (3.25), we obtain (3.14). +□ +Now it suffers to substitute (3.14) into (3.1) to obtain the formulas (1.42). The +proof of Lemma 1.2 is complete. +References +[1] J. R. Alexander, J. Beck, W. W. L. Chen, Geometric discrepancy theory and uniform dis- +tributions, in Handbook of Discrete and Computational Geometry (J. E. Goodman and +J. O’Rourke eds.), Chapter 10, pages 185–207, CRC Press LLC, Boca Raton, FL, 1997. +[2] G. E. Andrews, R. Askey, R. Roy, Special functions, Cambridge Univ. Press, 2000. +[3] A. Barg, Stolarsky’s invariance principle for finite metric spaces, Mathematika, 67(1), +(2021), 158–186. +[4] A. Barg, M.M. Skriganov, Bounds for discrepancies in the Hemming space, J. of Complexity, +65, (2021), 101552. +[5] J. Beck, Sums of distances between points on a sphere: An application of the theory of +irregularities of distributions to distance geometry, Mathematika, 31, (1984), 33–41. + +18 +M.M. SKRIGANOV +[6] J. Beck, W. W. L. Chen, Irregularities of Distribution, Cambridge Tracts in Math., vol. 89, +Cambridge Univ. Press, 1987. +[7] D. Bilyk, M. Lacey, One bit sensing, discrepancy, and Stolarsky principle, Sbornik Math., +208(6), (2017), 744–763. +[8] D. Bilyk, F. Dai, R. Matzke, +Stolarsky principle and energy optimization on the sphere, +Constr. Approx., 48(1), (2018), 31–60. +[9] D. Bilyk, R. Matzke, O. Vlasiuk, Positive definiteness and the Stolarsky principle, J. of Math. +Analysis, 513(1), (2022), 126220. +[10] J. S. Brauchart, J. Dick, A simple proof of Stolarsky’s invariance principle, Proc. Amer. +Math. Soc., 141, (2013), 2085–2096. +[11] J. S. Brauchart, P. J. Grabner, Weighted L2-norms of Gegenbauer polynomials, Aequat. +Math., 96, (2022), 741–762. +[12] P. Cartier, Introduction `a l’´etude des mouvements browniens `a plusieurs param`etres, +S´eminaire de Probabilit´es V, Lectures Notes in Math., 191, Springer—Verlag, 1971. +[13] M. M. Deza, M. Laurent, Geometry of cuts and metrics, Springer, 1997. +[14] F. Dai, Y. Xu, Approximation theory and harmonic analysis on spheres and balls, Springer, +2013. +[15] R. Gangolli, Positive definite kernels on homogeneous spaces and certain stochastic processes +related to L´evy’s Brownian motion of several parameters, Ann. Inst. Henri Poincar´e, vol. III, +No. 2, (1967), 121–325. +[16] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, +1978. +[17] S. Helgason, Groups and geometric analysis. Integral geometry, invariant differential opera- +tors, and spherical functions, Academic Press, 1984. +[18] V. I. Levenshtein, Universal bounds for codes and designs, in Handbook of Coding Theory +(V. S. Pless and W. C. Huffman eds.), Chapter 6, pages 499–648, Elsevier, 1998. +[19] C. M¨uller. Spherical Harmonics, Lecture Notes in Math., 17. Springer, 1966. +[20] A. V. Shchepetilov, Calculus and Mechanics on two-point homogeneous spaces, Springer, +2006. +[21] M. M. Skriganov, Point distributions in compact metric spaces, Mathematika, 63, (2017), +1152–1171. +[22] M. M. Skriganov, Point distributions in two-point homogeneous spaces, Mathematika, 65, +(2019), 557–587. +[23] M. M. Skriganov, Stolarsky’s invariance principle for projective spaces, J. of Complexity, 56, +(2020), 101428. +[24] L. J. Slater, Generalized hypergeometric functions, Cambridge Univ. Press, 1966. +[25] K. B. Stolarsky, Sums of distances between points on a sphere, II, Proc. Amer. Math. Soc., +41, (1973), 575–582. +[26] G. Szeg˝o , Orthogonal polynomials, Amer. Math. Soc., 1950. +[27] S. S. Tai, Minimum embeddings of compact symmetric spaces of rank one, J.Differential +Geometry, 2, (1968), 55–66. +[28] N. Ja. Vilenkin, A. U. Klimyk, Representation of Lie groups and special functions, vols. 1–3, +Kluwer Acad. Pub., Dordrecht, 1991–1992. +[29] J. A. Wolf, Spaces of constant curvature, Univ. Califormia, Berkley, 1972. +[30] J. A. Wolf, Harmonic analysis on commutative spaces, Math. Surveys and Monographs, +vol. 142, Amer. Math. Soc., 2007. +St. Petersburg Department of the Steklov Mathematical Institute of the Russian +Academy of Sciences, 27, Fontanka, St.Petersburg 191023, Russia +Email address: maksim88138813@mail.ru + diff --git a/AtAyT4oBgHgl3EQfRvdA/content/tmp_files/load_file.txt b/AtAyT4oBgHgl3EQfRvdA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe0b63c8e75ef9398861b6bb2633eae910344098 --- /dev/null +++ b/AtAyT4oBgHgl3EQfRvdA/content/tmp_files/load_file.txt @@ -0,0 +1,785 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf,len=784 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='00071v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='CO] 30 Dec 2022 SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' SKRIGANOV Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Stolarsky’s invariance principle, known for point distributions on the Euclidean spheres Sd [25], has been extended to the real RP n, complex CP n, and quaternionic HP n projective spaces and the octonionic OP 2 pro- jective plane in our previous paper [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Geometric features of such spaces as well as their models in terms of Jordan algebras have been used very essen- tially in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In the present paper, we give a new pure analytic proof of Stolarsky’s invariance principle relying on the theory of spherical functions on compact symmetric Riemannian manifolds of rank one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Introduction and main results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1 Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In 1973 Kenneth B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Stolarsky [25] established the following remarkable formula for point distributions on the Euclidean spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Let Sd = {x ∈ Rd+1 : ∥x∥ = 1} be the standard d-dimensional unit sphere in Rd+1 with the geodesic (great circle) metric θ and the Lebesgue measure µ normalized by µ(Sd) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' We write C(y, t) = {x ∈ Sd : (x, y) > t} for the spherical cap of height t ∈ [−1, 1] centered at y ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Here we write (·, ·) and ∥ · ∥ for the inner product and the Euclidean norm in Rd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For an N-point subset DN ⊂ Sd, the spherical cap quadratic discrepancy is defined by λcap[DN] = � 1 −1 � Sd ( #{|C(y, t) ∩ DN} − Nµ(C(y, t)) )2 dµ(y) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1) We introduce the sum of pairwise Euclidean distances between points of DN τ[DN] = 1 2 � x1,x2∈DN ∥x1 − x2∥ = � x1,x2∈DN sin 1 2θ(x1, x2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2) and write ⟨τ⟩ for the average value of the Euclidean distance on Sd, ⟨τ⟩ = 1 2 �� Sd×Sd ∥y1 − y2∥ dµ(y1) dµ(y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3) The study of the quantities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2) falls within the subjects of discrepancy theory and geometry of distances, see [1,6] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' It turns out that the quantities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2) are not independent and are intimately related by the following remarkable identity γ(Sd)λcap[DN] + τ[DN] = ⟨τ⟩N 2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4) 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 11K38, 22F30, 52C99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Geometry of distances, discrepancies, spherical functions, projective spaces, Jacobi polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' SKRIGANOV for an arbitrary N-point subset DN ⊂ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Here γ(Sd) is a positive constant inde- pendent of DN, γ(Sd) = d √π Γ(d/2) 2 Γ((d + 1)/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='5) The identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4) is known in the literature as Stolarsky’s invariance principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Its original proof given in [25] has been simplified in [7,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' simplified in [7,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In our previous paper [23] Stolarsky’s invariance principle (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4) has been ex- tended to the real RP n, the complex CP n, the quaternionic HP n projective spaces, and the octonionic OP 2 projective plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Geometric features of such spaces as well as their models in terms of Jordan algebras have been used very essentially in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The aim of the present paper is to give an alternative pure analytic proof relying on the theory of spherical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2 Discrepancies and metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' L1-invariance principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Let us consider Sto- larsky’s invariance principle in a broader context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Let M be a compact metric measure space with a fixed metric θ and a finite Borel measure µ, normalized, for convenience, by diam(M, θ) = π, µ(M) = 1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='6) where diam(E, ρ) = sup{ρ(x1, x2) : x1, x2 ∈ E} denotes the diameter of a subset E ⊆ M with respect to a metric ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' We write B(y, r) = {x ∈ M : θ(x, y) < r} for the ball of radius r ∈ I centered at y ∈ M and of volume v(y, r) = µ(B(y, r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Here I = {r = θ(x1, x2) : x1, x2 ∈ M} denotes the set of all possible radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' If the space M is connected, we have I = [ 0, π ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' We consider distance-invariant metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Recall that a metric space M is called distance-invariant, if the volume of any ball v(r) = v(y, r) is independent of y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The typical examples of distance-invariant spaces are homogeneous spaces M = G/H with G-invariant metrics θ and measures µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For an N-point subset DN ⊂ M, the ball quadratic discrepancy is defined by λ[ξ, DN] = � I � M ( #{B(y, r) ∩ DN} − Nv(r)) )2 dµ(y) dξ(r), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='7) where ξ is a finite measure on the set of radii I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Notice that for Sd spherical caps and balls are related by C(y, t) = B(y, r), t = cos r, and the discrepancies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='7) are related by λcap[DN] = λ[ξ♮, DN], where dξ♮(r) = sin r dr, r ∈ I = [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The ball quadratic discrepancy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='7) can be written in the form λ[ξ, DN] = � x1,x2∈DN λ(ξ, x1, x2) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='8) with the kernel λ(ξ, x1, x2) = � I � M Λ(B(y, r), x1) Λ(B(y, r), x2) dµ(y) dξ(r) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='9) where Λ(B(y, r), x) = χ(B(y, r), x) − v(r), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='10) and χ(E, ·) denotes the characteristic function of a subset E ⊆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For an arbitrary metric ρ on M we introduce the sum of pairwise distances ρ[DN] = � x1,x2∈DN ρ(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='11) SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE 3 and the average value ⟨ρ⟩ = � M×M ρ(y1, y2) dµ(y1) dµ(y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='12) We introduce the following symmetric difference metrics on the space M θ∆(ξ, y1, y2) = 1 2 � I µ(B(y1, r)∆B(y2, r)) dξ(r) = 1 2 � I � M χ(B(y1, r)∆B(y2, r), y) dµ(y) dξ(r), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='13) where B(y1, r)∆B(y2, r) = B(y1, r) ∪ B(y2, r) \\ B(y1, r) ∩ B(y2, r) is the symmetric difference of the balls B(y1, r) and B(y2, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In line with the definitions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='11) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='12), we put θ∆[ξ, DN] = � x1,x2∈DN θ∆(ξ, x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' and ⟨θ∆(ξ)⟩ = � M×M θ∆(ξ, y1, y2) dµ(y1) dµ(y2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' A direct calculation leads to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Let a compact metric measure space M be distance-invariant, then we have λ(ξ, y1, y2) + θ∆(ξ, y1, y2) = ⟨θ∆(ξ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='14) In particular, we have the following invariance principle λ[ ξ, DN ] + θ∆[ ξ, DN ] = ⟨θ∆(ξ)⟩ N 2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='15) for an arbitrary N-point subset DN ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In view of the symmetry of the metric θ, we have χ(B(x, r), y) = χ(B(y, r), x) = χ0(r − θ(y, x)) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='16) where χ0(·) is the characteristic function of the half-axis (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Therefore χ(B(y1, r)∆B(y2, r), y) = χ(B(y1, r), y) + χ(B(y2, r), y) −2χ(B(y1, r) ∩ B(y2, r), y) , and � M χ(B(x, r), y)dµ(x) = � M χ(B(x, r), y)dµ(y) = v(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Using these relations, we obtain λ(ξ, x1, x2) = � I � µ(B(x1, r) ∩ B(x2, r)) − v(r)2� dξ(r) , θ∆(ξ, y1, y2) = � I � v(r) − µ(B(y1, r) ∩ B(y2, r)) � dξ(r) , ⟨θ∆(ξ)⟩ = � I � v(r) − v(r)2� dξ(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' \uf8fc \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fe (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='17) These relations imply (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' □ 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' SKRIGANOV In the case of spheres Sd, relations of the type (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='14) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='15) were given in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Their extensions to more general metric measure spaces were given in [21, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1], [22, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='30)] and [23, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Notice that χ(B(y1, r)∆B(y2, r), y) = |χ(B(y1, r), y) − χ(B(y2, r), y)| , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='18) and hence θ∆(ξ, y1, y2) = 1 2 � I � M |χ(B(y1, r), y) − χ(B(y2, r), y)| dµ(y) dξ(r) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='19) is an L1-metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Recall that a metric space M with a metric ρ is called isometrically Lq-embeddable (q = 1 or 2), if there exists a mapping ϕ : M ∋ x → ϕ(x) ∈ Lq, such that ρ(x1, x2) = ∥ϕ(x1)−ϕ(x2)∥Lq for all x1, x2 ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Notice that the L2-embeddability is stronger and implies the L1-embeddability, see [13, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' It follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='19) that the space M with the symmetric difference metrics θ∆(ξ) is isometrically L1-embeddable by the formula M ∋ x → χ(B(x, r), y) ∈ L1(M × I) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='20) The identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='15) can be called the L1-invariance principle, while Stolarsky’s invariance principle (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4) should be called the L2-invariance principle, because it involves the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The identities of such a type including correspond- ingly L1 and L2 metrics could be also called weak and strong invariance principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3 L2-invariance principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Recall the definition and necessary facts on two- point homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Let G = G(M) be the group of isometries of a metric space M with a metric θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' θ(gx1, gx2) = θ(x1, x2) for all x1, x2 ∈ M and g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The space M is called two-point homogeneous, if for any two pairs of points x1, x2 and y1, y2 with θ(x1, x2) = θ(y1, y2) there exists an isometry g ∈ G, such that y1 = gx1, y2 = gx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In this case, the group G is obviously transitive on M and M = G/H is a homogeneous space, where the subgroup K ⊂ G is the stabilizer of a point x0 ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Furthermore, the homogeneous space M is symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' for any two points y1, y2 ∈ M there exists an isometry g ∈ G, such that gy1 = y2, gy2 = y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' There is a very large number of two-point homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For example, all Hamming spaces, known in the coding theory, are two-point homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' We will consider compact connected two-point homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The assumption that the space is connected turns out to be a strong restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' All compact connected two-point homogeneous spaces Q = G/H are known, and by Wang’s classifications they are the following, see [16,17,20,29,30]: (i) The d-dimensional Euclidean spheres Sd = SO(d + 1)/SO(d) × {1}, d ⩾ 2, and S1 = O(2)/O(1) × {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (ii) The real projective spaces RP n = O(n + 1)/O(n) × O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (iii) The complex projective spaces CP n = U(n + 1)/U(n) × U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (iv) The quaternionic projective spaces HP n = Sp(n + 1)/Sp(n) × Sp(1), (v) The octonionic projective plane OP 2 = F4/ Spin(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Here we use the standard notation from the theory of Lie groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' in particular, F4 is one of the exceptional Lie groups in Cartan’s classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' All these spaces are Riemannian symmetric manifolds of rank one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Geomet- rically, this means that all geodesic sub-manifolds in Q are one-dimensional and SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE 5 coincide with geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' From the spectral stand point, this also means that all operators on Q commuting with the action of the group G are functions of the Laplace–Beltrami operator on Q, see [16,17,29,30] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The spaces FP n as Riemannian manifolds have dimensions d, d = dimR FP n = nd0, d0 = dimR F, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='21) where d0 = 1, 2, 4, 8 for F = R, C, H, O, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For the spheres Sd we put d0 = d by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Projective spaces of dimension d0 (n = 1) are homeomorphic to the spheres Sd0: RP 1 ≈ S1, CP 1 ≈ S2, HP 1 ≈ S4, OP 1 ≈ S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' We can conveniently agree that d > d0 (n ⩾ 2) for projective spaces, while the equality d = d0 holds only for spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Under this convention, the dimen- sions d = nd0 and d0 define uniquely (up to homeomorphism) the corresponding homogeneous space which we denote by Q = Q(d, d0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' We consider Q(d, d0) as a metric measure space with the metric θ and measure µ proportional to the invariant Riemannian distance and measure on Q(d, d0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The coefficients of proportionality are defined to satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In what follows we always assume that n = 2 if F = O, since projective spaces OP n do not exist for n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Any space Q(d, d0) is distance-invariant and the volume of balls in the space is given by v(r) = κ � r 0 (sin 1 2u)d−1(cos 1 2u)d0−1 du r ∈ [ 0, π ] = κ 21−d/2−d0/2 � 1 cos r (1 − z) d 2 −1 (1 + z) d0 2 −1 dz, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='22) where κ = κ(d, d0) = B(d/2, d0/2)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' B(a, b) = Γ(a)Γ(b)/Γ(a + b) and Γ(a) are the beta and gamma functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Equivalent forms of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='22) can be found in the literature, see, for example, [15, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 177–178], [17, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 165–168], [18, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 508–510].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For even d0, the integrals (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='22) can be calculated explicitly that gives convenient expressions for v(r) in the case of CP n, HP n and OP 2, see, for example, [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The chordal metric on the spaces Q(d, d0) is defined by τ(x1, x2) = sin 1 2θ(x1, x2) = � 1 − cos(x1, x2) 2 , x1, x2 ∈ Q(d, d0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='23) The formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='23) defines a metric because the function ϕ(θ) = sin θ/2, 0 ⩽ θ ⩽ π, is concave, increasing, and ϕ(0) = 0, that implies the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For the sphere Sd we have τ(x1, x2) = sin 1 2θ(x1, x2) = 1 2 ∥x1 − x2∥, x1, x2 ∈ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='24) Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The space Q(d, d0), d = nd0, can be embedded into the unit sphere Π : Q(d, d0) ∋ x → Π(x) ∈ Sm−1 ⊂ Rm, m = 1 2(n + 1)(d + 2) − 1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='25) such that τ(x1, x2) = � d 2(d + d0) �1/2 ∥Π(x1) − Π(x2)∥, x1, x2 ∈ Q(d, d0), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='26) where ∥ · ∥ is the Euclidean norm in Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' SKRIGANOV Hence, the metric τ(x1, x2) is proportional to the Euclidean length of a segment joining the corresponding points Π(x1) and Π(x2) on the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The chordal metric τ on the complex projective space CP n is known as the Fubini–Study metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1 will be proved in Section 2, and the embedding (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='25) will be de- scribed explicitly in terms of spherical functions on the space Q(d, d0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Note that the embedding (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='25) can be described in different ways, see, for example, [23,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The following general result has been established in [23, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For each space Q = Q(d, d0), we have the equality τ(x1, x2) = γ(Q) θ∆(ξ♮, x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='27) where dξ♮(r) = sin r dr, r ∈ [0, π] and γ(Q) = √π 4 (d + d0) Γ(d0/2) Γ((d0 + 1)/2) = d + d0 2d0 γ(Sd0) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='28) where γ(Sd0) is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Comparing Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1 with Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1, we arrive to the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' We have the following L2-invariance principle γ(Q) λ[ξ♮, DN] + τ[DN] = ⟨τ⟩N 2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='29) for an arbitrary N-point subset DN ⊂ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The constant γ(Q) has the following geometric interpretation γ(Q) = ⟨τ⟩ ⟨θ∆(ξ♮)⟩ = diam(Q, τ) diam(Q, θ∆(ξ♮)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='30) Indeed, it suffices to calculate the average values (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='12) of both metrics in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='27) to obtain the first equality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Similarly, writing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='27) for any pair of antipodal points x1, x2, θ(x1, x2) = π, we obtain the second equality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The average value ⟨τ⟩ of the chordal metric τ can be easily calculated with the help of the formulas (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='12) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='22): ⟨τ⟩ = B(d/2, d0/2)−1 B((d + 1)/2, d0/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='31) In the case of spheres Sd, the identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='29) coincides with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='29) can be thought of as an extension of Stolarsky’s invariance principle to all projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Applications of L1- and L2-invariance principles and similar identities to the discrepancy theory, geometry of distances, and information theory have been given in many papers, see, for example, [1,3–10,21–23,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' It is worth noting that the equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='27) is of interest by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Since the integrand in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='19) takes the values 0 and 1 only, we can write θ∆(ξ, y1, y2) = � θ∆ p (ξ, y1, y2) �p , p > 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='32) where θ∆ p (ξ, y1, y2) = �1 2 � π 0 � Q |χ(B(y1, r), y) − χ(B(y2, r), y)|p) dµ(y) dξ(r) �1/p , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='33) is an Lp-metric for p ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE 7 Comparing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='27) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='32), we see that the chordal metric τ is proportional to the p-th power of the metric θ∆ p (ξ♮) for all p ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' This is a nontrivial fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For example, we have for p = 2 τ(x1, x2) = γ(Q) 2 � π 0 � Q |χ(B(x1, r), y) − χ(B(x2, r), y)|2 dµ(y) dξ♮(r), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='34) and the equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='34) implies the existence of Gaussian random fields on the spaces Q(d, d0), see [12,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' However, a detailed considerations of these questions is beyond the scope of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In the context of our discussion, the following open problems are of interest: Do there exist measures ξ on the set of radii for spaces Q(d, d0) (for spheres Sd, say) other than the measure ξ♮ such that the corresponding symmetric difference metrics θ∆(ξ) are the L2-metrics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Do there exist compact measure metric spaces other than spheres Sd and pro- jective spaces FP n for which the L2-invariance principle is also true?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In the present paper we use the theory of spherical functions to prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='27) is equivalent to the following series of formulas for Jacobi polynomials � 1 −1 � P (d/2,d0/2) l−1 (t) �2 (1 − t)d (1 + t)d0 dt = 2d+d0+1 (1/2)l−1 ((l − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' )2 B(d + 1, d0 + 1) Tl−1(d/2, d0/2) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='35) for all l ⩾ 1, where Tl−1(d/2, d0/2) == Γ(d/2 + l) Γ(d0/2 + l) Γ(d/2 + d0/2 + 3/2)) Γ(d/2 + 1) Γ(d0/2 + 1) Γ(d/2 + d0/2 + 1/2 + l) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='36) Here P (α,β) n (t), t ∈ [−1, 1], α > −1, β > −1, are the standard Jacobi polynomials of degree n normalized by P (α,β) n (1) = �α + n n � = Γ(α + n + 1) Γ(n + 1)Γ(α + 1) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='37) and P (α,β) n can be given by Rodrigues’ formula P (α,β) n (t) = (−1)n 2nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1 − t)−α(1 + t)−β dn dtn � (1 − t)n+α(1 + t)n+β� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='38) Notice that |P (α,β) n (t)| ⩽ P (α,β) n (1) for t ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Recall also that P (α,β) n are orthogonal polynomials with the following orthogonality relations π � 0 P (α,β) l (cos u)P (α,β) l′ (cos u)(sin 1 2u)2α+1(cos 1 2u)2β+1 du = 2−α−β−1 1 � −1 P (α,β) l (z)P (α,β) l′ (z)(1 − z)α(1 + z)β dz = M −1 l δll′, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='39) 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' SKRIGANOV where M0 = B(α + 1, β + 1)−1 and Ml = Ml(α, β) = (2l + α + β + 1)Γ(l + 1)Γ(l + α + β + 1) Γ(l + α + 1)Γ(l + β + 1), l ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='40) All necessary facts about Jacobi polynomials can be found in [2, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' We also use the notation (a)0 = 1, (a)k = a(a + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (a + k − 1) = Γ(a + k) Γ(a) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='41) for the rising factorial powers (Pochhammer’s symbol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2 reduces the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1 to the proof of the formulas (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Perhaps such formulas are known but I could not find them in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For spheres Jacobi polynomials P (d/2,d/2) n with equal parameters coincide (up to con- stant factors) with Gegenbauer polynomials, and in this case very general formulas for weighted L2-norms of Gegenbauer polynomials are given in the paper [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In the present paper we will prove the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For all n ⩾ 0, Re α > −1/2 and Re β > −1/2, we have � 1 −1 � P (α,β) n (t) �2 (1 − t)2α(1 + t)2β dt = 22α+2β+1 (1/2)n (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' )2 B(2α + 1, 2β + 1) Tn(α, β), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='42) where Tn(α, β) = (α + 1)n (β + 1)n (α + β + 3/2)n = Γ(α + n + 1) Γ(β + n + 1) Γ(α + β + 3/2)) Γ(α + 1) Γ(β + 1) Γ(α + β + 3/2 + n) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='43) is a rational function of α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The integral (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='42) converges for Re α > −1/2 and Re β > −1/2, and represents in this region a holomorphic function of two complex variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='42) defines an analytic continuation of the integral (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='42) to α ∈ C and β ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For α = d/2, β = d0/2 and n = l − 1 the equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='42) coincides with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' This proves Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2 will be proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The crucial point in the proof is Watson’s theorem on the value of hypergeometric series 3F2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Spherical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Proofs of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Invariant kernels and spherical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The general theory of spherical functions on homogeneous spaces can be found in [16,17,28,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The homogeneous spaces Q(d, d0) of interest to us belong to the class of so-called commutative spaces or symmetric Gelfand pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In this case the theory becomes significantly simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For Euclidean spheres Sd this theory is well known, see, for example, [14, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' However, the theory of spherical functions on general spaces Q(d, d0) is probably not commonly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In this section we describe the basic facts about spherical functions on spaces Q(d, d0) in a form convenient for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE 9 Let us consider the quasi-regular representation U(g)f(x) = f(g−1x), f ∈ L2(Q), x ∈ Q, g ∈ G, and its decomposition into the orthogonal sum U(g) = � � l⩾0 Ul(g), L2(Q) = � � l⩾0 Vl , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1) of irreducible representations Ul(g) in mutually orthogonal subspaces Vl of dimen- sions ml < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Let A denote the algebra of Hilbert–Schmidt operators in L2(Q) commuting with the representation U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Each K ∈ A is an integral operator Kf(x) = � Q K(x, y) f(y) dµ(y), with the invariant kernel: K(gx1, gx2) = K(x1, x2), x1, x2 ∈ Q, g ∈ G, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2) which satisfies the condition ||K||2 HS = Tr KK∗ = � Q×Q |K(x, y)|2 dµ(x)dµ(y) = � Q |K(x, y)|2 dµ(x) < ∞, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3) where Tr denotes the trace of an operator, and the second integral is independent of y in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Since the space Q is two-point homogeneous, the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2) implies that the kernel K(x1, x2) depends only on the distance θ(x1, x2), and can be written as K(x1, x2) = K(θ(x1, x2)) = k(cos θ(x1, x2)), x1, x2 ∈ Q, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4) with function K(z), z ∈ [0, π] and k(z), z ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The cosine is presented here for convenience in further calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4) can be also written as K(x1, x2) = K(θ(x, x0)) = k(cos θ(x, x0)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='5) where x1 = g1x0, x2 = g2x0, x = g−1 2 g1x0, g1, g2 ∈ G and x0 ∈ Q is the fixed point of the subgroup H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Moreover, K(hx, x0) = K(x, x0), h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Therefore, invariant kernels can be thought of as functions on the double co-sets H \\ G/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' In terms of the function K(·) and k(·), the Hilbert-Schmidt norm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3) takes the form ||K||2 HS = � π 0 |K(u)|2 dv(u) =κ � π 0 |k(cos u)|2(sin 1 2u)d−1(cos 1 2u)d0−1 du =κ 21−d/2−d0/2 � 1 −1 |k(z)|2 (1 − z) d 2 −1 (1 + z) d0 2 −1 dz, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='6) where v(·) is the volume function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' We conclude from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4) that for K ∈ A its kernel K(x1, x2) = K(x2, x1), the value K(x, x) = k(1) is independent of x ∈ Q, and if an opera- tor K is self-adjoint, then its kernel is real-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' SKRIGANOV It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4) that the algebra A is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Indeed, (K1K2)(x1, x2) = � Q K1(x1, x)K2(x, x2)dµ(x) = � Q K2(x2, x)K1(x, x1)dµ(x) = (K2K1)(x2, x1) = (K2K1)(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Therefore, the decomposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1) is multiplicity-free, that is any two representa- tions Ul and Ul′, l ̸= l′, are non-equivalent, because otherwise the algebras A could not be commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Let Pl denote orthogonal projectors in L2(Q) onto the subspaces Vl in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1), P ∗ l = Pl , Pl Pl′ = δl,l′ Pl , � l⩾0 Pl = I , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='7) where δl,l′ is Kronecker’s symbol and I is the identity operator in L2(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' By Schur’s lemma, we have for K ∈ A Pl K Pl′ = δl,l′ cl(K) Pl, , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='8) where cl(K) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Calculating the trace of both sides of the equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='8), we find cl(K) = m−1 l Tr KPl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Therefore, we have the expansions K = � l,l′⩾0 Pl K Pl′ = � l⩾0 cl(K) Pl, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='9) with Parseval’s identity ||K||2 HS = � l⩾0 ml |cl(K)|2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='10) and for K1, K2 ∈ A, we have K1 K2 = � l⩾0 cl(K1) cl(K2) Pl, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='11) The equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='10) implies that the series (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='11) converges in the Hilbert-Schmidt norm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3), while the series (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='11) converges in the norm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3) for the subclass of nuclear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Since Vl are invariant subspaces, Pl ∈ A, their kernels Pl(·, ·) are symmetric and real-valued, and can be written as follows Pl(x1, x2) = pl(cos θ(x1, x2)) = �ml 1 ψl,j(x1) ψl,j(x2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='12) where {ψl,j(·)}ml 1 is an orthonormal and real-valued basis in Vl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Hence, subspace Vl and irreducible representations Ul in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1) can be thought of as defined over the field of reals, this means that all representations Ul in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1) are of the real type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='12), we obtain the formulas ||Pl||2 HS = ml, Tr Pl = pl(1) = ml > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='13) Furthermore, Pl(x, x) = pl(1) = �ml 1 ψl,j(x)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='14) is independent of x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Applying Cauchy-Schwartz inequality to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='12) and taking (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='14) into account, we obtain the bound |Pl(x1, x2)| = |pl(cos θ(x1, x2))| ⩽ pl(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='15) It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='13) that the mapping Πl : Q ∋ x → (m−1/2 l ψl,1(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' m−1/2 l ψl,ml(x)) ∈ Sml−1 ⊂ Rml (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='16) SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE 11 defines an embedding of the space Q into the unite sphere in Rml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' By definition the (zonal) spherical function are kernels of the operators Φl = m−1 l Pl: Φl(x1, x2) = φl(cos θ(x1, x2)) = pl(cos θ(x1, x2)) pl(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='17) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='17) we conclude that |φl(cos θ(x1, x2))| ⩽ φl(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Comparing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='13), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='17), we find the formulas for dimensions ml = ||Φl||−2 HS = � κ � π 0 |φl(cos u)|2(sin 1 2u)d−1(cos 1 2u)d0−1 du �−1 = � κ 21−d/2−d0/2 � 1 −1 |φl(z)|2 (1 − z) d 2 −1 (1 + z) d0 2 −1 dz �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='18) In terms of spherical functions the formulas (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='11) take the form k(cos θ(x1, x2)) = � l⩾0 cl(K) ml φl(cos θ(x1, x2)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='19) where cl(K) = Tr KΦl = � Q K(x1, x2) Φ(x1, x2) dµ(x1)dµ(x2) = κ � π 0 k(cos u) φl(cos u) (sin 1 2u)d−1(cos 1 2u)d0−1 du = κ 21−d/2−d0/2 � 1 −1 k(z) φl(z) (1 − z) d 2 −1 (1 + z) d0 2 −1 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='20) and � Q k1(cos θ(x1, y)) k2( cos θ(y, x2)) dµ(y) = � l⩾0 cl(K1) cl(K2) ml φl(cos θ(x1, x2)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='21) for K1, K2 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='11) with K1 = K and K2 = K∗ that ||K||2 HS = � l⩾0 ml |cl(K)|2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='22) The above facts are valid for all compact two-point homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Since spaces Q are also symmetric Riemannian manifolds of rank one, the invariant ker- nels pl(cos θ(x, x0)) are eigenfunctions of the radial part of the Laplace–Beltrami operator on Q (in the spherical coordinates centered at x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' This leads to the following explicit formula for spherical functions Φ(x1, x2) = φl(cos θ(x1, x2)) = P ( d 2 −1, d0 2 −1) l (cos θ(x1, x2)) P ( d 2 −1, d0 2 −1) l (1) , l ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='23) where P (α,β) n (t), t ∈ [−1, 1], are Jacobi polynomials (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' We refer to [15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 178], [17, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' V, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='5], [18, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 514–512, 543–544], [28, Chapters 2 and 17]: [30, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='21] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='37) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='40) we obtain P ( d 2 −1, d0 2 −1) n (1) = Γ(n + d/2) Γ(n + 1)Γ(1 + d/2) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='24) 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' SKRIGANOV and Ml = Ml(d/2 − 1, d0/2 − 1), where M0 = B(d/2, d0/2)−1 and Ml = (2l − 1 + (d + d0)/2)Γ(l + 1)Γ(l − 1 + (d + d0)/2) Γ(l + d/2)Γ(l + d0/2) , l ⩾ 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='25) Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='23), into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='18) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='24) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='25), we obtain the following explicit formulas for dimensions of irreducible representations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1) : m0 = 1 and ml =Ml B(d/2, d0/2) � P ( d 2 −1, d0 2 −1)(1) �2 =(2l − 1 + (d + d0)/2) Γ(l − 1 + (d + d0)/2)Γ(l + d/2)Γ(d0/2) Γ((d + d0)/2)Γ(l + d0/2)Γ(d/2)Γ(l + 1) l ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='26) The formulas (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='19) for invariant kernels coincide with Fourier-Jacobi expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Suppose that a function k(t), t ∈ [−1, 1], has the expansion k(cos r) = � l⩾0 Ml Cl(F) P ( d 2 −1, d0 2 −1) l (cos r), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='27) with Fourier-Jacobi coefficients Cl(k) = � π 0 k(cos u) P ( d 2 −1, d0 2 −1) l (cos u) (sin 1 2u)d−1 (cos 1 2u)d0−1 du, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='28) then the corresponding invariant kernel k(cos θ(x1, x2)) has the expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='19) with coefficients cl(k) = Cl(k) κ(d, d0) P ( d 2 −1, d0 2 −1) l (1) , l ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='29) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (i) For the chordal metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='23), we have τ(x1, x2) = 1 2 � l⩾1 Ml Cl [ 1 − φl(x1, x2) ] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='30) where Cl = B((d + 1)/2, l + d0/2) Γ(l + 1)−1 (1/2)l−1 P (( d 2 −1, d0 2 −1)) l (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='31) (ii) For the symmetric difference metrics (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='13), we have θ∆(ξ, x1, x2) = κ(d, d0) � l⩾1 l−2MlAl(ξ) [ 1 − φl(x1, x2) ] , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='32) where Al(ξ) = � π 0 � P ( d 2 , d0 2 ) l−1 (cos r) �2 (sin 1 2r)2d(cos 1 2r)2d0 dξ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='33) The series (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='30) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='32) converge absolutely and uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The expansions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='30) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='32) have been established in [23, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1] and [22, Theorema 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1(ii)], correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3 Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Let us consider the embedding (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='16) for l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' From the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='26) we find m1 = d(d + d0 + 2) 2d0 = (n + 1)(d + 2) 2 − 1, d = nd0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='34) and for x1, x2 ∈ Q, we have ∥Π1(x1) − Π1(x2)∥2 = 2 − 2(Π1(x1), Π1(x2)) = 2(1 − φ1(cos θ(x1, x2)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='35) where ∥ · ∥ and (·, ·) are the Euclidean norm and inner product in Rm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE 13 On the other hand, from Rodrigues’ formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='38) we obtain P ( d 2 −1, d0 2 −1) 1 (t) = ((d + d0)t + d − d0)/4, P ( d 2 −1, d0 2 −1) 1 (1) = d/2, and 1 − t 2 = d d + d0 \uf8ee \uf8f01 − P ( d 2 −1, d0 2 −1) 1 (t) P ( d 2 −1, d0 2 −1) 1 (1) \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Therefore, 1 − cos θ(x1, x2) 2 = d d + d0 � 1 − φ1(cos θ(x1, x2)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='36) Comparing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='23), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='35) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='36), we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Since zonal spherical functions are mutually orthogonal, we conclude from the expansions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='30) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='32) that the equality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='27) is equiv- alent to the formulas γ(Q) l−2 B(d/2, d0/2)−1 Al(ξ♮) = Cl/2 , l ⩾ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='37) The integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='33) with the special measure dξ♮(r) = sin r dr takes the form Al(ξ♮) = � π 0 � P ( d 2 , d0 2 ) l−1 (cos r) �2 (sin 1 2r)2d(cos 1 2r)2d0 sin r dr = 2−d−d0 � 1 −1 � P (d/2,d0/2) l−1 (t) �2 (1 − t)d (1 + t)d0 dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='38) Hence, the formulas (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='37) can be written as follows � 1 −1 � P (d/2,d0/2) l−1 (t) �2 (1 − t)d (1 + t)d0 dt = 2d+d0+1 (1/2)l−1 ((l − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' )2 B(d + 1, d0 + 1) T ∗, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='39) where T ∗ = (l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' )2 B(d/2, d0/2) Cl 4 (1/2)l−1 B(d + 1, d0 + 1) γ(Q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='40) On the other hand, using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='37) and (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' ), we find Cl = (l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' )−1 (1/2)l−1 Γ(d/2 + 1/2) Γ(l + d/2) Γ(l + d0/2) Γ(l + 1/2 + d/2 + d0/2) Γ(d/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='41) Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='41) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='28) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='40), we obtain T ∗ =π−1/2 (d + d0)−1 Γ(d + d0 + 2) Γ(d + 1) Γ(d0 + 1) × × Γ(d/2 + 1/2) Γ(l + d/2) Γ(d0/2 + 1/2) Γ(l + d0/2) Γ(d/2 + d0/2) Γ(l + d/2 + d0/2 + 1/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='42) Applying the duplication formula for gamma function Γ(2z) = π−1/2 22z−1 Γ(z) Γ(z + 1/2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='43) 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' SKRIGANOV to the first co-factor in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='42), we find π−1/2 (d + d0)−1 Γ(d + d0 + 2) Γ(d + 1) Γ(d0 + 1) = Γ(d/2 + d0/2) Γ(d/2 + d0/2 + 3/2) Γ(d/2 + 1/2) Γ(d0/2 + 1) Γ(d0/2 + 1/2) Γ(d0/2 + 1) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='44) where the relation Γ(z + 1) = zΓ(z) with z = d/2 + d0/2 has been also used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='44) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='42), we find that T ∗ = Tl−1(d/2, d0/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2 follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2 given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For all n ⩾ 0, Re α > −1/2 and Re β > −1/2, we have � 1 −1 � P (α,β) n (t) �2 (1 − t)2α(1 + t)2β dt = 22α+2β+1 (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' )2 B(2α + 1, 2β + 1) Wn(α, β) (2α + 2β + 2)2n , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1) where Wn(α, β) = �2n k=0 (−1)n+k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' ⟨2n⟩k ⟨α + n⟩k ⟨β + n⟩2n−k (2α + 1)2n−k (2β + 1)k (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2) is a polynomial of α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Using Rodrigues’ formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='38), we can write � 1 −1 � P (α,β) n (t) �2 (1 − t)2α(1 + t)2β dt = � 1 2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' �2 In(α, β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3) where In(α, β) = � 1 −1 � dn dtn � (1 − t)n+α(1 + t)n+β� �2 dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4) Integrating in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='4) n times by part, we obtain In(α, β) = (−1)n � 1 −1 � (1 − t)n+α(1 + t)n+β� d2n dt2n � (1 − t)n+α(1 + t)n+β� dt , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='5) since all terms outside the integral vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' By Leibniz’s rule, d2n dt2n � (1 − t)n+α (1 + t)n+β� = �2n k=0 �2n k � dk dtk (1 − t)n+α d2n−k dt2n−k (1 + t)n+β , SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE 15 where �2n k � = ⟨2n⟩k/k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' and dk dtk (1 − t)n+α = (−1)k ⟨α + n⟩k (1 − t)n−k+α , d2n−k dt2n−k (1 + t)n+β = ⟨β + n⟩2n−k (1 + t)−n+k+β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Substituting these formulas into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='5), we obtain In(α, β) = 22α+2β+2n+1 �2n k=0 (−1)n+k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' ⟨2n⟩k ⟨α + n⟩k ⟨β + n⟩2n−k I(k) n (α, β) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='6) where I(k) n (α, β) = B(2α + 2n − k + 1, 2β + k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='7) Here we have used the following Euler’s integral 21−a−b � 1 −1 (1 − t)a−1 (1 + t)b−1 dt = B(a, b) = Γ(a)Γ(b) Γ(a + b) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='8) with Re a > 0, Re b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='7) can be written as follows I(k) n (α, β) = Γ(2α + 2n − k + 1) Γ(2β + k + 1) Γ(2α + 2β + 2n + 2) =Γ(2α + 2n − k + 1) Γ(2α + 1) Γ(2β + k + 1) Γ(2β + 1) Γ(2α + 1) Γ(2β + 1) Γ(2α + 2β + 2) Γ(2α + 2β + 2) Γ(2α + 2β + 2n + 2) =(2α + 1)2n−k (2β + 1)k (2α + 2β + 2)2n B(2α + 1, 2β + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='9) Combining the formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='9), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' □ The next Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2 is more specific, it relies on Watson’s theorem for general- ized hypergeometric series, see [2,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' We consider the series of the form 3F2(a, b, c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' d, e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' z) = � k⩾0 (a)k (b)k (c)k (d)k (e)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' z , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='10) where neither d nor e are negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The series absolutely converges for |z| ⩽ 1, if Re(d + e) > Re(a + b + c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='10) terminates, if one of the numbers a, b, c is a negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Watson’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='We have 3F2(a,b, c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (a + b + 1)/2, 2c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 1) = Γ(1/2) Γ(c + 1/2) Γ((a + b + 1)/2) Γ(c − (a + b − 1)/2) Γ((a + 1)/2) Γ((b + 1)/2) Γ(c − (a − 1)/2) Γ(c − (b − 1)/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='11) provided that Re (2c − a − b + 1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='12) The condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='12) ensures the convergence of hypergeometric series in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Furthermore, this condition is necessary for the truth of equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='11) even in the case of terminated series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The proof of Watson’s theorem can be found in [2, Therem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='5], [24, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='54, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='13)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' SKRIGANOV Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For all n ⩾ 0, α ∈ C and β ∈ C, the polynomial (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2) is equal to Wn(α, β) =22n (α + 1)n (β + 1)n (α + β + 1)n =22n Γ(α + 1 + n) Γ(β + 1 + n) Γ(α + β + 1 + n) Γ(α + 1) Γ(β + 1) Γ(α + β + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='13) In particular, Wn(α, β) (2α + 2β + 2)2n = (α + 1)n (β + 1)n (α + β + 3/2)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Since Wn(α, β) is a polynomial, it suffers to check the equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='13) for α and β in an open subset in C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' As such a subset we shall take the following region O = { α, β : Re α < 0, Re β < 0, Im α > 0, Im β > 0 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='15) For α and β in O, the co-factors in terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2) may be rearranged as follows: ⟨2n⟩k = (−1)k (−2n)k , ⟨α + n⟩k = (−1)k (−α − n)k , ⟨β + n⟩2n−k = (−1)k (−β − n)2n−k = (−β − n)2n (β + 1 − n)k , (2α + 1)2n−k = (−1)k(2α + 1)2n (−2α − 2n)k , \uf8fc \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fe (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='16) Here we have used the following elementary relation for the rising factorial powers (a)m−k = (−1)k (a)m (1 − a − m)k , m ⩾ 0 , 0 ⩽ k ⩽ m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='17) Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='16) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2), we find that Wn(α, β) = (−1)n (2α + 1)2n (−β − n)2n Fn(α, β) = (−1)n Γ(2α + 1 + 2n) Γ(−β + n) Γ(2α + 1) Γ(−β − n) Fn(α, β) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='18) where Fn(α, β) = �2n k=0 (−2n)k (2β + 1)k (−α − n)k (β + 1 − n)k (−2α − 2n)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='19) In view of the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='10), we have Fn(α, β) = 3F2 (−2n, 2β + 1, −α − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' β + 1 − n, −2α − 2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='20) The parameters in hypergeometric series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='20) are identical with those in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='11) for a = −2n, b = 2β + 1, c = −α − n, and in this case, (a + b + 1)/2 = 2β + 1 + n, 2c = −2α − 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='12) also holds for α and β in the region O, since Re (2c − a − b + 1) = Re (−2α − 2β) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Therefore, Watson’s theorem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='11) can be applied to obtain Fn(α, β) = Γ(1/2) Γ(−α − n − 1/2) Γ(β + 1 − n) Γ(−α − β) Γ(−n + 1/2) Γ(β + 1) Γ(−α + 1/2) Γ(−α − β − n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='21) Substituting the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='21) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='18) , we may write Wn(α, β) = c0 c1(α) c2(β) c3(α + β) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='22) SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE 17 where c0 = (−1)n Γ(1/2) Γ(−n + 1/2) , c1(α) = Γ(2α + 2n + 1) Γ(−α − n + 1/2) Γ(2α + 1) Γ(−α + 1/2) , c2(β) = Γ(β + 1 − n) Γ(−β + n) Γ(β + 1) Γ(−β − n) , c3(α + β) = Γ(−α − β) Γ(−α − β − n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' \uf8fc \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fe (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='23) Using the duplication formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='43) and reflection formulas, see [2, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2], Γ(1 − z)Γ(z) = π sin πz , Γ(1/2 − z)Γ(1/2 + z) = π cos πz , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='24) we may rearrange the expressions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='23) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For c0, we have c0 = (−1)n Γ(1/2)2 Γ(−n + 1/2) Γ(n + 1/2) Γ(n + 1/2) Γ(1/2) = (1/2)n , since Γ(1/2) = √π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' For c1(α) and c2(β), we have c1(α) =22n Γ(α + n + 1) Γ(α + n + 1/2) Γ(−α − n + 1/2) Γ(α + 1) Γ(α + 1/2) Γ(−α + 1/2) =22n cos πα Γ(α + n + 1) cos π(α + n) Γ(α + 1) = 22n (−1)n (α + 1)n and c2(β) = Γ(β + 1 − n) Γ(−β + n) Γ(β + 1) Γ(−β − n) = sin π(β + n) Γ(β + 1 + n) sin π(β − n) Γ(β + 1) = (β + 1)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Finally, c3(α + β) = sin π(α + β) Γ(α + β + 1 + n) sin π(α + β + n) Γ(α + β + 1) = (−1)n (α + β + 1)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' Substituting these expressions into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='22), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='31) and the duplication formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='43) that (2α + 2β + 2)2n = 22n (α + β + 1)n (α + β + 3/2)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='25) Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='13) together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='25), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' □ Now it suffers to substitute (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='14) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='1) to obtain the formulas (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' The proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content='2 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf'} +page_content=' R.' metadata={'source': 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P/V/Sℓ+νℓ with the SU(3) Flavor +Symmetry/Breaking +Ru-Min Wang1,†, +Yue-Xin Liu1, +Chong Hua1, +Jin-Huan Sheng2,§, +Yuan-Guo Xu1,♯ +1College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China +2School of Physics and Engineering, Henan University of Science and Technology, Luoyang, Henan 471000, China +†ruminwang@sina.com +§jinhuanwuli@126.com +♯yuanguoxu@jxnu.edu.cn +Many exclusive c → d/sℓ+νℓ (ℓ = e, µ, τ) transitions have been well measured, and they can be +used to test the theoretical calculations. Motivated by this, we study the D → P/V/Sℓ+νℓ decays +induced by the c → d/sℓ+νℓ transitions with the SU(3) flavor symmetry approach, where P denotes +the pseudoscalar meson, V denotes the vector meson, and S denotes the scalar meson with a mass +below 1 GeV . The different decay amplitudes of the D → Pℓ+νℓ, D → V ℓ+νℓ or D → Sℓ+νℓ +decays can be related by using the SU(3) flavor symmetry and by considering the SU(3) flavor +breaking. Using the present data of D → P/V/Sℓ+νℓ, we predict the not yet measured or not yet +well measured processes in the D → P/V/Sℓ+νℓ decays. We find that the SU(3) flavor symmetry +approach works well in the semileptonic D → P/V ℓ+νℓ decays. For the D → Sℓ+νℓ decays, only +the decay D+ +s → f0(980)e+νe has been measured, the branching ratios of the D+ +s → f0(980)e+νe +and D → S(S → P1P2)ℓ+νℓ decays are used to constrain the nonperturbative parameters and then +predict not yet measured D → Sℓ+νℓ decays, in addition, the two quark and the four quark scenarios +for the light scalar mesons are analyzed. The SU(3) flavor symmetry predictions of the D → Sℓ+νℓ +decays need to be further tested, and our predictions of the D → Sℓ+νℓ decays are useful for probing +the structure of light scalar mesons. Our results in this work could be used to test the SU(3) flavor +symmetry approach in the semileptonic D decays by the future experiments at BESIII, LHCb and +BelleII. +I. +Introduction +Semileptonic heavy meson decays dominated by tree-level exchange of W-bosons in the standard model have +attracted a lot of attention in testing the stand model and in searching for the new physics beyond the stand model. +Many semileptonic D → P/V ℓ+νℓ decays and one D → Sℓ+νℓ decay have been observed [1], and present experimental +measurements give us an opportunity to additionally test theoretical approaches. +In theory, the description of semileptonic decays are relatively simple, and the weak and strong dynamics can be +separated in these processes since leptons do not participate in the strong interaction. All the strong dynamics in +the initial and final hadrons is included in the hadronic form factors, which are important for testing the theoretical +calculations of the involved strong interaction. The form factors of the D decays have been calculated, for examples, +by quark model [2–7], QCD sum rules [8], light-cone sum rules [9–11], covariant light-front quark models [12–14], and +lattice QCD [15, 16]. +The SU(3) flavor symmetry approach is independent of the detailed dynamics offering us an opportunity to relate +different decay modes, nevertheless, it cannot determine the sizes of the amplitudes or the form factors by itself. +arXiv:2301.00079v1 [hep-ph] 31 Dec 2022 + +2 +However, if experimental data are enough, one may use the data to extract the amplitudes or the form factors, which +can be viewed as predictions based on symmetry, has a smaller dependency on estimated form factors, and can provide +some very useful information about the decays. The SU(3) flavor symmetry works well in the b-hadron decays [17–30], +and the c-hadron decays [29–45]. +Semileptonic decays of D mesons have been studied extensively in the standard model and its various extensions, for +instance, in Refs. [3, 46–56]. In this work, we will systematically study the D → P/V/Sℓ+νℓ decays with the SU(3) +flavor symmetry. We will firstly construct the amplitude relations between different decay modes of D → Pℓ+νℓ, +D → V ℓ+νℓ or D → Sℓ+νℓ decays by the SU(3) flavor symmetry and the SU(3) flavor breaking. We use the available +data to extract the SU(3) flavor symmetry/breaking amplitudes and the form factors, and then predict the not yet +measured modes for further tests in experiments. The forward-backward asymmetries Aℓ +F B, the lepton-side convexity +parameters Cℓ +F , the longitudinal polarizations of the final charged lepton P ℓ +L, the transverse polarizations of the final +charged lepton P ℓ +T , the lepton spin asymmetries Aλ and the longitudinal polarization fractions FL of the final vector +mesons with two ways of integration have also been predicted in the D → P/V ℓ+νℓ decays. In addition, the q2 +dependence of some differential observables for the D → P/V ℓ+νℓ decays are shown in figures. +This paper will be organized as follows. In Sec. II, the theoretical framework in this work is presented, including the +effective hamiltonian, the hadronic helicity amplitude relations, the observables and the form factors. The numerical +results of the D → P/V/Sℓ+νℓ semileptonic decays will be given in Sec. III. Finally, we give the summary and +conclusion in Sec. IV. +II. +Theoretical Frame +A. +The effective Hamiltonian +In the standard model, the four-fermion charged-current effective Hamiltonian below the electroweak scale for the +decays D → Mℓ+νℓ (M = P, V, S) can be written as +Heff(c → qℓ+νℓ) = GF +√ +2 V ∗ +cq¯qγµ(1 − γ5)c ¯νℓγµ(1 − γ5)ℓ, +(1) +with q = s, d. +The helicity amplitudes of the decays D → Mℓ+νℓ can be written as +M(D → Mℓ+νℓ) = GF +√ +2 Vcb +� +mm′ +gmm′Lλℓλν +m +HλM +m′ , +(2) +with +Lλℓλν +m += ϵα(m) ¯νℓγα(1 − γ5)ℓ, +(3) +HλM +m′ += +� +� +� +ϵ∗ +β(m′)⟨P/S(pP/S)|¯qγβ(1 − γ5)c|D(pD)⟩ +ϵ∗ +β(m′)⟨V (pV , ϵ∗)|¯qγβ(1 − γ5)c|D(pD)⟩ +, +(4) +where the particle helicities λM = 0 for M = P/S, λM = 0, ±1 for M = V, λℓ = ± 1 +2 and λν = + 1 +2, as well as ϵµ(m) +is the polarization vectors of the virtual W with m = 0, t, ±1. + +3 +The form factors of the D → P, D → S and D → V transitions are given by [2, 3, 13] +� +P(p) +�� ¯dkγµc +�� D(pD) +� += f P ++ (q2)(p + pD)µ + +� +f P +0 (q2) − f P ++ (q2) +� m2 +D − m2 +P +q2 +qµ, +(5) +� +S(p) +�� ¯dkγµγ5c +�� D(pD) +� += −i +� +f S ++(q2)(p + pD)µ + +� +f S +0 (q2) − f S ++(q2) +� m2 +D − m2 +S +q2 +qµ +� +, +(6) +� +V (p, ε∗) +�� ¯dkγµ(1 − γ5)c +�� D(pD) +� += +2V V (q2) +mD + mV +ϵµναβε∗νpα +Dpβ +−i +� +ε∗ +µ(mD + mV )AV +1 (q2) − (pD + p)µ(ε∗.pD) AV +2 (q2) +mD + mV +� ++iqµ(ε∗.pD)2mV +q2 [AV +3 (q2) − AV +0 (q2)], +(7) +where s = q2 (q = pD − pM), and ε∗ is the polarization of vector meson. The hadronic helicity amplitudes can be +written as +H± = 0, +(8) +H0 = 2mDq|⃗pP | +� +q2 +f P ++ (q2), +(9) +Ht = +m2 +Dq − m2 +P +� +q2 +f P +0 (q2), +(10) +for D → Pℓ+νℓ decays, +H± = 0, +(11) +H0 = i2mDq|⃗pS| +� +q2 +f S ++(q2), +(12) +Ht = +im2 +Dq − m2 +S +� +q2 +f S +0 (q2), +(13) +for D → Sℓ+νℓ decays, and +H± = (mDq + mV )A1(q2) ∓ +2mDq|⃗pV | +(mDq + mV )V (q2), +(14) +H0 = +1 +2mV +� +q2 +� +(m2 +Dq − m2 +V − q2)(mDq + mV )A1(q2) − +4m2 +Dq|⃗pV |2 +mDq + mV +A2(q2) +� +, +(15) +Ht = 2mDq|⃗pV | +� +q2 +A0(q2), +(16) +for D → V ℓ+νℓ decays, where |⃗pM| ≡ +� +λ(m2 +Dq, m2 +M, q2)/2mDq with λ(a, b, c) = a2 + b2 + c2 − 2ab − 2ac − 2bc. +B. +Hadronic helicity amplitude relations by the SU(3) flavor symmetry +Charmed mesons containing one heavy c quark are flavor SU(3) anti-triplets +Di = +� +D0(c¯u), D+(c ¯d), D+ +s (c¯s) +� +. +(17) + +4 +Light pseudoscalar P and vector V meson octets and singlets under the SU(3) flavor symmetry of u, d, s quarks are +[57] +P = +� +� +� +� +π0 +√ +2 + η8 +√ +6 + η1 +√ +3 +π+ +K+ +π− +− π0 +√ +2 + η8 +√ +6 + η1 +√ +3 +K0 +K− +K +0 +− 2η8 +√ +6 + η1 +√ +3 +� +� +� +� , +(18) +V += +� +� +� +� +ρ0 +√ +2 + ω8 +√ +6 + ω1 +√ +3 +ρ+ +K∗+ +ρ− +− ρ0 +√ +2 + ω8 +√ +6 + ω1 +√ +3 +K∗0 +K∗− +K +∗0 +− 2ω8 +√ +6 + ω1 +√ +3 +� +� +� +� , +(19) +where ω and φ mix in an ideal form, and the η and η′ ( ω and φ) are mixtures of η1(ω1) = u¯u+d ¯d+s¯s +√ +3 +and η8(ω8) = +u¯u+d ¯d−2s¯s +√ +6 +with the mixing angle θP (θV ). η and η′ (ω and φ) are given by +� +� η +η′ +� +� = +� +� cosθP −sinθP +sinθP +cosθP +� +� +� +� η8 +η1 +� +� , +� +� φ +ω +� +� = +� +� cosθV +−sinθV +sinθV +cosθV +� +� +� +� ω8 +ω1 +� +� , +(20) +where θP = [−20◦, −10◦] and θV = 36.4◦ from Particle Data Group (PDG) [1] will be used in our numerical analysis. +The structures of the light scalar mesons are not fully understood yet. Many suggestions are discussed, such as +ordinary two quark states, four quark states, meson-meson bound states, molecular states, glueball states or hybrid +states, for examples, in Refs. [58–66]. In this work, we will consider the two quark and the four quark scenarios for +the scalar mesons below or near 1 GeV . In the two quark picture, the light scalar mesons can be written as [67] +S = +� +� +� +� +a0 +0 +√ +2 + +σ +√ +2 +a+ +0 +K+ +0 +a− +0 +− a0 +0 +√ +2 + +σ +√ +2 +K0 +0 +K− +0 +K +0 +0 +f0 +� +� +� +� . +(21) +The two isoscalars f0(980) and f0(500) are obtained by the mixing of σ = u¯u+d ¯d +√ +2 +and f0 = s¯s +� +� f0(980) +f0(500) +� +� = +� +� cosθS +sinθS +−sinθS cosθS +� +� +� +� f0 +σ +� +� , +(22) +where the three possible ranges of the mixing angle, 25◦ < θS < 40◦, 140◦ < θS < 165◦ and +− 30◦ < θS < 30◦ +[58, 68] will be analyzed in our numerical results. In the four quark picture, the light scalar mesons are given as [1, 69] +σ = u¯ud ¯d, +f0 = (u¯u + d ¯d)s¯s/ +√ +2, +a0 +0 = (u¯u − d ¯d)s¯s/ +√ +2, +a+ +0 = u ¯ds¯s, +a− +0 = d¯us¯s, +K+ +0 = u¯sd ¯d, +K0 +0 = d¯su¯u, +¯K0 +0 = s ¯du¯u, +K+ +0 = s¯ud ¯d, +(23) +and the two isoscalars are expressed as +� +� f0(980) +f0(500) +� +� = +� +� cosφS +sinφS +−sinφS cosφS +� +� +� +� f0 +σ +� +� , +(24) +where the constrained mixing angle φS = (174.6+3.4 +−3.2)◦ [59]. + +5 +In terms of the SU(3) flavor symmetry, meson states and quark operators can be parameterized into SU(3) tensor +forms, while the leptonic helicity amplitudes Lλℓ,λν +m +are invariant under the SU(3) flavor symmetry. And the hadronic +helicity amplitude relations of the D → Mℓ+νℓ(M = P, V, S) decays can be parameterized as +H(D → Mℓ+νℓ) = cM +0 DiM i +jHj, +(25) +where H2 ≡ V ∗ +cd and H3 ≡ V ∗ +cs are the CKM matrix elements, and cM +0 +are the nonperturbative coefficients of the +D → Mℓ+νℓ decays under the SU(3) flavor symmetry. Noted that the hadronic helicity amplitudes for the D → Sℓ+νℓ +decays in Eq. (25) are given in the two quark picture of the light scalar mesons, and ones in the four quark picture +of the light scalar mesons will be given later. +The SU(3) flavor breaking effects mainly come from different masses of u, d and s quarks. Following Ref. [70], the +SU(3) breaking amplitudes of the D → Mℓ+νℓ decays can be give as +∆H(D → Mℓ+νℓ) = cM +1 DaW a +i M i +jHj + cM +2 DiM i +aW a +j Hj, +(26) +with +W = +� +W i +j +� += +� +� +� +� +1 0 +0 +0 1 +0 +0 0 −2 +� +� +� +� , +(27) +where cM +1,2 are the nonperturbative SU(3) flavor breaking coefficients. +In the four quark picture of the light scalar mesons, the hadronic helicity amplitudes of the D → Sℓ+νℓ decays +under the SU(3) flavor symmetry are +H(D → Sℓ+νℓ)4q = c′S +0 DiSim +jmHj. +(28) +And the corresponding SU(3) flavor breaking amplitudes of the D → Sℓ+νℓ decays are +∆H(D → Sℓ+νℓ)4q = c′S +1 DaW a +i Sim +jmHj + c′S +2 DiSim +amW a +j Hj + c′S +1 DiSim +ja W a +mHj. +(29) +In terms of the SU(3) flavor symmetry, the hadronic helicity amplitude relations for the D → Pℓ+νℓ, D → V ℓ+νℓ +and D → Sℓ+νℓ decays are summarized in later Tab. I, Tab. IV and Tab. VIII, respectively. +C. +Observables for the D → Mℓ+νℓ decays +The double differential branching ratios of the D → Mℓ+νℓ decays are [56] +dB(D → Mℓ+νℓ) +dq2d(cos θ) += τDG2 +F |Vcq|2λ1/2(q2 − m2 +ℓ)2 +64(2π)3M 3 +D(s)q2 +� +(1 + cos2 θ)HU + 2 sin2 θHL + 2 cos θHP ++m2 +ℓ +q2 (sin2 θHU + 2 cos2 θHL + 2HS − 4 cos θHSL) +� +, +(30) +where λ ≡ λ(m2 +Dq, m2 +M, q2), m2 +ℓ ≤ q2 ≤ (mDq − mM)2, and +HU = |H+|2 + |H−|2, +HL = |H0|2, +HP = |H+|2 − |H−|2, +HS = |Ht|2, +HSL = ℜ(H0H† +t ). +(31) + +6 +The differential branching ratios integrated over cos θ are [56] +dB(D(s) → Mℓ+νℓ) +dq2 += τDG2 +F |Vcq|2λ1/2(q2 − m2 +ℓ)2 +24(2π)3M 3 +D(s)q2 +Htotal, +(32) +with +Htotal ≡ (HU + HL) +� +1 + m2 +ℓ +2q2 +� ++ 3m2 +ℓ +2q2 HS. +(33) +The lepton flavor universality in D(s) → Mℓ+νℓ is defined in a manner identical Rµ/e as +Rµ/e = +� qmax +qmin dB(D(s) → Mµ+νµ)/dq2 +� qmax +qmin dB(D(s) → Me+νe)/dq2 . +(34) +The forward-backward asymmetries are defined as [56] +Aℓ +F B(q2) = +� 0 +−1 dcosθℓ +dB(D→Mℓν) +dq2dcosθℓ +− +� 1 +0 dcosθℓ +dB(D→Mℓν) +dq2dcosθℓ +� 0 +−1 dcosθℓ +dB(D→Mℓν) +dq2dcosθℓ ++ +� 1 +0 dcosθℓ +dB(D→Mℓν) +dq2dcosθℓ +(35) += 3 +4 +HP − 2m2 +ℓ +q2 HSL +Htotal +. +(36) +The lepton-side convexity parameters are given by [56] +Cℓ +F (q2) = 3 +4 +� +1 − m2 +ℓ +q2 +� HU − 2HL +Htotal +. +(37) +The longitudinal polarizations of the final charged lepton ℓ are defined by [56] +P ℓ +L(q2) = +(HU + HL) +� +1 − m2 +ℓ +2q2 +� +− 3m2 +ℓ +2q2 HS +Htotal +, +(38) +and its transverse polarizations are +P ℓ +T (q2) = − 3πmℓ +8 +� +q2 +HP + 2HSL +Htotal +. +(39) +The lepton spin asymmetry in the ℓ − ¯νℓ center of mass frame is defined by [71–74] +Aλ(q2) = dB(D → Mℓ+νℓ)[λℓ = − 1 +2]/dq2 − dB(D → Mℓ+νℓ)[λℓ = + 1 +2]/dq2 +dB(D → Mℓ+νℓ)[λℓ = − 1 +2]/dq2 + dB(D → Mℓ+νℓ)[λℓ = + 1 +2]/dq2 +(40) += +Htotal − 6m2 +ℓ +2q2 HS +Htotal +. +(41) +For the D → V ℓ+νℓ decays, the longitudinal polarization fractions of the final vector mesons are given by [56] +FL(q2) = +HL +� +1 + m2 +ℓ +2q2 +� ++ 3m2 +ℓ +2q2 HS +Htotal +, +(42) +then its transverse polarization fraction FT (q2) = 1 − FL(q2). +Noted that, for q2-integration of X(q2) = Aℓ +F B, Cℓ +F , P ℓ +L, P ℓ +T , Aλ and FL, following Ref. [75], two ways of integration +are considered. The normalized q2-integrated observables ⟨X⟩ are calculated by separately integrating the numerators +and denominators with the same q2 bins. The “naively integrated” observables are obtained by +X = +1 +q2max − q2 +min +� q2 +max +q2 +min +dq2X(q2). +(43) + +7 +D. +Form factors +In order to obtain more precise observables, one also need considering the q2 dependence of the form factors for +the D → Pℓ+νℓ, D → V ℓ+νℓ and D → Sℓ+νℓ decays. The following cases will be considered in our analysis of +D → P/V ℓ+νℓ decays. +C1: All form factors are treated as constants without the hadronic momentum-transfer q2 dependence, and different +form factors are related by the SU(3) flavor symmetry, i.e., the SU(3) flavor breaking terms such as cM +1,2 and +c′S +1,2,3 in later Tabs. I, IV and VIII are ignored. +C2: With the SU(3) flavor symmetry, the modified pole model for the q2-dependence of Fi(q2) is used [76] +Fi(q2) = +Fi(0) +� +1 − +q2 +m2 +pole +� � +1 − αi +q4 +m4 +pole +�, +(44) +where mpole = mD∗+ for c → dℓ+νℓ transitions and mpole = mD∗+ +s +for c → sℓ+νℓ transitions, and αi are free +parameters and are different for f P ++ (q2), f P +0 (q2), V (q2), A1(q2) and A2(q2), we will take αi ∈ [−1, 1] in our +analysis. +C3: With the SU(3) flavor symmetry, following Ref. [2] +Fi(q2) = +Fi(0) +� +1 − +q2 +m2 +pole +� � +1 − σ1i +q2 +m2 +pole + σ2i +q4 +m4 +pole +� +for f P ++ (q2) and V (q2), +(45) +Fi(q2) = +Fi(0) +� +1 − σ1i +q2 +m2 +pole + σ2i +q4 +m4 +pole +� +for f P +0 (q2), A1(q2) and A2(q2), +(46) +where σ1,2 for the D → π and D → K∗ transitions from Ref. [2] will be used in our results. +C4: Considering the SU(3) flavor breaking terms such as cM +1,2 and c′S +1,2,3 in later Tabs. I, IV and VIII, the form factors +in C3 case are used. +As for the form factors of the D → Sℓ+νℓ decays, we find that the vector dominance model [77] and the double +pole model [78] give the similar SU(3) flavor symmetry predictions for the branching ratios of the D → Sℓ+νℓ decays. +The following form factors from the vector dominance model will be used in the numerical results, +Fi(q2) = +Fi(0) +� +1 − q2/m2 +pole +� +for f S ++(q2) and f S +0 (q2). +(47) +After considering above q2 dependence, we only need to focus on the Fi(0). Since these form factors Fi(0) also +preserve the SU(3) flavor symmetry, the same relations in Tabs. I, IV and VIII will be used for Fi(0). If considering +the form factors ratios f+(0)/f0(0) = 1 for D → P/Sℓ+νℓ decays, rV ≡ V (0)/A1(0) = 1.46±0.07, r2 ≡ A2(0)/A1(0) = +0.68 ± 0.06 in D0 → K∗−ℓ+νℓ decays from PDG [1] and the SU(3) flavor symmetry, there is only one free form factor +f P,S ++ +(0) and A1(0) for the D → P/Sℓ+νℓ and D → V ℓ+νℓ decays, respectively. As a result, the branching ratios only +depend on one form factor f P ++ (0), f S ++(0) or A1(0) and the CKM matrix element Vcq. + +8 +III. +Numerical results +The theoretical input parameters and the experimental data within the 2σ errors from PDG [1] will be used in our +numerical results. +A. +D → Pℓ+νℓ decays +Considering both the SU(3) flavor symmetry and the SU(3) flavor breaking contributions, the hadronic helicity +amplitudes for the D → Pℓ+νℓ decays are given in Tab. I, in which we keep the CKM matrix element Vcs and Vcd +information for comparing conveniently. In addition, H(D+ +s → π0ℓ+νℓ) are obtained by neutral meson mixing with +δ2 = (5.18 ± 0.71) × 10−4 in Ref. [76]. From Tab. I, we can easily see the hadronic helicity amplitude relations +of the D → Pℓ+νℓ decays. +There are four nonperturbative parameters A1,2,3,4 in the D → Pℓ+νℓ decays with +A1 ≡ cP +0 + cP +1 − 2cP +2 , A2 ≡ cP +0 − 2cP +1 − 2cP +2 , A3 ≡ cP +0 + cP +1 + cP +2 and A4 ≡ cP +0 − 2cP +1 + cP +2 . If neglecting the SU(3) flavor +breaking cP +1 and cP +2 terms, A1 = A2 = A3 = A4 = cP +0 , and then all hadronic helicity amplitudes are related by only +one parameter cP +0 . +Many decay modes of the D → Pe+νe, Pµ+νµ decays have been measured, and the experimental data with 2σ +errors are listed in the second column of Tab. II. One can constrain the parameters Ai by the present experimental +data within 2σ errors and then predict other not yet measured branching ratios. Four cases C1,2,3,4 will be considered +in our analysis. The numerical results of B(D → Pℓ+νℓ) in the C1, C2, C3 and C4 cases are given in the third, forth, +fifth and sixth columns of Tab. II, respectively. And our comments on the results are as follows. +• Results in C1 case: +From the third column of Tab. II, one can see that the SU(3) flavor symmetry predictions +of B(D → Pℓ+νℓ) in the C1 case are entirely consistent with all present experiential data. The not yet measured +branching ratios of the D+ +s → π0e+νe, D+ +s → π0µ+νµ, D+ → η′µ+νµ and D+ +s → K0µ+νµ decays are predicted +TABLE I: The hadronic helicity amplitudes for the D → Pℓ+ν decays including both the SU(3) flavor symmetry and the +SU(3) flavor breaking contributions. A1 ≡ cP +0 + cP +1 − 2cP +2 , A2 ≡ cP +0 − 2cP +1 − 2cP +2 , A3 ≡ cP +0 + cP +1 + cP +2 , A4 ≡ cP +0 − 2cP +1 + cP +2 . +A1 = A2 = A3 = A4 = cP +0 if neglecting the SU(3) flavor breaking cP +1 and cP +2 terms. +Hadronic helicity amplitudes +SU(3) flavor amplitudes +H(D0 → K−ℓ+νℓ) +A1V ∗ +cs +H(D+ → K +0ℓ+νℓ) +A1V ∗ +cs +H(D+ +s → ηℓ+νℓ) +� +− cosθP +� +2/3 − sinθP / +√ +3� +A2V ∗ +cs +H(D+ +s → η′ℓ+νℓ) +� +− sinθP +� +2/3 + cosθP / +√ +3� +A2V ∗ +cs +H(D+ +s → π0ℓ+νℓ) +−δ� +− cosθP +� +2/3 − sinθP / +√ +3� +A2V ∗ +cs +H(D0 → π−ℓ+νℓ) +A3V ∗ +cd +H(D+ → π0ℓ+νℓ) +− 1 +√ +2 A3V ∗ +cd +H(D+ → ηℓ+νℓ) +� +cosθP / +√ +6 − sinθP / +√ +3� +A3V ∗ +cd +H(D+ → η′ℓ+νℓ) +� +sinθP / +√ +6 + cosθP / +√ +3� +A3V ∗ +cd +H(D+ +s → K0ℓ+νℓ) +A4V ∗ +cd + +9 +TABLE II: Branching ratios of the D → Pℓ+ν decays. †Denotes that the corresponding experimental data from PDG [1] are +not used to constrain Ai in this case. +Branching ratios +Exp. data +Ones in C1 +Ones in C2 +Ones in C3 +Ones in C4 +Previous ones +B(D+ → K +0e+νe)(×10−2) +8.72 ± 0.18 +8.84 ± 0.06 +8.83 ± 0.07 +8.84 ± 0.06 +8.83 ± 0.07 +B(D+ → π0e+νe)(×10−3) +3.72 ± 0.34 +3.75 ± 0.05 +5.40 ± 1.33† +5.04 ± 0.12† +3.70 ± 0.11 +B(D+ → ηe+νe)(×10−3) +1.11 ± 0.14 +1.15 ± 0.05 +1.20 ± 0.05 +1.20 ± 0.05 +0.92 ± 0.08 +B(D+ → η′e+νe)(×10−4) +2.0 ± 0.8 +2.59 ± 0.14 +2.22 ± 0.34 +2.09 ± 0.14 +1.50 ± 0.20 +B(D0 → K−e+νe)(×10−2) +3.549 ± 0.052 +3.52 ± 0.02 +3.52 ± 0.03 +3.52 ± 0.03 +3.52 ± 0.02 +B(D0 → π−e+νe)(×10−3) +2.91 ± 0.08 +2.95 ± 0.03 +4.23 ± 1.03† +3.97 ± 0.09† +2.89 ± 0.06 +B(D+ +s → ηe+νe)(×10−2) +2.32 ± 0.16 +2.37 ± 0.11 +2.34 ± 0.14 +2.36 ± 0.12 +2.32 ± 0.16 +B(D+ +s → η′e+νe)(×10−3) +8.0 ± 1.4 +9.05 ± 0.04 +8.25 ± 1.13 +8.04 ± 0.43 +8.02 ± 1.38 +B(D+ +s → K0e+νe)(×10−3) +3.4 ± 0.8 +3.10 ± 0.08 +3.56 ± 0.39 +3.54 ± 0.12 +3.40 ± 0.80 +B(D+ +s → π0e+νe)(×10−5) +· · · +1.51 ± 0.07 +2.10 ± 0.56 +1.96 ± 0.10 +1.92 ± 0.13 +2.65 ± 0.38 [76] +B(D+ → K +0µ+νµ)(×10−2) +8.76 ± 0.38 +8.56 ± 0.06 +8.69 ± 0.15 +8.61 ± 0.06 +8.61 ± 0.06 +B(D+ → π0µ+νµ)(×10−3) +3.50 ± 0.30 +3.67 ± 0.05 +5.32 ± 1.31† +4.96 ± 0.12† +3.64 ± 0.10 +B(D+ → ηµ+νµ)(×10−3) +1.04 ± 0.22 +1.11 ± 0.05 +1.18 ± 0.07 +1.17 ± 0.05 +0.90 ± 0.08 +1.21 [7] +0.75±0.15 [79] +B(D+ → η′µ+νµ)(×10−4) +· · · +2.42 ± 0.13 +2.10 ± 0.33 +1.96 ± 0.13 +1.41 ± 0.19 +2.11 [7] +1.06±0.20 [79] +B(D0 → K−µ+νµ)(×10−2) +3.41 ± 0.08 +3.41 ± 0.02 +3.44 ± 0.05 +3.43 ± 0.02 +3.43 ± 0.02 +B(D0 → π−µ+νµ)(×10−3) +2.67 ± 0.24 +2.89 ± 0.02 +4.17 ± 1.01† +3.90 ± 0.09† +2.85 ± 0.06 +B(D+ +s → ηµ+νµ)(×10−2) +2.4 ± 1.0 +2.30 ± 0.10 +2.30 ± 0.17 +2.31 ± 0.12 +2.26 ± 0.16 +B(D+ +s → η′µ+νµ)(×10−2) +1.1 ± 1.0 +0.86 ± 0.03 +0.79 ± 0.11 +0.77 ± 0.04 +0.76 ± 0.13 +B(D+ +s → K0µ+νµ)(×10−3) +· · · +3.01 ± 0.08 +3.51 ± 0.38 +3.46 ± 0.11 +3.33 ± 0.78 +3.9 [7] +3.85±0.76 [79] +B(D+ +s → π0µ+νµ)(×10−5) +· · · +1.48 ± 0.07 +2.09 ± 0.53 +1.93 ± 0.10 +1.89 ± 0.13 +B(D+ +s → π0τ +ντ)(×10−10) +· · · +3.45 ± 0.21 +160.34 ± 149.53 +4.20 ± 0.26 +4.08 ± 0.34 +(27 ∼ 36) [76] +Rµ/e(D+ → K +0ℓ+νℓ) +0.969 +0.984 ± 0.013 +0.974 +0.974 +Rµ/e(D+ → π0ℓ+νℓ) +0.977 +1.009 ± 0.026 +0.984 +0.984 +Rµ/e(D+ → ηℓ+νℓ) +0.967 +0.984 ± 0.014 +0.973 +0.973 +Rµ/e(D+ → η′ℓ+νℓ) +0.935 +0.948 ± 0.012 +0.940 +0.940 +Rµ/e(D0 → K−ℓ+νℓ) +0.969 +0.984 ± 0.013 +0.974 +0.974 +Rµ/e(D0 → π−ℓ+νℓ) +0.977 +1.008 ± 0.026 +0.984 +0.984 +Rµ/e(D+ +s → ηℓ+νℓ) +0.971 +0.987 ± 0.013 +0.976 +0.976 +Rµ/e(D+ +s → η′ℓ+νℓ) +0.946 +0.958 ± 0.011 +0.952 +0.952 +Rµ/e(D+ +s → K0ℓ+νℓ) +0.973 +0.992 ± 0.016 +0.978 +0.978 +Rµ/e(D+ +s → π0ℓ+νℓ) +0.980 +1.010 ± 0.025 +0.985 +0.985 + +10 +on the order of O(10−3 − 10−5), nevertheless, B(D+ +s → π0τ +ντ) is predicted on the order of O(10−10) due to +its narrow phase space and (q2 − m2 +τ)2 suppression of the differential branching ratios in Eq. (32). +• Results in C2,3 cases: +The numerical results in C2,3 cases are similar. The experimental upper limits of +B(D+ → π0ℓ+νℓ) and B(D0 → π−ℓ+νℓ) have not been used to constrain the predictions of B(D → Pℓ+νℓ), since +the upper limits of the predictions of B(D+ → π0ℓ+νℓ) and B(D0 → π−ℓ+νℓ) by the SU(3) flavor symmetry +in C2,3 cases are slightly larger than their experimental data. Other SU(3) flavor symmetry predictions are +consistent with their experimental data within 2σ errors. +• Results in C4 case: +As given in the sixth column of Tab. II, if considering both the hadronic momentum- +transfer q2 dependence of the form factors and the SU(3) flavor breaking contributions, all SU(3) flavor symmetry +predictions are consistent with their experimental data within 2σ errors. For some decays, the errors of the +theoretical predictions are much smaller than ones of their experimental data. +• The previous predictions for the not yet measured branching ratios are listed in the last column of Tab. II, our +predictions are in the same order of magnitude as previous ones for the D → Pe+νe, Pµ+νµ decays. And our +prediction of B(D+ +s → π0τ +ντ) is one order smaller than previous one in Ref. [76]. +• In addition, the lepton flavor universality parameters Rµ/e(D → Pℓ+νℓ) are also given in Tab. II, since many +terms are canceled in the ratios, these predictions are quite accurate, and all processes have similar results. +For the q2 dependence of the differential branching ratios of the D → Pℓ+νℓ decays with present experimental +bounds, we only show the not yet measured processes D+ → η′µ+νµ, D+ +s → K0µ+νµ, D+ +s → π0µ+νµ and D+ +s → +π0τ +ντ in Fig. 1. We do not show dB(D+ +s → π0e+νe)/dq2, since it is similar to dB(D+ +s → π0µ+νµ)/dq2 in Fig. 1 +C +1 +C +2 +C +3 +C +4 +dB(D ++ +s +0 ++ +)/dq +2 + ( x10 +-10 + ) +q +2 + + + + +FIG. 1: The q2 dependence of the differential branching ratios for some D → Pℓ+νℓ with present experimental bounds. + +qB(D +0.0 +S.0 +.0 +0 +文 +→>,")qd +Se +8(C) +d. +S +3 +4 +Q +00 +0C +Sc +c +(ε) +d.4 +a.0 +8.0 +0.10.8 +1.8 +0S.8 +3' +001 +5 +3 +4qB(D→>K^")/qd +0.0 +2.0 +0 +xXoX +S3(q) +3'S +3'3 +3 +003 +3'4 +0 ++ix +Q +0 +(p) +d.0.1 +2.1 +s'o +20 +0 +0'42.0 +1 +gB +0.0 +2 +个← +2.00.1 +3 +2.1 +s'o11 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +-20 +-15 +-10 +-5 +0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +-1.6 +-1.2 +-0.8 +-0.4 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0.0 +0.3 +0.6 +0.9 +1.2 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +-1.0 +-0.8 +-0.6 +-0.4 +-0.2 +0.0 +( d ) +( c ) +( b ) +( a ) + + + e +in unit of 10 +-6 + +in unit of 10 +-2 +dA +FB +(D ++ +s +K +0 +l ++ +l +)/dq +2 +q +2 + + + e + +dC +l +F +(D ++ +s +K +0 +l ++ +l +)/dq +2 +q +2 + + + e + +dP +l +L +(D ++ +s +K +0 +l ++ +l +)/dq +2 +q +2 + + + e + +dP +l +T +(D ++ +s +K +0 +l ++ +l +)/dq +2 +q +2 +FIG. 2: +The differential forward-backward asymmetries, differential lepton-side convexity parameters, differential longitudinal +lepton polarizations and differential transverse lepton polarizations for the D+ +s → K0ℓ+νℓ decays in the C3 case. +(c). From Fig. 1, one can see that present experimental measurements give quite strong bounds on the differential +branching ratios of D+ → η′µ+νµ, D+ +s → π0µ+νµ and D+ +s → π0τ +ντ decays in the C1, C3 and C4 cases as well as +D+ +s → K0µ+νµ decays in the C1 and C3 cases, and all predictions of the four differential branching ratios in the C2 +case have large error due to the form factor choice. Comparing with dB(D+ +s → π0µ+νµ)/dq2 in Fig. 1 (c), as shown +in Fig. 1 (d), dB(D+ +s → π0τ +ντ)/dq2 is suppressed about the order of O(10−4) by mτ. +The forward-backward asymmetries Aℓ +F B, the lepton-side convexity parameters Cℓ +F , the longitudinal polarizations +of the final charged leptons P ℓ +L and the transverse polarizations of the final charged leptons P ℓ +T with two ways of +integration for the D → Pℓ+νℓ decays could also be obtained. These predictions are very accurate, and they are +similar to each other in the four C1,2,3,4 cases. So we only give the predictions within the C3 case in Tab. III for +examples. From Tab. III, one can see that the predictions are obviously different between two ways of q2 integration, +and the slight difference in the same way of q2 integration is due to the different decay phase spaces. For displaying +the differences between the D → Pe+νe and D → Pµ+νµ decays, we take D+ +s → K0e+νe and D+ +s → K0µ+νµ +as examples. The differential forward-backward asymmetries, the differential lepton-side convexity parameters, the +differential longitudinal lepton polarizations and the differential transverse lepton polarizations of D+ +s → K0e+νe and +D+ +s → K0µ+νµ decays within the C3 case are displayed in Fig. 2. And one can see that differential observables +between ℓ = e and ℓ = µ are obviously different, specially in the low and high q2 ranges. + +12 +TABLE III: Quantities ⟨X⟩ and X of the D → Pℓ+ν in C3 case. +Decay modes +⟨Aℓ +F B⟩ +Ae +F B(×10−6) +Aµ,τ +F B(×10−2) +⟨Cℓ +F ⟩ +Cℓ +F +⟨P ℓ +L⟩ +P ℓ +L +⟨P ℓ +T ⟩ +P e +T (×10−3) +P µ,τ +T +D+ → K +0e+νe +−0.087 +−3.254 ± 0.001 +−1.239 +−1.500 +0.768 +1.000 +−0.273 +−2.442 ± 0.001 +D+ → π0e+νe +−0.083 +−2.054 ± 0.000 +−1.252 +−1.500 +0.780 +1.000 +−0.260 +−1.730 ± 0.000 +D+ → ηe+νe +−0.087 +−3.476 ± 0.001 +−1.239 +−1.500 +0.768 +1.000 +−0.273 +−2.490 ± 0.000 +D+ → η′e+νe +−0.093 +−7.075 ± 0.003 +−1.222 +−1.500 +0.753 +1.000 +−0.290 +−3.890 ± 0.001 +D0 → K−e+νe +−0.087 +−3.259 ± 0.001 +−1.239 +−1.500 +0.768 +1.000 +−0.273 +−2.446 ± 0.001 +D0 → π−e+νe +−0.083 +−2.077 ± 0.000 +−1.252 +−1.500 +0.779 +1.000 +−0.260 +−1.751 ± 0.000 +D+ +s → ηe+νe +−0.086 +−3.033 ± 0.001 +−1.242 +−1.500 +0.770 +1.000 +−0.270 +−2.300 ± 0.001 +D+ +s → η′e+νe +−0.091 +−5.829 ± 0.003 +−1.226 +−1.500 +0.757 +1.000 +−0.286 +−3.484 ± 0.001 +D+ +s → K0e+νe +−0.085 +−2.814 ± 0.001 +−1.245 +−1.500 +0.773 +1.000 +−0.267 +−2.118 ± 0.000 +D+ +s → π0e+νe +−0.082 +−1.850 ± 0.001 +−1.254 +−1.500 +0.781 +1.000 +−0.258 +−1.634 ± 0.001 +D+ → K +0µ+νµ +−0.226 +−4.278 ± 0.001 +−0.822 +−1.352 +0.394 +0.851 +−0.655 +−0.414 +D+ → π0µ+νµ +−0.201 +−2.810 ± 0.000 +−0.897 +−1.405 +0.462 +0.907 +−0.602 +−0.310 +D+ → ηµ+νµ +−0.227 +−4.490 ± 0.001 +−0.819 +−1.347 +0.391 +0.846 +−0.657 +−0.419 +D+ → η′µ+νµ +−0.263 +−8.097 ± 0.003 +−0.708 +−1.213 +0.287 +0.703 +−0.725 +−0.581 +D0 → K−µ+νµ +−0.226 +−4.285 ± 0.001 +−0.822 +−1.352 +0.393 +0.850 +−0.656 +−0.414 +D0 → π−µ+νµ +−0.201 +−2.844 ± 0.001 +−0.895 +−1.407 +0.461 +0.910 +−0.603 +−0.313 +D+ +s → ηµ+νµ +−0.221 +−4.001 ± 0.001 +−0.836 +−1.364 +0.406 +0.864 +−0.646 +−0.394 +D+ +s → η′µ+νµ +−0.254 +−6.952 ± 0.003 +−0.736 +−1.254 +0.314 +0.747 +−0.709 +−0.540 +D+ +s → K0µ+νµ +−0.215 +−3.701 ± 0.001 +−0.856 +−1.377 +0.425 +0.879 +−0.632 +−0.367 +D+ +s → π0µ+νµ +−0.197 +−2.571 ± 0.001 +−0.907 +−1.417 +0.472 +0.920 +−0.594 +−0.295 +D+ +s → π0τ +ντ +−0.281 +−27.429 ± 0.105−0.211 ± 0.003−0.212 ± 0.003−0.868 ± 0.001−0.873 ± 0.001−0.447 ± 0.002−0.437 ± 0.002 +B. +D → V ℓ+νℓ decays +The hadronic helicity amplitudes for the D → V ℓ+νℓ decays are given in Tab. IV. There are four nonperturbative +parameters B1,2,3,4 in the D → V ℓ+νℓ decay modes. +If neglecting the SU(3) flavor breaking cV +1 and cV +2 terms, +B1 = B2 = B3 = B4 = cV +0 , and then all hadronic helicity amplitudes of D → V ℓ+νℓ are related by only one parameter +cV +0 . +Among the D → V ℓ+νℓ decay modes, 13 branching ratios have been measured, and 2 branching ratios have been +upper limited by the experiments. The experimental data with 2σ errors are listed in the second column of Tab. V. +Now we use the listed experimental data to constrain the parameters Bi and then predict other not yet measured and +not yet well measured branching ratios. The numerical results of B(D → V ℓ+νℓ) in the C1, C2, C3 and C4 cases are +given in the third, forth, fifth and sixth columns of Tab. V, respectively. +The results in the C1, C2 and C3 cases are very similar. Since the SU(3) flavor symmetry predictions of B(D+ → +ωe+νe) and B(D0 → ρ−µ+νµ) are slightly larger than their experimental data within 2σ errors in the three cases, we + +13 +TABLE IV: The hadronic helicity amplitudes for D → V ℓ+ν decays including both the SU(3) flavor symmetry and the SU(3) +flavor breaking contributions. B1 = cV +0 +cV +1 −2cV +2 , B2 = cV +0 −2cV +1 −2cV +2 , B3 = cV +0 +cV +1 +cV +2 , B4 = cV +0 −2cV +1 +cV +2 . If neglecting +the SU(3) flavor breaking cV +1 and cV +2 terms, B1 = B2 = B3 = B4 = cV +0 . +Hadronic helicity amplitudes +SU(3) IRA amplitudes +H(D0 → K∗−ℓ+νℓ) +B1V ∗ +cs +H(D+ → K +∗0ℓ+νℓ) +B1V ∗ +cs +H(D+ +s → φℓ+νℓ) +� +− cosθV +� +2/3 − sinθV / +√ +3� +B2V ∗ +cs +H(D+ +s → ωℓ+νℓ) +� +− sinθV +� +2/3 + cosθV / +√ +3� +B2V ∗ +cs +H(D0 → ρ−ℓ+νℓ) +B3V ∗ +cd +H(D+ → ρ0ℓ+νℓ) +− 1 +√ +2 B3V ∗ +cd +H(D+ → φℓ+νℓ) +� +cosθV / +√ +6 − sinθV / +√ +3� +B3V ∗ +cd +H(D+ → ωℓ+νℓ) +� +sinθV / +√ +6 + cosθV / +√ +3� +B3V ∗ +cd +H(D+ +s → K∗0ℓ+νℓ) +B4V ∗ +cd +do not use them to constrain the nonperturbative parameter cV +0 . One can see that the prediction of B(D0 → ρ−µ+νµ) +is agree with its experimental data within 3σ errors, nevertheless, the prediction of B(D+ → ωe+νe) still slightly +larger than experimental data within 3σ errors. B(D+ +s → K∗0µ+νµ) and B(D+ +s → ωe+νe, ωµ+νµ) are predicted +on the order of O(10−3) and O(10−5), respectively. +And they could be measured in BESIII, LHCb and BelleII +experiments. In the C4 case, as given in the sixth column of Tab. V, after considering both the hadronic momentum- +transfer q2 dependence of the form factors and the SU(3) flavor breaking contributions, all SU(3) flavor symmetry +predictions are consistent with their experimental data within 2σ errors. Among relevant not yet measured decays, +B(D+ +s → K∗0µ+νµ) is calculated in the SM using light-cone sum rules [79] and in the relativistic quark model [7], +B(D+ +s → K∗0µ+νµ) = (2.23 ± 0.32) × 10−3 [79] and 2.0 × 10−3 [7], and our predictions of B(D+ +s → K∗0µ+νµ) in the +C1, C2, C3 and C4 cases are coincident with previous ones in Refs. [7, 79]. In addition, the lepton flavor universality +parameters Rµ/e(D → V ℓ+νℓ) are also given in Tab. V. Since many terms are canceled in the ratios, these predictions +of the lepton flavor universality parameters are quite accurate, and our predictions in all four cases are similar to each +other. +For the q2 dependence of the differential branching ratios of the D → V ℓ+νℓ decays with present experimental +bounds, we only show the not yet measured processes D+ → φµ+νµ, D+ +s → ωµ+νµ and D+ +s → K∗0µ+νµ in Fig. 3. +The differential branching ratios of D+ → φe+νe (D+ +s → ωe+νe) is similar to D+ → φµ+νµ (D+ +s → ωµ+νµ), so we +do not shown them in Fig. 3. From Fig. 3, one can see that present experiment data give quite strong bounds on all +differential branching ratios of D+ → φµ+νµ, D+ +s → ωµ+νµ and D+ +s → K∗0µ+νµ decays in the C1, C2 and C3 cases. +The prediction of dB(D+ → φµ+νµ)/dq2 in the C4 case could be distinguished from ones in the C1,2,3 cases within +the middle range of q2. And the error of dB(D+ +s → K∗0µ+νµ)/dq2 in the C4 case is obviously larger than ones in +C1,2,3 cases. +The forward-backward asymmetries Aℓ +F B, the lepton-side convexity parameters Cℓ +F , the longitudinal polarizations +P ℓ +L, the transverse polarizations P ℓ +T , the lepton spin asymmetries Aλ and the longitudinal polarization fractions of the +final vector mesons FL with two ways of integration have also been predicted in the four cases. Since many theoretical + +14 +TABLE V: Branching ratios of the D → V ℓ+ν within 2σ errors. +†The experimental data of B(D+ → ωe+νe) and B(D0 → +ρ−µ+νµ) from PDG [1] are not used in the C1,2,3 cases. +Branching ratios +Exp. data +Ones in C1 +Ones in C2 +Ones in C3 +Ones in C4 +B(D+ → K +∗0e+νe)(×10−2) +5.40 ± 0.20 +5.44 ± 0.15 +5.42 ± 0.18 +5.36 ± 0.08 +5.44 ± 0.16 +B(D+ → ρ0e+νe)(×10−3) +2.18+0.34 +−0.50 +2.31 ± 0.07 +2.39 ± 0.13 +2.33 ± 0.05 +1.83 ± 0.15 +B(D+ → ωe+νe)(×10−3) +1.69 ± 0.22 +2.24 ± 0.07† +2.33 ± 0.12† +2.26 ± 0.04† +1.77 ± 0.14 +B(D+ → φe+νe)(×10−7) +< 130 +3.13 ± 0.12 +3.11 ± 0.19 +3.07 ± 0.07 +2.38 ± 0.23 +B(D0 → K∗−e+νe)(×10−2) +2.15 ± 0.32 +2.12 ± 0.09 +2.13 ± 0.10 +2.08 ± 0.06 +2.13 ± 0.10 +B(D0 → ρ−e+νe)(×10−3) +1.50 ± 0.24 +1.79 ± 0.08 +1.86 ± 0.11 +1.80 ± 0.06 +1.41 ± 0.13 +B(D+ +s → φe+νe)(×10−2) +2.39 ± 0.32 +2.46 ± 0.12 +2.43 ± 0.14 +2.40 ± 0.10 +2.39 ± 0.32 +B(D+ +s → ωe+νe)(×10−5) +< 200 +2.45 ± 0.13 +2.56 ± 0.20 +2.47 ± 0.10 +2.49 ± 0.38 +B(D+ +s → K∗0e+νe)(×10−3) +2.15 ± 0.56 +2.17 ± 0.10 +2.25 ± 0.13 +2.17 ± 0.08 +2.15 ± 0.56 +B(D+ → K +∗0µ+νµ)(×10−2) +5.27 ± 0.30 +5.12 ± 0.15 +5.13 ± 0.16 +5.05 ± 0.08 +5.12 ± 0.15 +B(D+ → ρ0µ+νµ)(×10−3) +2.4 ± 0.8 +2.19 ± 0.07 +2.29 ± 0.13 +2.22 ± 0.04 +1.74 ± 0.14 +B(D+ → ωµ+νµ)(×10−3) +1.77 ± 0.42 +2.13 ± 0.06 +2.23 ± 0.12 +2.15 ± 0.04 +1.68 ± 0.13 +B(D+ → φµ+νµ)(×10−7) +· · · +2.89 ± 0.11 +2.89 ± 0.17 +2.84 ± 0.07 +2.20 ± 0.21 +B(D0 → K∗−µ+νµ)(×10−2) +1.89 ± 0.48 +1.99 ± 0.09 +2.01 ± 0.09 +1.96 ± 0.06 +2.01 ± 0.10 +B(D0 → ρ−µ+νµ)(×10−3) +1.35 ± 0.26 +1.70 ± 0.07† +1.78 ± 0.11† +1.72 ± 0.06† +1.34 ± 0.13 +B(D+ +s → φµ+νµ)(×10−2) +1.9 ± 1.0 +2.30 ± 0.12 +2.29 ± 0.12 +2.25 ± 0.09 +2.24 ± 0.30 +B(D+ +s → ωµ+νµ)(×10−5) +· · · +2.34 ± 0.12 +2.47 ± 0.19 +2.37 ± 0.09 +2.38 ± 0.36 +B(D+ +s → K∗0µ+νµ)(×10−3) +· · · +2.06 ± 0.10 +2.15 ± 0.13 +2.07 ± 0.08 +2.05 ± 0.53 +Rµ/e(D+ → K +∗0ℓ+νℓ) +0.939 ± 0.001 +0.944 ± 0.004 +0.941 ± 0.001 +0.941 ± 0.001 +Rµ/e(D+ → ρ0ℓ+νℓ) +0.950 ± 0.001 +0.956 ± 0.005 +0.952 ± 0.001 +0.952 ± 0.001 +Rµ/e(D+ → ωℓ+νℓ) +0.950 ± 0.001 +0.956 ± 0.005 +0.952 ± 0.001 +0.952 ± 0.001 +Rµ/e(D+ → φℓ+νℓ) +0.923 ± 0.001 +0.928 ± 0.005 +0.925 ± 0.001 +0.925 ± 0.001 +Rµ/e(D0 → K∗−ℓ+νℓ) +0.939 ± 0.001 +0.944 ± 0.004 +0.941 ± 0.001 +0.941 ± 0.001 +Rµ/e(D0 → ρ−ℓ+νℓ) +0.950 ± 0.001 +0.956 ± 0.005 +0.952 ± 0.001 +0.952 ± 0.001 +Rµ/e(D+ +s → φℓ+νℓ) +0.937 ± 0.001 +0.942 ± 0.004 +0.939 ± 0.001 +0.939 ± 0.001 +Rµ/e(D+ +s → ωℓ+νℓ) +0.957 ± 0.001 +0.963 ± 0.004 +0.959 ± 0.001 +0.959 ± 0.001 +Rµ/e(D+ +s → K∗0ℓ+νℓ) +0.949 ± 0.001 +0.955 ± 0.005 +0.951 ± 0.001 +0.951 ± 0.001 +uncertainties are canceled in the ratios, these predictions are very accurate. These predictions are similar to each +other in the four cases, and we only list the results in the C3 case in Tabs. VI-VII for examples. One can see that +the predictions are obviously different between two ways of q2 integration, and they are also quite different between +D → V e+νe and D → V µ+νµ decays. +The differential observables of D+ +s → K∗0ℓ+νℓ decays in the C3 case are displayed in Fig. 4. One can see that, +in the low q2 ranges, the differential observables expect dFL(D+ +s → K∗0ℓ+νℓ)/dq2 are obviously different between +decays with ℓ = e and ℓ = µ. + +15 +C +3 +C +4 +dB(D ++ +s +K +*0 ++ +)/dq +2 + ( x10 +-3 + ) +q +2 +FIG. 3: The q2 dependence of the differential branching ratios for some not yet measured D → V µ+νµ decays with present +experimental bounds. +0.0 +0.4 +0.8 +1.2 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0.0 +0.0 +0.4 +0.8 +1.2 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +0.0 +0.4 +0.8 +1.2 +0.0 +0.4 +0.8 +1.2 +0.0 +0.4 +0.8 +1.2 +-6 +-4 +-2 +0 +0.0 +0.4 +0.8 +1.2 +0.0 +0.4 +0.8 +1.2 +0.0 +0.4 +0.8 +1.2 +0.0 +0.4 +0.8 +1.2 +( f ) +( e ) +( d ) +( c ) +( b ) +( a ) + + + e + +dA +FB +(D ++ +s +K +*0 +l ++ +l +)/dq +2 +q +2 + + + e + +dC +l +F +(D ++ +s +K +*0 +l ++ +l +)/dq +2 +q +2 + + + e + +dP +l +L +(D ++ +s +K +*0 +l ++ +l +)/dq +2 +q +2 + + + e: in unit of 10 +-3 + +dP +l +T +(D ++ +s +K +*0 +l ++ +l +)/dq +2 +q +2 + + + e + +dA +(D ++ +s +K +*0 +l ++ +l +)/dq +2 +q +2 + + + e + +dF +L +(D ++ +s +K +*0 +l ++ +l +)/dq +2 +q +2 +FIG. 4: The differential forward-backward asymmetries, differential lepton-side convexity parameters, differential longitudinal +lepton polarizations and differential transverse lepton polarizations for the D+ +s → K0ℓ+νℓ decays in the C3 case. + +d. +S +0'4 +a.0 +8.0cqB(D) +0.0 +8.0 +0 +0 +2 +8←S +3(p) +d +S +a. +e.0 +S.1C2. +0.0 +8.0 +0 +H +0 +X +x"S +3(c) +a.0 +e.000 +S.0 +qB(D) +S(0rx) "pbl( +7 +e16 +TABLE VI: The forward-backward asymmetries Aℓ +F B, the lepton-side convexity parameters Cℓ +F , the longitudinal polarizations +P ℓ +L of the D → V ℓ+ν decays in the C3 case. +Decay modes +⟨Aℓ +F B⟩ +Aℓ +F B +⟨Cℓ +F ⟩ +Cℓ +F +⟨P ℓ +L⟩ +P ℓ +L +D+ → K +∗0e+νe +−0.125 ± 0.006 +−0.190 ± 0.020 +−1.046 ± 0.019 +−0.500 ± 0.032 +0.786 ± 0.004 +1.000 +D+ → ρ0e+νe +−0.130 ± 0.008 +−0.222 ± 0.024 +−1.052 ± 0.023 +−0.496 ± 0.041 +0.789 ± 0.004 +1.000 +D+ → ωe+νe +−0.130 ± 0.008 +−0.220 ± 0.024 +−1.052 ± 0.023 +−0.497 ± 0.041 +0.789 ± 0.004 +1.000 +D+ → φe+νe +−0.121 ± 0.005 +−0.164 ± 0.017 +−1.037 ± 0.015 +−0.500 ± 0.025 +0.784 ± 0.003 +1.000 +D0 → K∗−e+νe +−0.125 ± 0.006 +−0.191 ± 0.020 +−1.046 ± 0.019 +−0.500 ± 0.032 +0.786 ± 0.004 +1.000 +D0 → ρ−e+νe +−0.130 ± 0.008 +−0.221 ± 0.024 +−1.052 ± 0.023 +−0.497 ± 0.041 +0.789 ± 0.004 +1.000 +D+ +s → φe+νe +−0.122 ± 0.006 +−0.176 ± 0.018 +−1.043 ± 0.016 +−0.500 ± 0.028 +0.786 ± 0.003 +1.000 +D+ +s → ωe+νe +−0.130 ± 0.008 +−0.229 ± 0.025 +−1.057 ± 0.025 +−0.496 ± 0.044 +0.790 ± 0.004 +1.000 +D+ +s → K∗0e+νe +−0.128 ± 0.007 +−0.207 ± 0.022 +−1.049 ± 0.021 +−0.495 ± 0.036 +0.789 ± 0.004 +1.000 +D+ → K +∗0µ+νµ +−0.284 ± 0.009 +−0.226 ± 0.019 +−0.466 ± 0.021 +−0.395 ± 0.028 +0.514 ± 0.017 +0.886 ± 0.002 +D+ → ρ0µ+νµ +−0.292 ± 0.011 +−0.252 ± 0.023 +−0.491 ± 0.027 +−0.405 ± 0.037 +0.524 ± 0.020 +0.903 ± 0.002 +D+ → ωµ+νµ +−0.292 ± 0.011 +−0.251 ± 0.022 +−0.490 ± 0.027 +−0.405 ± 0.037 +0.524 ± 0.020 +0.902 ± 0.002 +D+ → φµ+νµ +−0.277 ± 0.008 +−0.206 ± 0.016 +−0.433 ± 0.016 +−0.376 ± 0.021 +0.503 ± 0.014 +0.864 ± 0.002 +D0 → K∗−µ+νµ +−0.284 ± 0.009 +−0.226 ± 0.019 +−0.466 ± 0.021 +−0.395 ± 0.029 +0.514 ± 0.017 +0.886 ± 0.002 +D0 → ρ−µ+νµ +−0.292 ± 0.011 +−0.252 ± 0.023 +−0.490 ± 0.027 +−0.405 ± 0.037 +0.524 ± 0.020 +0.902 ± 0.002 +D+ +s → φµ+νµ +−0.277 ± 0.008 +−0.213 ± 0.017 +−0.459 ± 0.018 +−0.391 ± 0.024 +0.514 ± 0.015 +0.882 ± 0.002 +D+ +s → ωµ+νµ +−0.291 ± 0.012 +−0.257 ± 0.024 +−0.509 ± 0.029 +−0.414 ± 0.041 +0.531 ± 0.021 +0.913 ± 0.002 +D+ +s → K∗0µ+νµ +−0.286 ± 0.010 +−0.239 ± 0.021 +−0.485 ± 0.024 +−0.402 ± 0.033 +0.525 ± 0.018 +0.900 ± 0.002 +TABLE VII: The transverse polarizations P ℓ +T , the lepton spin asymmetries Aλ and the longitudinal polarization fractions of +the final vector mesons FL of the D → V ℓ+ν decays in the C3 case. +Decay modes +⟨P ℓ +T ⟩ +P e +T (×10−3) +P µ +T +⟨Aλ⟩ +Aλ +⟨FL⟩ +FL +D+ → K +∗0e+νe +−0.251 ± 0.004 +−1.205 ± 0.066 +1.000 +1.000 +0.905 ± 0.010 +0.556 ± 0.014 +D+ → ρ0e+νe +−0.249 ± 0.005 +−1.040 ± 0.072 +1.000 +1.000 +0.907 ± 0.012 +0.554 ± 0.018 +D+ → ωe+νe +−0.249 ± 0.005 +−1.049 ± 0.073 +1.000 +1.000 +0.907 ± 0.012 +0.554 ± 0.018 +D+ → φe+νe +−0.254 ± 0.003 +−1.417 ± 0.061 +1.000 +1.000 +0.902 ± 0.008 +0.556 ± 0.011 +D0 → K∗−e+νe +−0.251 ± 0.004 +−1.206 ± 0.067 +1.000 +1.000 +0.905 ± 0.010 +0.556 ± 0.014 +D0 → ρ−e+νe +−0.249 ± 0.005 +−1.045 ± 0.073 +1.000 +1.000 +0.907 ± 0.012 +0.554 ± 0.018 +D+ +s → φe+νe +−0.251 ± 0.004 +−1.255 ± 0.060 +1.000 +1.000 +0.904 ± 0.009 +0.555 ± 0.012 +D+ +s → ωe+νe +−0.247 ± 0.005 +−0.953 ± 0.071 +1.000 +1.000 +0.908 ± 0.013 +0.554 ± 0.020 +D+ +s → K∗0e+νe +−0.248 ± 0.004 +−1.075 ± 0.066 +1.000 +1.000 +0.905 ± 0.011 +0.553 ± 0.016 +D+ → K +∗0µ+νµ +−0.454 ± 0.022 +−0.156 ± 0.012 +0.935 ± 0.005 +0.928 ± 0.002 +0.775 ± 0.019 +0.557 ± 0.014 +D+ → ρ0µ+νµ +−0.452 ± 0.026 +−0.139 ± 0.014 +0.944 ± 0.006 +0.937 ± 0.002 +0.782 ± 0.023 +0.555 ± 0.018 +D+ → ωµ+νµ +−0.452 ± 0.026 +−0.140 ± 0.014 +0.944 ± 0.006 +0.937 ± 0.002 +0.782 ± 0.023 +0.555 ± 0.018 +D+ → φµ+νµ +−0.455 ± 0.018 +−0.175 ± 0.011 +0.924 ± 0.005 +0.915 ± 0.002 +0.763 ± 0.015 +0.557 ± 0.011 +D0 → K∗−µ+νµ +−0.454 ± 0.022 +−0.156 ± 0.012 +0.935 ± 0.005 +0.927 ± 0.002 +0.775 ± 0.019 +0.557 ± 0.014 +D0 → ρ−µ+νµ +−0.452 ± 0.026 +−0.140 ± 0.014 +0.944 ± 0.006 +0.937 ± 0.002 +0.782 ± 0.023 +0.555 ± 0.018 +D+ +s → φµ+νµ +−0.454 ± 0.019 +−0.162 ± 0.011 +0.934 ± 0.005 +0.925 ± 0.002 +0.771 ± 0.016 +0.557 ± 0.012 +D+ +s → ωµ+νµ +−0.452 ± 0.027 +−0.131 ± 0.014 +0.950 ± 0.005 +0.943 ± 0.002 +0.788 ± 0.024 +0.555 ± 0.019 +D+ +s → K∗0µ+νµ +−0.451 ± 0.023 +−0.143 ± 0.012 +0.943 ± 0.005 +0.936 ± 0.002 +0.779 ± 0.021 +0.555 ± 0.016 + +17 +C. +D → Sℓ+νℓ decays +For D → Sℓ+νℓ decays, the two quark and the four quark scenarios for the scalar mesons below or near 1 GeV are +considered. The hadronic helicity amplitudes for the D → Sℓ+νℓ decays are given in Tab. VIII, in which the CKM +matrix element Vcs and Vcd information are kept for comparing conveniently. There are four (five) nonperturbative +parameters E1,2,3,4 (E′ +1,2,3,4,5) in the two quark (four quark) picture. +After ignoring the SU(3) flavor breaking +contributions, only one nonperturbative parameter E1 = E2 = E3 = E4 = cS +0 or E′ +1 = E′ +2 = E′ +3 = E′ +4 = E′ +5 = c′S +0 +relates all decay amplitudes in the two quark or the four quark picture, respectively. +Unlike many measured decay modes in the D → Pℓ+νℓ and D → V ℓ+νℓ decays, among these D → Sℓ+νℓ decays, +only D+ +s → f0(980)e+νe decay has been measured, and its branching ratio with 2σ errors is [1] +B(D+ +s → f0(980)e+νe) = (2.3 ± 0.8) × 10−3. +(48) +In addition, the branching ratios of the D → P1P2ℓ+νℓ decays with the light scalar resonances can be obtained by +using B(D → Sℓ+νℓ) and B(S → P1P2), and the detail analysis can been found in Ref. [80]. Five branching ratios +TABLE VIII: The hadronic helicity amplitudes for D → Sℓ+ν decays including both the SU(3) flavor symmetry and the SU(3) +flavor breaking contributions. In the two quark picture of the scalar mesons, E1 ≡ cS +0 + cS +1 − 2cS +2 , E2 ≡ cS +0 − 2cS +1 − 2cS +2 , +E3 ≡ cS +0 + cS +1 + cS +2 , E4 ≡ cS +0 − 2cS +1 + cS +2 . E1 = E2 = E3 = E4 = cS +0 if neglecting the SU(3) flavor breaking cS +1 and cS +2 terms. In +the four quark picture of the scalar mesons, E′ +1 ≡ c′S +0 + c′S +1 − 2c′S +2 + c′S +3 , E′ +2 ≡ c′S +0 − 2c′S +1 − 2c′S +2 + c′S +3 , E′ +3 ≡ c′S +0 + c′S +1 + c′S +2 − 2c′S +3 , +E′ +4 ≡ c′S +0 + c′S +1 + c′S +2 + c′S +3 , E′ +5 ≡ c′S +0 − 2c′S +1 + c′S +2 + c′S +3 , E′ +1 = E′ +2 = E′ +3 = E′ +4 = E′ +5 = c′S +0 if neglecting the SU(3) flavor breaking +c′S +1 , c′S +2 and c′S +3 terms. +Hadronic helicity amplitudes +ones for two-quark scenario +ones for four-quark scenario +H(D0 → K− +0 ℓ+νℓ) +E1V ∗ +cs +E′ +1V ∗ +cs +H(D+ → K +0 +0ℓ+νℓ) +E1V ∗ +cs +E′ +1V ∗ +cs +H(D+ +s → f0ℓ+νℓ) +E2V ∗ +cs +√ +2E′ +2V ∗ +cs +H(D+ +s → f0(980)ℓ+νℓ) +cosθS E2V ∗ +cs +√ +2cosφS E′ +2V ∗ +cs +H(D+ +s → f0(500)ℓ+νℓ) +−sinθS E2V ∗ +cs +− +√ +2sinφS E′ +2V ∗ +cs +H(D0 → a− +0 ℓ+νℓ) +E3V ∗ +cd +E′ +3V ∗ +cd +H(D+ → a0 +0ℓ+νℓ) +− 1 +√ +2E3V ∗ +cd +− 1 +√ +2E′ +3V ∗ +cd +H(D+ → f0ℓ+νℓ) +0 +1 +√ +2E′ +3V ∗ +cd +H(D+ → σℓ+νℓ) +1 +√ +2E3V ∗ +cd +E′ +4V ∗ +cd +H(D+ → f0(980)ℓ+νℓ) +1 +√ +2sinθS E3V ∗ +cd +( 1 +√ +2E′ +3cosφS + E′ +4sinφS)V ∗ +cd +H(D+ → f0(500)ℓ+νℓ) +1 +√ +2cosθS E3V ∗ +cd +(− 1 +√ +2E′ +3sinφS + E′ +4cosφS)V ∗ +cd +H(D+ +s → K0 +0ℓ+νℓ) +E4V ∗ +cd +E′ +5V ∗ +cd + +18 +and two upper limits of B(D → Sℓ+νℓ, S → P1P2) have been measured, and the data within 2σ errors are +B(D+ +s → f0(980)e+νe, f0(980) → π+π−) = (1.30 ± 0.63) × 10−3 [81], +B(D+ +s → f0(980)e+νe, f0(980) → π0π0) = (7.9 ± 2.9) × 10−4 [82], +B(D0 → a0(980)−e+νe, a0(980)− → ηπ−) = (1.33+0.68 +−0.60) × 10−4 +[1], +B(D+ → a0(980)0e+νe, a0(980)0 → ηπ0) = (1.7+1.6 +−1.4) × 10−4 +[1], +B(D+ → f0(500)e+νe, f0(500) → π+π−) = (6.3 ± 1.0) × 10−4 +[1], +B(D+ → f0(980)e+νe, f0(980) → π+π−) < 2.8 × 10−5 +[83], +B(D+ +s → f0(500)e+νe, f0(500) → π0π0) < 6.4 × 10−4 [82]. +(49) +Two cases S1 and S2 will be considered in the D → Sℓ+νℓ decays. +In S1 case, only experimental datum of +B(D+ +s → f0(980)e+νe) is used to constrain one parameter cS +0 or c′S +0 and then predict other not yet measured branching +ratios. The numerical results of B(D → Sℓ+ν) in S1 case are given in the 2-4th and 8th columns of Tab. IX. In the +S2 case, the experimental data of both B(D+ +s → f0(980)e+νe) in Eq. (48) and B(D → Sℓ+νℓ, S → P1P2) in Eq. (49) +will be used to constrain the parameter cS +0 or c′S +0 . The predictions of B(D → Sℓ+ν) in S2 case are listed in the 5-7th +and 9th columns of Tab. IX. Our comments on the results in the S1,2 cases are as follows. +• Results in the two quark picture: In the two quark picture, the three possible ranges of the mixing angle, +25◦ < θS < 40◦, 140◦ < θS < 165◦ and −30◦ < θS < 30◦ [58, 68] have been analyzed. In S1 case, using the +data of B(D+ +s → f0(980)e+νe), many predictions of B(D → Sℓ+ν) are obtained. As given in the 2-4th columns +of Tab. IX, one can see that the predictions with 25◦ < θS < 40◦ are similar to ones with 140◦ < θS < 165◦, +the predictions with −30◦ < θS < 30◦ are slightly different from the first two, and the errors of predictions are +quite large. After adding the experimental bounds of B(D → Sℓ+νℓ, S → P1P2), as given in the 5-7th columns +of Tab. IX, the three possible ranges of the mixing angle θS are obviously constrained, and they reduce to +25◦ < θS < 35◦, 144◦ < θS < 158◦ and 22◦ ≤ |θS| ≤ 30◦, respectively. In addition, the error of every prediction +become smaller by adding the experimental bounds of B(D → Sℓ+νℓ, S → P1P2). +• Results in the four quark picture: The predictions in the four quark picture are listed in the 8-9th columns +of Tab. IX. The majority of predictions in four quark picture are smaller than corresponding ones in two quark +picture. Strong coupling constants g′ +4 and g4 are appeared in S → P1P2 decays with the four quark picture +of light scalar mesons. At present, we only can determine +�� g′ +4 +g4 +�� from the S → P1P2 decays. The results of +involved decays with both g′ +4 +g4 > 0 and g′ +4 +g4 < 0 are given in the 9th column of Tab. IX, and one can see that, +except B(D+ +s → f0(500)e+νe) and B(D+ +s → f0(980)µ+νµ), the other involved branching ratios are not obviously +affected by the choice of g′ +4 +g4 > 0 or g′ +4 +g4 < 0. The errors of the branching ratio predictions are obviously reduced +by the experimental bounds of B(D → Sℓ+νℓ, S → P1P2). +• Comparing with previous predictions: Previous predictions are listed in the last column of Tab. +IX. +B(D+ +s → f0(500)e+νe), B(D+ +s → f0(500)µ+νµ) and B(D+ → f0(500)µ+νµ) are predicted for the first time. Our +predictions of B(D+ +s → f0(980)µ+νµ), B(D+ → a0 +0e+νe), B(D+ → f0(980)e+νe), B(D+ → f0(500)e+νe) and +B(D+ → a0 +0µ+νµ) are consistent with previous predictions in Refs. [78, 84, 85]. Our other predictions are about +one order smaller or one order larger than previous ones in Refs. [67, 86]. + +19 +TABLE IX: +Branching ratios of D → Sℓ+ν decays within 2σ errors. As given in Ref. [80], g′ +4 and g4 are strong coupling constants obtained by the SU(3) flavor +symmetry in S → P1P2 decays, adenotes the results with +g′ +4 +g4 > 0, and bdenotes ones with +g′ +4 +g4 < 0, †denotes the results with two quark picture, and ‡denotes the results +with four quark picture. +Branching ratios +ones for 2q state in S1 +ones for 2q state in S2 +ones for 4q +ones for 4q +Previous ones +[25◦, 40◦] +[140◦, 165◦] +[−30◦, 30◦] +[25◦, 35◦] +[144◦, 158◦] +22◦ ≤ |θS| ≤ 30◦ +state in S1 +state in S2 +B(D0 → K− +0 e+νe)(×10−3) +3.38 ± 2.12 +3.18 ± 2.05 +2.57 ± 1.58 +3.02 ± 1.11 +3.00 ± 1.10 +2.98 ± 1.05 +1.11 ± 0.63 +1.25 ± 0.45 +0.103 ± 0.115† [67] +B(D+ → K +0 +0e+νe)(×10−3) +8.66 ± 5.55 +7.99 ± 5.02 +7.02 ± 4.48 +7.74 ± 2.88 +7.78 ± 2.77 +7.68 ± 2.78 +2.85 ± 1.65 +3.36 ± 1.25 +38.8 ± 5.6† [67] +B(D+ +s → f0(980)e+νe)(×10−3) +2.30 ± 0.80 +2.30 ± 0.80 +2.30 ± 0.80 +2.58 ± 0.52 +2.57 ± 0.53 +2.71 ± 0.39 +2.30 ± 0.80 +2.49±0.61a +2.54±0.56b +2.1 ± 0.2† [78], 2+0.5† +−0.4 +[84] +B(D+ +s → f0(500)e+νe)(×10−3) +6.73 ± 6.11 +5.98 ± 5.75 +3.25 ± 3.25 +1.49 ± 0.43 +1.45 ± 0.46 +1.42 ± 0.50 +0.37 ± 0.37 +0.31±0.31a +0.17±0.17b +B(D0 → K− +0 µ+νµ)(×10−3) +2.90 ± 1.84 +2.73 ± 1.77 +2.20 ± 1.36 +2.59 ± 0.97 +2.57 ± 0.96 +2.56 ± 0.92 +0.95 ± 0.54 +1.09 ± 0.39 +0.103 ± 0.115† [67] +B(D+ → K +0 +0µ+νµ)(×10−3) +7.46 ± 4.81 +6.87 ± 4.33 +6.04 ± 3.88 +6.65 ± 2.52 +6.69 ± 2.43 +6.59 ± 2.43 +2.45 ± 1.43 +2.89 ± 1.09 +38.8 ± 5.6† [67] +B(D+ +s → f0(980)µ+νµ)(×10−3) +1.95 ± 0.70 +1.95 ± 0.70 +1.95 ± 0.69 +2.20 ± 0.45 +2.20 ± 0.45 +2.32 ± 0.33 +1.95 ± 0.70 +2.12±0.54a +2.16±0.49b +2.1 ± 0.2† [78] +B(D+ +s → f0(500)µ+νµ)(×10−3) +6.21 ± 5.66 +5.53 ± 5.32 +3.01 ± 3.01 +1.33 ± 0.39 +1.31 ± 0.43 +1.28 ± 0.46 +0.34 ± 0.34 +0.29±0.29a +0.16±0.16b +B(D0 → a− +0 e+νe)(×10−5) +9.99 ± 6.54 +9.56 ± 6.50 +8.34 ± 5.67 +9.22 ± 3.98 +9.09 ± 3.65 +9.17 ± 3.58 +3.42 ± 2.06 +4.32 ± 1.17 +16.8±1.5† [78], 40.8+13.7† +−12.2 [86], +24.4±3.0† [67] +B(D+ → a0 +0e+νe)(×10−5) +13.09 ± 8.62 +12.62 ± 8.67 +10.89 ± 7.35 +12.09 ± 5.19 +11.81 ± 4.71 +11.97 ± 4.66 +4.49 ± 2.71 +5.68 ± 1.52 +21.8±3.8† [78], 54.0+17.8† +−15.9 [86] +6∼8†[85], 5∼5.4‡[85] +B(D+ → f0(980)e+νe)(×10−5) +3.92 ± 2.92 +3.48 ± 3.13 +1.59 ± 1.59 +2.62 ± 0.82 +2.52 ± 0.94 +2.40 ± 0.80 +3.14 ± 1.98 +3.35±1.80a +3.89±1.35b +7.78±0.68† [78], 5.7±1.3† [87] +0.4∼3.5†[85], 1.9∼6.3‡[85] +B(D+ → f0(500)e+νe)(×10−4) +4.05 ± 3.20 +4.08 ± 3.10 +4.21 ± 3.28 +2.16 ± 0.96 +2.59 ± 1.38 +2.70 ± 1.28 +4.97 ± 4.13 +4.97±3.34a +4.95±3.36b +0.4 ∼ 0.6†[85], 0.88 ∼ 1.4‡[85] +B(D+ +s → K0 +0e+νe)(×10−4) +3.73 ± 2.37 +3.41 ± 2.13 +2.99 ± 1.88 +3.35 ± 1.21 +3.32 ± 1.20 +3.35 ± 1.15 +1.25 ± 0.71 +1.43 ± 0.51 +26.5 ± 2.8† [67] +B(D0 → a− +0 µ+νµ)(×10−5) +8.25 ± 5.45 +7.89 ± 5.42 +6.91 ± 4.75 +7.61 ± 3.37 +7.51 ± 3.10 +7.57 ± 3.04 +2.83 ± 1.72 +3.57 ± 0.99 +16.3 ± 1.4† [78], 24.4 ± 3.0† [67] +B(D+ → a0 +0µ+νµ)(×10−5) +10.83 ± 7.19 +10.44 ± 7.23 +9.04 ± 6.16 +10.00 ± 4.41 +9.76 ± 4.00 +9.89 ± 3.97 +3.73 ± 2.28 +4.69 ± 1.30 +21.2 ± 3.7† [78] +B(D+ → f0(980)µ+νµ)(×10−5) +3.23 ± 2.41 +2.88 ± 2.60 +1.32 ± 1.32 +2.15 ± 0.70 +2.09 ± 0.78 +1.99 ± 0.66 +2.56 ± 1.62 +2.74±1.49a +3.20±1.14b +7.87 ± 0.67† [78] +B(D+ → f0(500)µ+νµ)(×10−4) +3.69 ± 2.96 +3.71 ± 2.86 +3.84 ± 3.04 +1.92 ± 0.88 +2.32 ± 1.27 +2.42 ± 1.19 +4.54 ± 3.81 +4.52±3.10a +4.49±3.12b +B(D+ +s → K0 +0µ+νµ)(×10−4) +3.28 ± 2.10 +3.00 ± 1.88 +2.62 ± 1.66 +2.94 ± 1.08 +2.91 ± 1.06 +2.94 ± 1.02 +1.10 ± 0.63 +1.26 ± 0.45 +26.5 ± 2.8† [67] + +20 +IV. +Summary +Many semileptonic D → P/V/Sℓ+νℓ decays have been measured, and these processes could be used to test the +SU(3) flavor symmetry approach. In terms of the SU(3) flavor symmetry and the SU(3) flavor breaking, the amplitude +relations have been obtained. Then using the present data of B(D → P/V/Sℓ+νℓ), we have presented a theoretical +analysis of the D → P/V/Sℓ+νℓ decays. Our main results can be summarized as follows. +• D → Pℓ+νℓ decays: Our predictions with the SU(3) flavor symmetry in the C1 case and the predictions after +adding SU(3) flavor breaking contributions in the C4 case are quite consistent with all present experimental data +of B(D → Pℓ+νℓ) within 2σ errors. In the C2 and C3 cases, our SU(3) flavor symmetry predictions are consistent +with all present experimental data except B(D+ → π0ℓ+νℓ) and B(D0 → π−ℓ+νℓ), which are slight larger +than their experiential upper limits. The not yet measured B(D+ +s → π0e+νe), B(D+ → η′µ+νµ), B(D+ +s → +K0µ+νµ), B(D+ +s +→ π0µ+νµ), B(D+ +s +→ π0τ +ντ) and the lepton flavor universality parameters have been +obtained. Moreover, the forward-backward asymmetries, the lepton-side convexity parameters, the longitudinal +(transverse) polarizations of the final charged leptons with two ways of integration for the D → Pℓ+νℓ decays +have been predicted. The q2 dependence of corresponding differential quantities of the D → Pℓ+νℓ decays in +the C3 case have been displayed. +• D → V ℓ+νℓ decays: As given in the C1, C2 and C3 cases, our SU(3) flavor symmetry predictions of B(D+ → +ωe+νe) and B(D0 → ρ−µ+νµ) are slightly larger than its experimental upper limits, and other SU(3) flavor +symmetry predictions are consistent with present data. After considering the SU(3) flavor breaking effects, as +given in the C4 case, all predictions are consistent with present data. The not yet measured or not yet well +measured branching ratios of D+ → φe+νe, D+ +s → ωe+νe, D+ → φµ+νµ, D+ +s → ωµ+νµ, and D+ +s → K∗0µ+νµ +have been predicted. The q2 dependence of corresponding differential quantities of the D → V ℓ+νℓ decays in +the C3 case have also been displayed. +• D → Sℓ+νℓ decays: +Among 18 D → Sℓ+νℓ decay modes, only B(D+ +s → f0(980)e+νe) has been measured, and +this experimental datum has been used to constrain the SU(3) flavor symmetry parameter and then predict other +not yet measured branching ratios. Furthermore, the relevant experimental bounds of B(D → Sℓ+νℓ, S → P1P2) +have also been added. The two quark and the four quark scenarios for the light scalar mesons are considered, +and the three possible ranges of the mixing angle θS in the two quark picture have been analyzed. +The SU(3) flavor symmetry is approximate approach, and it can still provide very useful information. 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D 80 (2009), 074030 [arXiv:0907.5465 [hep-ph]]. + diff --git a/DdAzT4oBgHgl3EQfif0c/content/tmp_files/2301.01499v1.pdf.txt b/DdAzT4oBgHgl3EQfif0c/content/tmp_files/2301.01499v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..19ca75defc8785c4aac06b1b77ec29eca3e14049 --- /dev/null +++ b/DdAzT4oBgHgl3EQfif0c/content/tmp_files/2301.01499v1.pdf.txt @@ -0,0 +1,1739 @@ +Received: Added at production +Revised: Added at production +Accepted: Added at production +DOI: xxx/xxxx +ARTICLE TYPE +Thermodynamic and transport properties of plasmas: low-density +benchmarks +G. Röpke +Institut für Physik, Universität Rostock, +18051 Rostock, Germany +Correspondence +Email: gerd.roepke@uni-rostock.de +Abstract +Physical properties of plasmas such as equations of state and transport coefficients +are expressed in terms of correlation functions, which can be calculated using various +approaches (analytical theory, numerical simulations). The method of Green’s func- +tions provides benchmark values for these properties in the low-density limit. For +the equation of state and electrical conductivity, expansions with respect to density +(virial expansions) are considered. Comparison of analytical results with numerical +simulations is used to verify theory, to prove the accuracy of simulations, and to +establish interpolation formulas. +KEYWORDS: +plasma equation of state, electrical conductivity, virial expansion, DFT-MD simulations, PIMC simula- +tions +1 +PLASMA PROPERTIES AND CORRELATION FUNCTIONS +Plasmas consist of charged particles, number 푁푖 of species 푖 in the volume Ω, which interact via the Coulomb law. If we denote +the charge of the component 푖 by 푍푖푒, we obtain (휖0 is the permittivity of the vacuum) +푉 Coul +푖푗 +(푟) = +푍푖푍푗푒2 +4휋휖0푟 . +(1) +In general, an additional short-range interaction may occur. Examples are the homogeneous electron gas (uniform electron +gas UEG), where the electrons move over a positively charged background to realize charge neutrality, or the two-component +Hydrogen plasma, consisting of electrons and protons, where the particle density is 푛푒 = 푛푝 to maintain charge neutrality. In +thermodynamic equilibrium, the state of the plasma is determined by the temperature 푇 in addition to the densities 푛푖 = 푁푖∕Ω +of the components or the corresponding chemical potentials 휇푖. The relationships between the various state variables such as +internal energy 푈, free energy 퐹, entropy 푆, pressure 푃 , etc. are called equations of state (EoS). All thermodynamic properties +can be derived from a thermodynamic potential, for example is 퐹(Ω, 푁푖, 푇 ) as function of Ω, 푁푖, 푇 a thermodynamic potential. +Statistical physics allows to calculate the thermodynamic properties from the microscopic properties, i.e. from the Hamiltonian +퐻 = 퐻kin + 푉 , with kinetic energy 퐻kin = ∑ +푖 +∑푁푖 +푘 푝2 +푖,푘∕2푚푖 and potential energy 푉 = (1∕2) ∑ +푖,푘≠푗,푙 푉 (퐫푖,푘 − 퐫푗,푙). To calculate +physical quantities, various expressions can be used. For example, for classical systems we can start from the well-known +partition function 푍can(Ω, 푁푖, 푇 ) with 퐹(Ω, 푁푖, 푇 ) = −푘퐵푇 ln 푍can(Ω, 푁푖, 푇 ). For quantum systems, it is convenient to work +with the grand canonical ensemble defined by 훽 = 1∕푘퐵푇 and the chemical potentials 휇푖. Then the single-particle distribution +functions for the ideal quantum system (푉 = 0) have a simple form, the Fermi or Bose distribution. In second quantization, +we introduce 푎+ +푖,푘, 푎푖,푘 as a creation or annihilation operator for particles of species 푖 in the quantum state 푘 = {ℏ퐤, 휎}, which +arXiv:2301.01499v1 [physics.plasm-ph] 4 Jan 2023 + +2 +denotes momentum vector and spin. The occupation number of this quantum state is given as +푓푖,푘 = ⟨푎+ +푖,푘푎푖,푘⟩ = +1 +푍gr.can +Tr +{ +푒−훽(퐻−∑ +푖 휇푖푁푖)푎+ +푖,푘푎푖,푘 +} +, +푍gr.can = Tr푒−훽(퐻−∑ +푖 휇푖푁푖) +(2) +with 퐻 as the Hamiltonian in second quantization and 푁푖 = ∑ +푘 푎+ +푖,푘푎푖,푘. The relation between the densities and the chemical +potentials is given as follows. +푛푖(푇 , {휇푗}) = 1 +Ω⟨푁푖⟩ = 1 +Ω +∑ +푘 +푓푖,푘. +(3) +It is convenient to introduce a 휏-dependent correlation function as a generalization of (2) that contains dynamic information, +⟨푎+ +푗,푙푒휏(퐻−∑ +푖 휇푖푁푖)푎푖,푘푒−휏(퐻−∑ +푖 휇푖푁푖)⟩ = +∞ +∫ +−∞ +푑휔 +2휋 푒−휔휏퐼푖푘,푗푙(휔). +(4) +The spectral density 퐼푖푘,푗푙(휔) is related to the spectral function 퐴푖푘,푗푙(휔) = (1+푒훽휔)퐼푖푘,푗푙(휔) (Fermi statistics). An exact expression +for the EoS is found if the spectral function is known, +푛푖(푇 , {휇푗}) = 1 +Ω +∑ +푘 +∞ +∫ +−∞ +푑휔 +2휋 +1 +푒훽휔 + 1퐴푖푘,푖푘(휔). +(5) +The spectral function which is diagonal with respect to {푖, 푘} for a homogeneous system, is related to the self-energy Σ푖,푘(푧) for +which a systematic evaluation applying diagram techniques is possible, see1,2: +퐴푖푘(휔) = +2ImΣ푖,푘(휔 − 푖0) +[휔 − 휖푖,푘 − ReΣ푖,푘(휔)]2 + [ImΣ푖,푘(휔 − 푖0)]2 , +(6) +휖푖,푘 = ℏ2푘2∕2푚푖 − 휇푖 is the kinetic energy shifted by the chemical potential. +The electrical conductivity 휎(푇 , 푛) of low-density plasmas was first calculated in the framework of kinetic theory. In a seminal +work3, Spitzer and Härm determined 휎 of the fully ionized Hydrogen plasma by solving a Fokker-Planck equation. To calculate +휎(푇 , 푛) in a wide range of temperature 푇 and particle density 푛, a quantum statistical many-particle theory is needed that +describes screening, correlations, and degeneracy effects in a systematic way. A generalized linear response theory4,5,6 has been +elaborated that expresses transport coefficients in terms of equilibrium correlation functions (fluctuation-dissipation theorems). +An example is the Kubo formula7 which relates the transport coefficient 휎 to the electron current-current correlation function, +휎(푇 , 푛) = +푒2 +푚2 +푒푘퐵푇 Ω⟨푃 ; 푃 ⟩푖휖 +(7) +with the total momentum of the electrons 푃 = ∑ +푘 ℏ푘푥푎+ +푒,푘푎푒,푘 in 푥 direction (the small ion contribution to the electrical current +may be added). The thermodynamic correlation function is the Laplace transform of the Kubo scalar product (the particle number +is assumed to commute with the observables), +⟨퐴; 퐵⟩푧 = +∞ +∫ +0 +푑푡 푒푖푧푡 1 +훽 +훽 +∫ +0 +푑휏⟨푒(푖∕ℏ)(푡−푖ℏ휏)퐻퐴푒−(푖∕ℏ)(푡−푖ℏ휏)퐻퐵⟩ . +(8) +For more details on generalized linear response theory and the evaluation of correlation functions using the method of thermo- +dynamic Green’s functions, see2. For the relationship between generalized linear response theory and kinetic theory, see8 and +references therein. +2 +EVALUATION OF CORRELATION FUNCTIONS +The properties of plasmas are expressed in terms of correlation functions in thermodynamic equilibrium. Examples are thermo- +dynamic properties (2) and transport properties (7). There are several methods to calculate these correlation functions. Exact +solutions are known only for ideal quantum gases where there is no interaction potential 푉 . The equations of state are known, +e.g., the pressure 푃 is expressed by Fermi integrals. At fixed temperature, the equation of state for ideal classical gases 푃 = 푛푘퐵푇 +is approximated by considering the limiting case of low density. For electrical conductivity, 휎 = ∞ is obtained because of +conservation of total momentum. The resistivity follows as 휌 = 1∕휎 = 0 for charged ideal Fermi gases. + +3 +Correlations appear for the plasma Hamiltonian with complete interaction 푉 . No closed-form solutions are known, and we +must perform approximations to solve this many-body problem. Here we discuss three possibilities: +1. Perturbation expansion with respect to 푉 . We obtain analytic expressions for arbitrary orders of 푉 in terms of nonin- +teracting equilibrium correlation functions, which can be easily evaluated using Wick’s theorem. However, we have no +proof of the convergence of this series expansion and no error estimate. In order to make this analytical approach more +efficient, the method of thermodynamic Green’s functions and Feynman diagram technique were elaborated1,2,9. Conver- +gence is improved by performing partial summations corresponding to special concepts such as the introduction of the +quasiparticle picture (self-energy Σ), screening of the potential (polarization function Π), or formation of bound states +(Bethe-Salpeter equation). This leads to useful results for the properties of the plasma in a wide range of 푇 and 푛. However, +as characteristic for perturbative approaches, exact results can be found only in some limiting cases. +2. This drawback is eliminated by numerical simulations of the correlation functions that apply to arbitrary interaction +strength. In Born-Oppenheimer approximation, density functional theory (DFT) for the electron system with given ion +configuration and molecular dynamics (MD) for the ion system are applied to evaluate the correlation functions. Single- +electron states are calculated by solving the Kohn-Sham equations. The total energy is obtained from the kinetic energy +of a non-interacting reference system, the classical electron-electron interaction, and an exchange-correlation energy that +includes, to a certain approximation, all unknown contributions. +The DFT-MD approach has been successfully applied to calculate the thermodynamic properties of complex materials in +a wide range of 푇 and 푛, which will not be reported here, see, e.g.,10,11,12,13 and the references given there. For electrical +conductivity (7), the Kubo-Greenwood formula7,14 +Re [휎(휔)] = +2휋푒2 +3푚2 +푒휔Ω +∑ +푘 +푤푘 +푁 +∑ +푗=1 +푁 +∑ +푖=1 +3 +∑ +훼=1 +[푓(휖푗,푘) − 푓(휖푖,푘)]|⟨Ψ푗,푘| ̂푝훼|Ψ푖,푘⟩|2훿(휖푖,푘 − 휖푗,푘 − ℏ휔) +(9) +was used to calculate the frequency-dependent dynamic electrical conductivity 휎(휔) in the long-wavelength +limit16,17,18,19,20,15. Kohn-Sham wave functions Ψ푖,푘 from density functional theory calculations are used to calculate the +transition matrix elements of the momentum operator ̂푝훼. The Fermi-Dirac distribution 푓(휖) accounts for the average +occupation at energy 휖, and the summation over momentum space 푘 contains the 푘-point weights 푤푘. +Due to the finite size of the simulation box, the delta function in equation (9) must be approximated by a finite-width +Gaussian, which also prevents the direct calculation of the dc conductivity at 휔 = 0. Therefore, the dynamic conductivity +is extrapolated to the limit 휔 → 0 by a Drude fit, +Re [휎(휔)] = +푛푒2휈 +휈2 + 휔2 , +(10) +where 휈 is the collision frequency. Thus, the calculated direct current conductivity depends on choosing the appropriate +width for the Gaussian and finding a suitable range for the Drude-fitting to 휎(휔) calculated from equation (9). The last +point can be improved by using a frequency-dependent collision frequency21. +One of the main shortcomings of the DFT-MD approach is that the many-particle interaction is replaced by a mean-field +potential. When using product wave functions for the many-electron system, correlations are excluded. The exchange- +correlation energy density functional reflects the Coulomb interaction to some approximation, e.g., as it exists in the +homogeneous electron gas, but becomes problematic in the low-density limit where correlations are important. +3. In principle, an accurate evaluation of equilibrium correlation functions is possible using path-integral Monte Carlo +(PIMC) simulations, see22,23,24 and references therein. The shortcomings of this approach at present are the relatively +small number of particles (a few dozen), the sign problem for fermions, and the computational challenges in accurately +computing path integrals. Instead of using an exchange-correlation energy density functional, 푒 − 푒 collisions are treated +accurately. However, at present accurate calculations have only been performed for the uniform electron gas model in +which the charge-compensating ion subsystem is replaced by a homogeneously charged jellium. The results presented +in25 are shown below in sec. 5. High-precision calculations for the two-component Hydrogen plasma would be of interest +for both thermodynamics and transport properties. + +4 +3 +GREEN’S FUNCTIONS AND FEYNMAN DIAGRAMS +In quantum statistics, the method of thermodynamic Green’s functions has been worked out to evaluate correlation functions in +thermodynamic equilibrium. For the ideal quantum gas, in which there is no interaction, all equilibrium correlation functions +can be calculated using Wick’s theorem. For plasmas, we can perform a power series expansion with respect to the interaction +strength according to the Dyson series. The terms of this perturbation expansion are represented by Feynman diagrams. +The problem of the perturbation expansion is that the convergence property remains open, and we cannot anticipate that for +the correlation functions a power series expansion with respect to the interaction strength is possible. A predetermined wrong +analytical behavior near the singular case of ideal gases leads to divergencies which are avoided performing partial summations +that can modify the analytic behavior. The most important partial summations are the quasiparticle concept associated with the +introduction of the self-energy, the screening associated with the introduction of the polarization function, and the introduction +of bound states performing partial summation of ladder diagrams. For instance, the Bethe-Salpeter equation for the two-particle +Green function in ladder approximation corresponds to the solution of the two-body problem. +From classical statistics, the Mayer cluster expansion is well known for short-range potentials is well known for the partition +function, and the virial expansion in powers in 푛 is obtained. Because of the long-range nature of the Coulomb potential, this +expansion in powers in 푛 is not possible for plasmas, the virial coefficients are divergent. Screening, i.e. partial summation of the +so-called ring diagrams in quantum statistics, solves this convergence problem, and the expansion in powers of 푛1∕2 is possible. +When considering the spectral function, the contribution of the free particles is replaced by the contribution of the quasiparticles, +with the energies containing the Debye shift. To obtain the thermodynamic potentials 퐹 or 푃 Ω from the equation of state (5) +we must perform integration over 휇 or 푛, respectively, and logarithmic terms may appear. In particular, for the free energy of +the Hydrogen plasma, the virial expansion reads +퐹(푇 , Ω, 푁) = Ω푘퐵푇 {푛 ln 푛 + [3∕2 ln(2휋ℏ2∕(푚푘퐵푇 )) − 1]푛 +−퐹0(푇 )푛3∕2 − 퐹1(푇 )푛2 ln 푛 − 퐹2(푇 )푛2 − 퐹3(푇 )푛5∕2 ln 푛 − 퐹4(푇 )푛5∕2 + (푛3 ln 푛)} . +(11) +see9,25 where expressions for the lowest virial coefficients 퐹푖 are also given. Details on the calculation of the EoS for Coulomb +systems can be found in Ref.9 and will not be repeated here. The virial expansion for the uniform electron gas is discussed below +in Sec. 5. +Perturbation expansion and partial summations also apply to the evaluation of the correlation function (7) which is related to +the electrical conductivity. In the lowest order of perturbation theory, where interactions are neglected, the total momentum of +the electrons is conserved. As a consequence, the expression (7) becomes divergent, the ideal plasma shows no finite value for the +conductivity. Partial summations, in particular the self-energy and vertex corrections, lead to finite values for the conductivity, +see26. Analytical evaluation of the Kubo formula remains difficult and cumbersome. +In contrast, it is possible to perform a virial expansion for the inverse conductivity 푅 = 1∕휎, expressed as a correlation function +of the stochastic forces26. A generalized linear response theory was worked out that takes into account correlation functions +of higher moments of the occupation number distribution4. In this way the relation to the kinetic theory was shown21. These +correlation functions are also treated by the methods of Green functions, Feynman diagram techniques and partial summations, +so that virial expansions can be carried out. +The dc conductivity 휎(푛, 푇 ) is usually associated with a dimensionless function 휎∗(푛, 푇 ) according to +휎(푛, 푇 ) = (푘퐵푇 )3∕2(4휋휖0)2 +푚1∕2 +푒 +푒2 +휎∗(푛, 푇 ). +(12) +We consider both 휎 and 휎∗ as a function of density 푛 at fixed temperature 푇 . In the limiting case of low density, the following +virial expansion for the inverse conductivity 휌∗(푛, 푇 ) = 1∕휎∗(푛, 푇 ) was obtained from kinetic theory and generalized linear +response theory4,5,6: +휌∗(푛, 푇 ) = 휌1(푇 ) ln 1 +푛 + 휌2(푇 ) + 휌3(푇 ) 푛1∕2 ln 1 +푛 + (푛1∕2), +(13) +which begins with a logarithmic term. Values for the virial coefficients 휌푖(푇 ) are given below in Sec. 6. + +5 +4 +VIRIAL PLOTS +Equilibrium properties, such as the correlation functions considered here, depend on a limited number of state variables. For +the Hydrogen plasma, this are the temperature 푇 and the electron number density 푛 (for charge neutral plasmas, the ion (proton) +number density is also 푛). For the uniform electron gas, we have the same variables. Instead of the ion subsystem a homo- +geneously charged background (jellium model) is considered to establish charge neutrality. In the case of a many-component +plasma, the independent partial densities 푛푖 (not connected by chemical reactions and charge neutrality) of the components are +the state variables in addition to 푇 . We focus here on the two simple cases where the state variables are 푇 , 푛, and we study +the correlation energy ̄푉 (푇 , 푛) of the uniform electron gas and the electrical conductivity 휎(푇 , 푛) of the Hydrogen plasma, in +particular the resistivity 푅(푇 , 푛) = 1∕휎(푇 , 푛). +It is convenient to introduce dimensionless variables instead of 푇 , 푛. We use atomic units with the Hartree energy +퐸Ha = +( +푒2 +4휋휖0 +)2 푚 +ℏ2 = 27, 21137 eV = 2 Ry +(14) +and the Bohr radius +푎퐵 = 4휋휖0 +푒2 +ℏ2 +푚 = 5.2918 × 10−11 m. +(15) +The density in atomic units is usually represented by the radius of a sphere containing an electron, +푟푠 = +( 3 +4휋푛 +)1∕3 1 +푎퐵 +. +(16) +The temperature is related to the energy 푘퐵푇 , so that 1 eV corresponds to 11604.6 K. We denote 푇eV as 푘퐵푇 measured in units +of eV, 푇Ha in units of 퐸Ha, and 푇Ry in units of Ry so that +푇Ha = 푘퐵푇 +퐸Ha += 2푇Ry = 27, 21137 푇eV. +(17) +Another well-known choice of dimensionless parameters is +Γ = +푒2 +4휋휖0푘퐵푇 +(4휋 +3 푛 +)1∕3 +, +Θ = 2푚푘퐵푇 +ℏ2 +(3휋2푛)−2∕3. +(18) +The plasma parameter Γ characterises the ratio of potential to kinetic energy in the non-degenerate case, and the electron degen- +eracy parameter Θ characterises the range in which the electrons are degenerate. Different sets of dimensionless parameters are +related. Thus, PIMC calculations for specific parameter values of 푟푠, Θ are discussed in the following section, the corresponding +plasma parameters 푛, 푇 are determined as follows, +푛 = 3 +4휋 +1 +(푟푠푎퐵)3 , +푘퐵푇 = 퐸Ha +1 +2 +(9휋 +4 +)2∕3 Θ +푟2 +푠 +(19) +with 퐸Ha∕푘퐵 = 315777.1 K. +The dc conductivity 휎(푛, 푇 ) is also associated with a dimensionless function 휎∗(푛, 푇 ) according to +휎(푛, 푇 ) = (푘퐵푇 )3∕2(4휋휖0)2 +푚1∕2 +푒 +푒2 +휎∗ = 0.0258883 푇 3∕2 휎∗(Ωm K3∕2)−1 = 32405.4 푇 3∕2 +eV 휎∗(Ωm)−1 . +(20) +As with thermodynamic relations, the dimensionless conductivity 휎∗ can be expressed as a function of dimensionless variables +푟푠, 푇Ha or Γ, Θ. These functions are now to be specified. Exact results are currently known only for limiting cases, in particular +virial expansions. +The analysis of a virial expansion is sometimes not easy because trivial terms dominate in limiting cases so that interesting +terms remain hidden. In the example of the thermodynamic EoS considered in Sec. 5, one dominant term is the Debye shift, +which covers the contribution of higher virial coefficients. We introduce reduced virial expansions where these exactly known +contributions are suppressed, and quantities are introduced that anticipate a linear relationship in special cases. The virial plot +is the representation of this asymptotic linear relationship and allows us to extrapolate virial coefficients from simulations. We +demonstrate this procedure for two cases, the mean potential energy of the uniform electron gas in Sec. 5 and the electrical +conductivity of the Hydrogen plasma in Sec. 6. + +6 +If we express 휎∗(푛, 푇 ) in terms of dimensionless parameters Γ, Θ and use the Born parameter Γ∕Θ, which is of interest in the +range 푘퐵푇 ≫ 1 Ry, from Eq. (13) we obtain a modified virial expansion where the argument of the logarithm is dimensionless, +1 +휎∗(Γ, Θ) = 휌∗(Γ, Θ) = ̃휌1(Γ2Θ) ln +(Θ +Γ +) ++ ̃휌2(Γ2Θ) + … , +Γ2Θ = +27∕3 +34∕3휋3∕3 +1 +푇Ha +, +Θ +Γ = +21∕3 +31∕3휋5∕3 +푇 2 +Ha +푛푎3 +퐵 +(21) +We define the reduced effective virial coefficient ̃휌eff +2 (푇 ) according to +̃휌eff +2 (푛, 푇 ) = +32405.4 +휎(푛, 푇 )[Ωm]푇 3∕2 +eV − ̃휌1(푇 ) ln +(Θ +Γ +) +, +(22) +with lim푛→0 ̃휌eff +2 (푛, 푇 ) = ̃휌2(푇 ), see also Eq. (45) below. The plot of 휌∗∕ ln(Θ∕Γ) as a function of 푥 = 1∕ ln(Θ∕Γ) at given 푇 is +called a virial plot. It directly allows the determination the virial coefficients 휌1(푇 ), 휌2(푇 ), as it is shown in Sec. 6. +As will be demonstrated in this work, virial plots are very sensitive to diverse approaches, including the results of numerical +simulations, in the low density domain. Since trivial dominant terms, which are known exactly, are suppressed, they have no +effects due to possible approximations, and the extrapolation of numerical simulations into the low-density domain becomes +immediately possible. +5 +VIRIAL EXPANSION OF THE EOS OF THE UEG, COMPARISON WITH PIMC +SIMULATIONS +The problem of the second virial coefficient for the mean correlation energy ̄푉 was considered in a recent work25. There was +a controversy about the high-temperature limit of the second virial coefficient, i.e. the term ∝ 1∕ +√ +푇 27. This controversy dis- +appears in charge-neutral two-component plasmas, but not in the uniform electron gas (UEG), where interacting electrons are +moving in front of a positively charged jellium-like background to neutralize the Coulomb field at large distances. Accurate +PIMC simulations have been available at low densities and high temperatures25, so that it was possible to confirm the correct +limiting behavior. In this section, we not only show the virial plot method to confirm the correct limiting law, but consider the +full second virial coefficient and discuss deviations from this expansion. +The virial expansion of the free energy 퐹(푇 , Ω, 푁) of the UEG is obtained from the general formula for a multi-component +plasma given in9,25. The mean potential energy 푉 is determined by +푉 (푇 , Ω, 푁) = 푒2 +휕 +휕(푒2)퐹(푇 , Ω, 푁) +(23) +(for the relation to the internal energy see28). +From the virial expansion of 퐹(푇 , Ω, 푁), we get the following virial expansion of 푉 +푉 +푁푘퐵푇 = − 휅3 +8휋푛 − 휋푛휆3휏3 ln(휅휆) +−휋푛휆3 +[ +휏 +2 − +√ +휋 +2 (1 + ln(2))휏2 + +( +퐶 +2 + ln(3) − 1 +3 + 휋2 +24 +) +휏3 ++ +√ +휋 +∞ +∑ +푚=4 +(−1)푚푚 +2푚Γ(푚∕2 + 1) +[2휁(푚 − 2) − (1 − 4∕2푚)휁(푚 − 1)] 휏푚 +] +−휋푛휆4휏4휅 ln(휅휆) + 푉4(푇 ) +푁푘퐵푇 푛3∕2 + (푛2 ln(푛)) +(24) +with the variables +휅2 = +푛푒2 +휖0푘퐵푇 , +휆2 = +ℏ2 +푚푘퐵푇 , +휏 = +푒2√ +푚 +4휋휖0 +√ +푘퐵푇 ℏ +. +(25) +휁(푥) denotes the Riemann zeta function, and 퐶 = 0.57721 … is Euler’s constant. We express this expansion in terms of 푇 , 푛 +and introduce atomic units ℏ = 푚 = 푒2∕4휋휖0 = 1 so that 푘퐵푇 is measured in Hartree (Ha) and 푛 in electrons per 푎3 +퐵. +The virial expansion of the specific mean potential energy 푣 = 푉 ∕푁 is as follows +푣(푇 , 푛) = 푣0(푇 )푛1∕2 + 푣1(푇 )푛 ln (휅2휆2) + 푣2(푇 )푛 + 푣3(푇 )푛3∕2 ln (휅2휆2) + 푣4(푇 )푛3∕2 + (푛2 ln(푛)). +(26) + +7 +If atomic units are used, this results in (휅2휆2 = 4휋푛∕푇 2) +푣0(푇 ) = − +√ +휋 +푇 1∕2 , +푣1(푇 ) = − 휋 +2푇 2 , +푣2(푇 ) = − 휋 +푇 +[ +1 +2 − +√ +휋 +2 (1 + ln(2)) 1 +푇 1∕2 + +( +퐶 +2 + ln(3) − 1 +3 + 휋2 +24 +) +1 +푇 +− +√ +휋 +∞ +∑ +푚=4 +푚 +2푚Γ(푚∕2 + 1) +( −1 +푇 1∕2 +)푚−1 +[2휁(푚 − 2) − (1 − 4∕2푚)휁(푚 − 1)] +] +, +푣3(푇 ) = − 3휋3∕2 +2푇 7∕2 . +(27) +In ref.25, a virial plot was presented to study the behavior of the second virial coefficient. We consider the lowest orders of +the virial expansion, +푣(1)(푇 , 푛) = − +√ +휋 +푇 1∕2 푛1∕2 − +휋 +2푇 2 푛 ln +(4휋푛 +푇 2 +) +, +(28) +as exactly known and subtract them from the data obtained from the PIMC simulations, 푣PIMC = 푉 PIMC∕푁. These exactly +known terms may become very large, hiding the higher virial coefficients. (Note that the logarithmic term contains a factor to +become dimensionless. This factor can be moved to the next virial coefficient.) +In25 we introduced the reduced potential energy (휏 = 푇 −1∕2, atomic units) +푣red +2 (푇 , 푛) = [푣PIMC − 푣(1)(푇 , 푛)] −푇 +휋푛 = −푇 +휋 푣2(푇 ) + (푛1∕2 ln(푛)) += 1 +2 − +√ +휋 +2 (1 + ln(2))휏 + +( +퐶 +2 + ln(3) − 1 +3 + 휋2 +24 +) +휏2 + (휏3) + (푛1∕2 ln(푛)). +(29) +Table 1 PIMC calculations for the uniform electron gas: 푣PIMC and 푣red +2 , eq. (29), for special parameter values 푟푠, Θ and the +corresponding values of 푇 , 휏, 푛. +푟푠 +Θ +푣PIMC [Ha] +푇Ha +휏 +푣red +2 +푇 [K] +푛 [cm−3] +0.5 +128 +-0.0826214 +942.891 +0.0325664 +0.453524 +2.97742e8 +1.28882e25 +64 +-0.1180456 +471.446 +0.0460558 +0.420822 +1.48871e8 +1.28882e25 +32 +-0.169272 +235.723 +0.0651327 +0.398701 +7.44354e7 +1.28882e25 +16 +-0.2423993 +117.861 +0.0921116 +0.356465 +3.72177e7 +1.28882e25 +8 +-0.3447641 +58.9307 +0.130265 +0.294433 +1.86089e7 +1.28882e25 +2 +128 +-0.0402248 +58.9307 +0.130265 +0.290766 +1.8609e7 +2.01378e23 +64 +-0.0568062 +29.4653 +0.184223 +0.257047 +9.30448e6 +2.01378e23 +32 +-0.0797147 +14.7327 +0.260531 +0.207038 +4.65224e6 +2.01378e23 +16 +-0.1101257 +7.36634 +0.368446 +0.126496 +2.32612e6 +2.01378e23 +8 +-0.1486611 +3.68317 +0.521062 +0.0596564 +1.16306e6 +2.01378e23 +20 +128 +-0.0119299 +0.589307 +1.30265 +1.50247 +186090. +2.01378e20 +64 +-0.0160051 +0.294653 +1.84223 +3.48031 +93044.8 +2.01378e20 +32 +-0.0207112 +0.147327 +2.60531 +6.67878 +46522.4 +2.01378e20 +16 +-0.0256337 +0.0736634 +3.68446 +10.2475 +23261.2 +2.01378e20 +8 +-0.0302098 +0.0368317 +5.21062 +9.50255 +11630.6 +2.01378e20 +In Tab. 1, the parameter values of the uniform electron gas are given for which PIMC calculations were presented in Ref.25, +together with the values for 푣red +2 +(29). The results for 푣red +2 +are also shown in Figs. 1, 2. +In Fig. 1 all calculated PIMC data of25 are considered and the corresponding value of 푣red +2 +is shown as function of 휏, see Tab. +1. In addition, three expressions for (29) are shown: up to order 휏, i.e., 1∕2 − +√ +휋(1 + ln(2))휏∕2, up to order 휏2, and the full 휏 +dependence. This Fig. 1 shows in which interval of 휏 the linear or quadratic approximation is applicable. The PIMC data are + +8 +0 +1 +2 +3 +4 +5 +6 +T +-1/2 [Ha] +0 +5 +10 +15 +20 +v +red +rs = 0.5 +rs = 2 +rs = 20 +T +-1/2 +T +-1 +2 +nd virial +3 +rd virial, rs=20 +Figure 1 Reduced potential energy 푣red +2 (푇 , 푛), Eq. (29), as function of 휏 = 1∕ +√ +푇 for different densities, 푟푠 = 0, 5; 2; 20. For +comparison, the reduced second virial coefficient 푣red +2 (푇 ) = −(푇 ∕휋)푣2(푇 ) [2nd virial, according Eq. (27)] as well as the lowest +orders in 1∕푇 are shown. In addition, the curve 3rd virial given by Eq. (30) is also shown. (Atomic units are used.) +very different. The lowest density, 푟푠 = 20, should be most relevant to the low-density limit, where higher virial coefficients are +less important. However, the inverse temperature 휏 = 푇 −1∕2 +Ha +is too large to reach the limit 휏 → 0. Close to this limit are PIMC +simulation data for 푟푠 = 0.5. The relatively large density is compensated by the very high temperature, see Tab. 1. +A part of Fig. 1 is shown enlarged in Fig. 2. It was a main result of Ref.25 to show that the PIMC simulation data confirm the +limit 푣red +2 (휏 = 0) = 1∕2. Linear fit to the data for 푟푠 = 0.5 is possible, and extrapolation to 푣red +2 (휏 = 0) gives 1/2. At the same +time, one gets an idea of the accuracy of the simulation, which shows up as scatter around the analytical behavior. The PIMC +data for 푟푠 = 2 are not described by the linear approximation but almost well by the quadratic approximation. Finally, we have +to make a comparison with the full second virial coefficient and will find that good agreement is obtained in all three density +cases, given by the parameter 푟푠, only for the lowest values of 휏 (an exception is the lowest 휏 parameter calculation for 푟푠 = 2, +which needs to be checked). As 휏 increases, the PIMC data are systematically below the second virial curve. We assume that +the PIMC simulations are very accurate, so this deviation indicates the contribution of higher virial coefficients. +Deviations from the second virial coefficient −(푇 ∕휋)푣2(푇 ) indicate the contribution of higher orders to the virial expansion. +We expect a significant next order contribution from the low-density calculations, i.e., 푟푠 = 20. We consider the expression +푣red +2+3(푇 , 푛) = −푇 +휋 +[ +푣2(푇 ) + 푣3(푇 )푛1∕2 ln +(4휋푛 +푇 2 +)] +, +(30) +which accounts for the contribution of the third virial coefficient. For 푟푠 = 20, the data are well reproduced for the lowest values +of 휏, see also Fig. 1. Deviations for larger 휏 indicate the contributions of higher virial coefficients. +The deviation +Δ푣red +2 (푇 , 푛) = [푣PIMC − 푣(1)(푇 , 푛) − 푣2(푇 )푛] 푇 +휋푛 +(31) +is shown in Tab. 2, together with the deviation +Δ푣red +3 (푇 , 푛) = +[ +푣PIMC − 푣(1)(푇 , 푛) − 푣2(푇 )푛 − 푣3(푇 )푛3∕2 ln +(4휋푛 +푇 2 +)] 푇 +휋푛. +(32) +As mentioned before, the inclusion of the third virial coefficient 푣3(푇 ) improves the agreement of the PIMC simulations with +the virial expansion, as also shown in Fig. 1. The remaining difference Δ푣red +3 (푇 , 푛) is related to the fourth-order and higher-order +virial coefficient, +푣eff +4 (푇 , 푛) = Δ푣red +3 (푇 , 푛) +휋 +푇 푛1∕2 = 푣4(푇 ) + (푛1∕2 ln(푛)). +(33) + +9 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +T +-1/2 [Ha] +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +v +red +rs = 0.5 +rs = 2 +T +-1/2 +T +-1 +2 +nd virial +Figure 2 Detail of Fig. 1. +The fourth virial coefficient results when higher-order virial coefficients are neglected, lim푛→0 푣eff +4 (푇 , 푛) = 푣4(푇 ). This should +be possible in the low-density limit, where the contributions of higher orders of the density expansion become small. However, +high-precision calculations are required to extract the higher-order coefficients, and the accuracy of the present calculations25 +is not sufficient to determine precisely the fourth- and higher-order virial coefficients. We give here only a discussion of the +present data. +From the virial expansion of the free energy9, the fourth virial coefficient 푣4(푇 ) contains contributions with temperature +dependence ∝ 푇 −2 = 휏4 and higher orders in 휏, as well as contributions ∝ 푇 −7∕2. The coefficient of the 휏4 term follows as +3휋 +√ +4휋. We expect a high-temperature limit behavior ∝ 푇 −2, and we show in Fig. 3 the quantity 푣eff +4 (푇 , 푛) × 푇 2. +We see that the lowest value of density, 푟푠 = 20, exhibits behavior at small 휏 values that can be compared to a curve 3휋 +√ +4휋 − +6 × 휏3. However, the exact determination of the fourth virial coefficient 푣4(푇 ) is not possible from the available data. At the +higher densities corresponding to smaller 푟푠, the accuracy of the numerical PIMC simulations may not be sufficient to extract +higher-order virial coefficients. In the context of our analysis, in addition to the dependence on 푇 , the dependence on 푛 would be +of interest to perform the virial plot as a function of 푛. Further calculations for density parameter values in the range of 푟푠 = 20 +would be required. Since we are investigating the differences between large numbers, high accuracy is necessary. +The study of the uniform electron gas is not only of interest for the discussion of the exchange-correlation term of the energy- +density functional in DFT calculations, for which Dornheim, Groth, and Bonitz derived analytical formulas29,30. It is also a +prerequisite to treat the more interesting case of a two-component plasma, e.g., the Hydrogen plasma. The equation of state at +low densities is of interest, for example, in helioseismology31, where the fourth virial coefficient 푣4(푇 ) is important32. In this +context, the high-temperature limit of 푣red +2 (휏 = 0) was discussed in27,25. For a discussion of the fourth virial coefficient 푣4(푇 ) of +Hydrogen plasma, see also Alastuey and Ballenegger33,34. +6 +VIRIAL EXPANSION OF THE INVERSE CONDUCTIVITY OF H PLASMAS, +COMPARISON TO DFT-MD SIMULATIONS +Numerous studies have been performed to calculate the electrical conductivity 휎(푛, 푇 ) of Hydrogen plasma in a wide range of +parameters, a recent review can be found in Ref.35. A comparative study36 was also recently published that considered different +approaches and showed large differences in the calculated conductivities. Analytical calculations in the framework of generalized + +10 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +τ = (THa) +-1/2 +-300 +-250 +-200 +-150 +-100 +-50 +0 +50 +v4 +eff(T,n)* (THa) +2 +6π +3/2− 6τ +3 +rs=0.5 +rs=2 +rs=20 +Figure 3 Effective reduced fourth virial coefficient 푣eff +4 (푇 , 푛) × 푇 2, Eq. (33), plotted as function of 휏 = 1∕ +√ +푇Ha for different +densities, 푟푠 = 0, 5; 2; 20. For comparison, a curve 3휋 +√ +4휋 − 6휏3 is seen. (Atomic units used.) +Table 2 PIMC calculations for the UEG: 푣 and 푣red. The calculation with the second virial coefficient, Eq. (32), is denoted by +푣vir and 푣red +vir . +푟푠 +Θ +푣 [Ha] +푇Ha +푛 푎3 +퐵 +휏 +푣red +2 +Δ푣red +2 +Δ푣red +3 +0.5 +128 +-0.082621 +942.891 +1.90986 +0.0325664 +0.453524 +-0.000818266 +-0.000819682 +64 +-0.118045 +471.446 +1.90986 +0.0460558 +0.420822 +0.0132298 +0.0132228 +32 +-0.169272 +235.723 +1.90986 +0.0651327 +0.398701 +0.00992642 +0.00989306 +16 +-0.242399 +117.861 +1.90986 +0.0921116 +0.356465 +0.0181564 +0.0180015 +8 +-0.344764 +58.9307 +1.90986 +0.130265 +0.294433 +0.0360992 +0.0354136 +2 +128 +-0.040224 +58.9307 +0.0298416 +0.130265 +0.290766 +0.039767 +0.0396097 +64 +-0.056806 +29.4653 +0.0298416 +0.184223 +0.257047 +0.0194226 +0.0186676 +32 +-0.079714 +14.7327 +0.0298416 +0.260531 +0.207038 +0.0103138 +0.00680714 +16 +-0.110125 +7.36634 +0.0298416 +0.368446 +0.126496 +0.043972 +0.0284584 +8 +-0.148661 +3.68317 +0.0298416 +0.521062 +0.0596564 +0.12329 +0.0599871 +20 +128 +-0.011929 +0.589307 +0.0000298416 +1.30265 +1.50247 +0.440148 +0.0680086 +64 +-0.016005 +0.294653 +0.0000298416 +1.84223 +3.48031 +1.60138 +-0.0765312 +32 +-0.020711 +0.147327 +0.0000298416 +2.60531 +6.67878 +5.91999 +-1.155 +16 +-0.025633 +0.0736634 +0.0000298416 +3.68446 +10.2475 +19.7824 +-6.56867 +8 +-0.030209 +0.0368317 +0.0000298416 +5.21062 +9.50255 +60.1201 +-11.6087 +linear response theory were performed for simple systems such as the Hydrogen plasma. For more complex plasmas, the DFT- +MD approach16,37,38,19 was elaborated to evaluate the Kubo-Greenwood formula. However, as discussed in21, electron-electron +collisions are not correctly described in this approach. In a recent study15, the low-density limit of the electrical conductivity +휎(푛, 푇 ) of Hydrogen as the simplest ionic plasma is presented as a function of temperature 푇 and particle density 푛 in terms + +11 +0 +0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 +1 +1.1 +1/ln[Θ/Γ] +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +32405.4 (T/eV) +3/2/σ[S/m] / ln(Θ/Γ] +tilde ρ1Spitzer +tilde ρ1Lorentz +Karakhtanov +QLB, Ronald +64 +125 +216 +95 +T = 2000 eV +T = 200 eV +n = 40 g/ccm +n = 2 g/ccm +Figure 4 Reduced resistivity ̃휌(푥, 푇 ) (42) for hydrogen plasma as a function of 푥 = 1∕ ln(Θ∕Γ): DFT-MD simulations from +Ref.15, and Lenard-Balescu results (QLB, Ronald) of Desjarlais et al.37 and Karakhtanov40. 휌Spitzer +1 += 0.846 and 휌Lorentz +1 += 0.492 +are defined in the text. The green line represents a linear extrapolation of the converged DFT-MD results. Data are given in the +Supplemental material of15. +of a virial expansion of resistivity. The non-consideration of the contribution of electron-electron collisions in other transport +coefficients such as thermopower and thermal conductivity has also been discussed recently37,39. +The virial expansion of the dimensionless resistivity 휌∗, Eq. (13), contains the logarithmic term ln(1∕푛). To make its argument +dimensionless we use the Born parameter, see Ref.15, +Θ +Γ = +푇 2 +Ry +푛Bohr +(96휋5)−1∕3 , +(34) +where the temperature is measured in Rydberg units, 푇Ry = 2푇Ha = 푘퐵푇 ∕13.6 eV. As discussed in Sec. 4 in connection with +the logarithmic term, we use a modified virial expansion and rewrite (13) +휌∗(푛, 푇 ) = ̃휌1(푇 ) ln +(Θ +Γ +) ++ ̃휌2(푇 ) + … . +(35) +The modified virial coefficients ̃휌푖 are related to 휌푖 replacing in Eq. (35) the variables Θ, Γ by 푛, 푇 according to Eq. (34). +Comparing with Eq. (13), ̃휌1 = 휌1 is obtained and +̃휌2 = 휌2 + 휌1 ln[(96휋5)1∕3∕푇 2 +Ry] . +(36) +A highlight of plasma transport theory is that the exact value of the first virial coefficient for Coulomb systems is known from +the seminal paper of Spitzer and Härm3, +휌1 = ̃휌1 = 휌Spitzer +1 += 0.846024, +(37) +which does not depend on 푇 . Note that Eq. (37) accounts for the contribution of the electron-electron (푒 − 푒) interaction. In +contrast, for the Lorentz plasma model where the 푒 − 푒 collisions are neglected so that only the electron-ion interaction is +considered, the first virial coefficient is4 +휌Lorentz +1 += 1 +16(2휋3)1∕2 = 0.492126 . +(38) +Although 푒−푒 collisions do not contribute to a change of the total momentum of the electrons due to conservation of momentum, +the distribution in momentum space is changed by 푒−푒 collisions ("reshaping"), and higher moments of the electron distribution + +12 +are not conserved by 푒−푒 collisions. The indirect influence of 푒−푒 collisions on the dc conductivity becomes clear in generalized +linear response theory where these higher moments are considered, see4,6. +No exact value is known for the second virial coefficient 휌2(푇 ) or ̃휌2(푇 ). It depends on the treatment of the many-body effects, +in particular on the screening of the Coulomb potential. In a quantum statistical approach, the static (Debye) screening by +electrons and ions should be replaced by dynamical screening. For the Hydrogen plasma considered here, the Born approximation +for the collision integral at high temperatures 푇Ry ≫ 1 is justified. Consideration of screening in the random phase approximation +(RPA), leads to the quantum Lenard-Balescu (QLB) expression. Thus, at very high temperatures, where the dynamically screened +Born approximation becomes valid, we obtain the QLB result, see37,40, +lim +푇 →∞ ̃휌2(푇 ) = ̃휌QLB +2 += 0.4917 . +(39) +As 푇 decreases, strong binary collisions (represented by ladder diagrams) become important and must be treated in the +calculation of the second virial coefficient ̃휌2(푇 ) beyond the Born approximation. According to Spitzer and Härm3, the classical +treatment of strong collisions with a statically screened potential gives for 휌∗ = 1∕휎∗ the result +휌∗ +Sp = 0.846 ln +[3 +2Γ−3] +. +(40) +Interpolation formulas have been proposed that link the high-temperature limit ̃휌QLB +2 +with the low-temperature Spitzer +limit45,41,42,5,43,6,4,44. Based on a T-matrix calculation in quasiclassical (Wentzel-Kramers-Brillouin, WKB) approximation45,46, +the expression (푇eV = 푘퐵푇 ∕eV) +̃휌2(푇eV) ≈ 0.4917 + 0.846 ln +[ +1 + 8.492∕푇eV +1 + 25.83∕푇eV + 167.2∕푇 2 +eV +] +(41) +is a simple interpolation that combines the QLB result with the Spitzer limit in WKB approximation. However, the exact +analytical form of the temperature dependence of the second virial coefficient ̃휌2(푇 ) remains an open problem. +Thus, the available exact results for the virial expansion (35) of the inverse conductivity of fully ionized Hydrogen plasma are: +(i) the value of the first virial coefficient is ̃휌1 = 0.846; +(ii) the second virial coefficient has the high-temperature limit lim푇 →∞ ̃휌2(푇 ) = 0.4917; +(iii) the second virial coefficient is temperature dependent, an approximation is given by Eq. (41). +To extract the first and second virial coefficient from calculated or measured dc conductivities, we plot the expression +̃휌(푥, 푇 ) = +휌∗ +ln(Θ∕Γ) = +32405.4 +휎(푛, 푇 )(Ωm)푇 3∕2 +eV +1 +ln(Θ∕Γ) +(42) +as a function of 푥 = 1∕ ln(Θ∕Γ) and 푇 in Fig. 4 which is called virial plot. According to Eqs. (13), (35), the behavior of any +isotherm (fixed 푇 ) is linear near 푛 → 0, +̃휌(푥, 푇 ) = ̃휌1(푇 ) + ̃휌2(푇 )푥 + … , +(43) +with ̃휌1(푇 ) as the value at 푥 = 0 and ̃휌2(푇 ) as the slope of the isotherm. In this way, the extraction of virial coefficients becomes +immediately possible. For 푥 > 1∕ ln(100) = 0.217, the contributions of higher order virial coefficients have to be taken into +account15. For fixed 푇 and low density, where 휃 ≫ 1, a classical plasma is present and the effects of degeneracy contribute to +the higher order virial coefficients. +In Fig. 4 two cases for the first virial coefficient 휌1 on the ordinate axis are shown, see also4,5,6: +(i) 휌Spitzer +1 +from kinetic theory when 푒 − 푒 collisions are taken into account, +(ii) when 푒 − 푒-collisions are neglected, 휌Lorentz +1 +is obtained for the Lorentz plasma model. +Moreover, the second virial coefficient ̃휌QLB +2 +of the Lenard-Balescu approximation. (39) is shown, which is correct in the high +temperature limit. The QLB calculations of Desjarlais et al.37 are shown in Fig. 4. The 푒 − 푒 collisions are taken into account, +yielding the same asymptote (푥 → 0) as in Karakhtanov40. With increasing 푥 = 1∕ ln(Θ∕Γ) small deviations from linear +behavior are observed. When isotherms are presented, this deviation indicates the contribution of higher virial coefficients. +Virial plots are presented in15 to investigate two problems: Which of the various approaches that give us analytical expressions +for the electrical conductivity of Hydrogen plasmas are accurate in the low density limit? The virial expansion of the inverse +conductivity serves as an exact benchmark for theoretical approaches, so that the accuracy and consistency of semi-empirical +results for conductivity, such as those collected in Ref.36, can be checked. A more fundamental problem is whether numerical +results from molecular dynamics simulations based on density functional theory (DFT-MD) correctly contain the contribution +of electron-electron collisions. The virial plot confirms the position that DFT-MD simulations in the low-density limit describe +a Lorentz plasma with only electron-ion collisions, the contribution of electron-electron collisions to 휌1 is missing15. + +13 +Here we discuss some details of the virial expansion for the inverse conductivity and the corresponding virial plots, see Fig. +4. DFT-MD simulations are given in Ref.15, see the tables of data in the supplementary material. These data have sufficiently +high accuracy, as can be seen from the small deviations from the fit line in Fig. 4. In addition to the precise solution of the Kubo- +Greenwood formula, this is achieved by good control of convergence with increasing particle number, as shown by comparison +of calculations with different numbers of particles. The number of particles must be sufficiently large to ensure convergence. +In the parameter range considered in the figure, about 100 particles in the box are necessary to achieve convergence. Further +calculations with 216 electrons were not possible due to limited computer capacity. For 푇 = 150 eV, even 125 electrons exceed +the currently available computer capacity. This point was also discussed in a recent work39, where earlier calculations37 were +improved to achieve convergence. Another problem is the determination of the value of the dc conductivity 휎(0) from the calcu- +lation of the optical conductivity 휎(휔) at finite frequencies. Because of the discretisation in a finite box, the energy eigenvalues +have a minimum spacing and the energy-conserving 훿 function must be smeared by a parameter 휖 to allow for transitions, see +also section 3 above. To reach the limit 휔 → 0, an extrapolation is performed according to the Drude formula (10). This was +discussed also in Ref.39. Instead, one can use the dynamic collision frequency to perform this extrapolation. +The results shown in Fig. 4 allow the extraction of virial coefficients 휌1(푇 ), ̃휌2(푇 ). Compared to other approaches, including +interpolation formulas, see15, as well as the QLB calculation, we assume that we are in the linear region of the virial curve. +Deviations from linearity can be observed for QLB already at 푥 = 0.2, since the density is high (40 g/cm3). For DFT-MD +simulations with density about 2 g/cm3, the deviation from linearity for the last point is observed at 푥 ≈ 1. +As pointed out in15, the extrapolated value of 휌1 in the virial plot at 푥 = 0 points to the Lorentz value (38) but misses the +Spitzer value (37). This means that electron-electron collisions are not considered in the DFT-MD calculations for the electrical +conductivity. Also of interest is the value of ̃휌2(푇 ) given by the slope in the virial plot near 푥 = 0. Fitting it to the data gives a slope +of 0.9886 for the DFT-MD calculations. This is about twice the slope ̃휌QLB +2 +given above. From analytical approaches, it appears +that the slope is determined by various effects such as dynamical screening and strong collisions. In the limiting case of high +temperatures, the Born approximation should be possible, but the Coulomb potential must be replaced by a screened potential. +Static screening of the proton scatterer with both electrons and protons would lead to the following result (퐶 = 0.57721 … is +Euler’s constant). +lim +푇 →∞ ̃휌2(푇 ) = 휋3∕2 +24 +√ +2 +[11 +2 − 3퐶 + ln +(3 +2휋2)] += 1.06036 +(44) +which is close to the observed slope of the DFT-MD simulations. However, it remains unclear to what extent the screening is +included in the simulations. We assume that the ionic structure factor, which is the ionic contribution to the screening, is well +described, and that the electron screening is also captured by the exchange-correlation functional. However, we need to consider +dynamical screening, a problem that has been discussed in previous work5 on virial expansion. +We return to the long-debated question of whether or not 푒 − 푒 collisions are accounted for in the DFT-MD formalism. For +example, it was pointed out in Ref.21 that a mean-field approach is not able to describe two-particle correlations, in particular +푒 − 푒 collisions. However, to some approximation, the 푒 − 푒 interaction is accounted for by the exchange-correlation energy. +DFT-MD simulations, which are mean-field theories that account for the 푒−푒 interaction only through the exchange-correlation +part of the energy density, cannot account for the effect of 푒 − 푒 collisions on the conductivity, so that 휌1(푇 ) corresponds to the +Lorentz plasma, but ̃휌2(푇 ) is determined by screening. The question arises to what extent dynamical screening, as implemented +in the QLB calculations, is also described by the exchange-correlation part of the energy density functional. We would like to +mention that in the case of thermal conductivity it has been shown that the contribution of 푒 − 푒 collisions is not taken into +account in DFT-MD simulations37,35,39 and yields an additional term. Other approaches such as generalized linear response +theory may be considered to indicate appropriate approaches. +Our analysis has shown that the simulation results with virial evolution are extrapolated to the low-density region, where +DFT-MD simulations are no longer feasible. The current simulations, while computationally expensive, are still not very close +to 푥 = 0, so extrapolation to the 푥 = 0 limit is not very accurate. Better data for DFT-MD simulations would be of interest to +confirm our results. Conversely, the benchmark capability of virial expansion discussed in this work can also serve as a criterion +to verify the accuracy of numerical approaches such as DFT-MD simulations to evaluate conductivity. +Another application of the virial plot is experiments to measure electrical conductivity. Assuming that the value 0.846024, +Eq. (37), for 휌1 is exact, an effective second virial coefficient +̃휌eff +2 (푛, 푇 ) = +32405.4 +휎(푛, 푇 )[Ωm] +( 푇 +eV +)3∕2 +− 0.846024 ln +(Θ +Γ +) +(45) + +14 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +1/T [eV] +-3 +-2 +-1 +0 +1 +2 +3 +second virial coefficient tilde ρ2(T) +H: Guenther, Radtke +Ar, Xe, Ne: Ivanov +Ar, Xe: Popovic +Lenard-Balescu value +ERR interpolation +QLB/Desjarlais +interpolation (7) +Figure 5 Second virial coefficients ̃휌2(푇 ) and ̃휌eff +2 (푛, 푇 ) for the dc conductivity of Hydrogen plasmas. Analytical interpolation +formulas (41) and Ref.45 are compared with experiments of Günther and Radtke47 for H plasmas as well as of Ivanov et al.48 +and Popovic et al.49 for rare gas plasmas. The black dashed line corresponds to the high temperature limit that is given by +the quantum Lenard-Balescu value. The broken blue line is the interpolation formula of Ref.45, the red full line represents the +interpolation formula (41) for the second virial coefficient. +has been introduced which gives the second virial coefficient in the low-density limit, lim푛→0 ̃휌eff +2 (푛, 푇 ) = ̃휌2(푇 ). A dependence +of ̃휌eff +2 (푥, 푇 ) on density shows that higher orders of the virial expansion are relevant. We anticipate that at very high 푇 , i.e., +1∕푇 → 0, the Lenard-Balescu value is approximated. The deviations at increasing 1∕푇 , shown in the interpolation formula and +the DFT-MD simulations, indicate that already below temperatures of the order of 100 eV, the effect of strong collisions beyond +the Born approximation should be taken into account. +Ultimately, the virial expansion (35) must be verified experimentally, but accurate data for the conductivity of Hydrogen +plasma in the low-density limit and/or at high temperatures are scarce. Accurate conductivity data for dense Hydrogen plasma +were derived by Günther and Radtke47. They are close to the benchmark data of the virial expansion. It should be noted that there +are systematic errors associated with the analysis of such experiments. For example, the appearance of bound states requires +a realistic treatment of the plasma composition and the influence of neutrals on electron mobility. Alternatively, conductivity +measurements in highly compressed noble gas plasmas were carried out by Ivanov et al.48 and Popovic et al.49,45, but the +interaction of the electrons with the ions deviates from the pure Coulomb potential due to the core of bound electrons. The +corresponding virial plot is close to the data of Hydrogen plasma, see15, but requires a more detailed discussion on the role of +bound electrons. +It should also be mentioned that the densities are quite high, and extrapolation to zero density must be performed to obtain the +second virial coefficient. This tendency can be seen in Fig. 5, especially for the experiments with Ar, Xe49, where low-density +data point to ̃휌2(푇 ). +Quantum statistical methods provide accurate values for the lowest virial coefficients, which serve as benchmarks for an- +alytical approaches to electrical conductivity as well as for numerical results from molecular dynamics simulations based on +density functional theory (DFT-MD) or path integral Monte Carlo (PIMC) simulations. While these simulations are well suited +to compute 휎(푛, 푇 ) in a wide range of densities and temperatures, especially for the warm dense matter region, they become com- +putationally expensive in the low density limit, and virial expansions can be used to compensate for this drawback. Interpolation +formulas that take both approaches into account would be very useful for calculating the conductivity of plasmas. + +15 +Table 3 Experimental data for the electrical conductivity. Günther and Radtke: H47; Ivanov et al.: Ar, Xe, Ne48; Popovic et al.: +Ar, Xe49. +Plasma +̂푛푒 × 1025 +푛 × 10−6 +푇 × 104 +푇 +Γ +Θ +1∕ ln(Θ∕Γ) +휎 × 103 +̃휌(푥, 푇 ) +̃휌eff +2 +[m3] +[g/cm3] +[K] +[eV] +[(Ωm)−1] +H +0.1 +1.67262 +1.54 +1.32706 +0.174914 +363.932 +0.130883 +6.2 +1.04579 +1.52647 +H +0.15 +2.50893 +1.87 +1.61143 +0.164892 +337.249 +0.131177 +9.1 +0.955544 +0.835097 +H +0.25 +4.18155 +2.15 +1.85271 +0.170041 +275.832 +0.13529 +11.4 +0.969821 +0.915228 +Ar +2.8 +46.8334 +2.22 +1.91303 +0.36845 +56.8959 +0.198426 +19.0 +0.895459 +0.249255 +Ar +5.5 +91.9942 +2.03 +1.74931 +0.504626 +33.1707 +0.238914 +15.5 +1.15565 +1.29607 +Ar +8.1 +135.482 +1.93 +1.66313 +0.603878 +24.3632 +0.270456 +17.0 +1.10575 +0.960407 +Ar +14 +234.167 +1.9 +1.63728 +0.736152 +16.6533 +0.320623 +25.5 +0.853604 +0.0237179 +Ar +17 +284.346 +1.78 +1.53387 +0.838316 +13.7074 +0.357872 +24.5 +0.899216 +0.148701 +Xe +25 +418.155 +3.01 +2.5938 +0.563757 +17.9242 +0.289077 +45 +0.869607 +0.081664 +Ne +1.1 +18.3988 +1.98 +1.70622 +0.302559 +94.6027 +0.174059 +13 +0.966995 +0.695135 +Ne +1.9 +31.7798 +1.96 +1.68899 +0.366725 +65.0509 +0.193113 +16.5 +0.832499 +-0.0699113 +air +0.13 +2.17441 +1.1 +0.9479 +0.26726 +218.238 +0.14914 +6 +0.743367 +-0.688167 +Ar +0.06 +1.00357 +1.64 +1.41323 +0.138532 +544.807 +0.120816 +8.3 +0.792469 +-0.443077 +Ar +0.1 +1.67262 +1.64 +1.41323 +0.164248 +387.564 +0.128762 +7.9 +0.887358 +0.321199 +Ar +0.13 +2.17441 +1.64 +1.41323 +0.17926 +325.373 +0.133264 +7.6 +0.954636 +0.815191 +Ar +0.15 +2.50893 +1.64 +1.41323 +0.188017 +295.767 +0.135855 +6.4 +1.15567 +2.27941 +Xe +0.06 +1.00357 +1.24 +1.06854 +0.18322 +411.928 +0.129569 +4.6 +1.0082 +1.25185 +Xe +0.12 +2.00715 +1.24 +1.06854 +0.18322 +411.928 +0.13529 +4.1 +1.0082 +2.20078 +Xe +0.07 +1.17083 +1.26 +1.08578 +0.189819 +377.693 +0.131652 +4.8 +1.00558 +1.21211 +Xe +0.14 +2.34167 +1.26 +1.08578 +0.239157 +237.931 +0.144873 +4.4 +1.20715 +2.49289 +To obtain the correct values for the thermoelectric transport coefficients of Hydrogen plasma in the low-density limit, where +the inclusion of 푒 − 푒 collisions is essential, different solutions can be considered. PIMC simulations, as successfully performed +for the uniform electron gas25, should also be performed for the two-component plasmas. First steps of this ambitious project are +recently in progress12,50. The study of such PIMC calculations with the virial plot would be of great interest. From generalized +linear response theory, we also learn that higher order correlation functions, such as force-force correlation functions associated +with the dynamic collision frequency, may be a suitable approach to include the contribution of 푒 − 푒 collisions in the transport +coefficients4,5,6. +7 +CONCLUSIONS +We have from quantum statistics exact expressions for thermodynamic and transport properties of plasmas by equilibrium cor- +relation functions, but the evaluation is a complex problem in many-particle physics. Numerical simulations are becoming more +accurate as computer capacity increases. However, they need to be controlled with respect to their limits such as size effects, +but also fundamental problems such as the correct description of electron-electron collisions in the context of DFT or the sign +problem in PIMC. It is expected that PIMC simulations will provide an adequate description of electron-electron interactions, +but they are currently unable to solve complex plasmas such as multiply charged ions in the low-temperature range. +The comparison of analytical results for the virial expansion of thermodynamic properties with PIMC calculations for the +uniform electron gas has been performed. In particular, we show that high-precission PIMC simulations confirm the correct +form of the virial expansion, which has been debated recently. It seems to be possible to give also numerical values for higher +virial coefficients, in particular the interesting 푛5∕2 coefficient. These values can be considered as exact results in plasma physics. +Numerical values for higher virial coefficients would also be of great interest for transport properties. + +16 +Analytical theory gives us exact results in limiting cases. This can be used to obtain results for parameter ranges where +numerical simulations are not efficient, e.g. in the low density range. Virial expansions are used to control theories and numerical +simulations. They are of interest to construct interpolation formulas. +It was indicated that the evaluation of the Kubo-Greenwood formula using DFT-MS simulations does not take into account +the effects of electron-electron scattering and cannot reproduce the low-density limit of the electrical conductivity of Hydrogen +plasmas. Similar results were recently reported by French et al.39 for other thermoelectric transport coefficients. It would be of +interest to perform PIMC simulations that can accurately describe electron-electron collisions. +The theory of virial expansion must be extended if the formation of bound states is of importance, i.e. for 푇 ∕푇Ha ≤ 1, see +appendix. New approaches are needed. The approach described here is also applicable to other correlation functions such as +the dynamic structure factor or to other transport properties such as thermal conductivity, thermopower, viscosity, and diffusion +coefficients. Also of interest is the extension of virial expansion to elements other than Hydrogen, where different ions can be +formed and the electron-ion interaction is no longer purely Coulombic. +ACKNOWLEDGMENTS +Thanks to M. Schörner, R. Redmer, M. Bethkenhagen, M. French, H. Reinholz, T. Dornheim, J. Vorberger, Z. Moldabekov, and +W.-D. Kraeft for collaboration and discussions. This work was supported by the German Research Foundation (DFG), Grant # +RO 905/37-1 AOBJ 655625. +Author contributions +This is an author contribution text. It is based on a contribution to the SCCS22 conference. +Financial disclosure +None reported. +Conflict of interest +The author declares no potential conflict of interests. +APPENDIX +A BOUND STATE FORMATION +A special problem of plasmas is the formation of bound states (atoms, charged ions: clusters with a certain number of elemen- +tary particles, i.e., nuclei and electrons) which can dominate the properties in the low-density and low-temperature region. A +simple approach is the chemical picture9, where the bound states are considered as new constituents. The interaction between +the different constituents is neglected except for reactive collisions. Thus, a chemical equilibrium is achieved in which the com- +position of the plasma is described by the law of mass action. For a systematic approach including bound state formation see +Refs.33,34 and references given there. We will not present here an exhaustive discussion of the chemical picture, but only discuss +some aspects in the context of our work. For a recent review, see51,52, where further references can be found. +Within the chemical picture, several issues arise that need to be discussed in order to improve this simple approximation, +using the concept of virial expansions. +(i) In addition to the ground state, excited states (푠) with excitation energy 퐸훼,푠 can occur, which can also be treated as new +species. It is more convenient to introduce the intrinsic partition function of the cluster 훼, which is summed over all excited +bound states by the statistical factor exp[−훽퐸훼,푠]. + +17 +(ii) In addition to bound states, there are also scattering states that must be included in the calculation of virial coefficients. +This leads to the Beth-Uhlenbeck formula, in which the scattering phase shifts appear. Sometimes resonances can appear in +the spectrum of excited states. In the resonance gas approximation, the intrinsic partition function is improved by extending +the summation over all excitations 푠 to the resonances in the continuum. Moreover, the contribution of scattering phase shifts +should be included. +iii) We arrive at higher virial coefficients and need to include density effects. In the framework of a quasiparticle approach, the +intrinsic partition functions are calculated with shifted energies due to screening, mean-field effects, Pauli blocking and other +effects. +As example, let us consider the H plasma and give the intrinsic partition function in the simplest approximation +푧H = +∑ +푠 +2푠2푒−퐸H,푠∕푘퐵푇 +(A1) +with the known energy levels 퐸H,푠 = −퐸Ha∕(2푠2) (퐸Ha = 27.2 eV is the Hartree energy). The factor 2푠2 denotes the degeneracy +of the excitation 푠 including the spin factor. As specific for the Coulomb interaction, we have infinitely many bound states +near the continuum edge for 푠 → ∞. Expression (A1) is not applicable because it is divergent. A convergent expression is the +Planck-Brillouin-Larkin partition function, see9, +푧H = +∑ +푠 +2푠2 +[ +푒−퐸H,푠∕푘퐵푇 − 1 + +퐸H,푠 +푘퐵푇 +] +. +(A2) +The subtraction of 1 is explained as follows: We need to include the contribution of the scattering states which compensate for +the most divergent term of the contribution of the bound states. For the short-range interaction, this has been discussed in detail, +and generalized phase shifts have been introduced to avoid separating the bound and scattering parts of the intrinsic partition +function53,54. +More complex is the explanation of the subtraction of 퐸Ha∕(2푠2푘퐵푇 ). Because of the long-range character of the Coulomb +interaction, phase shifts cannot be defined in the usual form, and the contribution of the scattering states is not well defined +when scattering phase shifts are used. This fundamental problem of the Coulomb interaction is solved introducing the concept +of screening. In the framework of a quantum statistical approach, we have to perform the partial sum of so-called ring diagrams +and introduce quasiparticles. We must, however, avoid double counting. This has already been discussed in detail for the Hartree- +Fock approximation55,56. Of interest is the generalization to partially ionized plasmas with multiply charged ions52. +A systematic approach arises from consideration of the spectral function. We can identify a quasiparticle contribution and +perform a cluster decomposition of the self-energy. For the cluster decomposition of the self-energy, we can introduce different +channels. To avoid double counting, diagrams used for the single-particle self-energy must be subtracted from the ladder sums +defining the cluster states. +A related problem is the definition of the ionization degree in dense plasmas, since the separation of the bound state contri- +bution from the intrinsic partition function is arbitrary, see57,58,59 and references given there. A possible solution would be the +definition of the single-quasiparticle contribution which is extracted from the spectral function. Thus it can be performed by +considering the compressibility or the dynamical conductivity. +The inclusion of bound states and the corresponding generalization of the chemical picture, involving quasiparticle concepts +for the free and bound states, is a difficult problem in plasma theory. Of course, at fixed temperature there is always a low- +density limit at which bound states are dissolved (because of entropy) but this regime can be very limited, for instance it is not +applicable to gases under normal conditions. A realistic description is often based on the chemical picture where bound states +are considered, i.e. for temperatures below the binding energies. A generalized quasiparticle approach is well defined at low +densities, but has to be generalized considering the spectral function (6) if densities are increasing. The formation of bound states +is not only important for the thermodynamic properties, as discussed above for the second virial coefficient of the Hydrogen +plasma. It also determines the transport properties, and the consideration of bound states as additional scatterers remains a +complex problem if we want to go beyond the simple chemical picture. +References +1. A. L. Fetter and J. D. 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Research 2, 023260 (2020). + diff --git a/DdAzT4oBgHgl3EQfif0c/content/tmp_files/load_file.txt b/DdAzT4oBgHgl3EQfif0c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fffcb23bf304a571663631370debe41d1d589c39 --- /dev/null +++ b/DdAzT4oBgHgl3EQfif0c/content/tmp_files/load_file.txt @@ -0,0 +1,1412 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf,len=1411 +page_content='Received: Added at production Revised: Added at production Accepted: Added at production DOI: xxx/xxxx ARTICLE TYPE Thermodynamic and transport properties of plasmas: low-density benchmarks G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Röpke Institut für Physik, Universität Rostock, 18051 Rostock, Germany Correspondence Email: gerd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='roepke@uni-rostock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='de Abstract Physical properties of plasmas such as equations of state and transport coefficients are expressed in terms of correlation functions, which can be calculated using various approaches (analytical theory, numerical simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The method of Green’s func- tions provides benchmark values for these properties in the low-density limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For the equation of state and electrical conductivity, expansions with respect to density (virial expansions) are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Comparison of analytical results with numerical simulations is used to verify theory, to prove the accuracy of simulations, and to establish interpolation formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' KEYWORDS: plasma equation of state, electrical conductivity, virial expansion, DFT-MD simulations, PIMC simula- tions 1 PLASMA PROPERTIES AND CORRELATION FUNCTIONS Plasmas consist of charged particles, number 푁푖 of species 푖 in the volume Ω, which interact via the Coulomb law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' If we denote the charge of the component 푖 by 푍푖푒, we obtain (휖0 is the permittivity of the vacuum) 푉 Coul 푖푗 (푟) = 푍푖푍푗푒2 4휋휖0푟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (1) In general, an additional short-range interaction may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Examples are the homogeneous electron gas (uniform electron gas UEG), where the electrons move over a positively charged background to realize charge neutrality, or the two-component Hydrogen plasma, consisting of electrons and protons, where the particle density is 푛푒 = 푛푝 to maintain charge neutrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In thermodynamic equilibrium, the state of the plasma is determined by the temperature 푇 in addition to the densities 푛푖 = 푁푖∕Ω of the components or the corresponding chemical potentials 휇푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The relationships between the various state variables such as internal energy 푈, free energy 퐹, entropy 푆, pressure 푃 , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' are called equations of state (EoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' All thermodynamic properties can be derived from a thermodynamic potential, for example is 퐹(Ω, 푁푖, 푇 ) as function of Ω, 푁푖, 푇 a thermodynamic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Statistical physics allows to calculate the thermodynamic properties from the microscopic properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' from the Hamiltonian 퐻 = 퐻kin + 푉 , with kinetic energy 퐻kin = ∑ 푖 ∑푁푖 푘 푝2 푖,푘∕2푚푖 and potential energy 푉 = (1∕2) ∑ 푖,푘≠푗,푙 푉 (퐫푖,푘 − 퐫푗,푙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' To calculate physical quantities, various expressions can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For example, for classical systems we can start from the well-known partition function 푍can(Ω, 푁푖, 푇 ) with 퐹(Ω, 푁푖, 푇 ) = −푘퐵푇 ln 푍can(Ω, 푁푖, 푇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For quantum systems, it is convenient to work with the grand canonical ensemble defined by 훽 = 1∕푘퐵푇 and the chemical potentials 휇푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Then the single-particle distribution functions for the ideal quantum system (푉 = 0) have a simple form, the Fermi or Bose distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In second quantization, we introduce 푎+ 푖,푘, 푎푖,푘 as a creation or annihilation operator for particles of species 푖 in the quantum state 푘 = {ℏ퐤, 휎}, which arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='01499v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='plasm-ph] 4 Jan 2023 2 denotes momentum vector and spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The occupation number of this quantum state is given as 푓푖,푘 = ⟨푎+ 푖,푘푎푖,푘⟩ = 1 푍gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='can Tr { 푒−훽(퐻−∑ 푖 휇푖푁푖)푎+ 푖,푘푎푖,푘 } , 푍gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='can = Tr푒−훽(퐻−∑ 푖 휇푖푁푖) (2) with 퐻 as the Hamiltonian in second quantization and 푁푖 = ∑ 푘 푎+ 푖,푘푎푖,푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The relation between the densities and the chemical potentials is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 푛푖(푇 , {휇푗}) = 1 Ω⟨푁푖⟩ = 1 Ω ∑ 푘 푓푖,푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (3) It is convenient to introduce a 휏-dependent correlation function as a generalization of (2) that contains dynamic information, ⟨푎+ 푗,푙푒휏(퐻−∑ 푖 휇푖푁푖)푎푖,푘푒−휏(퐻−∑ 푖 휇푖푁푖)⟩ = ∞ ∫ −∞ 푑휔 2휋 푒−휔휏퐼푖푘,푗푙(휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (4) The spectral density 퐼푖푘,푗푙(휔) is related to the spectral function 퐴푖푘,푗푙(휔) = (1+푒훽휔)퐼푖푘,푗푙(휔) (Fermi statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' An exact expression for the EoS is found if the spectral function is known, 푛푖(푇 , {휇푗}) = 1 Ω ∑ 푘 ∞ ∫ −∞ 푑휔 2휋 1 푒훽휔 + 1퐴푖푘,푖푘(휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (5) The spectral function which is diagonal with respect to {푖, 푘} for a homogeneous system, is related to the self-energy Σ푖,푘(푧) for which a systematic evaluation applying diagram techniques is possible, see1,2: 퐴푖푘(휔) = 2ImΣ푖,푘(휔 − 푖0) [휔 − 휖푖,푘 − ReΣ푖,푘(휔)]2 + [ImΣ푖,푘(휔 − 푖0)]2 , (6) 휖푖,푘 = ℏ2푘2∕2푚푖 − 휇푖 is the kinetic energy shifted by the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The electrical conductivity 휎(푇 , 푛) of low-density plasmas was first calculated in the framework of kinetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In a seminal work3, Spitzer and Härm determined 휎 of the fully ionized Hydrogen plasma by solving a Fokker-Planck equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' To calculate 휎(푇 , 푛) in a wide range of temperature 푇 and particle density 푛, a quantum statistical many-particle theory is needed that describes screening, correlations, and degeneracy effects in a systematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A generalized linear response theory4,5,6 has been elaborated that expresses transport coefficients in terms of equilibrium correlation functions (fluctuation-dissipation theorems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' An example is the Kubo formula7 which relates the transport coefficient 휎 to the electron current-current correlation function, 휎(푇 , 푛) = 푒2 푚2 푒푘퐵푇 Ω⟨푃 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 푃 ⟩푖휖 (7) with the total momentum of the electrons 푃 = ∑ 푘 ℏ푘푥푎+ 푒,푘푎푒,푘 in 푥 direction (the small ion contribution to the electrical current may be added).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The thermodynamic correlation function is the Laplace transform of the Kubo scalar product (the particle number is assumed to commute with the observables), ⟨퐴;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 퐵⟩푧 = ∞ ∫ 0 푑푡 푒푖푧푡 1 훽 훽 ∫ 0 푑휏⟨푒(푖∕ℏ)(푡−푖ℏ휏)퐻퐴푒−(푖∕ℏ)(푡−푖ℏ휏)퐻퐵⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (8) For more details on generalized linear response theory and the evaluation of correlation functions using the method of thermo- dynamic Green’s functions, see2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For the relationship between generalized linear response theory and kinetic theory, see8 and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 2 EVALUATION OF CORRELATION FUNCTIONS The properties of plasmas are expressed in terms of correlation functions in thermodynamic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Examples are thermo- dynamic properties (2) and transport properties (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' There are several methods to calculate these correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Exact solutions are known only for ideal quantum gases where there is no interaction potential 푉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The equations of state are known, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=', the pressure 푃 is expressed by Fermi integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' At fixed temperature, the equation of state for ideal classical gases 푃 = 푛푘퐵푇 is approximated by considering the limiting case of low density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For electrical conductivity, 휎 = ∞ is obtained because of conservation of total momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The resistivity follows as 휌 = 1∕휎 = 0 for charged ideal Fermi gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 3 Correlations appear for the plasma Hamiltonian with complete interaction 푉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' No closed-form solutions are known, and we must perform approximations to solve this many-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Here we discuss three possibilities: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Perturbation expansion with respect to 푉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We obtain analytic expressions for arbitrary orders of 푉 in terms of nonin- teracting equilibrium correlation functions, which can be easily evaluated using Wick’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' However, we have no proof of the convergence of this series expansion and no error estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In order to make this analytical approach more efficient, the method of thermodynamic Green’s functions and Feynman diagram technique were elaborated1,2,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Conver- gence is improved by performing partial summations corresponding to special concepts such as the introduction of the quasiparticle picture (self-energy Σ), screening of the potential (polarization function Π), or formation of bound states (Bethe-Salpeter equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This leads to useful results for the properties of the plasma in a wide range of 푇 and 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' However, as characteristic for perturbative approaches, exact results can be found only in some limiting cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This drawback is eliminated by numerical simulations of the correlation functions that apply to arbitrary interaction strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In Born-Oppenheimer approximation, density functional theory (DFT) for the electron system with given ion configuration and molecular dynamics (MD) for the ion system are applied to evaluate the correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Single- electron states are calculated by solving the Kohn-Sham equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The total energy is obtained from the kinetic energy of a non-interacting reference system, the classical electron-electron interaction, and an exchange-correlation energy that includes, to a certain approximation, all unknown contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The DFT-MD approach has been successfully applied to calculate the thermodynamic properties of complex materials in a wide range of 푇 and 푛, which will not be reported here, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=',10,11,12,13 and the references given there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For electrical conductivity (7), the Kubo-Greenwood formula7,14 Re [휎(휔)] = 2휋푒2 3푚2 푒휔Ω ∑ 푘 푤푘 푁 ∑ 푗=1 푁 ∑ 푖=1 3 ∑ 훼=1 [푓(휖푗,푘) − 푓(휖푖,푘)]|⟨Ψ푗,푘| ̂푝훼|Ψ푖,푘⟩|2훿(휖푖,푘 − 휖푗,푘 − ℏ휔) (9) was used to calculate the frequency-dependent dynamic electrical conductivity 휎(휔) in the long-wavelength limit16,17,18,19,20,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Kohn-Sham wave functions Ψ푖,푘 from density functional theory calculations are used to calculate the transition matrix elements of the momentum operator ̂푝훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The Fermi-Dirac distribution 푓(휖) accounts for the average occupation at energy 휖, and the summation over momentum space 푘 contains the 푘-point weights 푤푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Due to the finite size of the simulation box, the delta function in equation (9) must be approximated by a finite-width Gaussian, which also prevents the direct calculation of the dc conductivity at 휔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Therefore, the dynamic conductivity is extrapolated to the limit 휔 → 0 by a Drude fit, Re [휎(휔)] = 푛푒2휈 휈2 + 휔2 , (10) where 휈 is the collision frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Thus, the calculated direct current conductivity depends on choosing the appropriate width for the Gaussian and finding a suitable range for the Drude-fitting to 휎(휔) calculated from equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The last point can be improved by using a frequency-dependent collision frequency21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' One of the main shortcomings of the DFT-MD approach is that the many-particle interaction is replaced by a mean-field potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' When using product wave functions for the many-electron system, correlations are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The exchange- correlation energy density functional reflects the Coulomb interaction to some approximation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=', as it exists in the homogeneous electron gas, but becomes problematic in the low-density limit where correlations are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In principle, an accurate evaluation of equilibrium correlation functions is possible using path-integral Monte Carlo (PIMC) simulations, see22,23,24 and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The shortcomings of this approach at present are the relatively small number of particles (a few dozen), the sign problem for fermions, and the computational challenges in accurately computing path integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Instead of using an exchange-correlation energy density functional, 푒 − 푒 collisions are treated accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' However, at present accurate calculations have only been performed for the uniform electron gas model in which the charge-compensating ion subsystem is replaced by a homogeneously charged jellium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The results presented in25 are shown below in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' High-precision calculations for the two-component Hydrogen plasma would be of interest for both thermodynamics and transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 4 3 GREEN’S FUNCTIONS AND FEYNMAN DIAGRAMS In quantum statistics, the method of thermodynamic Green’s functions has been worked out to evaluate correlation functions in thermodynamic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For the ideal quantum gas, in which there is no interaction, all equilibrium correlation functions can be calculated using Wick’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For plasmas, we can perform a power series expansion with respect to the interaction strength according to the Dyson series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The terms of this perturbation expansion are represented by Feynman diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The problem of the perturbation expansion is that the convergence property remains open, and we cannot anticipate that for the correlation functions a power series expansion with respect to the interaction strength is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A predetermined wrong analytical behavior near the singular case of ideal gases leads to divergencies which are avoided performing partial summations that can modify the analytic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The most important partial summations are the quasiparticle concept associated with the introduction of the self-energy, the screening associated with the introduction of the polarization function, and the introduction of bound states performing partial summation of ladder diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For instance, the Bethe-Salpeter equation for the two-particle Green function in ladder approximation corresponds to the solution of the two-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' From classical statistics, the Mayer cluster expansion is well known for short-range potentials is well known for the partition function, and the virial expansion in powers in 푛 is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Because of the long-range nature of the Coulomb potential, this expansion in powers in 푛 is not possible for plasmas, the virial coefficients are divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Screening, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' partial summation of the so-called ring diagrams in quantum statistics, solves this convergence problem, and the expansion in powers of 푛1∕2 is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' When considering the spectral function, the contribution of the free particles is replaced by the contribution of the quasiparticles, with the energies containing the Debye shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' To obtain the thermodynamic potentials 퐹 or 푃 Ω from the equation of state (5) we must perform integration over 휇 or 푛, respectively, and logarithmic terms may appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In particular, for the free energy of the Hydrogen plasma, the virial expansion reads 퐹(푇 , Ω, 푁) = Ω푘퐵푇 {푛 ln 푛 + [3∕2 ln(2휋ℏ2∕(푚푘퐵푇 )) − 1]푛 −퐹0(푇 )푛3∕2 − 퐹1(푇 )푛2 ln 푛 − 퐹2(푇 )푛2 − 퐹3(푇 )푛5∕2 ln 푛 − 퐹4(푇 )푛5∕2 + \ue23b(푛3 ln 푛)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (11) see9,25 where expressions for the lowest virial coefficients 퐹푖 are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Details on the calculation of the EoS for Coulomb systems can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='9 and will not be repeated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The virial expansion for the uniform electron gas is discussed below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Perturbation expansion and partial summations also apply to the evaluation of the correlation function (7) which is related to the electrical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In the lowest order of perturbation theory, where interactions are neglected, the total momentum of the electrons is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' As a consequence, the expression (7) becomes divergent, the ideal plasma shows no finite value for the conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Partial summations, in particular the self-energy and vertex corrections, lead to finite values for the conductivity, see26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Analytical evaluation of the Kubo formula remains difficult and cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In contrast, it is possible to perform a virial expansion for the inverse conductivity 푅 = 1∕휎, expressed as a correlation function of the stochastic forces26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A generalized linear response theory was worked out that takes into account correlation functions of higher moments of the occupation number distribution4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In this way the relation to the kinetic theory was shown21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' These correlation functions are also treated by the methods of Green functions, Feynman diagram techniques and partial summations, so that virial expansions can be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The dc conductivity 휎(푛, 푇 ) is usually associated with a dimensionless function 휎∗(푛, 푇 ) according to 휎(푛, 푇 ) = (푘퐵푇 )3∕2(4휋휖0)2 푚1∕2 푒 푒2 휎∗(푛, 푇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (12) We consider both 휎 and 휎∗ as a function of density 푛 at fixed temperature 푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In the limiting case of low density, the following virial expansion for the inverse conductivity 휌∗(푛, 푇 ) = 1∕휎∗(푛, 푇 ) was obtained from kinetic theory and generalized linear response theory4,5,6: 휌∗(푛, 푇 ) = 휌1(푇 ) ln 1 푛 + 휌2(푇 ) + 휌3(푇 ) 푛1∕2 ln 1 푛 + \ue23b(푛1∕2), (13) which begins with a logarithmic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Values for the virial coefficients 휌푖(푇 ) are given below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 5 4 VIRIAL PLOTS Equilibrium properties, such as the correlation functions considered here, depend on a limited number of state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For the Hydrogen plasma, this are the temperature 푇 and the electron number density 푛 (for charge neutral plasmas, the ion (proton) number density is also 푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For the uniform electron gas, we have the same variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Instead of the ion subsystem a homo- geneously charged background (jellium model) is considered to establish charge neutrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In the case of a many-component plasma, the independent partial densities 푛푖 (not connected by chemical reactions and charge neutrality) of the components are the state variables in addition to 푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We focus here on the two simple cases where the state variables are 푇 , 푛, and we study the correlation energy ̄푉 (푇 , 푛) of the uniform electron gas and the electrical conductivity 휎(푇 , 푛) of the Hydrogen plasma, in particular the resistivity 푅(푇 , 푛) = 1∕휎(푇 , 푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It is convenient to introduce dimensionless variables instead of 푇 , 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We use atomic units with the Hartree energy 퐸Ha = ( 푒2 4휋휖0 )2 푚 ℏ2 = 27, 21137 eV = 2 Ry (14) and the Bohr radius 푎퐵 = 4휋휖0 푒2 ℏ2 푚 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='2918 × 10−11 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (15) The density in atomic units is usually represented by the radius of a sphere containing an electron, 푟푠 = ( 3 4휋푛 )1∕3 1 푎퐵 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (16) The temperature is related to the energy 푘퐵푇 , so that 1 eV corresponds to 11604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We denote 푇eV as 푘퐵푇 measured in units of eV, 푇Ha in units of 퐸Ha, and 푇Ry in units of Ry so that 푇Ha = 푘퐵푇 퐸Ha = 2푇Ry = 27, 21137 푇eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (17) Another well-known choice of dimensionless parameters is Γ = 푒2 4휋휖0푘퐵푇 (4휋 3 푛 )1∕3 , Θ = 2푚푘퐵푇 ℏ2 (3휋2푛)−2∕3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (18) The plasma parameter Γ characterises the ratio of potential to kinetic energy in the non-degenerate case, and the electron degen- eracy parameter Θ characterises the range in which the electrons are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Different sets of dimensionless parameters are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Thus, PIMC calculations for specific parameter values of 푟푠, Θ are discussed in the following section, the corresponding plasma parameters 푛, 푇 are determined as follows, 푛 = 3 4휋 1 (푟푠푎퐵)3 , 푘퐵푇 = 퐸Ha 1 2 (9휋 4 )2∕3 Θ 푟2 푠 (19) with 퐸Ha∕푘퐵 = 315777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The dc conductivity 휎(푛, 푇 ) is also associated with a dimensionless function 휎∗(푛, 푇 ) according to 휎(푛, 푇 ) = (푘퐵푇 )3∕2(4휋휖0)2 푚1∕2 푒 푒2 휎∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='0258883 푇 3∕2 휎∗(Ωm K3∕2)−1 = 32405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4 푇 3∕2 eV 휎∗(Ωm)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (20) As with thermodynamic relations, the dimensionless conductivity 휎∗ can be expressed as a function of dimensionless variables 푟푠, 푇Ha or Γ, Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' These functions are now to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Exact results are currently known only for limiting cases, in particular virial expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The analysis of a virial expansion is sometimes not easy because trivial terms dominate in limiting cases so that interesting terms remain hidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In the example of the thermodynamic EoS considered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 5, one dominant term is the Debye shift, which covers the contribution of higher virial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We introduce reduced virial expansions where these exactly known contributions are suppressed, and quantities are introduced that anticipate a linear relationship in special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The virial plot is the representation of this asymptotic linear relationship and allows us to extrapolate virial coefficients from simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We demonstrate this procedure for two cases, the mean potential energy of the uniform electron gas in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 5 and the electrical conductivity of the Hydrogen plasma in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 6 If we express 휎∗(푛, 푇 ) in terms of dimensionless parameters Γ, Θ and use the Born parameter Γ∕Θ, which is of interest in the range 푘퐵푇 ≫ 1 Ry, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (13) we obtain a modified virial expansion where the argument of the logarithm is dimensionless, 1 휎∗(Γ, Θ) = 휌∗(Γ, Θ) = ̃휌1(Γ2Θ) ln (Θ Γ ) + ̃휌2(Γ2Θ) + … , Γ2Θ = 27∕3 34∕3휋3∕3 1 푇Ha , Θ Γ = 21∕3 31∕3휋5∕3 푇 2 Ha 푛푎3 퐵 (21) We define the reduced effective virial coefficient ̃휌eff 2 (푇 ) according to ̃휌eff 2 (푛, 푇 ) = 32405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4 휎(푛, 푇 )[Ωm]푇 3∕2 eV − ̃휌1(푇 ) ln (Θ Γ ) , (22) with lim푛→0 ̃휌eff 2 (푛, 푇 ) = ̃휌2(푇 ), see also Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (45) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The plot of 휌∗∕ ln(Θ∕Γ) as a function of 푥 = 1∕ ln(Θ∕Γ) at given 푇 is called a virial plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It directly allows the determination the virial coefficients 휌1(푇 ), 휌2(푇 ), as it is shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' As will be demonstrated in this work, virial plots are very sensitive to diverse approaches, including the results of numerical simulations, in the low density domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Since trivial dominant terms, which are known exactly, are suppressed, they have no effects due to possible approximations, and the extrapolation of numerical simulations into the low-density domain becomes immediately possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 5 VIRIAL EXPANSION OF THE EOS OF THE UEG, COMPARISON WITH PIMC SIMULATIONS The problem of the second virial coefficient for the mean correlation energy ̄푉 was considered in a recent work25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' There was a controversy about the high-temperature limit of the second virial coefficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' the term ∝ 1∕ √ 푇 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This controversy dis- appears in charge-neutral two-component plasmas, but not in the uniform electron gas (UEG), where interacting electrons are moving in front of a positively charged jellium-like background to neutralize the Coulomb field at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Accurate PIMC simulations have been available at low densities and high temperatures25, so that it was possible to confirm the correct limiting behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In this section, we not only show the virial plot method to confirm the correct limiting law, but consider the full second virial coefficient and discuss deviations from this expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The virial expansion of the free energy 퐹(푇 , Ω, 푁) of the UEG is obtained from the general formula for a multi-component plasma given in9,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The mean potential energy 푉 is determined by 푉 (푇 , Ω, 푁) = 푒2 휕 휕(푒2)퐹(푇 , Ω, 푁) (23) (for the relation to the internal energy see28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' From the virial expansion of 퐹(푇 , Ω, 푁), we get the following virial expansion of 푉 푉 푁푘퐵푇 = − 휅3 8휋푛 − 휋푛휆3휏3 ln(휅휆) −휋푛휆3 [ 휏 2 − √ 휋 2 (1 + ln(2))휏2 + ( 퐶 2 + ln(3) − 1 3 + 휋2 24 ) 휏3 + √ 휋 ∞ ∑ 푚=4 (−1)푚푚 2푚Γ(푚∕2 + 1) [2휁(푚 − 2) − (1 − 4∕2푚)휁(푚 − 1)] 휏푚 ] −휋푛휆4휏4휅 ln(휅휆) + 푉4(푇 ) 푁푘퐵푇 푛3∕2 + \ue23b(푛2 ln(푛)) (24) with the variables 휅2 = 푛푒2 휖0푘퐵푇 , 휆2 = ℏ2 푚푘퐵푇 , 휏 = 푒2√ 푚 4휋휖0 √ 푘퐵푇 ℏ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (25) 휁(푥) denotes the Riemann zeta function, and 퐶 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='57721 … is Euler’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We express this expansion in terms of 푇 , 푛 and introduce atomic units ℏ = 푚 = 푒2∕4휋휖0 = 1 so that 푘퐵푇 is measured in Hartree (Ha) and 푛 in electrons per 푎3 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The virial expansion of the specific mean potential energy 푣 = 푉 ∕푁 is as follows 푣(푇 , 푛) = 푣0(푇 )푛1∕2 + 푣1(푇 )푛 ln (휅2휆2) + 푣2(푇 )푛 + 푣3(푇 )푛3∕2 ln (휅2휆2) + 푣4(푇 )푛3∕2 + \ue23b(푛2 ln(푛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (26) 7 If atomic units are used, this results in (휅2휆2 = 4휋푛∕푇 2) 푣0(푇 ) = − √ 휋 푇 1∕2 , 푣1(푇 ) = − 휋 2푇 2 , 푣2(푇 ) = − 휋 푇 [ 1 2 − √ 휋 2 (1 + ln(2)) 1 푇 1∕2 + ( 퐶 2 + ln(3) − 1 3 + 휋2 24 ) 1 푇 − √ 휋 ∞ ∑ 푚=4 푚 2푚Γ(푚∕2 + 1) ( −1 푇 1∕2 )푚−1 [2휁(푚 − 2) − (1 − 4∕2푚)휁(푚 − 1)] ] , 푣3(푇 ) = − 3휋3∕2 2푇 7∕2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (27) In ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='25, a virial plot was presented to study the behavior of the second virial coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We consider the lowest orders of the virial expansion, 푣(1)(푇 , 푛) = − √ 휋 푇 1∕2 푛1∕2 − 휋 2푇 2 푛 ln (4휋푛 푇 2 ) , (28) as exactly known and subtract them from the data obtained from the PIMC simulations, 푣PIMC = 푉 PIMC∕푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' These exactly known terms may become very large, hiding the higher virial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (Note that the logarithmic term contains a factor to become dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This factor can be moved to the next virial coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=') In25 we introduced the reduced potential energy (휏 = 푇 −1∕2, atomic units) 푣red 2 (푇 , 푛) = [푣PIMC − 푣(1)(푇 , 푛)] −푇 휋푛 = −푇 휋 푣2(푇 ) + \ue23b(푛1∕2 ln(푛)) = 1 2 − √ 휋 2 (1 + ln(2))휏 + ( 퐶 2 + ln(3) − 1 3 + 휋2 24 ) 휏2 + \ue23b(휏3) + \ue23b(푛1∕2 ln(푛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (29) Table 1 PIMC calculations for the uniform electron gas: 푣PIMC and 푣red 2 , eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (29), for special parameter values 푟푠, Θ and the corresponding values of 푇 , 휏, 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 푟푠 Θ 푣PIMC [Ha] 푇Ha 휏 푣red 2 푇 [K] 푛 [cm−3] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 128 0.' metadata={'source': 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+page_content='50255 11630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='01378e20 In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 1, the parameter values of the uniform electron gas are given for which PIMC calculations were presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='25, together with the values for 푣red 2 (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The results for 푣red 2 are also shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 1 all calculated PIMC data of25 are considered and the corresponding value of 푣red 2 is shown as function of 휏, see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In addition, three expressions for (29) are shown: up to order 휏, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=', 1∕2 − √ 휋(1 + ln(2))휏∕2, up to order 휏2, and the full 휏 dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 1 shows in which interval of 휏 the linear or quadratic approximation is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The PIMC data are 8 0 1 2 3 4 5 6 T 1/2 [Ha] 0 5 10 15 20 v red rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 rs = 2 rs = 20 T 1/2 T 1 2 nd virial 3 rd virial, rs=20 Figure 1 Reduced potential energy 푣red 2 (푇 , 푛), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (29), as function of 휏 = 1∕ √ 푇 for different densities, 푟푠 = 0, 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For comparison, the reduced second virial coefficient 푣red 2 (푇 ) = −(푇 ∕휋)푣2(푇 ) [2nd virial, according Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (27)] as well as the lowest orders in 1∕푇 are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In addition, the curve 3rd virial given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (30) is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (Atomic units are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=') very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The lowest density, 푟푠 = 20, should be most relevant to the low-density limit, where higher virial coefficients are less important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' However, the inverse temperature 휏 = 푇 −1∕2 Ha is too large to reach the limit 휏 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Close to this limit are PIMC simulation data for 푟푠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The relatively large density is compensated by the very high temperature, see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 1 is shown enlarged in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It was a main result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='25 to show that the PIMC simulation data confirm the limit 푣red 2 (휏 = 0) = 1∕2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Linear fit to the data for 푟푠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 is possible, and extrapolation to 푣red 2 (휏 = 0) gives 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' At the same time, one gets an idea of the accuracy of the simulation, which shows up as scatter around the analytical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The PIMC data for 푟푠 = 2 are not described by the linear approximation but almost well by the quadratic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Finally, we have to make a comparison with the full second virial coefficient and will find that good agreement is obtained in all three density cases, given by the parameter 푟푠, only for the lowest values of 휏 (an exception is the lowest 휏 parameter calculation for 푟푠 = 2, which needs to be checked).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' As 휏 increases, the PIMC data are systematically below the second virial curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We assume that the PIMC simulations are very accurate, so this deviation indicates the contribution of higher virial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Deviations from the second virial coefficient −(푇 ∕휋)푣2(푇 ) indicate the contribution of higher orders to the virial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We expect a significant next order contribution from the low-density calculations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=', 푟푠 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We consider the expression 푣red 2+3(푇 , 푛) = −푇 휋 [ 푣2(푇 ) + 푣3(푇 )푛1∕2 ln (4휋푛 푇 2 )] , (30) which accounts for the contribution of the third virial coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For 푟푠 = 20, the data are well reproduced for the lowest values of 휏, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Deviations for larger 휏 indicate the contributions of higher virial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The deviation Δ푣red 2 (푇 , 푛) = [푣PIMC − 푣(1)(푇 , 푛) − 푣2(푇 )푛] 푇 휋푛 (31) is shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 2, together with the deviation Δ푣red 3 (푇 , 푛) = [ 푣PIMC − 푣(1)(푇 , 푛) − 푣2(푇 )푛 − 푣3(푇 )푛3∕2 ln (4휋푛 푇 2 )] 푇 휋푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (32) As mentioned before, the inclusion of the third virial coefficient 푣3(푇 ) improves the agreement of the PIMC simulations with the virial expansion, as also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The remaining difference Δ푣red 3 (푇 , 푛) is related to the fourth-order and higher-order virial coefficient, 푣eff 4 (푇 , 푛) = Δ푣red 3 (푇 , 푛) 휋 푇 푛1∕2 = 푣4(푇 ) + \ue23b(푛1∕2 ln(푛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (33) 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='6 T 1/2 [Ha] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='6 v red rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 rs = 2 T 1/2 T 1 2 nd virial Figure 2 Detail of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The fourth virial coefficient results when higher-order virial coefficients are neglected, lim푛→0 푣eff 4 (푇 , 푛) = 푣4(푇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This should be possible in the low-density limit, where the contributions of higher orders of the density expansion become small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' However, high-precision calculations are required to extract the higher-order coefficients, and the accuracy of the present calculations25 is not sufficient to determine precisely the fourth- and higher-order virial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We give here only a discussion of the present data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' From the virial expansion of the free energy9, the fourth virial coefficient 푣4(푇 ) contains contributions with temperature dependence ∝ 푇 −2 = 휏4 and higher orders in 휏, as well as contributions ∝ 푇 −7∕2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The coefficient of the 휏4 term follows as 3휋 √ 4휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We expect a high-temperature limit behavior ∝ 푇 −2, and we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 3 the quantity 푣eff 4 (푇 , 푛) × 푇 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We see that the lowest value of density, 푟푠 = 20, exhibits behavior at small 휏 values that can be compared to a curve 3휋 √ 4휋 − 6 × 휏3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' However, the exact determination of the fourth virial coefficient 푣4(푇 ) is not possible from the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' At the higher densities corresponding to smaller 푟푠, the accuracy of the numerical PIMC simulations may not be sufficient to extract higher-order virial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In the context of our analysis, in addition to the dependence on 푇 , the dependence on 푛 would be of interest to perform the virial plot as a function of 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Further calculations for density parameter values in the range of 푟푠 = 20 would be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Since we are investigating the differences between large numbers, high accuracy is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The study of the uniform electron gas is not only of interest for the discussion of the exchange-correlation term of the energy- density functional in DFT calculations, for which Dornheim, Groth, and Bonitz derived analytical formulas29,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It is also a prerequisite to treat the more interesting case of a two-component plasma, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=', the Hydrogen plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The equation of state at low densities is of interest, for example, in helioseismology31, where the fourth virial coefficient 푣4(푇 ) is important32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In this context, the high-temperature limit of 푣red 2 (휏 = 0) was discussed in27,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For a discussion of the fourth virial coefficient 푣4(푇 ) of Hydrogen plasma, see also Alastuey and Ballenegger33,34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 6 VIRIAL EXPANSION OF THE INVERSE CONDUCTIVITY OF H PLASMAS, COMPARISON TO DFT-MD SIMULATIONS Numerous studies have been performed to calculate the electrical conductivity 휎(푛, 푇 ) of Hydrogen plasma in a wide range of parameters, a recent review can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A comparative study36 was also recently published that considered different approaches and showed large differences in the calculated conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Analytical calculations in the framework of generalized 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 τ = (THa) 1/2 300 250 200 150 100 50 0 50 v4 eff(T,n)* (THa) 2 6π 3/2− 6τ 3 rs=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 rs=2 rs=20 Figure 3 Effective reduced fourth virial coefficient 푣eff 4 (푇 , 푛) × 푇 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (33), plotted as function of 휏 = 1∕ √ 푇Ha for different densities, 푟푠 = 0, 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For comparison, a curve 3휋 √ 4휋 − 6휏3 is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (Atomic units used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=') Table 2 PIMC calculations for the UEG: 푣 and 푣red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The calculation with the second virial coefficient, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (32), is denoted by 푣vir and 푣red vir .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 푟푠 Θ 푣 [Ha] 푇Ha 푛 푎3 퐵 휏 푣red 2 Δ푣red 2 Δ푣red 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 128 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='8 2 32405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4 (T/eV) 3/2/σ[S/m] / ln(Θ/Γ] tilde ρ1Spitzer tilde ρ1Lorentz Karakhtanov QLB, Ronald 64 125 216 95 T = 2000 eV T = 200 eV n = 40 g/ccm n = 2 g/ccm Figure 4 Reduced resistivity ̃휌(푥, 푇 ) (42) for hydrogen plasma as a function of 푥 = 1∕ ln(Θ∕Γ): DFT-MD simulations from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='15, and Lenard-Balescu results (QLB, Ronald) of Desjarlais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='37 and Karakhtanov40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 휌Spitzer 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='846 and 휌Lorentz 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='492 are defined in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The green line represents a linear extrapolation of the converged DFT-MD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Data are given in the Supplemental material of15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' of a virial expansion of resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The non-consideration of the contribution of electron-electron collisions in other transport coefficients such as thermopower and thermal conductivity has also been discussed recently37,39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The virial expansion of the dimensionless resistivity 휌∗, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (13), contains the logarithmic term ln(1∕푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' To make its argument dimensionless we use the Born parameter, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='15, Θ Γ = 푇 2 Ry 푛Bohr (96휋5)−1∕3 , (34) where the temperature is measured in Rydberg units, 푇Ry = 2푇Ha = 푘퐵푇 ∕13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 4 in connection with the logarithmic term, we use a modified virial expansion and rewrite (13) 휌∗(푛, 푇 ) = ̃휌1(푇 ) ln (Θ Γ ) + ̃휌2(푇 ) + … .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (35) The modified virial coefficients ̃휌푖 are related to 휌푖 replacing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (35) the variables Θ, Γ by 푛, 푇 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Comparing with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (13), ̃휌1 = 휌1 is obtained and ̃휌2 = 휌2 + 휌1 ln[(96휋5)1∕3∕푇 2 Ry] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (36) A highlight of plasma transport theory is that the exact value of the first virial coefficient for Coulomb systems is known from the seminal paper of Spitzer and Härm3, 휌1 = ̃휌1 = 휌Spitzer 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='846024, (37) which does not depend on 푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (37) accounts for the contribution of the electron-electron (푒 − 푒) interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In contrast, for the Lorentz plasma model where the 푒 − 푒 collisions are neglected so that only the electron-ion interaction is considered, the first virial coefficient is4 휌Lorentz 1 = 1 16(2휋3)1∕2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='492126 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (38) Although 푒−푒 collisions do not contribute to a change of the total momentum of the electrons due to conservation of momentum, the distribution in momentum space is changed by 푒−푒 collisions ("reshaping"), and higher moments of the electron distribution 12 are not conserved by 푒−푒 collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The indirect influence of 푒−푒 collisions on the dc conductivity becomes clear in generalized linear response theory where these higher moments are considered, see4,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' No exact value is known for the second virial coefficient 휌2(푇 ) or ̃휌2(푇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It depends on the treatment of the many-body effects, in particular on the screening of the Coulomb potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In a quantum statistical approach, the static (Debye) screening by electrons and ions should be replaced by dynamical screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For the Hydrogen plasma considered here, the Born approximation for the collision integral at high temperatures 푇Ry ≫ 1 is justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Consideration of screening in the random phase approximation (RPA), leads to the quantum Lenard-Balescu (QLB) expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Thus, at very high temperatures, where the dynamically screened Born approximation becomes valid, we obtain the QLB result, see37,40, lim 푇 →∞ ̃휌2(푇 ) = ̃휌QLB 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4917 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (39) As 푇 decreases, strong binary collisions (represented by ladder diagrams) become important and must be treated in the calculation of the second virial coefficient ̃휌2(푇 ) beyond the Born approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' According to Spitzer and Härm3, the classical treatment of strong collisions with a statically screened potential gives for 휌∗ = 1∕휎∗ the result 휌∗ Sp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='846 ln [3 2Γ−3] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (40) Interpolation formulas have been proposed that link the high-temperature limit ̃휌QLB 2 with the low-temperature Spitzer limit45,41,42,5,43,6,4,44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Based on a T-matrix calculation in quasiclassical (Wentzel-Kramers-Brillouin, WKB) approximation45,46, the expression (푇eV = 푘퐵푇 ∕eV) ̃휌2(푇eV) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4917 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='846 ln [ 1 + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='492∕푇eV 1 + 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='83∕푇eV + 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='2∕푇 2 eV ] (41) is a simple interpolation that combines the QLB result with the Spitzer limit in WKB approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' However, the exact analytical form of the temperature dependence of the second virial coefficient ̃휌2(푇 ) remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Thus, the available exact results for the virial expansion (35) of the inverse conductivity of fully ionized Hydrogen plasma are: (i) the value of the first virial coefficient is ̃휌1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='846;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (ii) the second virial coefficient has the high-temperature limit lim푇 →∞ ̃휌2(푇 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4917;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (iii) the second virial coefficient is temperature dependent, an approximation is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' To extract the first and second virial coefficient from calculated or measured dc conductivities, we plot the expression ̃휌(푥, 푇 ) = 휌∗ ln(Θ∕Γ) = 32405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4 휎(푛, 푇 )(Ωm)푇 3∕2 eV 1 ln(Θ∕Γ) (42) as a function of 푥 = 1∕ ln(Θ∕Γ) and 푇 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 4 which is called virial plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' According to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (13), (35), the behavior of any isotherm (fixed 푇 ) is linear near 푛 → 0, ̃휌(푥, 푇 ) = ̃휌1(푇 ) + ̃휌2(푇 )푥 + … , (43) with ̃휌1(푇 ) as the value at 푥 = 0 and ̃휌2(푇 ) as the slope of the isotherm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In this way, the extraction of virial coefficients becomes immediately possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For 푥 > 1∕ ln(100) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='217, the contributions of higher order virial coefficients have to be taken into account15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For fixed 푇 and low density, where 휃 ≫ 1, a classical plasma is present and the effects of degeneracy contribute to the higher order virial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 4 two cases for the first virial coefficient 휌1 on the ordinate axis are shown, see also4,5,6: (i) 휌Spitzer 1 from kinetic theory when 푒 − 푒 collisions are taken into account, (ii) when 푒 − 푒-collisions are neglected, 휌Lorentz 1 is obtained for the Lorentz plasma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Moreover, the second virial coefficient ̃휌QLB 2 of the Lenard-Balescu approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (39) is shown, which is correct in the high temperature limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The QLB calculations of Desjarlais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='37 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The 푒 − 푒 collisions are taken into account, yielding the same asymptote (푥 → 0) as in Karakhtanov40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' With increasing 푥 = 1∕ ln(Θ∕Γ) small deviations from linear behavior are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' When isotherms are presented, this deviation indicates the contribution of higher virial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Virial plots are presented in15 to investigate two problems: Which of the various approaches that give us analytical expressions for the electrical conductivity of Hydrogen plasmas are accurate in the low density limit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The virial expansion of the inverse conductivity serves as an exact benchmark for theoretical approaches, so that the accuracy and consistency of semi-empirical results for conductivity, such as those collected in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='36, can be checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A more fundamental problem is whether numerical results from molecular dynamics simulations based on density functional theory (DFT-MD) correctly contain the contribution of electron-electron collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The virial plot confirms the position that DFT-MD simulations in the low-density limit describe a Lorentz plasma with only electron-ion collisions, the contribution of electron-electron collisions to 휌1 is missing15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 13 Here we discuss some details of the virial expansion for the inverse conductivity and the corresponding virial plots, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' DFT-MD simulations are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='15, see the tables of data in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' These data have sufficiently high accuracy, as can be seen from the small deviations from the fit line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In addition to the precise solution of the Kubo- Greenwood formula, this is achieved by good control of convergence with increasing particle number, as shown by comparison of calculations with different numbers of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The number of particles must be sufficiently large to ensure convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In the parameter range considered in the figure, about 100 particles in the box are necessary to achieve convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Further calculations with 216 electrons were not possible due to limited computer capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For 푇 = 150 eV, even 125 electrons exceed the currently available computer capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This point was also discussed in a recent work39, where earlier calculations37 were improved to achieve convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Another problem is the determination of the value of the dc conductivity 휎(0) from the calcu- lation of the optical conductivity 휎(휔) at finite frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Because of the discretisation in a finite box, the energy eigenvalues have a minimum spacing and the energy-conserving 훿 function must be smeared by a parameter 휖 to allow for transitions, see also section 3 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' To reach the limit 휔 → 0, an extrapolation is performed according to the Drude formula (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This was discussed also in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Instead, one can use the dynamic collision frequency to perform this extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 4 allow the extraction of virial coefficients 휌1(푇 ), ̃휌2(푇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Compared to other approaches, including interpolation formulas, see15, as well as the QLB calculation, we assume that we are in the linear region of the virial curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Deviations from linearity can be observed for QLB already at 푥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='2, since the density is high (40 g/cm3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For DFT-MD simulations with density about 2 g/cm3, the deviation from linearity for the last point is observed at 푥 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' As pointed out in15, the extrapolated value of 휌1 in the virial plot at 푥 = 0 points to the Lorentz value (38) but misses the Spitzer value (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This means that electron-electron collisions are not considered in the DFT-MD calculations for the electrical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Also of interest is the value of ̃휌2(푇 ) given by the slope in the virial plot near 푥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Fitting it to the data gives a slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='9886 for the DFT-MD calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This is about twice the slope ̃휌QLB 2 given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' From analytical approaches, it appears that the slope is determined by various effects such as dynamical screening and strong collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In the limiting case of high temperatures, the Born approximation should be possible, but the Coulomb potential must be replaced by a screened potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Static screening of the proton scatterer with both electrons and protons would lead to the following result (퐶 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='57721 … is Euler’s constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' lim 푇 →∞ ̃휌2(푇 ) = 휋3∕2 24 √ 2 [11 2 − 3퐶 + ln (3 2휋2)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='06036 (44) which is close to the observed slope of the DFT-MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' However, it remains unclear to what extent the screening is included in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We assume that the ionic structure factor, which is the ionic contribution to the screening, is well described, and that the electron screening is also captured by the exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' However, we need to consider dynamical screening, a problem that has been discussed in previous work5 on virial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We return to the long-debated question of whether or not 푒 − 푒 collisions are accounted for in the DFT-MD formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For example, it was pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='21 that a mean-field approach is not able to describe two-particle correlations, in particular 푒 − 푒 collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' However, to some approximation, the 푒 − 푒 interaction is accounted for by the exchange-correlation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' DFT-MD simulations, which are mean-field theories that account for the 푒−푒 interaction only through the exchange-correlation part of the energy density, cannot account for the effect of 푒 − 푒 collisions on the conductivity, so that 휌1(푇 ) corresponds to the Lorentz plasma, but ̃휌2(푇 ) is determined by screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The question arises to what extent dynamical screening, as implemented in the QLB calculations, is also described by the exchange-correlation part of the energy density functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We would like to mention that in the case of thermal conductivity it has been shown that the contribution of 푒 − 푒 collisions is not taken into account in DFT-MD simulations37,35,39 and yields an additional term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Other approaches such as generalized linear response theory may be considered to indicate appropriate approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Our analysis has shown that the simulation results with virial evolution are extrapolated to the low-density region, where DFT-MD simulations are no longer feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The current simulations, while computationally expensive, are still not very close to 푥 = 0, so extrapolation to the 푥 = 0 limit is not very accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Better data for DFT-MD simulations would be of interest to confirm our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Conversely, the benchmark capability of virial expansion discussed in this work can also serve as a criterion to verify the accuracy of numerical approaches such as DFT-MD simulations to evaluate conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Another application of the virial plot is experiments to measure electrical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Assuming that the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='846024, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (37), for 휌1 is exact, an effective second virial coefficient ̃휌eff 2 (푛, 푇 ) = 32405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4 휎(푛, 푇 )[Ωm] ( 푇 eV )3∕2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='846024 ln (Θ Γ ) (45) 14 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='9 1 1/T [eV] 3 2 1 0 1 2 3 second virial coefficient tilde ρ2(T) H: Guenther, Radtke Ar, Xe, Ne: Ivanov Ar, Xe: Popovic Lenard-Balescu value ERR interpolation QLB/Desjarlais interpolation (7) Figure 5 Second virial coefficients ̃휌2(푇 ) and ̃휌eff 2 (푛, 푇 ) for the dc conductivity of Hydrogen plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Analytical interpolation formulas (41) and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='45 are compared with experiments of Günther and Radtke47 for H plasmas as well as of Ivanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='48 and Popovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='49 for rare gas plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The black dashed line corresponds to the high temperature limit that is given by the quantum Lenard-Balescu value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The broken blue line is the interpolation formula of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='45, the red full line represents the interpolation formula (41) for the second virial coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' has been introduced which gives the second virial coefficient in the low-density limit, lim푛→0 ̃휌eff 2 (푛, 푇 ) = ̃휌2(푇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A dependence of ̃휌eff 2 (푥, 푇 ) on density shows that higher orders of the virial expansion are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We anticipate that at very high 푇 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=', 1∕푇 → 0, the Lenard-Balescu value is approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The deviations at increasing 1∕푇 , shown in the interpolation formula and the DFT-MD simulations, indicate that already below temperatures of the order of 100 eV, the effect of strong collisions beyond the Born approximation should be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Ultimately, the virial expansion (35) must be verified experimentally, but accurate data for the conductivity of Hydrogen plasma in the low-density limit and/or at high temperatures are scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Accurate conductivity data for dense Hydrogen plasma were derived by Günther and Radtke47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' They are close to the benchmark data of the virial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It should be noted that there are systematic errors associated with the analysis of such experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For example, the appearance of bound states requires a realistic treatment of the plasma composition and the influence of neutrals on electron mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Alternatively, conductivity measurements in highly compressed noble gas plasmas were carried out by Ivanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='48 and Popovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='49,45, but the interaction of the electrons with the ions deviates from the pure Coulomb potential due to the core of bound electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The corresponding virial plot is close to the data of Hydrogen plasma, see15, but requires a more detailed discussion on the role of bound electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It should also be mentioned that the densities are quite high, and extrapolation to zero density must be performed to obtain the second virial coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This tendency can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 5, especially for the experiments with Ar, Xe49, where low-density data point to ̃휌2(푇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Quantum statistical methods provide accurate values for the lowest virial coefficients, which serve as benchmarks for an- alytical approaches to electrical conductivity as well as for numerical results from molecular dynamics simulations based on density functional theory (DFT-MD) or path integral Monte Carlo (PIMC) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' While these simulations are well suited to compute 휎(푛, 푇 ) in a wide range of densities and temperatures, especially for the warm dense matter region, they become com- putationally expensive in the low density limit, and virial expansions can be used to compensate for this drawback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Interpolation formulas that take both approaches into account would be very useful for calculating the conductivity of plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 15 Table 3 Experimental data for the electrical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Günther and Radtke: H47;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Ivanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' : Ar, Xe, Ne48;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Popovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' : Ar, Xe49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Plasma ̂푛푒 × 1025 푛 × 10−6 푇 × 104 푇 Γ Θ 1∕ ln(Θ∕Γ) 휎 × 103 ̃휌(푥, 푇 ) ̃휌eff 2 [m3] [g/cm3] [K] [eV] [(Ωm)−1] H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='67262 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='20715 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='49289 To obtain the correct values for the thermoelectric transport coefficients of Hydrogen plasma in the low-density limit, where the inclusion of 푒 − 푒 collisions is essential, different solutions can be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' PIMC simulations, as successfully performed for the uniform electron gas25, should also be performed for the two-component plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' First steps of this ambitious project are recently in progress12,50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The study of such PIMC calculations with the virial plot would be of great interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' From generalized linear response theory, we also learn that higher order correlation functions, such as force-force correlation functions associated with the dynamic collision frequency, may be a suitable approach to include the contribution of 푒 − 푒 collisions in the transport coefficients4,5,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 7 CONCLUSIONS We have from quantum statistics exact expressions for thermodynamic and transport properties of plasmas by equilibrium cor- relation functions, but the evaluation is a complex problem in many-particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Numerical simulations are becoming more accurate as computer capacity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' However, they need to be controlled with respect to their limits such as size effects, but also fundamental problems such as the correct description of electron-electron collisions in the context of DFT or the sign problem in PIMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It is expected that PIMC simulations will provide an adequate description of electron-electron interactions, but they are currently unable to solve complex plasmas such as multiply charged ions in the low-temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The comparison of analytical results for the virial expansion of thermodynamic properties with PIMC calculations for the uniform electron gas has been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In particular, we show that high-precission PIMC simulations confirm the correct form of the virial expansion, which has been debated recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It seems to be possible to give also numerical values for higher virial coefficients, in particular the interesting 푛5∕2 coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' These values can be considered as exact results in plasma physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Numerical values for higher virial coefficients would also be of great interest for transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 16 Analytical theory gives us exact results in limiting cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This can be used to obtain results for parameter ranges where numerical simulations are not efficient, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' in the low density range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Virial expansions are used to control theories and numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' They are of interest to construct interpolation formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It was indicated that the evaluation of the Kubo-Greenwood formula using DFT-MS simulations does not take into account the effects of electron-electron scattering and cannot reproduce the low-density limit of the electrical conductivity of Hydrogen plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Similar results were recently reported by French et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='39 for other thermoelectric transport coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It would be of interest to perform PIMC simulations that can accurately describe electron-electron collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The theory of virial expansion must be extended if the formation of bound states is of importance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' for 푇 ∕푇Ha ≤ 1, see appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' New approaches are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The approach described here is also applicable to other correlation functions such as the dynamic structure factor or to other transport properties such as thermal conductivity, thermopower, viscosity, and diffusion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Also of interest is the extension of virial expansion to elements other than Hydrogen, where different ions can be formed and the electron-ion interaction is no longer purely Coulombic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' ACKNOWLEDGMENTS Thanks to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Schörner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Redmer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Bethkenhagen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' French, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Reinholz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Dornheim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Vorberger, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Moldabekov, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Kraeft for collaboration and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This work was supported by the German Research Foundation (DFG), Grant # RO 905/37-1 AOBJ 655625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Author contributions This is an author contribution text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It is based on a contribution to the SCCS22 conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Financial disclosure None reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Conflict of interest The author declares no potential conflict of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' APPENDIX A BOUND STATE FORMATION A special problem of plasmas is the formation of bound states (atoms, charged ions: clusters with a certain number of elemen- tary particles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=', nuclei and electrons) which can dominate the properties in the low-density and low-temperature region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A simple approach is the chemical picture9, where the bound states are considered as new constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The interaction between the different constituents is neglected except for reactive collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Thus, a chemical equilibrium is achieved in which the com- position of the plasma is described by the law of mass action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For a systematic approach including bound state formation see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='33,34 and references given there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We will not present here an exhaustive discussion of the chemical picture, but only discuss some aspects in the context of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For a recent review, see51,52, where further references can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Within the chemical picture, several issues arise that need to be discussed in order to improve this simple approximation, using the concept of virial expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (i) In addition to the ground state, excited states (푠) with excitation energy 퐸훼,푠 can occur, which can also be treated as new species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It is more convenient to introduce the intrinsic partition function of the cluster 훼, which is summed over all excited bound states by the statistical factor exp[−훽퐸훼,푠].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 17 (ii) In addition to bound states, there are also scattering states that must be included in the calculation of virial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This leads to the Beth-Uhlenbeck formula, in which the scattering phase shifts appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Sometimes resonances can appear in the spectrum of excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In the resonance gas approximation, the intrinsic partition function is improved by extending the summation over all excitations 푠 to the resonances in the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Moreover, the contribution of scattering phase shifts should be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' iii) We arrive at higher virial coefficients and need to include density effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In the framework of a quasiparticle approach, the intrinsic partition functions are calculated with shifted energies due to screening, mean-field effects, Pauli blocking and other effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' As example, let us consider the H plasma and give the intrinsic partition function in the simplest approximation 푧H = ∑ 푠 2푠2푒−퐸H,푠∕푘퐵푇 (A1) with the known energy levels 퐸H,푠 = −퐸Ha∕(2푠2) (퐸Ha = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='2 eV is the Hartree energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The factor 2푠2 denotes the degeneracy of the excitation 푠 including the spin factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' As specific for the Coulomb interaction, we have infinitely many bound states near the continuum edge for 푠 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Expression (A1) is not applicable because it is divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A convergent expression is the Planck-Brillouin-Larkin partition function, see9, 푧H = ∑ 푠 2푠2 [ 푒−퐸H,푠∕푘퐵푇 − 1 + 퐸H,푠 푘퐵푇 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' (A2) The subtraction of 1 is explained as follows: We need to include the contribution of the scattering states which compensate for the most divergent term of the contribution of the bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For the short-range interaction, this has been discussed in detail, and generalized phase shifts have been introduced to avoid separating the bound and scattering parts of the intrinsic partition function53,54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' More complex is the explanation of the subtraction of 퐸Ha∕(2푠2푘퐵푇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Because of the long-range character of the Coulomb interaction, phase shifts cannot be defined in the usual form, and the contribution of the scattering states is not well defined when scattering phase shifts are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This fundamental problem of the Coulomb interaction is solved introducing the concept of screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' In the framework of a quantum statistical approach, we have to perform the partial sum of so-called ring diagrams and introduce quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We must, however, avoid double counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' This has already been discussed in detail for the Hartree- Fock approximation55,56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Of interest is the generalization to partially ionized plasmas with multiply charged ions52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A systematic approach arises from consideration of the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' We can identify a quasiparticle contribution and perform a cluster decomposition of the self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' For the cluster decomposition of the self-energy, we can introduce different channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' To avoid double counting, diagrams used for the single-particle self-energy must be subtracted from the ladder sums defining the cluster states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A related problem is the definition of the ionization degree in dense plasmas, since the separation of the bound state contri- bution from the intrinsic partition function is arbitrary, see57,58,59 and references given there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A possible solution would be the definition of the single-quasiparticle contribution which is extracted from the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Thus it can be performed by considering the compressibility or the dynamical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The inclusion of bound states and the corresponding generalization of the chemical picture, involving quasiparticle concepts for the free and bound states, is a difficult problem in plasma theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Of course, at fixed temperature there is always a low- density limit at which bound states are dissolved (because of entropy) but this regime can be very limited, for instance it is not applicable to gases under normal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A realistic description is often based on the chemical picture where bound states are considered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' for temperatures below the binding energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A generalized quasiparticle approach is well defined at low densities, but has to be generalized considering the spectral function (6) if densities are increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' The formation of bound states is not only important for the thermodynamic properties, as discussed above for the second virial coefficient of the Hydrogen plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' It also determines the transport properties, and the consideration of bound states as additional scatterers remains a complex problem if we want to go beyond the simple chemical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A.' metadata={'source': 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G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Röpke, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' E 85, 036401 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Kraeft, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Kremp, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Ebeling, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Röpke, Quantum Statistics of Charged Particle Systems ( Plenum Press, New York and London, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Lorenzen, B.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Zhang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Plasmas 20, 092703 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Bonitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' London 71, 585 (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Röpke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Schörner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Redmer, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Bethkenhagen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' E 104, 045204 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Hummer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Kresse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Furthmüller, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Bechstedt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' B 73, 045112 (2006).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' E 91, 043105 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Dornheim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Groth, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Bonitz, Physics Reports 744, 1 (2018).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Bonitz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Dornheim, Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Moldabekov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Hamann, 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Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' E 95, 033203 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Holst, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' French, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Redmer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Karakhtanov, Contrib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 56, 343 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Gould and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' DeWitt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 155, 68 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Williams and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' DeWitt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Fluids 12, 2326 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Redmer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Röpke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Morales, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Kilimann, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Fluids B 2, 390 (1990).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Esser, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Redmer, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Röpke, Contrib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 43, 33 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Reinholz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Redmer, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Tamme, Contrib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 29, 395 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Popovic, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Vitel, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Mihajlov, in Strongly Coupled Plasmas, edited by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Ichimaru (Elsevier, Yamada 1990), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Böhme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 129, 066402 (2022).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Special Topics 229, 3403 (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Contrib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 61, e202100085 (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='1002/ctpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content='202100085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Ebeling and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Röpke, Plasma , 6, 1 (2023);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' https://doi.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' E 99, 038201 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Bethkenhagen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} +page_content=' Research 2, 023260 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfif0c/content/2301.01499v1.pdf'} diff --git a/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf b/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2266be4ec7d37dcdd7c0b499a29b7930c0925d10 --- /dev/null +++ b/F9FKT4oBgHgl3EQfbS5P/content/2301.11811v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d92740ea1e5892ae00330447020bd3748a63fd73a8817e45706f4e0b8dd6622a +size 552666 diff --git a/FtE3T4oBgHgl3EQftQtg/content/tmp_files/load_file.txt b/FtE3T4oBgHgl3EQftQtg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b6b9f5baf897e2650a8155524265dfa1c7945ef9 --- /dev/null +++ b/FtE3T4oBgHgl3EQftQtg/content/tmp_files/load_file.txt @@ -0,0 +1,881 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf,len=880 +page_content='Prepared for submission to JHEP IPARCOS-23-002 Late vacuum choice and slow roll approximation in gravitational particle production during reheating Jose A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Cembranos,a Luis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Garay,a Álvaro Parra-Lópeza and Jose M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Sánchez Velázquezb aDepartamento de Física Teórica and IPARCOS, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Ciudad Universitaria, 28040 Madrid, Spain bInstituto de Física Teórica UAM/CSIC, c/ Nicolás Cabrera 13-15, Cantoblanco, 28049, Madrid, Spain E-mail: cembra@fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='es, luisj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='garay@ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='es, alvaparr@ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='es, jm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='sanchez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='velazquez@csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='es Abstract: In the transition between inflation and reheating, the curvature scalar typically undergoes oscillations which have significant impact on the density of gravitationally produced particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The commonly used adiabatic vacuum prescription for the extraction of produced particle spectra becomes a non-reliable definition of vacuum in the regimes for which this oscillatory behavior is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In this work, we study particle production for a scalar field non-minimally coupled to gravity, taking into account the complete dynamics of spacetime during inflation and reheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We derive an approximation for the solution to the mode equation during the slow-roll of the inflaton and analyze the importance of Ricci scalar oscillations in the resulting spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Additionally, we propose a prescription for the vacuum that allows to safely extrapolate the result to the present, given that the test field interacts only gravitationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Lastly, we calculate the abundance of dark matter this mechanism yields and compare it to observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Keywords: Cosmology of Theories beyond the SM, Effective Field Theories, Classical Theories of Gravity arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='04674v1 [gr-qc] 11 Jan 2023 Contents 1 Introduction 1 2 Dynamics of a scalar field in flat FLRW cosmologies 3 3 Background dynamics 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 Inflationary era - Slow-roll approximation 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 Late reheating 7 4 Particle production 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 Solution to the mode equation 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 Choice of reference vacua 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='3 Slow-roll approximation for the solution to the mode equation 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='4 Adiabaticity and oscillations 16 5 Spectra of particles and total density 17 6 Conclusions 21 A Parameters 23 1 Introduction The theory of quantum fields in curved spacetimes accommodates a plethora of unexpected phenomena such as Hawking radiation [1], the Unruh effect [2], or entanglement across horizons [3–6], that have changed our perspective on the interplay between quantum fields and gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Gravitational particle production due to the spacetime dynamics [7, 8] is one of these phenomena and can be particularly important during the early stages of the universe, since it may be able to explain the dark matter abundance, as it has been extensively discussed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The rapidly evolving spacetime during inflation [9–11] and the consequent transient to reheating [12–16] produce a significant abundance of particles for any field that is coupled to the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Since this is the only requirement for this process to occur, it is of particular interest to analyze it from the perspective of dark matter production mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Because of the absence of interactions with other fields, the abundance of dark matter produced in the early universe due to the expansion of spacetime is not diluted as a consequence of thermalization with other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' It remains then as a relic abundance, so that this mechanism alone can in fact explain current observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This has been mostly explored for scalar fields that are non-minimally coupled to gravity in a myriad of works, such as [17–20] for supermassive dark matter candidates (WIMPZillas), or, more recently, in references [21–24], where the importance of the oscillatory behavior of – 1 – the background geometry was incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' On the other hand, gravitational production of more general fields, such as fermion and vector fields, has also been analyzed in [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Usually, the dark matter candidate is regarded as a spectator field [27, 28] which does not source gravity, and with no direct coupling to the inflationary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' However, it is generally non-minimally coupled to the geometry via the curvature scalar, and interactions with other fields are disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In all these works, it is customary to make use of the adiabatic prescription to define the vacuum state of the dark matter field in order to calculate the gravitational production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This definition seems to hold after a few oscillations of the inflaton in the reheating stage, but only in the case of very large masses of the dark matter candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In the regime of low masses, however, this vacuum provides a correct prediction only when considering very late times, after many oscillations have occured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Importantly, this oscillating behavior influences gravitational production [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' It is worth mentioning that the type of dark matter produced in this way is adiabatic [23, 29], and therefore the observational constraints on isocurvature perturbations [30] do not have to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In this work, we study the gravitational production of a massive scalar field ϕ described by a Klein-Gordon action that includes a non-minimal coupling to the Ricci curvature scalar R through a term of the form ξRϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The strength of this coupling is determined by the parameter ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In an attempt to accommodate the arguments put forward in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' [24, 31–33] concerning vacuum instability, overproduction, and quantum cosmology analyses, we restrict ourselves to the range 1/6 ≤ ξ ≤ 1 for the coupling constant ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' As inflationary model, we consider a single inflaton field φ that slowly rolls down a quadratic potential and starts oscillating around its minimum, leading then to a reheating phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The dynamics for the inflaton is analytically solved at the onset of inflation, while the transition to the reheating epoch is modeled numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Our scalar field is assumed to be in the Bunch-Davies vacuum state when inflation starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In order to extract the gravitational production, the Klein-Gordon equation of the field ϕ is solved from that point in time until well inside the reheating era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This moment effectively corresponds to the time where the dynamics enters the adiabatic regime and particle production becomes negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Moreover, one also needs to provide a definition of vacuum for this instant, for which the adiabatic prescription is usually adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We discuss its validity and introduce as well an averaged vacuum that produces the same density of particles but allows to obtain the correct result much earlier than the time at which adiabaticity is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This is particularly helpful when considering masses way below the inflaton mass for our scalar field, which are of great interest concerning dark matter candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Also, we stress the importance of taking into account the first few hundreds of oscillations of the inflaton in the final prediction and present the results in the form of spectra and total density of produced particles for different values of the scalar field mass m and its coupling ξ to the Ricci scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In section 2, we introduce the field that is coupled to the expanding geometry, and work out the formalities of Bogoliubov- like particle production in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In order to determine the complete form of the mode equation, we need to provide the background dynamics coming from the particular inflationary model in consideration, which we do in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' With all these ingredients, we explore the gravitational production for the scalar field in section 4, analyzing the solution – 2 – to the mode equation in the different regimes and studying the influence of the oscillations of the curvature scalar in the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Moreover, we discuss the importance of the vacuum choice when obtaining the numer density of produced particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Lastly, we present our results in the form of spectra and total density of particles in section 5 and elaborate our conclusions in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We set MP = √ G, ℏ = c = kB = 1, and use the metric signature (−, +, +, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Furthermore, greek indices µ, ν run from 0 to 3, while latin indices i, j run from 1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 2 Dynamics of a scalar field in flat FLRW cosmologies We will consider a massive scalar field ϕ non-minimally coupled to gravity in a Friedmann- Lemaître-Robertson-Walker (FLRW) spacetime with vanishing spatial curvature [34–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We will not consider here any coupling of the derivatives of the scalar field (see [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The dynamics of our scalar field is encoded in the action S = −1 2 � d4x√−g � ∂µϕ∂µϕ + � m2 + ξR � ϕ2� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1) where g is the determinant of the metric, m is the bare mass of the field, and ξ is the coupling to the Ricci curvature scalar R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The geometry is determined by the spatially flat FLRW line element ds2 = a2(η) � −dη2 + dx2 + dy2 + dz2� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2) where we have considered Cartesian coordinates for the flat spatial sections, and η is the conformal time, related to cosmological time by a(η)dη = dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' It is convenient to work with the auxiliary field χ(η, x) = a(η)ϕ(η, x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='3) whose equation of motion can be obtained from the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1), χ′′(η, x) − � ∆ + a′′(η) a(η) − a2(η) � m2 + ξR �� χ(η, x) = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='4) where ∆ is the Laplace operator, the prime denotes derivative with respect to conformal time, and R = 6a′′/a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We can use the eigenfunctions of the Laplace operator, which in our case are Fourier modes, as a basis of functions to expand the scalar field χ, χ(η, x) = � d3k (2π)2/3 �akvk(η) + a∗ −kv∗ k(η) � eikx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5) where the coefficients ak, a∗ k become creation and annihilation operators upon quantization of the field, with the standard commutation relations [42–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The time-dependent mode functions vk(η) and v∗ k(η) satisfy a harmonic oscillator equation v′′ k(η) + ω2 k(η)vk(η) = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) – 3 – with k = √ k2 and a time-dependent frequency ω2 k(η) = k2 + a2(η) � m2 + (ξ − 1/6)R(η) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='7) The solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) have to fulfill the normalization condition vkv′ ∗ k − v′ kv∗ k = i, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='8) so that they are compatible with the standard commutation relations of creation and annihilation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For a given evolution of the background geometry, encoded in the scale factor a(η) and the Ricci scalar R(η), both (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) are sufficient to determine vk(η), v∗ k(η), which is a basis of the space of solutions of the mode equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Since any other solution can be expressed as a linear combination of vk(η) and v∗ k(η), any two sets of solutions vk(η) and uk(η) must be related by uk = αkvk + βkv∗ k, where normalization (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='8) on the temporal modes implies the relation |αk|2 − |βk|2 = 1 for the complex coefficients αk and βk, which are known as Bogoliubov coefficients [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Note that the expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5) can be carried out using either basis of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Upon quantization of the field, both sets of coefficients ak and bk (associated with the basis vk and uk, respectively) and their complex conjugates become operators that give rise to two different definitions on quanta and vacua [43], ˆak |0a⟩ = 0 and ˆbk |0b⟩ = 0, ∀ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='9) These two quantizations are related by the Bogoliubov transformation ˆbk = α∗ kˆak − β∗ kˆa† k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The mean number density of b-particles in the a-vacuum, which will be, in general, a non-vacuum state according to the ˆbk operators, is given by ⟨0a| ˆnb k |0a⟩ = |βk|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='10) Integrating over all modes, we find the total mean density � d3k |βk|2, which will remain finite as long as |βk|2 → 0 faster than k−3 for increasing k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Let us now associate each basis of solutions to two observers living at different times ta < tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' If spacetime is static, the frequency (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='7) is constant, so that the solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) takes the same form at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' As a consequence, observers at different times have the same notion of particle, and therefore βk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' However, if geometry undergoes an expansion, two observers living at different times (before and after the expansion) have different notions of vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Thus, βk ̸= 0 and therefore nb ̸= 0, which can be understood as the number density of particles produced out of the original vacuum state due to the expansion of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For the problem at hand, the goal is to extract the number of produced particles after the evolution of the universe during inflation and reheating, once these stages have finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Then, as long as the test particle is not (strongly) interacting, this will be related to the abundance one observer would measure today only by the expansion dilution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Hence, we will take the Bunch-Davies vacuum as initial state, as defined by the solution of the mode – 4 – equation at very early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In our case, we will take the geometry to approach de Sitter spacetime at the beginning of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' On the other hand, the notion of vacuum for an inertial observer after reheating will be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' If the evolution of spacetime is sufficiently adiabatic after this phase, we can assume this is the same vacuum we observe nowadays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Therefore, the corresponding operators will measure the number of particles created in the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The specific form of the scale factor and the Ricci scalar will be determined by the specific inflationary model under consideration, which we describe in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 3 Background dynamics We will describe the early epoch of the universe with a chaotic inflationary model consisting of a single scalar field φ with a quadratic potential of the form V (φ) = 1 2m2 φφ2, where mφ denotes the inflaton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The equation of motion for the inflaton is, if we assume homogeneity and isotropy, 0 = ¨φ + 3H(t) ˙φ + ∂φV (φ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1) where H(t) ≡ ˙a(t)/a(t) is the Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Note that in this context it is customary to work with cosmological time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We will assume that the inflaton contribution to the total energy-momentum tensor is dominant when deriving the corresponding Friedmann equation, H2 = 4π 3M2 P � ˙φ2 + 2V (φ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2) We will also need the Ricci curvature scalar in order to properly describe the frequency of the mode equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6), which in terms of the inflaton field reads R = 8π M2 P � 4V (φ) − ˙φ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='3) Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1), together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2), has no analytic solution in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' However, one can find approximations for certain regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' When this is not possible, we must rely on numerical computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We analyze two different regions which, in conformal time, correspond to η = � � � ηi ≤ η < η∗, Slow-roll approximation, η∗ ≤ η ≤ ηf, Numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='4) For the inflationary period, we can use the well-known slow-roll approximation to obtain a solution to the inflaton equation of motion, as we describe in subsection (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' However, during the transition between inflation and reheating, the dynamics of the inflaton has to be obtained numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Both the inflaton field φ and the Ricci scalar R start to oscillate with decreasing amplitude, as can be observed in figure 1, where φ(η) and R(η) are depicted for an interval of time during the transition phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This is the epoch in which most of the particles are produced and the inflaton dynamics is solved until a numerically accessible time ηf is reached, when production becomes negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For late times, deep in the reheating era, we can also use an analytic approximation for the solution of the inflaton equation of – 5 – 3 2 1 0 1 2 3 0 2 4 6 2 1 0 1 2 3 4 0 5 10 15 20 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Inflaton field φ(η) (left panel) and curvature scalar R(η) (right panel) as functions of conformal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The range of time corresponds to the end of inflation and the beginning of reheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The parameters used for all figures in this article are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' motion, given in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2, which —although not used in our calculations— will be used to make some remarks in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 Inflationary era - Slow-roll approximation We will choose the inflationary period to start at the negative, initial time ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Inflation requires that the inflaton field changes slowly in comparison to the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Within the slow-roll approximation [46, 47], we can neglect the derivative of the field in favor of the potential, namely ˙φ2 ≪ |V (φ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' When this condition is satisfied, the field slowly rolls over until it falls to a minimum and starts oscillating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' At this point, inflation ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' With this assumption, we can approximately write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2) during the slow roll as H ≃ � 8π 3M2 P V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5) A slowly-varying inflaton implies that H ∼ constant for this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Hence, the expansion of spacetime is said to be quasi-exponential, as it resembles the pure de Sitter solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Usually, one also assumes a small rate of change for the (already slow) velocity of φ, such that |¨φ| ≪ 3H| ˙φ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This allows the slow-roll condition to be maintained long enough to solve the flatness and horizon problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' With these assumptions, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1) becomes easily solvable, ˙φ ≃ −∂φV (φ) 3H ≃ −∂φV (φ) MP � 24πV (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) – 6 – For the particular potential V (φ) = 1 2m2 φφ2, the solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) is φSR(t) = φ0 − MP √ 12πmφt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='7) where t < 0 corresponds to the inflationary period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Note that t = 0 and φ0 are the ending time of inflation and the value of the field at this instant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' From here, it is straightforward to obtain an explicit expression for the Ricci scalar, introducing the solution into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The scale factor is obtained by integrating the Hubble rate, and in the slow-roll approximation it reads aSR(t) ≃ a0e −� φ(t) φ0 dφ 8π M2 P V (φ) ∂φV (φ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='8) which for the quadratic potential becomes aSR(t) = a0e − 2π M2 P [φ2 SR(t)−φ2 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='9) Lastly, we need the relation between cosmological and conformal time in order to write both a(η) and R(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This relation can be obtained numerically from η = η0 + � t 0 dt/a(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' These are the necessary ingredients for determining the frequency of the mode equation in this region, under the slow-roll approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This regime is valid as long as the slow-roll parameter, ϵH = − ˙H/H2, is much smaller than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' When this no longer holds, at, say, t > t∗ with t∗ < 0, the equation of motion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1) has to be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The field begins to exit the inflationary regime and t = η = 0 marks both the end of inflation and the beginning of reheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' At this point, the scale factor reaches the value a0, which merely sets the scale and hence we take it to be a0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 Late reheating For late times, well into the reheating epoch (η∗ ≪ η ≲ ηf), one can find an approximate solution to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We do not use it for obtaining our results, but it will be important for the discussion in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In this approximation, the Hubble rate reads H(t) ≃ 2 3t � 1 − sin (2mφt − 2ϕ) 2mφt + O(m−2 φ t−2) �−1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='10) whereas the inflaton field is given by the expression φ = Φ0 t sin mφt � 1 − cos 2mφt 2mφt + O(m−2 φ t−2) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='11) with Φ0 ≡ MP /( √ 3πmφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This solution is valid as long as mφt ≫ 1, condition which is fulfilled during reheating, since, as we will see, the scale factor behaves as that of a matter dominated universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Indeed, we can integrate H(t) in order to approximately obtain the scale factor a(t), a(t) = Ct2/3 � 1 + O(m−2 φ t−2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='12) – 7 – The constant C is determined by requiring that the value of the scale factor at late times coincides with the one obtained from the numerical simulation in the previous region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' One can now integrate the scale factor in order to obtain t(η) = (Cη/3)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Now that we have a solution for the inflaton field and the scale factor valid for late times, we can obtain the Ricci scalar from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='3) by taking the solution for φ(t) to first order in (mφt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We end up with R = 8 3t2 � �2 sin2 mφt − � cos mφt − sin mφt mφt �2 + O(m−3 φ t−3) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='13) With this, we are able to describe the frequency of the mode equation until very late times, for which the approximations derived in this subsection behave even better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The density of produced particles will be calculated at a sufficiently large time ηf, such that the particle production is negligible from that point in time onwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 4 Particle production Once we have determined the behavior of the background geometry during inflation and reheating, we can solve the mode equation in order to extract the Bogoliubov coefficients after the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 Solution to the mode equation In order to compute the gravitational production once reheating has ended, we need to solve equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) from the onset of inflation at ti until a time tf well inside the adiabatic regime at the end of reheating, with the frequency of the oscillator determined by the background geometry described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In a similar way as we did for the background dynamics in section 3, the mode equation is solved in the regions η = � � � ηi ≤ η ≤ η∗, Slow-roll approximation, η∗ ≤ η ≤ ηf, Numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1) Let us start with the slow-roll era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In a de Sitter geometry, the Hubble rate is exactly constant, H0, the Ricci scalar is R = 12H0, and the scale factor reads a(η) = 1/(1 − H0η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Therefore, the frequency (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='7) takes the form ω2 k,dS = k2 + µ2 (η − η0)2 , with µ2 = m2/H2 0 + 12(ξ − 1/6), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2) where H0 = H(ηi) = 1/η0 is the Hubble rate at the beginning of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The solution to equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) in this simplified scenario which assimptotically at η → −∞ behaves as a positive frequency plane wave is given by vk,dS(η) = � π|η − η0|/2 eiπνH(1) ν (k|η − η0|) , ν = � 1/4 − µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='3) This is the so-called Bunch-Davies solution [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Note that there is a critical value µ2 = 1/4 for which ν = 0, which separates the regimes of real and imaginary ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In particular, for – 8 – m2/H2 0 ≪ 1, we can approximately write µ2 ≈ 12 (ξ − 1/6), and therefore µ2 = 1/4 for ξ = 3/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' At this point, there is no gravitational pair production in a de Sitter geometry [41], and this fact will be important for the analysis in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' However, our background geometry is not exactly de Sitter, but given by the inflaton dynamics derived in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Within the slow-roll approximation, valid from the start of inflation at ηi until η∗, the mode equation to solve is v′′ k(η) + ω2 k,SR(η)vk(η) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='4) where the scale factor and the Ricci scalar in ωk,SR(η) correspond to the analysis in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Nevertheless, in the slow-roll regime, and for a certain range in k, m, and ξ, we can approximate the solution satisfying Bunch-Davies initial conditions by (see subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='3 for details) vk,SR(η) ≃ � π|τk|/2eiπνH(1) ν (k|τk|) , τk = ωk,SR(η) ωk,dS(η) (η − η∗,k) + η∗,k − η0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5) where η∗,k marks the limit of validity of the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' From this point on, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) has to be solved numerically, independently of the background dynamics being numerical or analytical, taking as initial condition solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5) and its derivative at η∗,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The frequency one has to use in this case is that in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 Choice of reference vacua The solution vk(η) to the mode equation is associated with a particular choice of vacuum: the one that behaves as a plane wave at η → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The procedure in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 allows us to evaluate vk(ηf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' However, in order to obtain the Bogoliubov coefficient βk, we also need uk(ηf), which is the solution to the mode equation associated with the vacuum at this point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Then, from the Bogoliubov coefficients αk and βk, we will be able to extract the number density of produced particles at ηf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This time is chosen such that particle production becomes negligible for later times, condition that is fulfilled in the adiabatic regime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=', when ����� ω′ k(ηf) ω2 k(ηf) ����� ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) The value of ηf highly depends on the parameters of the scalar field, and in particular, it becomes larger as the mass m decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This is why, for certain regions in parameter space, it may be convenient to use the late-time approximation for the background dynamics described in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2, instead of solving numerically the equation of motion of the inflaton field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' It is worth mentioning that at the same time, a smaller coupling ξ to the curvature implies that the Ricci scalar oscillations, which are the main source of non-adiabaticity, are less important, therefore resulting in an earlier ηf at which (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' As long as the background is not static, the meaning of vacuum will change in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Nevertheless, if the evolution is adiabatic enough, namely condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) is fulfilled, one can use the so-called adiabatic prescription to define the instantaneous vacuum at a given – 9 – instant ηf, uk(ηf) = 1 � ωk(ηf) , u′ k(ηf) = − 1 √ωk � iωk(ηf) + 1 2 ω′ k(ηf) ωk(ηf) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='7) In fact, it is this feature that allows us to extrapolate the results obtained at ηf to the present when considering fields that interact only gravitationally [22, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' When the mass of the field ϕ is above mφ, particle production is governed by the mass term of the frequency (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='7), namely ω2 k(η) ≃ k2 + a2(η)m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='8) Since the scale factor at late times behaves as a(η) ∼ η2, condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) is fulfilled after few oscillations of the inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In other words, in this case we have that ηf is small enough that we do not need to invoke the late-time solution for the background, since everything can be calculated numerically in an efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This is not the case for masses smaller than the inflaton, for which production stabilizes after many, many oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' As a consequence, if we want to use the adiabatic vacuum description, we need to go up to a very large ηf, and therefore we need to use the analytic approximation for the inflaton dynamics described in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Alternatively, we can take a different definition for the vacuum that allows us to calculate the number density of produced particles at ¯η ≪ ηf, even for m ≪ mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Because the oscillating term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='7) becomes negligible at sufficiently large (numerically accessible) ¯η, we can define the frequency ω(avg) 2 k (η) = k2 + a2(η) � m2 + (ξ − 1/6) ⟨R⟩ (η) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='9) where the Ricci scalar oscillations are averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We can take this frequency to calculate the averaged vacuum u(avg) k (¯η) = 1 � ω(avg) k (¯η) , u(avg) ′ k (¯η) = − 1 � ω(avg) k (η) � iω(avg) k (¯η) + 1 2 ω(avg) ′ k (¯η) ω(avg) k (¯η) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='10) This prescription of vacuum is such that the spectrum of produced particles obtained at ¯η essentially concides with the one given by the adiabatic vacuum at the time where we reach the adiabatic regime, ηf, namely n(avg) k ��� η=¯η ≃ n(ad) k ��� η=ηf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='11) The larger discrepancies will reside in low wavenumbers, for which k ∼ a2(η) ⟨R⟩, but this region of momentum space is supressed in the total density of produced particles by a factor k2 (for details see next subsection), since n(m, ξ) = � d3k (2π)3 ⟨0| ˆnk |0⟩ = � dk 2π2 k2|βk|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='12) – 10 – As a consequence, no differences are appreciated at the chosen ¯η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This procedure has a limitation: It is valid up to the smallest mass m for which the dynamics presented here remain the same until ηf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' If reheating ends before ηf for a particular mass, the result provided by the averaged vacuum will not be the particles produced after this period is over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Nevertheless, a simple estimation shows that masses above the order of m ∼ 10−30 eV would reach adiabaticity early enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This is many orders of magnitude below the mass of fuzzy cold dark matter, and hence all the interesting range of masses lie within the regime of validity of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='3 Slow-roll approximation for the solution to the mode equation During inflation, spacetime expands quasi-exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' More specifically, the number of e-folds a(t0) a(ti) = eN (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='13) is required to be such that N ≈ 50 − 60 [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Because eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) cannot be solved analytically, even considering a slowly rolling inflaton field, one would need to use numerical methods in order to find a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' However, the large amount of e-folds to cover makes it more interesting and feasible to rely on an analytic approximation, such as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We dedicate this subsection to formally develop the approximation and to test its validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For notational convenience, in the calculations that follow we will write η − η0 as η, and drop the mode index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Let us start by defining the following small parameters for given values of k, m and ξ which will be useful in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' First, we have ϵ(m, ξ) = max η∈I1 �����1 − ωSR(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' m, ξ) ωdS(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' m, ξ) �����, with I1 = (−∞, η1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='14) where η1 is chosen such that ϵ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Then, we can define f(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' m, ξ) by ωSR ωdS = 1 + ϵf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='15) By construction, |f(η)| ≤ 1 for η ∈ I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Moreover, f′(η) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' It will also be convenient to define σ(m, ξ) = max η∈I2 ���f′(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' m, ξ)η ���, with I2 = (−∞, η2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='16) and choose η2 such that σ ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Then, we introduce g(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' m, ξ) as f′(η) = σg(η) η , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='17) for which again we have that |gk(η)| ≤ 1 for η ∈ I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' – 11 – Similarly, we define ρ(m, ξ) = max η∈I3 ����� ω′ dS(η) ωdS(η)η �����, with I3 = (−∞, η3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='18) and choose η3 such that ρ ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Now, we take η∗ = min(η1, η2, η3) and I = (−∞, η∗), where I is the interval for which the three parameters ϵ, σ, ρ are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Note that η∗ < 0 since inflation ends at η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We also need |η∗/η0| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The task is to solve equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='4), for which we define a new time coordinate ζ within the interval I, dζ = ωSR(η) ωdS(η) dη = [1 + ϵf(η)] dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='19) After integration until η ∈ I and taking the absolute value, this becomes |(ζ − ζ∗) − (η − η∗)| = ϵ ����� � η∗ η f(t)dt ����� = O(ϵ)(η − η∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='20) Then, choosing ζ∗ = η∗, this can be expressed as ζ = η [1 + O(ϵ)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='21) We change time coordinates η → ζ in the mode equation, which takes the form ¨w(ζ) + ω2 dS [η(ζ)] w(ζ) + ϵf′ [η(ζ)] ω2 dS [η(ζ)] ω2 SR [η(ζ)] ˙w(ζ) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='22) where w(ζ) = v [η(ζ)] and the dot denotes here derivative with respect to ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Let us analyze the last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' With this aim, we introduce the dimensionless time ¯ζ = ζ/η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Then, in terms of ¯ζ, the equation above has the same form except for the last term that acquires an extra factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Using the definition of f′ and σ above, the coefficient of this term is ϵf′ ω2 dS ω2 SR η0 = ϵσg(1 + ϵf)η0 η = O(ϵ2)η0 η (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='23) If we choose η∗ such that |η∗/η0| > 1, as mentioned above, this coefficient is of order O(ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Furthermore, the frequency in the second term of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='22) is ω2 dS(η(ζ)) = ω2 dS (ζ [1 + O(ϵ)]) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='24) = ω2 dS(ζ) � �1 + 2ω′ dS ωdS ����� ζ ζ O(ϵ) � � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='25) = ω2 dS (ζ) � 1 + O(ϵ2) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='26) provided that |ζ ω′ dS(ζ)/ωdS(ζ)| ≤ ρ = O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This is satisfied for ζ = η [1 + O(ϵ)] < η∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=', for η < η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Thus, the equation for w can finally be written as ¨w(ζ) + ω2 dS(ζ)w(ζ) = O(ϵ2 k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='27) – 12 – We can perturbatively solve the differential equation by writting w = w0 + ϵw1 + O(ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The solution to order ϵ0 is nothing but the de Sitter modes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='3), w0(ζ) = � π|ζ| eiπνH(1) ν (k|ζ|) , ν = � 1/4 − µ2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='28) and as a consequence, wk,0 behaves asymptotically (ζ → −∞) as a plane wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' On the other hand, the coefficients of the solution to order ϵ1 will satisfy the same original equation but with the initial conditions that w1(−∞) = 0 and therefore w1 is identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We can then write w as w(ζ) = w0(ζ) � 1 + O(ϵ2) � = � π|ζ| eiπνH(1) ν (k|ζ|) � 1 + O(ϵ2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='29) In order to undo the coordinate transformation ζ → η while keeping the error up to O(ϵ2), we need to consider the O(ϵ1) terms in ζ = η [1 + O(ϵ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For this, we note that ����� (ζ − η∗) − ωSR(η) ωdS(η) (η − η∗) ����� = ����� (η − η∗) + ϵ � η η∗ f(t)dt − [1 + ϵf(η)] (η − η∗) ����� = ϵ ����� � η η∗ f(t)dt − � η η∗ f(η)dt ����� ≤ ϵ � η η∗ |f(t) − f(η)|dt = ϵ � η η∗ ����f′(η)(t − η) + 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='f′′(η)(t − η)2 + · · · ���� dt ≤ ϵ ����� 1 2f′(η)(η − η∗)2 ���� + ���� 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='f′′ (η − η∗)3 ���� + · · · � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='30) This means that, as long as the terms in curly brackets are of order O(ϵ), we can write ζ = η∗ + �ωSR(η) ωdS(η) + O(ϵ2) � (η − η∗) = η∗ + ωSR(η) ωdS(η) (η − η∗) � 1 + O(ϵ2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='31) The first term is equal to 1 2σ ����g(η)η − η∗ η ���� = O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='32) The next terms are of the form f(n) (η − η∗)n+1 /n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=', which numerically can be seen to be smaller than the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Therefore, undoing the translation of η to η − η0 that we did at the beginning of this calculation, the solution to the mode equation can be written as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5) up to terms of order O(ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' With fixed ξ, and choosing η∗ independent of k, the error ϵk increases with increasing m and decreasing k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' When we numerically solve the mode equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) from η∗, the error in the initial condition coming from the slow-roll solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5) carries through as vk(η) = vk,0(η) � 1 + O(ϵ2 k) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='33) – 13 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5 0 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Maximum of the errors squared as function of the wave number k and the field mass m, for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 (left) and ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='8 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We take η∗ = −500mφ for all values of k, m and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5 0 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Maximum of the errors squared times k2 as function of the wave number k and the field mass m, for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 (left) and ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='8 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We take η∗ = −500mφ for all values of k, m and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' such that vk(η) → vk,SR(η) as η → η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Therefore, we have for the total density defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='12) that n(m, ξ) = � ∞ 0 dk 2π2 k2|βk|2 = n0 � 1 + 1 n0 � ∞ 0 dk 2π2 k2|βk,0|2O(ϵ2 k) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='34) where n0 = � ∞ 0 dk 2π2 k2|βk,0|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Although the error ϵk increases as k decreases, the factor k2 compensates this increase for low k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Essentially, although ϵ2 k increases for k < mφ, the quantity k2ϵ2 k remains small, whereas |βk,0|2 is roughly of the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' More explicitly, for the calculations in this paper, we take η∗ = −500mφ, for which the maximum of the – 14 – 800 600 400 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='010 800 600 400 10-8 10-6 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='010 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Relative error in the absolute value (left panel) and the phase (right panel) of the numerical solution to the exact mode equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) compared to the analytical approximation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5), for wavenumbers ranging from k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01mφ to k = 100mφ, and m = mφ, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Here, we take ηdS = −1000/mφ and η∗ = −500/mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' three small parameters squared, ϵ2 k, σ2 k, ρ2 k, as function of mass and wavenumber, for two different choices of coupling ξ, is shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For m ≤ mφ and k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1mφ, the error is of order O(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01) or smaller for the various values of ξ considered, and thus the approximation is controlled in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' At the same time, we can observe in figure 3 that k2ϵ2 k decreases as we move to the low-part of the momentum range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This guarantees that this region of the spectrum is robust against errors in the mode equation approximation we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' On the other hand, from figure 3 we observe that the quantity k2ϵ2 k grows with k for k > mφ, since the decrease in ϵ2 k (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' figure 2) can not compensate the power k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' However, gravitational production for high-momentum particles is very small, namely |βk|2 ≈ 0 for k ≫ mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' As a consequence, n(m, ξ) ≈ n0 approximates well the total number density of particles produced, since the weight of wavenumbers k ≫ mφ is very small when compared to the rest of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Furthermore, we can test the validity of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5) when compared to the numerical solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) by putting ourselves in the following scenario: Let us assume that the geometry can be approximated by a de Sitter spacetime during the early stages of inflation, such that the solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='3) is valid for a region ηi ≤ η < ηdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' At ηdS, slow-roll starts to matter, and deviations from the de Sitter solution vk,dS(η) occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In this scenario, we explore two different paths to continue continuing solving the equation: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We assume slow-roll inflation is a good description for the background dynamics in the region ηdS ≤ η < η∗, and take as solution the approximation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We solve numerically the exact equation of motion for the inflaton, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1), obtaining the frequency corresponding to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6), equation which we again solve numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This – 15 – solution, vk(η), will be valid even for η ≥ η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In figure 4, we compare the analytical slow-roll solution with the exact numerical solution by plotting the relative difference between their absolute values, ∆rAbs [vk,SR(η)] ≡ ����� Abs [vk(η)] − Abs [vk,SR(η)] Abs [vk(η)] �����, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='35) as well as their phase difference, ∆rArg [vk,SR(η)] ≡ ����� Arg [vk(η)] − Arg [vk,SR(η)] π �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='36) We do so for different wavenumbers, ranging from k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01mφ to k = 100mφ, denoted by the different shapes in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We have taken ηdS = −1000/mφ as start of the slow-roll and η∗ = −500/mφ as the time when the slow-roll approximation breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For k = mφ, the relative error is very small, of order ∼ 10−4 at η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For wavenumbers larger than the mass of the inflaton, k > mφ, the approximation is still good, although it worsens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' On the other hand, the error for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01mφ starts becoming significant, and gets worse for k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' However, the corresponding region of the spectrum of produced particles is highly suppressed, as discussed above, and therefore the contribution to the total density of particles is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Similarly, particle production is very small for wavenumbers larger than k > 100mφ, and therefore the range of interest in k is under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Hence, we can assume the approximation is valid in the region ηdS ≤ η < η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Note that if this solution behaves well in this region, it has to become an even better approximation before ηdS, since the further towards the past we go, the more de Sitter-like is the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Thus, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5) can be taken as well as a solution to the mode equation in the region ηi ≤ η < ηdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Under this approximations, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6) can be solved analytically from the start of inflation, ηi, until η∗, for which the slow-roll approximation starts to fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' From there, the mode equation is solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='4 Adiabaticity and oscillations In order to illustrate the importance of the choice of vacuum, we studied the evolution of spectra when calculated using prescription (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='7) before the dynamics has entered the adiabatic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' As an example, we plotted in figure 5 the spectra of particles with mass m = 10−3mφ obtained at two different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The dots correspond to η = 40/mφ, whereas the solid lines denote η = ηf = 100/mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For this particular choice of mass, the latter time lies within the adiabatic regime, and this is the reason why the non-adiabatic dots relax to their final value as we approach this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' As expected, the effect is less noticeable the lower the coupling to the geometry is, as it is the main source of non-adiabaticity in the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' At the same time, we also characterized the importance of the first oscillations of the curvature scalar in the final spectrum of produced particles, obtained with the averaged vacuum defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' As can be seen in figure 6, even after several oscillations of – 16 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='8 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Spectra of produced particles of mass m = 10−3mφ and different values of ξ, obtained with the adiabatic prescription of the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The dots correspond to η = 40/mφ, before the adiabatic regime has been reached for this value of the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The solid lines correspond to η = ηf = 100/mφ, when most of the particles have been produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' R(η) (for example, at η = 2/mφ), the production changes greatly if one compares with the obtained spectra at ¯η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Even when looking only at the total number of produced particles in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='12), differences are still significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We observe that the spectrum does not stabilize until η ≃ 5/mφ, which for our model means after hundreds of oscillations of the curvature scalar R(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' With this, we want to stress that obtaning the particle production after one or two oscillations does not account for the whole process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 5 Spectra of particles and total density Let us finally give the results for the spectra of produced particles as function of the parameters of the field, the mass m, and the coupling to the curvature ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We explore first the regime of masses below the inflaton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Represented by the solid line in figure 7, we have masses m ≤ 10−4mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For these values, the mass contribution to the frequency becomes negligible, and the dynamics is entirely given by the coupling to the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The spectra lie on top of each other, with very small differences in the low values of k ∼ a(η)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We observe, however, slight differences in the shape of the spectrum when increasing the mass, especially for small wavenumbers, as the rest of the curves in figure 7 show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We can choose a mass in this regime, m = 10−1mφ, and explore the influence of the coupling ξ in the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This is shown in figure 8, where one observes increasing production of particles with larger values of the coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Lastly, let us come to the mass of the inflaton, whose corresponding spectra are shown in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In such a case, it is – 17 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 0 2 4 6 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Spectra for m = 10−4mφ and ξ = 1, obtained with the averaged vacuum prescription, for different instants of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The spectrum stabilises after very many oscillations of the curvature scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Spectrum of particles for masses below the mass of the inflaton, with ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For very small masses (m ≤ 10−4mφ), production is dominated by curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In the region 10−3 ≤ m ≤ 10−1mφ, differences in production due to the mass can be noticed, especially for low values of k/mφ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' harder to characterize the behavior with ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' It is clear, nevertheless, that particle production decreases as the mass of particles becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' – 18 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Spectra for m = 10−1mφ and several values of the coupling ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Particle production increases when the curvature term becomes more important, and the maximum of the spectrum is shifted towards higher values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1 1 10 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='02 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Spectra of particles with the mass of the inflaton, for different values of the coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In this particular case, increasing the coupling does not translate directly into an increase of particle production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This can be more clearly seen by examining the total density of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' It is easier to characterize particle production in this regime using the total number density of particles (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='12), which we show in figure 10 as function of the two parameters of the field, m and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Here, one clearly sees that the prediction is independent of the value of – 19 – Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Logarithm of the total density of produced particles for different values of m and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In order to give the mass and density in units of GeV, we took mφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 × 1013 GeV for the mass of the inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We explore a wide range of masses in the left panel while we focus on a smaller region close to the mass of the inflaton on the right panel in order to appreciate the dependence of the total density with the coupling ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' the mass as long as it is below m ∼ 10−2mφ, in particular for a sufficiently high value of the coupling, ξ ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In this case, the mass is completely negligible when compared to the dynamics of the curvature scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Only when the coupling to the curvature is close to ξ ∼ 1/6, the production of particles is still sensible to m, up to m ∼ 10−7mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For this value, even in the conformal case, the relevant wavenumbers, k ∼ a(η)m, are too suppressed to make a difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In all these regime of low masses, the number of produced particles increases with larger coupling ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Closer to the mass of the inflaton, 10−2mφ < m < mφ, the fact that a heavier particle translates into a lower production becomes apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Lastly, in the region around the mass of the inflaton, m ∼ mφ, the behavior with the coupling is different, and production may even decrease when raising the value of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In fact, there appears to exist a critical value ξc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='22 which separates two qualitatively different regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' As we commented previously, this value is related to the parameter µ2 = 1/4 of the Hankel functions, which were a good approximation of the mode functions of our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For m < mφ, the number density drops very rapidly if ξ < ξc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For m ∼ mφ, ξc is the value below which production decreases with ξ, and above which it increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This is also illustrated in figure 9, where production for ξ = 1/6 is larger than for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='26, and from there it increases again with the coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Moreover, we observe the expected strong suppression in the number density of produced particles for masses above the mass of the inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We can confirm this behaviour by calculating the spectra for even higher masses, provided we select a negative enough η∗ — and therefore leading to a very heavy computation — in this case, as explained in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Note that we took mφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 × 1013 GeV for the mass of the inflaton, and as a consequence, the density in figure 10 is given in units of GeV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' – 20 – 10-3 101 105 109 1013 1/6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='0 10-3 101 105 109 1013 1/6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='0 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Logarithm of the predicted abundance of dark matter today for different values of m and ξ, and a reheating temperature of Treh = 1015 GeV (left) and Treh = 1013 GeV (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In order to give the mass and density in units of GeV, we took mφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 × 1013 GeV for the mass of the inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Finally, one can consider these gravitationally produced scalar particles as dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In this case, it is interesting to compare the resulting abundance with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Assuming that the scalar field is non-interacting, the evolution of the particle density showed in figure 10 is only due to the expansion of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Then, the predicted abundance can be written in terms of the background radiation temperature [24] as Ω(m, ξ) = 8π 3M2 P H2 today gtoday ∗S grh ∗S �Ttoday Trh �3 m n(m, ξ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1) where Ttoday and Trh are the radiation temperature today and at the end of reheating, respectively, and gtoday ∗S and grh ∗S are the corresponding relativistic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This is represented in figure 11 for two different reheating temperatures, together with the observed abundance given by the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We observe that the proposed mechanism can explain observations if the dark matter candidate is light enough (m ≲ 108 GeV), independently of the value of the coupling ξ for the range that we considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In addition, heavier particles can also reach the observed dark matter abundance since their production is strongly suppressed above the inflaton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' 6 Conclusions Gravitational particle production is a very interesting process due to its universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' It only requires the studied field to interact with gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Even without a direct coupling to the inflaton, as it is the case of spectator fields such as the one we have studied, it can – 21 – give rise to a significant abundance for the species considered after the heavy expansion of spacetime during the early stages of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' However, predictions need for a definition of vacuum after reheating, since the non-static geometry leads to certain ambiguity in the meaning of particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In this manuscript, we studied the production of massive, scalar particles whose dynamics is described by a non-minimally coupled to gravity action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' However, the discussion on the validity of the definition of vacuum is pertinent when considering any other field as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' First, we have provided a method for solving in a complete form the background dynamics, governed by a single scalar inflaton field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' For this, we did not have to assume a de Sitter geometry of spacetime, which would significantly change the amount of particles produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Although we make a choice of potential, this procedure can be extended to other cases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' We provided an analytic approximation to the solution of the slow-roll mode equation where the error is well under control in our parameter region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' More importantly, we showed that, for masses smaller than the inflaton mass, the commonly used adiabatic prescription for the vacuum determines correctly the production of particles after reheating only when calculated at very late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Moreover, we define an alternative vacuum choice that allows one to obtain the right abundance when calculating particle production at a much earlier time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' This allowed us to explore the contribution of the first oscillations to the total number of produced particles, revealing that the spectra only stabilizes after hundreds of periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Lastly, after all these considerations have been taken into account, we analyzed both the spectra and the total density of particles for different values of the mass of the field and its coupling to the curvature scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' When regarded as dark matter, the production of the spectator field can be directly related to the abundance that would be observed today if one assumes no couplings to any other fields also after reheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' In particular, we find agreement with the observed dark matter abundance for a certain range of masses and couplings of the spectator field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Moreover, this analysis can be used to constrain the values of the field parameters by demanding that the predicted dark matter abundance does not exceed observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Acknowledgements This work was partially supported by the MICINN (Ministerio de Ciencia e Innovación, Spain) projects PID2019-107394GB-I00/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='13039/501100011033 (AEI/FEDER, UE) and PID2020-118159GBC44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Additionally, Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' is supported by the MIU (Ministerio de Universidades, Spain) fellowship FPU20/0560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Finally, JARC acknowledges support by Institut Pascal at Université Paris-Saclay during the Paris-Saclay Astroparticle Symposium 2022, with the support of the P2IO Laboratory of Excellence (program “Investissements d’avenir” ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038), the P2I axis of the Graduate School of Physics of Université Paris-Saclay, as well as IJCLab, CEA, APPEC, IAS, OSUPS, and the IN2P3 master projet UCMN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' – 22 – A Parameters In the majority of the analyses, we have left all the quantities expressed in terms of the mass of the inflaton, mφ, which sets up the scale of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' When it has been necessary to assume a numerical value for such a mass, we have taken mφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='2 × 1013 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Accordingly, the Planck mass MP has the value MP = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='02 × 106mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The initial value for the inflaton field, under the slow-roll assumption, is taken to be φSR(ti) = φi = 3MP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' When inflation ends, at t = 0, the field value is φSR(t = 0) = φ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5MP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The slow-roll approximation can then be used to extract ti ≃ −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='35/mφ as the time when inflation starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Equation of motion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='1) can also be solved numerically taking as initial conditions the same as for slow-roll, φ(ti) = φi, and the derivative of the approximate solution at this point, φ′(ti) = φ′ SR(ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Both solutions will be very close up to t∗, where the slow-roll approximation starts to break down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Then, φ(t = 0) slightly deviates from φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The scale factor is chosen such that a(t = 0) = a0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Slow-roll is a assumed to be a good approximation until η∗ = −500/mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' Unless the contrary is expressly stated, particle production is calculated using the averaged vacuum prescription at ¯η = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='33/mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content=' The range of masses explored is 10−7mφ ≤ m ≤ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE3T4oBgHgl3EQftQtg/content/2301.04674v1.pdf'} +page_content='5mφ, although for obtaining figure 11 it is assumed that production is the same for m ≤ 10−7mφ.' 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sha256:a18208be2c66510da30a4c953d7e0e7ec4d04c47aa59147b39f7071c889fce9a +size 166192 diff --git a/ItFJT4oBgHgl3EQfFyzw/content/tmp_files/2301.11445v1.pdf.txt b/ItFJT4oBgHgl3EQfFyzw/content/tmp_files/2301.11445v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cfed98470129b7361685a3e87042edfe389e16ba --- /dev/null +++ b/ItFJT4oBgHgl3EQfFyzw/content/tmp_files/2301.11445v1.pdf.txt @@ -0,0 +1,1817 @@ +3DShape2VecSet: A 3D Shape Representation for Neural Fields and +Generative Diffusion Models +BIAO ZHANG, KAUST, Saudi Arabia +JIAPENG TANG, TU Munich, Germany +MATTHIAS NIESSNER, TU Munich, Germany +PETER WONKA, KAUST, Saudi Arabia +Input +Reconstruction +Input +Reconstruction +Condition +Generation +“the tallest chair” +car +Fig. 1. Left: Shape autoencoding results (surface reconstruction from point clouds) Right: the various down-stream applications of 3DShape2VecSet (from +top to down): (a) category-conditioned generation; (b) point clouds conditioned generation (shape completion from partial point clouds); (c) image conditioned +generation (shape reconstruction from single-view images); (d) text-conditioned generation. +We introduce 3DShape2VecSet, a novel shape representation for neural fields +designed for generative diffusion models. Our shape representation can en- +code 3D shapes given as surface models or point clouds, and represents them +as neural fields. The concept of neural fields has previously been combined +with a global latent vector, a regular grid of latent vectors, or an irregu- +lar grid of latent vectors. Our new representation encodes neural fields on +top of a set of vectors. We draw from multiple concepts, such as the ra- +dial basis function representation and the cross attention and self-attention +function, to design a learnable representation that is especially suitable for +processing with transformers. Our results show improved performance in +3D shape encoding and 3D shape generative modeling tasks. We demon- +strate a wide variety of generative applications: unconditioned generation, +category-conditioned generation, text-conditioned generation, point-cloud +completion, and image-conditioned generation. +Additional Key Words and Phrases: 3D Shape Generation, 3D Shape Repre- +sentation, Diffusion Models, Shape Reconstruction, Generative models +Authors’ addresses: Biao Zhang, KAUST, Saudi Arabia, biao.zhang@kaust.edu.sa; Jia- +peng Tang, TU Munich, Germany, jiapeng.tang@tum.de; Matthias Nießner, TU Munich, +Germany, niessner@tum.de; Peter Wonka, KAUST, Saudi Arabia, peter.wonka@kaust. +edu.sa. +1 +INTRODUCTION +The ability to generate realistic and diverse 3D content has many +potential applications, including computer graphics, gaming, and vir- +tual reality. To this end, many generative models have been explored, +e.g., generative adversarial networks, variational autoencoders, nor- +malizing flows, and autoregressive models. Recently, diffusion mod- +els have emerged as one of the most popular method with fantastic +results in the 2D image domain [Ho et al. 2020; Rombach et al. 2022] +and have shown their superiority over other generative methods. For +instance, it is possible to do unconditional generation [Karras et al. +2022; Rombach et al. 2022], text conditioned generation [Rombach +et al. 2022; Saharia et al. 2022], and generative image inpainting [Lug- +mayr et al. 2022]. However, the success in the 2D domain has not +yet been matched in the 3D domain. +In this work, we will study diffusion models for 3D shape genera- +tion. One major challenge in adapting 2D diffusion models to 3D is +the design of a suitable shape representation. The design of such a +shape representation is the major focus of our work, and we will +discuss several design choices that lead to the development of our +proposed representation. +, Vol. 1, No. 1, Article . Publication date: January 2023. +arXiv:2301.11445v1 [cs.CV] 26 Jan 2023 + +Different from 2D images, there are several predominant ways +to represent 3D data, e.g., voxels, point clouds, meshes, and neu- +ral fields. In general, we believe that surface-based representations +are more suitable for downstream applications than point clouds. +Among the available choices, we choose to build on neural fields as +they have many advantages: they are continuous, represent com- +plete surfaces and not only point samples, and they enable many +interesting combinations of traditional data structure design and +representation learning using neural networks. +Two major approaches for 2D diffusion models are to either use a +compressed latent space, e.g., latent diffusion [Rombach et al. 2022], +or to use a sequence of diffusion models of increasing resolution, +e.g., [Ramesh 2022; Saharia et al. 2022]. While both of these ap- +proaches seem viable in 3D, our initial experiments indicated that it +is much easier to work with a compressed latent space. We therefore +follow the latent diffusion approach. +A subsequent design choice for a latent diffusion approach is to de- +cide between a learned representation or a manually designed repre- +sentation. A manually designed representation such as wavelets [Hui +et al. 2022] is easier to design and more lightweight, but in many con- +texts learned representations have shown to outperform manually +designed ones. We therefore opt to explore learned representations. +This requires a two-stage training strategy. The first stage is an +autoencoder (variational autoencoder) to encode 3D shapes into a +latent space. The second stage is training a diffusion model in the +learned latent space. +In the case of training diffusion models for 3D neural fields, it +is even more necessary to generate in latent space. First, diffusion +models often work with data of fixed size (e.g., images of a given +fixed resolution). Second, a neural field is a continuous real-valued +function that can be seen as an infinite-dimensional vector. For both +reasons, we decide to find a way to encode shapes into latent space +before all else (as well as a decoding method for reverting latents +back to shapes). +Finally, we have to design a suitable learned neural field rep- +resentation that provides a good trade-off between compression +and reconstruction quality. Such a design typically requires three +components: a spatial data structure to store the latent information, +a spatial interpolation method, and a neural network architecture. +There are multiple options proposed in the literature shown in Fig. 2. +Early methods used a single global latent vector in combination +with an MLP network [Mescheder et al. 2019; Park et al. 2019]. This +concept is simple and fast but generally struggles to reconstruct +high-quality shapes. Better shape details can be achieved by using a +3D regular grid of latents [Peng et al. 2020] together with tri-linear +interpolation and an MLP. However, such a representation is too +large for generative models and it is only possible to use grids of +very low resolution (e.g., 8×8×8). By introducing sparsity, e.g., [Yan +et al. 2022; Zhang et al. 2022], latents are arranged in an irregular +grid. The latent size is largely reduced, but there is still a lot of +room for improvement which we capitalize on in the design of +3DShape2VecSet. +The design of 3DShape2VecSet combines ideas from neural fields, +radial basis functions, and the network architecture of attention +layers. Similar to radial basis function representation for continuous +functions, we can also re-write existing methods in a similar form +(linear combination). Inspired by cross attention in the transformer +network [Vaswani et al. 2017], we derived the proposed latent rep- +resentation which is a fixed-size set of latent vectors. There are +two main reasons that we believe contribute to the success of the +representations. First, the representation is well-suited for the use +with transformer-based networks. As transformer-based networks +tend to outperform current alternatives, we can better benefit from +this network architecture. Instead of only using MLPs to process +latent information, we use a linear layer and cross-attention. Sec- +ond, the representation no longer uses explicitly designed positional +features, but only gives the network the option to encode positional +information in any form it considers suitable. This is in line with our +design principle of favoring learned representations over manually +designed ones. See Fig. 2 e) for the proposed latent representation. +Using our novel shape representation, we can train diffusion mod- +els in the learned 3D shape latent space. Our results demonstrate an +improved shape encoding quality and generation quality compared +to the current state of the art. While pioneering work in 3D shape +generation using diffusion models already showed unconditional +3D shape generation, we show multiple novel applications of 3D dif- +fusion models: category-conditioned generation, text-conditioned +shape generation, shape reconstruction from single-view image, and +shape reconstruction from partial point clouds. +To sum up, our contributions are as follows: +(1) We propose a new representation for 3D shapes. Any shape +can be represented by a fixed-length array of latents and +processed with cross-attention and linear layers to yield a +neural field. +(2) We propose a new network architecture to process shapes +in the proposed representation, including a building block to +aggregate information from a large point cloud using cross- +attention. +(3) We improve the state of the art in 3D shape autoencoding to +yield a high fidelity reconstruction including local details. +(4) We propose a latent set diffusion. We improve the state of +the art in 3D shape generation as measured by FID, KID, FPD, +and KPD. +(5) We show 3D shape diffusion for category-conditioned gener- +ation, text-conditioned generation, point-cloud completion, +and image-conditioned generation. +2 +RELATED WORK +In this section, we briefly review the literature of 3D shape learning +with various data representations and 3D shape generative models. +2.1 +3D Shape Representations +We mainly discuss the following representations for 3D shapes, +including voxels, point clouds, and neural fields. +Voxels. Voxel grids, extended from 2D pixel grids, simply repre- +sent a 3D shape as a discrete volumetric grid. Due to their regular +structure, early works take advantage of 3D transposed convolution +operators for shape prediction [Brock et al. 2016; Choy et al. 2016; +Dai et al. 2017; Girdhar et al. 2016; Wu et al. 2016, 2015]. A draw- +back of the voxels-based decoders is that the computational and +memory costs of neural networks cubicly increases with respect to +2 + +(x𝑖, 𝜆𝑖 ) +x +𝜙 (x, x𝑖 ) +(a) RBF +(b) Global Latent +x +(x𝑖, f𝑖 ) +(c) Latent Grid +(x𝑖, f𝑖 ) +x +𝜙 (x, x𝑖 ) +(d) Irregular Latent Grid +x +f𝑖 +𝜙 (x, f𝑖 ) = exp +� +q(x)⊺k(f𝑖 )/ +√ +𝑑 +� +(e) Latent Set (Ours) +Fig. 2. Continuous function representation. Scalars are represented with spheres while vectors are cubes. The arrows show how spatial interpolation is +computed. x𝑖 and x are the coordinates of an anchor and a querying point respectively. 𝜆𝑖 is the SDF value of the anchor point x𝑖 in (a). f𝑖 is the associate +feature vector located in x𝑖 in (c)(d). The queried SDF/feature of x is based on the distance function 𝜙 (x, x𝑖) in (a)(c)(d), while our proposed latent set +representation (e) utilizes the similarity 𝜙 (x, f𝑖) between querying coordinate and anchored features via cross attention mechanism. +# Latents +Latent Position +Methods +OccNet [Mescheder et al. 2019] +DeepSDF [Park et al. 2019] +Single +Global +IM-Net [Chen and Zhang 2019] +ConvOccNet [Peng et al. 2020] +IF-Net [Chibane et al. 2020] +LIG [Jiang et al. 2020] +DeepLS [Chabra et al. 2020] +SA-ConvOccNet [Tang et al. 2021] +Multiple +Regular Grid +NKF [Williams et al. 2022] +LDIF [Genova et al. 2020] +Point2Surf [Erler et al. 2020] +DCC-DIF [Li et al. 2022] +3DILG [Zhang et al. 2022] +Multiple +Irregular Grid +POCO [Boulch and Marlet 2022] +Multiple +Global +Ours +Table 1. Neural Fields for 3D Shapes. We show different types how la- +tents are positioned. +the grid resolution. Thus, most voxel-based methods are limited to +low-resolution. Octree-based decoders [Häne et al. 2017; Meagher +1980; Riegler et al. 2017b,a; Tatarchenko et al. 2017; Wang et al. +2017, 2018] and sparse hash-based decoders [Dai et al. 2020] take +3D space sparsity into account, alleviating the efficiency issues and +supporting high-resolution outputs. +Point Clouds. Early works on neural-network-based point cloud +processing include PointNet [Qi et al. 2017a,b] and DGCNN [Wang +et al. 2019]. These works are built upon per-point fully connected +layers. More recently, transformers [Vaswani et al. 2017] were pro- +posed for point cloud processing, e.g., [Guo et al. 2021; Zhang et al. +2022; Zhao et al. 2021]. These works are inspired by Vision Trans- +formers (ViT) [Dosovitskiy et al. 2021] in the image domain. Points +are firstly grouped into patches to form tokens and then fed into +a transformer with self-attention. In this work, we also introduce +a network for processing point clouds. Improving upon previous +works, we compress a given point cloud to a small representation +that is more suitable for generative modeling. +Neural Fields. A recent trend is to use neural fields as a 3d data +representation. The key building block is a neural network which +accepts a 3D coordinate as input, and outputs a scalar [Chen and +Generative +3D +Models +Representation +3D-GAN [Wu et al. 2016] +GAN +Voxels +l-GAN [Achlioptas et al. 2018] +GAN★ +Point Clouds +IM-GAN [Chen and Zhang 2019] +GAN★ +Fields +PointFlow [Yang et al. 2019] +NF +Point Clouds +GenVoxelNet [Xie et al. 2020] +EBM +Voxels +PointGrow [Sun et al. 2020] +AR +Point Clouds +PolyGen [Nash et al. 2020] +AR +Meshes +GenPointNet [Xie et al. 2021] +EBM +Point Clouds +3DShapeGen [Ibing et al. 2021] +GAN★ +Fields +DPM [Luo and Hu 2021] +DM +Point Clouds +PVD [Zhou et al. 2021] +DM +Point Clouds +AutoSDF[Mittal et al. 2022] +AR★ +Voxels +CanMap [Cheng et al. 2022] +AR★ +Point Clouds +ShapeFormer[Yan et al. 2022] +AR★ +Fields +3DILG [Zhang et al. 2022] +AR★ +Fields +LION [Zeng et al. 2022] +DM★ +Point Clouds +SDF-StyleGAN [Zheng et al. 2022] +GAN +Fields +NeuralWavelet [Hui et al. 2022] +DM★ +Fields +TriplaneDiffusion [Shue et al. 2022]⋄ +DM★ +Fields +DiffusionSDF [Chou et al. 2022]⋄ +DM★ +Fields +Ours +DM★ +Fields +★ Generative models in latent space. +⋄ Works in submission. +Table 2. Generative models for 3d shapes. +Zhang 2019; Mescheder et al. 2019; Michalkiewicz et al. 2019; Park +et al. 2019] or a vector [Mildenhall et al. 2020]. A 3D object is then +implicitly defined by this neural network. Neural fields have gained +lots of popularity as they can generate objects with arbitrary topolo- +gies and infinite resolution. The methods are also called neural +implicit representations or coordinate-based networks. For neural +fields for 3d shape modeling, we can categorize methods into global +methods and local methods. 1) The global methods encode a shape +with a single global latent vector [Mescheder et al. 2019; Park et al. +2019]. Usually the capacity of these kind of methods is limited and +3 + +they are unable to encode shape details. 2) The local methods use +localized latent vectors which are defined for 3D positions defined +on either a regular [Chibane et al. 2020; Jiang et al. 2020; Peng et al. +2020; Tang et al. 2021] or irregular grid [Boulch and Marlet 2022; +Genova et al. 2020; Li et al. 2022; Zhang et al. 2022]. In contrast, we +propose a latent representation where latent vectors do not have +associated 3D positions. Instead, we learn to represent a shape as a +list of latent vectors. See Tab. 1. +2.2 +Generative models. +We have seen great success in different 2D image generative mod- +els in the past decade. Popular deep generative methods include +generative adversarial networks (GANs) [Goodfellow et al. 2014], +variational autoencoers (VAEs) [Kingma and Welling 2014], nor- +malizing flows (NFs) [Rezende and Mohamed 2015], energy-based +models [LeCun et al. 2006; Xie et al. 2016], autoregressive models +(ARs) [Esser et al. 2021; Van Den Oord et al. 2017] and more re- +cently, diffusion models (DMs) [Ho et al. 2020] which are the chosen +generative model in our work. +In 3D domain, GANs have been popular for 3D generation [Achliop- +tas et al. 2018; Chen and Zhang 2019; Ibing et al. 2021; Wu et al. +2016; Zheng et al. 2022], while only a few works are using NFs [Yang +et al. 2019] and VAEs [Mo et al. 2019]. A lot of recent work employs +ARs [Cheng et al. 2022; Mittal et al. 2022; Nash et al. 2020; Sun et al. +2020; Yan et al. 2022; Zhang et al. 2022]. DMs for 3D shapes are +relatively unexplored compared to other generative methods. +There are several DMs dealing with point cloud data [Luo and Hu +2021; Zeng et al. 2022; Zhou et al. 2021]. Due to the high freedom +degree of regressed coordinates, it is always difficult to obtain clean +manifold surfaces via post-processing. As mentioned before, we +believe that neural fields are generally more suitable than point +clouds for 3D shape generation. The area of combining DMs and +neural fields is still underexplored. +The recent NeuralWavelet [Hui et al. 2022] first encodes shapes +(represented as signed distance fields) into the frequency domain +with the wavelet transform, and then train DMs on the frequency +coefficients. While this formulation is elegant, generative models +generally work better on learned representations. Some concurrent +works [Chou et al. 2022; Shue et al. 2022] in submission also utilize +DMs in a latent space for neural field generation. The TriplaneDif- +fusion [Shue et al. 2022] trains an autodecoder first for each shape. +DiffusionSDF [Chou et al. 2022] runs a shape autoencoder based on +triplane features [Peng et al. 2020]. +Summary of 3D generation methods. We list several 3d generation +methods in Tab. 2, highlighting the choice of generative model (GAN, +DM, EBM, NF, or AR) and the choice of data structure to represent +3D shapes (point clouds, meshes, voxels or fields). +3 +PRELIMINARIES +An attention layer [Vaswani et al. 2017] has three types of inputs: +queries, keys, and values. Queries Q = [q1, q2, . . . , q𝑁𝑞] ∈ R𝑑×𝑁𝑞 +and keys K = [k1, k2, . . . , k𝑁𝑘 ] ∈ R𝑑×𝑁𝑘 are first compared to +produce coefficients q⊺ +𝑗 k𝑖/ +√ +𝑑 (they need to be normalized with the +softmax function), +𝐴𝑖,𝑗 = +q⊺ +𝑗 k𝑖/ +√ +𝑑 +�𝑁𝑘 +𝑖=1 exp +� +q⊺ +𝑗 k𝑖/ +√ +𝑑. +� +(1) +The coefficients are then used to (linearly) combine values V = +[v1, v2, . . . , v𝑁𝑘 ] ∈ R𝑑𝑣×𝑁𝑘 . We can write the output of an attention +layer as follows, +Attention(Q, K, V) += +�o1 +o2 +· · · +o𝑁𝑞 +� +∈ R𝑑𝑣×𝑁𝑞 += +� 𝑁𝑘 +∑︁ +𝑖=1 +𝐴𝑖,1v𝑖 +𝑁𝑘 +∑︁ +𝑖=1 +𝐴𝑖,2v𝑖 +· · · +𝑁𝑘 +∑︁ +𝑖=1 +𝐴𝑖,𝑁𝑞v𝑖 +� +(2) +Cross Attention. Given two sets A = +� +a1, a2, . . . , a𝑁𝑎 +� +∈ R𝑑𝑎×𝑁𝑎 +and B = +� +b1, b2, . . . , b𝑁𝑏 +� +∈ R𝑑𝑏×𝑁𝑏 , the query vectors Q are con- +structed with a linear function q(·) : R𝑑𝑎 → R𝑑 by taking elements +of A as input. Similarly, we construct K and V with k(·) : R𝑑𝑏 → R𝑑 +and v(·) : R𝑑𝑏 → R𝑑, respectively. The inputs of both k(·) and v(·) +are from B. Each column in the output of Eq. (2) can be written as, +o(a𝑗, B) = +𝑁𝑏 +∑︁ +𝑖=1 +v(b𝑖) · +1 +𝑍 (a𝑗, B) exp +� +q(a𝑗)⊺k(b𝑖)/ +√ +𝑑 +� +, +(3) +where 𝑍 (a𝑗, B) = �𝑁𝑏 +𝑖=1 exp +� +q(a𝑗)⊺k(b𝑖)/ +√ +𝑑 +� +is a normalizing fac- +tor. The cross attention operator between two sets is, +CrossAttn(A, B) = +�o(a1, B) +o(a2, B) +· · · +o(a𝑁𝑎, B)� +∈ R𝑑×𝑁𝑎 +(4) +Self Attention. In the case of self attention, we let the two sets be +the same A = B, +SelfAttn(A) = CrossAttn(A, A). +(5) +4 +LATENT REPRESENTATION FOR NEURAL FIELDS +Our representation is inspired by radial basis functions (RBFs). We +will therefore describe our surface representation design using RBFs +as a starting point, and how we extended them using concepts +from neural fields and the transformer architecture. A continuous +function can be represented with a set of weighted points in 3D +using RBFs: +ˆORBF(x) = +𝑀 +∑︁ +𝑖=1 +𝜆𝑖 · 𝜙(x, x𝑖) +(6) +where 𝜙(x, x𝑖) is a radial basis function (RBF) and typically repre- +sents the similarity (or dissimilarity) between two inputs, +𝜙(x, x𝑖) = 𝜙(∥x − x𝑖 ∥). +(7) +Given ground-truth occupancies of x𝑖, the values of 𝜆𝑖 can be ob- +tained by solving a system of linear equations. In this way, we +can represent the continuous function O(·) as a set of 𝑀 points +including their corresponding weights, +� +𝜆𝑖 ∈ R, x𝑖 ∈ R3�𝑀 +𝑖=1 . +(8) +However, in order to retain the details of a 3d shape, we often need +a very large number of points (e.g., 𝑀 = 80, 000 in [Carr et al. 2001]). +4 + +Shape Encoding (Sec. 5.1) +Shape Decoding (Sec. 5.3) +KL (Sec. 5.2) +latent queries +Point Cloud +Position Embeddings +Surface Sampling +Cross Attention +K, V +Q +... +... +latents +KL Regularization +Self Attention +... +Self Attention +... +· · · +... +Self Attention +Query Points +Position Embeddings +Cross Attention +K, V +Q +Target +· · · +Isosurface +Fig. 3. Shape autoencoding pipeline. Given a 3D ground-truth surface mesh as the input, we first sample a point cloud that is mapped to positional +embeddings and encode them into a set of latent codes through a cross-attention module (Sec. 5.1). Next, we perform (optional) compression and KL- +regularization in the latent space to obtain structured and compact latent shape representations (Sec. 5.2). Finally, the self-attention is carried out to aggregate +and exchange the information within the latent set. And a cross-attention module is designed to calculate the interpolation weights of query points. The +interpolated feature vectors are fed into a fully connected layer for occupancy prediction (Sec. 5.3). +This representation does not benefit from recent advances in repre- +sentation learning and cannot compete with more compact learned +representations. We therefore want to modify the representation to +change it into a neural field. +One approach to neural fields is to represent each shape as a +separate neural network (making the network weights of a fixed +size network the representation of a shape) and train a diffusion +process as hypernetwork. A second approach is to have a shared +encoder-decoder network for all shapes and represent each shape as +a latent computed by the encoder. We opt for the second approach, +as it leads to more compact representations because it is jointly +learned from all shapes in the data set and the network weights +themselves do not count towards the latent representation. Such a +neural field takes a tuple of coordinates x and 𝐶-dimensional latent +f as input and outputs occupancy, +ˆONN(x) = NN(x, f), +(9) +where NN : R3 × R𝐶 → [0, 1] is a neural network. A first approach +was to use a single global latent f, but a major limitation is the ability +to encode shape details [Mescheder et al. 2019]. Some follow-up +works study coordinate-dependent latents [Chibane et al. 2020; Peng +et al. 2020] that combine traditional data structures such as regular +grids with the neural field concept. Latent vectors are arranged in a +spatial data structure and then interpolated (trilinearly) to obtain +the coordinate-dependent latent fx. A recent work 3DILG [Zhang +et al. 2022] proposed a sparse representation for 3D shapes, using +latents f𝑖 arranged in an irregular grid at point locations x𝑖. The +final coordinate-dependent latent fx is then estimated by kernel +regression, +fx = ˆFKN(x) = +𝑀 +∑︁ +𝑖=1 +f𝑖 · +1 +𝑍 +� +x, {x𝑖}𝑀 +𝑖=1 +� 𝜙(x, x𝑖), +(10) +where 𝑍 +� +x, {x𝑖}𝑀 +𝑖=1 +� += �𝑀 +𝑖=1 𝜙(x, x𝑖) is a normalizing factor. Thus +the representation for a 3D shape can be written as +� +f𝑖 ∈ R𝐶, x𝑖 ∈ R3�𝑀 +𝑖=1 . +(11) +After that, an MLP : R𝐶 → [0, 1] is applied to project the approxi- +mated feature ˆFKN(x) to occupancy, +ˆO3DILG(x) = MLP +� +ˆFKN(x) +� +. +(12) +Neural networks with latent sets (proposed). We initially explored +many variations for 3D shape representation based on irregular +and regular grids as well as tri-planes, frequency compositions, and +other factored representations. Ultimately, we could not improve +on existing irregular grids. However, we were able to achieve a +significant improvement with the following chance. We aim to keep +the structure of an irregular grid and the interpolation, but without +representing the actual spatial position explicitly. We let the net- +work encode spatial information. Both the representations (RBF in +Eq. (6) and 3DILG in Eq. (10)) are composed by two parts, values and +similarities. We keep the structure of the interpolation, but elmini- +tate explicit point coordinates and integrate cross attention from +Eq. (3). The result is the following learnable function approximator, +ˆF (x) = +𝑀 +∑︁ +𝑖=1 +v(f𝑖) · +1 +𝑍 +� +x, {f𝑖}𝑀 +𝑖=1 +� 𝑒q(x)⊺k(f𝑖)/ +√ +𝑑, +(13) +where 𝑍 +� +x, {f𝑖}𝑀 +𝑖=1 +� += �𝑀 +𝑖=1 𝑒q(x)⊺k(f𝑖)/ +√ +𝑑 is a normalizing factor. +Similar to the MLP in Eq. 12, we apply a single fully connected layer +to get desired occupancy values, +ˆO(x) = FC +� +ˆF (x) +� +. +(14) +Compared to 3DILG and all other coordinate-latent-based methods, +we dropped the dependency of the coordinate set {x𝑖}𝑀 +𝑖=1, the new +5 + +Cross Attention +K, V +Q +Learnable +(a) Learnable Queries +Cross Attention +K, V +Q +Subsample and Copy +(b) Point Queries +Fig. 4. Two ways to encode a point cloud. (a) uses a learnable query set; +(b) uses a downsampled version of input point embeddings as the query set. +representation only contains a set of latents, +� +f𝑖 ∈ R𝐶�𝑀 +𝑖=1 . +(15) +An alternative view of our proposed function approximator is to +see it as cross attention between query points x and a set of latents. +5 +NETWORK ARCHITECTURE FOR SHAPE +REPRESENTATION LEARNING +In this section, we will discuss how we design a variational autoen- +coder based on the latent representation proposed in Sec. 4. The +architecture has three components discussed in the following: a 3D +shape encoder, KL regularization block, and a 3D shape decoder. +5.1 +Shape encoding +We sample the surfaces of 3D input shapes in a 3D shape dataset. +This results in a point clouds of size 𝑁 for each shape, {x𝑖 ∈ R3}𝑁 +𝑖=1 +or in matrix form X ∈ R3×𝑁 . While the dataset used in the paper +originally represents shapes as triangle meshes, our framework +is directly compatible with other surface representations, such as +scanned point clouds, spline surfaces, or implicit surfaces. +In order to learn representations in the form of Eq. (15), the first +challenge is to aggregate the information contained in a possibly +large point cloud {x𝑖}𝑁 +𝑖=1 into a smaller set of latent vectors {f𝑖}𝑀 +𝑖=1. +We design a set-to-set network to this effect. +A popular solution to this problem in previous work is to divide +the large point cloud into a smaller set of patches and to learn one +latent vector per patch. Although this is a very well researched +and standard component in many networks, we discovered a more +successful way to aggregate features from a large point cloud that is +better compatible with the transformer architecture. We considered +two options. +One way is to define a learnable query set. Inspired by DETR [Car- +ion et al. 2020] and Perceiver [Jaegle et al. 2021], we use the cross +attention to encode X, +Enclearnable(X) = CrossAttn(L, PosEmb(X)) ∈ R𝐶×𝑀, +(16) +where L ∈ R𝐶×𝑀 is a learnable query set where each entry is 𝐶- +dimensional, and PosEmb : R3 → R𝐶 is a column-wise positional +embedding function. +Another way is to utilize the point cloud itself. We first subsample +the point cloud X to a smaller one with furthest point sampling, +X0 = FPS(X) ∈ R3×𝑀. The cross attention is applied to X0 and X, +Encpoints(X) = CrossAttn(PosEmb(X0), PosEmb(X)), +(17) +which can also be seen as a “partial” self attention. See Fig. 4 for +an illustration of both design choices. Intuitively, the number 𝑀 +affects the reconstruction performance: the larger the 𝑀, the better +reconstruction. However, 𝑀 strongly affects the training time due +to the transformer architecture, so it should not be too large. In our +final model, the number of latents 𝑀 is set as 512, and the number +of channels 𝐶 is 512 to provide a trade off between reconstruction +quality and training time. +5.2 +KL regularization block +Latent diffusion [Rombach et al. 2022] proposed to use a variational +autoencoder (VAE) [Kingma and Welling 2014] to compress images. +We adapt this design idea for our 3D shape representation and +also regularize the latents with KL-divergence. We should note +that the KL regularization is optional and only necessary for the +second-stage diffusion model training. If we just want a method for +surface reconstruction from point clouds, we do not need the KL +regularization. +We first linear project latents to mean and variance by two net- +work branches, respectively, +FC𝜇 (f𝑖) = �𝜇𝑖,𝑗 +� +𝑗 ∈[1,2,···,𝐶0] +FC𝜎 (f𝑖) = +� +log𝜎2 +𝑖,𝑗 +� +𝑗 ∈[1,2,···,𝐶0] +(18) +where FC𝜇 : R𝐶 → R𝐶0 and FC𝜎 : R𝐶 → R𝐶0 are two linear +projection layers. We use a different size of output channels 𝐶0, +where 𝐶0 ≪ 𝐶. This compression enables us to train diffusion +models on smaller latents of total size 𝑀 · 𝐶0 ≪ 𝑀 · 𝐶. We can +write the bottleneck of the VAE formally, ∀𝑖 ∈ [1, 2, · · · , 𝑀], 𝑗 ∈ +[1, 2, · · · ,𝐶0], +𝑧𝑖,𝑗 = 𝜇𝑖,𝑗 + 𝜎𝑖,𝑗 · 𝜖, +(19) +where 𝜖 ∼ N (0, 1). The KL regularization can be written as, +Lreg +� +{f𝑖}𝑀 +𝑖=1 +� += +1 +𝑀 · 𝐶0 +𝑀 +∑︁ +𝑖=1 +𝐶0 +∑︁ +𝑗=1 +1 +2 +� +𝜇2 +𝑖,𝑗 + 𝜎2 +𝑖,𝑗 − log𝜎2 +𝑖,𝑗 +� +. +(20) +In practice, we set the weight for KL loss as 0.001 and report the +performance for different values of𝐶0 in Sec. 8.1. Our recommended +setting is 𝐶0 = 32. +5.3 +Shape decoding +To increase the expressivity of the network, we add a latent learning +network between the two parts. Because our latents are a set of +vectors, it is natural to use transformer networks here. Thus, the +proposed network here is a series of self attention blocks, +{f𝑖}𝑀 +𝑖=1 ← SelfAttn(𝑙) � +{f𝑖}𝑀 +𝑖=1 +� +, +for 𝑖 = 1, · · · , 𝐿. +(21) +The SelfAttn(·) with a superscript (𝑙) here means 𝑙-th block. The +latents {f𝑖}𝑀 +𝑖=1 obtained using either Eq. (16) or Eq. (17) are fed into +the self attention blocks. Given a query x, the corresponding latent +is interpolated using Eq. (13), and the occupancy is obtained with a +fully connected layer as shown in Eq. (14). +6 + +Fig. 5. KL regularization. Given a set of latents {f𝑖 ∈ R𝐶 }𝑀 +𝑖=1 obtained +from the shape encoding in Sec. 5.1, we employ two linear projection layers +FC𝜇, FC𝜎 to predict the mean and variance of a low-dimensional latent +space, where a KL regularization commonly used in VAE training is applied +to constrain the feature diversity. Then, we obtain smaller latents {z𝑖 ∈ +R𝐶0 } of size 𝑀 · 𝐶0 ≪ 𝑀 · 𝐶 via reparametrization sampling. Finally, the +compressed latents are mapped back to the original space by FCup to obtain +a higher dimensionality for the shape decoding in Sec. 5.3. +Forward Diffusion Process +Reverse Diffusion Process +Add Noise +Add Noise +Add Noise +Denoise +Denoise +Denoise +Condition +Fig. 6. Latent set diffusion models. The diffusion model operates on +compressed 3D shapes in the form of a regularized set of latent vectors +{z𝑖 }𝑀 +𝑖=1. +Self Attention +Self Attention +· · · +(a) Unconditional Denoising Network +Self Attention +Cross Attention +K V +Q +· · · +Condition +(b) Conditional Denoising Network +Fig. 7. Denoising network. Our denoising network is composed of several +denoising layers (a box in the figure denotes a layer). The denoising layer +for unconditional generation contains two sequential self attention blocks. +The denoising layer for conditional generation contains a self attention +and a cross attention block. The cross attention is for injecting condition +information such as categories, images or partial point clouds. +Loss. We optimize the binary cross entropy loss between our +approximated function and the ground-truth indicator function as +in prior works [Mescheder et al. 2019]. +Lrecon +� +{f𝑖}𝑀 +𝑖=1, O +� += Ex∈R3 +� +BCE +� +ˆO(x), O(x) +�� +. +(22) +Surface reconstruction. We sample query points in a grid of res- +olution 1283. The final surface is reconstructed with Marching +Cubes [Lorensen and Cline 1987]. +6 +SHAPE GENERATION +Our proposed diffusion model combines design decisions from latent +diffusion (the idea of the compressed latent space), EDM [Karras et al. +2022] (most of the training details), and our shape representation +design (the architecture is based on attention and self-attention +instead of convolution). +We train diffusion models in the latent space, i.e., the bottleneck +in Eq. (19). Following the diffusion formulation in EDM [Karras et al. +2022], our denoising objective is +En𝑖∼N(0,𝜎2I) +1 +𝑀 +𝑀 +∑︁ +𝑖=1 +���Denoiser +� +{z𝑖 + n𝑖}𝑀 +𝑖=1, 𝜎, C +� +𝑖 − z𝑖 +��� +2 +2 , +(23) +where Denoiser(·, ·, ·) is our denoising neural network, 𝜎 is the noise +level, and C is the optional conditional information (e.g., categories, +images, partial point clouds and texts). We denote the corresponding +output of z𝑖 +n𝑖 with the subscript 𝑖, i.e. Denoiser(·, ·, ·)𝑖. We should +minimize the loss for every noise level 𝜎. The sampling is done by +solving ordinary/stochastic differential equations (ODE/SDE). See +Fig. 6 for an illustration and EDM [Karras et al. 2022] for a detailed +description for both the forward (training) and reverse (sampling) +process. +The function Denoiser(·, ·, ·) is a set denoising network (set-to-set +function). The network can be easily modeled by a self-attention +transformer. Each layer consists of two attention blocks. The first +one is a self attention for attentive learning of the latent set. The +second one is for injecting the condition information C (Fig. 7 (b)) +as in prior works [Rombach et al. 2022]. For simple information +like categories, C is a learnable embedding vector (e.g., 55 different +embedding vectors for 55 categories). For a single-view image , we +use ResNet-18 [He et al. 2016] as the context encoder to extract +a global feature vector as condition C. For text conditioning, we +use BERT [Devlin et al. 2018] to learn a global feature vector as +C. For partial point clouds, we use the shape encoder introduced +in Sec. 5.1 to obtain a set of latent embeddings as C. In the case +of unconditional generation, the cross attention degrades to self +attention (Fig. 7 (a)). +7 +EXPERIMENTAL SETUP +We use the dataset of ShapeNet-v2 [Chang et al. 2015] as a bench- +mark, containing 55 categories of man-made objects. We use the +training/val splits in [Zhang et al. 2022]. We preprocess shapes as +in [Mescheder et al. 2019]. Each shape is first converted to a water- +tight mesh, and then normalized to its bounding box, from which we +further sample a dense surface point cloud of size 50,000. To learn +the neural fields, we randomly sample 50,000 points with occupan- +cies in the 3D space, and 50,000 points with occupancies in the near +surface region. For the single-view object reconstruction, we use +the 2D rendering dataset provided by 3D-R2N2 [Choy et al. 2016], +where each shape is rendered into RGB images of size of 224 × 224 +from 24 random viewpoints. For text-driven shape generation, we +use the text prompts of ShapeGlot [Achlioptas et al. 2019]. For data +preprocess of shape completion training, we create partial point +clouds by sampling point cloud patches. +7.1 +Baselines +For shape auto-encoding, we conduct experiments against state- +of-the-art methods for implicit surface reconstruction from point +clouds. We use OccNet [Mescheder et al. 2019], ConvOccNet [Peng +et al. 2020], IF-Net [Chibane et al. 2020], and 3DILG [Zhang et al. +2022] as baselines. The OccNet is the first work of learning neural +fields from a single global latent vector. ConvOccNet and IF-Net +7 + +OccNet +ConvOccNet +IF-Net +3DILG +Ours +Learned Queries +Point Queries +table +0.823 +0.847 +0.901 +0.963 +0.965 +0.971 +car +0.911 +0.921 +0.952 +0.961 +0.966 +0.969 +chair +0.803 +0.856 +0.927 +0.950 +0.957 +0.964 +airplane +0.835 +0.881 +0.937 +0.952 +0.962 +0.969 +sofa +0.894 +0.930 +0.960 +0.975 +0.975 +0.982 +rifle +0.755 +0.871 +0.914 +0.938 +0.947 +0.960 +lamp +0.735 +0.859 +0.914 +0.926 +0.931 +0.956 +mean (selected) +0.822 +0.881 +0.929 +0.952 +0.957 +0.967 +IoU ↑ +mean (all) +0.825 +0.888 +0.934 +0.953 +0.955 +0.965 +table +0.041 +0.036 +0.029 +0.026 +0.026 +0.026 +car +0.082 +0.083 +0.067 +0.066 +0.062 +0.062 +chair +0.058 +0.044 +0.031 +0.029 +0.028 +0.027 +airplane +0.037 +0.028 +0.020 +0.019 +0.018 +0.017 +sofa +0.051 +0.042 +0.032 +0.030 +0.030 +0.029 +rifle +0.046 +0.025 +0.018 +0.017 +0.016 +0.014 +lamp +0.090 +0.050 +0.038 +0.036 +0.035 +0.032 +mean (selected) +0.058 +0.040 +0.034 +0.032 +0.031 +0.030 +Chamfer ↓ +mean (all) +0.072 +0.052 +0.041 +0.040 +0.039 +0.038 +table +0.961 +0.982 +0.998 +0.999 +0.999 +0.999 +car +0.830 +0.852 +0.888 +0.892 +0.898 +0.899 +chair +0.890 +0.943 +0.990 +0.992 +0.994 +0.997 +airplane +0.948 +0.982 +0.994 +0.993 +0.994 +0.995 +sofa +0.918 +0.967 +0.988 +0.986 +0.986 +0.990 +rifle +0.922 +0.987 +0.998 +0.997 +0.998 +0.999 +lamp +0.820 +0.945 +0.970 +0.971 +0.970 +0.975 +mean (selected) +0.898 +0.951 +0.975 +0.976 +0.977 +0.979 +F-Score ↑ +mean (all) +0.858 +0.933 +0.967 +0.966 +0.966 +0.970 +Table 3. Shape autoencoding (surface reconstruction from point clouds) on ShapeNet. We show averaged metrics on all 55 categories and individual +metrics for the 7 largest categories. +𝑀 = 512 𝑀 = 256 𝑀 = 128 𝑀 = 64 +IoU ↑ +0.965 +0.956 +0.940 +0.916 +Chamfer ↓ +0.038 +0.039 +0.043 +0.049 +F-Score ↑ +0.970 +0.965 +0.953 +0.929 +Table 4. Results for different number of latents 𝑀 for +shape autoencoding +𝐶0 = 1 𝐶0 = 2 𝐶0 = 4 𝐶0 = 8 𝐶0 = 16 𝐶0 = 32 𝐶0 = 64 +IoU ↑ +0.727 +0.816 +0.957 +0.960 +0.962 +0.963 +0.964 +Chamfer ↓ +0.133 +0.087 +0.038 +0.038 +0.038 +0.038 +0.038 +F-Score ↑ +0.703 +0.815 +0.967 +0.967 +0.970 +0.969 +0.970 +Table 5. Ablation study of compression via the number of channels𝐶0 for shape +(variational) autoencoding. +Grid-83 +3DILG +Ours +𝐶0 = 8 𝐶0 = 16 𝐶0 = 32 𝐶0 = 64 +Surface-FPD ↓ +4.03 +1.89 +2.71 +1.87 +0.76 +0.97 +Surface-KPD (×103) ↓ +6.15 +2.17 +3.48 +2.42 +0.66 +1.11 +Rendering-FID ↓ +32.78 +24.83 +28.25 +27.26 +17.08 +24.24 +Rendering-KID (×103) ↓ +14.12 +10.51 +14.60 +19.37 +6.75 +11.76 +Table 6. Unconditional generation on full ShapeNet. +PVD +Ours +Surface-FPD ↓ +2.33 +0.63 +Surface-KPD (×103) ↓ +2.65 +0.53 +Rendering-FID ↓ +270.64 +17.08 +Rendering-KID (×103) ↓ +281.54 +6.75 +Table 7. Unconditional generation on full ShapeNet. +learn local neural fields based on latent vectors arranged in a regular +grid, while 3DILG uses latent vectors on an irregular grid. +For 3D shape generation, we compare against recent state-of-the- +art generative models, including PVD [Zhou et al. 2021], 3DILG [Zhang +8 + +Input +GT +OccNet +ConvONet +IF-Net +3DILG +Proposed +Learnable Queries +Point Queries +Fig. 8. Visualization of shape autoencoding results (surface reconstruction from point clouds from ShapeNet). +et al. 2022], and NeuralWavelet [Hui et al. 2022]. PVD is a diffusion +model for 3D point cloud generation, and 3DILG utilizes autore- +gressive models. NeuralWavelet utilized diffusion models in the +frequency domain of shapes. +9 + +Ours +3DILG +Grid-83 +PVD +Fig. 9. Unconditional generation. All models are trained on full ShapeNet. +airplane +chair +table +car +sofa +3DILG +NW +Ours +3DILG +NW +Ours +3DILG +NW +Ours +3DILG +NW +Ours +3DILG +NW +Ours +Surface-FID +0.71 +0.38 +0.62 +0.96 +1.14 +0.76 +2.10 +1.12 +1.19 +2.93 +- +2.04 +1.83 +- +0.77 +Surface-KID (×103) +0.81 +0.53 +0.83 +1.21 +1.50 +0.70 +3.84 +1.55 +1.87 +7.35 +- +3.90 +3.36 +- +0.70 +Table 8. Category conditioned generation. NW is short for NeuralWavelet. The dash sign “-” means the method NeuralWavelet does not release models +trained on these categories. +7.2 +Evaluation metrics +To evaluate the reconstruction accuracy of shape auto-encoding +from point clouds, we adopt Chamfer distance, volumetric Intersection- +over-Union (IoU), and F-score as primary evaluation metrics. IoU +is computed based on the occupancy predictions of 50𝑘 querying +points sampled in 3D space. Chamfer distance and F-score are cal- +culated between two sampled point clouds with the size of 50𝑘 +respectively from reconstructed and ground-truth surfaces. For IoU +and F-score, higher is better, while for Chamfer, lower is better. +To measure the mesh quality of unconditional and conditional +shape generation, we follow [Ibing et al. 2021; Shue et al. 2022; Zhang +et al. 2022] to adapt the Fréchet Inception Distance (FID) and Kernel +Inception Distance (KID) commonly used to assess the image gener- +ative models to rendered images of 3d shapes. To calculate FID and +KID of rendered images, we render each shape from 10 viewpoints. +The metrics are named as Rendering-FID and Rendering-KID. +The Rendering-FID is defined as, +Rendering-FID = ∥𝜇g − 𝜇r∥ +𝑇𝑟 +� +Σ𝑔 + Σ𝑟 − 2(Σ𝑔Σ𝑟)1/2� +(24) +where 𝑔 and 𝑟 denotes the generated and training datasets respec- +tively. 𝜇 and Σ are the statistical mean and covariance matrix of the +feature distribution extracted by the Inception network. +The Rendering-KID is defined as, +Rendering-KID = MMD +� +1 +|R| +∑︁ +x∈R +max +y∈G 𝐷(x, y) +�2 +(25) +where 𝐷(x, y) is a polynomial kernel function to evaluate the simi- +larity of two samples, G and R are feature distributions of generated +set and reference set, respectively. The function MMD(·) is Maxi- +mum Mean Discrepancy. However, the rendering-based FID and KID +are essentially designed to understand 3D shapes from 2D images. +Thus, they have the inherent issue of not accurately understanding +shape compositions in the 3D world. To compensate their draw- +backs, we also adapt the FID and KID to 3D shapes directly. For each +generated or groud-truth shape, we sample 4096 points (with nor- +mals) from the surface mesh and then feed them into a pre-trained +PointNet++ [Qi et al. 2017b] to extract a global latent vector, repre- +senting the global structure of the 3D shape. The PointNet++ is first +pretrained on shape classification on ShapeNet-55. As we use point +clouds, we call the FID and KID for 3D shapes as Fréchet PointNet++ +Distance (FPD) and Kernel PointNet++ Distance (KPD). The two +metrics are defined similarly as in Eq. (24) and Eq. (25), except that +the features are extracted from a PointNet++ network. +10 + +Ours +NW +3DILG +Grid-83 +Ours +NW +3DILG +Grid-83 +Ours +NW +3DILG +Grid-83 +Fig. 10. Category-conditional generation. From top to bottom, we show category (airplane, chair, table) conditioned generation results. +7.3 +Implementation +For the shape auto-encoder, we use the point cloud of size 2048 as +input. At each iteration, we individually sample 1024 query points +from the bounding volume ([−1, 1]3) and the other 1024 points +from near surface region for the occupancy values prediction. The +shape auto-encoder is trained on 8 A100, with batch size of 512 +for 𝑇 = 1, 600 epochs. The learning rate is linearly increased to +𝑙𝑟max = 5𝑒 − 5 in the first 𝑡0 = 80 epochs, and then gradually +decreased using the cosine decay schedule 𝑙𝑟max ∗ 0.51+𝑐𝑜𝑠 ( 𝑡−𝑡0 +𝑇 −𝑡0 ) +until reaching the minimum value of 1𝑒 − 6. The diffusion models +are trained on 4 A100 with batch size of 256 for 𝑇 = 8, 000 epochs. +The learning rate is linearly increased to 𝑙𝑟𝑚𝑎𝑥 = 1𝑒 − 4 in the first +𝑡0 = 800 epochs, and then gradually decreased using the above +mentioned decay schedule until reaching 1𝑒 − 6. We use the default +settings for the hyperparameters of EDM [Karras et al. 2022]. During +sampling, we obtain the final latent set via only 18 denoising steps. +8 +RESULTS +We present our results for multiple applications: 1) shape auto- +encoding, 2) unconditional generation, 3) category-conditioned +generation, 4) text-conditioned generation, 5) shape completion, +11 + +6) image-conditioned generation. Finally, we perform a shape nov- +elty analysis to validate that we are not overfitting to the dataset. +8.1 +Shape Auto-Encoding +We show the quantitative results in Tab. 3 for a deterministic au- +toencoder without the KL block described in Sec. 5.2. In particular, +we show results for the largest 7 categories as well as averaged re- +sults over the categories. The two design choices of shape encoding +described in Sec. 5.1 are also investigated. The case of using the +subsampled point cloud as queries is better than learnable queries in +all categories. Thus we use subsampled point clouds in our later ex- +periments. The visualization of reconstruction results can be found +in Fig. 8. We visualize some extremely difficult shapes from the +datasets (test split). These shapes often contain some thin structures. +However, our method still performs well. +Ablation study of the number of latents. The number 𝑀 is the +number of latent vectors used in the network. Intuitively, a larger +𝑀 leads to a better reconstruction. We show results of 𝑀 in Tab. 4. +Thus, in all of our experiments, 𝑀 is set to 512. We are limited by +computation time to work with larger 𝑀. +Ablation study of the KL block. We described the KL block in Sec. 5.2 +that leads to additional compression. In addition, this block changes +the deterministic shape encoding into a variational autoencoder. +The introduced hyperparameter is 𝐶0. A smaller 𝐶0 leads to a higher +compression rate. The choice of𝐶0 is ablated in Tab. 5. Clearly, larger +𝐶0 gives better results. The reconstruction results of𝐶0 = 8, 16, 32, 64 +are very close. However, they differ significantly in the second stage, +because a larger latent size could make the training of diffusion +models more difficult. This result is very encouraging for our model, +because it indicates that aggressively increasing the compression +in the KL block does not decrease reconstruction performance too +much. We can also see that compressing with the KL block by de- +creasing 𝐶0 is much better than compressing using fewer latent +vectors 𝑀. +8.2 +Unconditional Shape Generation +Comparison with surface generation. We evaluate the task of un- +conditional shape generation with the proposed metrics in Tab. 6. +We also compared our method with a baseline method proposed +in [Zhang et al. 2022]. The method is called Grid-83 because the +latent grid size is 83, which is exactly the same as in AutoSDF [Mittal +et al. 2022]. The table also shows the results of different 𝐶0. Our +results are best when 𝐶0 = 32 in all metrics. When 𝐶0 = 64 the +results become worse. This also aligns with our conjecture that a +larger latent size makes the training more difficult. +Comparison with point cloud generation. Additionally, we compare +our method with PVD [Zhou et al. 2021] which is a point cloud +diffusion method. We re-train PVD using the official released code +on our preprocessed dataset and splits. We use the same evaluation +protocol as before but with one major difference. Since PVD can only +generate point clouds without normals, we use another pretrained +PointNet++ (without normals) as the feature extractor to calculate +Surface-FPD and Surface-KPD. The Tab. 7 shows we can beat PVD +by a large margin. Additionally, we also show the metrics calculated +AutoSDF +Ours +“horizontal slats on top of back” +“one big hole between back and seat” +“this chair has wheels” +“vertical back ribs” +Fig. 11. Text conditioned generation. For each text prompt, we generate +3 shapes. Our results (Right) are compared with AutoSDF (Left). +on rendered images. Visualization of generated results can be found +in Fig. 9. +8.3 +Category-conditioned generation +We train a category-conditioned generation model using our method. +We evaluate our models in Tab. 8. We should note that the competitor +method NeuralWavelet [Hui et al. 2022] trains models for categories +separately; thus, NeuralWavelet is not a true category-conditioned +model. We also visualize some results (airplane, chair, and table) +in Fig. 10. Our training is more challenging, as we train on a dataset +that is an order of magnitude larger and we train for all classes +jointly. While NeuralWavelet already has good results, the joint +training is necessary / beneficial for many subsequent applications. +8.4 +Text-conditioned generation +The results of our text-conditioned generation model can be found +in Fig. 11. Since the model is a probabilistic model, we can sample +shapes given a text prompt. The results are very encouraging and +they constitute the first demonstration of text-conditioned 3D shape +generation using diffusion models. To the best of our knowledge, +there are no published competing methods at the point of submitting +this work. +8.5 +Probabilistic shape completion +We also extend our diffusion model for probablistic shape comple- +tion by using a partial point cloud as conditioning input. The compar- +ison against ShapeFormer [Yan et al. 2022] is depicted in Fig. 12. As +seen, our latent set diffusion can predict more accurate completion, +and we also have the ability to achieve more diverse generations. +12 + +GT +Condition +ShapeFormer +Ours +Fig. 12. Point cloud conditioned generation. We show three generated +results given a partial cloud. The ground-truth point cloud and the partial +point cloud used as condition are shown in Left. We compare our results +(Right) with ShapeFormer (Middle). +Condition +IM-Net +OccNet +Ours +Fig. 13. Image conditioned generation. In the left column we show the +condition image. In the middle we show results obtained by the method +IM-Net and OccNet. Our generated results are shown on the right. +8.6 +Image-conditioned shape generation. +We also provide comparisons on the task of single-view 3D object +reconstruction in Fig. 13. Compared to other deterministic methods +including OccNet [Mescheder et al. 2019] and IM-Net [Chen and +Zhang 2019], our latent set diffusion can not only reconstruct more +accurate surface details, (e.g. long rods and tiny holes in the back), +but also support multi-modal prediction, which is a desired property +to deal with severe occlusions. +Ref +Gen +Ref +Gen +Ref +Gen +Ref +Gen +Fig. 14. Shape generation novelty. For a generated shape, we retrieve +the top-1 similar shape in the training set. The similarity is measured using +Chamfer distance of sampled surface point clouds. In each pair, we show +the retrieved shape (left) and the generated shape (right). The generated +shapes are from our category-conditioned generation results. +8.7 +Shape novelty analysis +We use shape retrieval to demonstrate that we are not simply over- +fitting to the training set. Given a generated shape, we measure the +Chamfer distance between it and training shapes. The visualization +of retrieved shapes can be found in Fig. 14. Clearly, the model can +synthesize new shapes with realistic structures. +8.8 +Limitations +While our method shows convincing results on a variety of tasks, +our design choices also have drawbacks that we would like to dis- +cuss. For instance, we require a two stage training strategy. While +this leads to improved performance in terms of generation quality, +training the first stage is more time consuming than relying on +manually-designed features such as wavelets [Hui et al. 2022]. In +addition, the first stage might require retraining if the shape data in +consideration changes, and for the second stage – the core of our +diffusion architecture – training time is also relatively high. Overall, +we believe that there is significant potential for future research av- +enues to speed up training, in particular, in the context of diffusion +models. +9 +CONCLUSION +We have introduced 3DShape2VecSet, a novel shape representation +for neural fields that is tailored to generative diffusion models. To +this end, we combine ideas from radial basis functions, previous +neural field architectures, variational autoencoding, as well as cross +attention and self-attention to design a learnable representation. +Our shape representation can take a variety of inputs including +triangle meshes and point clouds and encode 3D shapes as neu- +ral fields on top of a set of latent vectors. As a result, our method +demonstrates improved performance in 3D shape encoding and 3D +shape generative modeling tasks, including unconditioned genera- +tion, category-conditioned generation, text-conditioned generation, +point-cloud completion, and image-conditioned generation. +In future work, we see many exciting possibilities. Most impor- +tantly, we believe that our model further advances the state of the +art in point cloud and shape processing on a large variety of tasks. +In particular, we would like to employ the network architecture of +3DShape2VecSet to tackle the problem of surface reconstruction +from scanned point clouds. In addition, we can see many applica- +tions for content-creation tasks, for example 3D shape generation +of textured models along with their material properties. Finally, we +13 + +would like to explore editing and manipulation tasks leveraging +pretrained diffusion models for prompt to prompt shape editing, +leveraging the recent advances in image diffusion models. +REFERENCES +Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, and Leonidas Guibas. 2018. Learn- +ing representations and generative models for 3d point clouds. In International +conference on machine learning. PMLR, 40–49. +Panos Achlioptas, Judy Fan, Robert Hawkins, Noah Goodman, and Leonidas J Guibas. +2019. 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In Proceedings of the IEEE/CVF International Conference on +Computer Vision. 5826–5835. +15 + diff --git a/KdAyT4oBgHgl3EQff_ia/content/tmp_files/2301.00351v1.pdf.txt b/KdAyT4oBgHgl3EQff_ia/content/tmp_files/2301.00351v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..887bdb52f0d7aad90294fe0f84b54b45fd376951 --- /dev/null +++ b/KdAyT4oBgHgl3EQff_ia/content/tmp_files/2301.00351v1.pdf.txt @@ -0,0 +1,1731 @@ +Article +Skew Class-balanced Re-weighting +for Unbiased Scene Graph Generation +Haeyong Kang +and Chang D. Yoo* +School of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST); +haeyong.kang@kaist.ac.kr +* +Correspondence: cd_yoo@kaist.ac.kr +† +Current address: Daejeon 34141, Republic of Korea. +Abstract: +An unbiased scene graph generation (SGG) algorithm referred to as Skew Class-balanced +Re-weighting (SCR) is proposed for considering the unbiased predicate prediction caused by the long- +tailed distribution. The prior works focus mainly on alleviating the deteriorating performances of the +minority predicate predictions, showing drastic dropping recall scores, i.e., losing the majority predicate +performances. It has not yet correctly analyzed the trade-off between majority and minority predicate +performances in the limited SGG datasets. In this paper, to alleviate the issue, the Skew Class-balanced Re- +weighting (SCR) loss function is considered for the unbiased SGG models. Leveraged by the skewness of +biased predicate predictions, the SCR estimates the target predicate weight coefficient and then re-weights +more to the biased predicates for better trading-off between the majority predicates and the minority +ones. Extensive experiments conducted on the standard Visual Genome dataset and Open Image V4 & +V6 show the performances and generality of the SCR with the traditional SGG models. The source code is +available at https://github.com/ihaeyong/Unbiased-SGG. +Keywords: Scene Graph Generation (SGG), Skew Class-balanced Re-weighting (SCR), Predicate Sample +Estimates, Skew Class-balanced Effective Number +0. Introduction +Scene Graph Generation (SGG) is receiving increased attention for improving image +understanding [1–15]. The core building blocks of SGG are the objects in the image. There +can be diverse relationships among the objects [16,17]. All relationships can be represented as +triplets subject, relation, object, which can be used for generating the scene graph. The precise +interpretation of an image depends on the core relations between the subject-object pairs. For +example, given an image of dog, woman, kite objects, the image can be interpreted in multiple +ways such as ⟨dog, has, leg⟩, ⟨dog, near, woman⟩, and ⟨woman, holding, kite⟩. +In general, however, the real-world large data sets often have shown long-tailed pred- +icate distributions as depicted in Fig.1a - the predicate proportion of the Visual Genome[1], +containing 51 predicates. One cause of long-tail distribution is the background predicates. The +background predicates account for more than 80% in total predicates since it is not trivial to +annotate all possible relationships, leading to possibly far more subject-object pairs without +annotating ground truth than ones with ground truth labels. Another reason is the visual event +frequency, as shown in Fig.1b. The majority predicates more frequently occur than the minority +ones given a visual subject-object pair in the real-world scene. For example, given a man-board +pair, the predicate on is more frequently observed than holding and riding. +This long-tailed label distribution causes the trained model to tend to be biased toward +the majority predicates. The SGG models [13,18–22] focus on increasing the minority predicate +performances for the unbiased scene graph generation, i.e., increasing mean recall scores. How- +ever, these methods lead to deteriorating performances of the majority predicates, drastically +dropping recall scores, i.e., losing the majority predicate performances. Recently, the issue has +arXiv:2301.00351v1 [cs.LG] 1 Jan 2023 + +2 of 16 +backgrounds +on +has +wearing +of +in +near +with +holding +behind +above +sitting on +wears +riding +under +in front of +standing on +at +carrying +attached to +walking on +for +over +laying on +looking at +hanging from +belonging to +parked on +eating +using +and +covering +part of +covered in +between +along +lying on +watching +on back of +to +walking in +mounted on +against +across +from +growing on +painted on +made of +playing +says +flying in +0 +2 +4 +6 +8 +10 +12 +14 +log(freq) +the predicates of all subject-object pairs +(a) The Long-tailed Predicate Distribution. +backgrounds +on +has +wearing +of +in +near +with +holding +behind +above +sitting on +wears +riding +under +in front of +standing on +at +carrying +attached to +walking on +for +over +laying on +looking at +hanging from +belonging to +parked on +eating +using +and +covering +part of +covered in +between +along +lying on +watching +on back of +to +walking in +mounted on +against +across +from +growing on +painted on +made of +playing +says +flying in +0 +1 +2 +3 +4 +5 +6 +7 +8 +log(freq) +the predicates of man-board pair +(b) The Predicate Distribution of the Man - Board Pair. +Figure 1. The Long-tailed Predicate Distribution: (a) The predicate proportion of the Visual Genome[1] +which is possibly far more predicates of subject - object pairs without annotating ground-truth than others +and (b) The majority predicates are more frequently observed than the minority ones given a subject - +object pair. Given a man - board pair, the possible predicates- backgrounds, on, holding, riding, etc. represents +a long-tail distribution. +been alleviated through resampling images and object instances [23]. However, this method +needs more computing power and time to train the model on the resampled samples. The +current state-of-the-art models focus on re-balancing biased loss [24] or correcting noisy labels +[25] to acquire an unbiased SGG model. Nevertheless, the prior works have not yet adequately +described and analyzed the trade-off performances between majority and minority predicates +for learning SGG models based on the given imbalanced datasets. +In this paper, the Skew Class-balanced Re-weighting (SCR) loss function is considered for +alleviating the issues and acquiring the best trade-off performances in the multiple SGG tasks. +Leveraged by the skewness of predicate predictions, the SCR estimates its weight coefficients +and then reweights more to biased predicate samples to adaptively be unbiased SGG models. +The extensive experimental results show that the SCR loss function gives more generalized +performances than priors in the multiple SGG tasks on the Visual Genome dataset[26] and the +Open Images dataset [27]. +Contributions of our SCR learning scheme to unbiased SGG models: +• +Leveraged by the skewness of biased predicate predictions, the Skew Class-balanced Re- +weighting (SCR) loss function is firstly proposed for the unbiased scene graph generation +(SGG) models. + +3 of 16 +• +The SCR is applied to the current state-of-the-art SGG models to show its effectiveness, +leading to more generalized performances: the SCR outperforms the prior reweighted +methods on both mean recall and recall measurements in the multiple SGG tasks. +This paper is organized as follows. The Related Work section provides discussions on +unbiased scene graph generation. The unbiased SGG section presents scene graph generation. +In the Skew Class-balanced Re-weighting (SCR) section, the SCR loss function is depicted in +detail. In the experimental section, the results of scene graph generation on the Visual Genome +benchmark dataset are examined, along with an analysis and ablation study on the SCR with +the current state-of-the-art SGG models. Finally, we conclude. +1. Related Works. +Unbiased Scene Graph Generation (SGG). Predicate distribution is much more long-tailed +than object distribution. For N objects and R predicates, the model has to address the fundamen- +tal challenge of learning O(N2R) relations with few [28,28,29]. To overcome the limited training +dataset, the linguistic external knowledge [8,28,30] was used by Yu et al. [31], regularizing the +deep neural network; using linguistic knowledge, the probabilistic model has also alleviated +the semantical ambiguity of visual relationships [32]. Furthermore, to alleviate the imbalanced +relationship distribution, Yin et al. [8] reformulated the conventional one-hot classification as a +n-hot multiclass hierarchical recognition via a novel Intra-Hierarchical trees (IH-trees) for each +label set in the triplet ⟨subject, predicate, object⟩. Recently, unbiased SGG [13,17–22,24,25,33–42] +has drawn unprecedented interest for more generalized SGG models. Occurrence-based Node +Priority Sensitive (NPS)-loss [17] was used for balancing predictions; the Total Direct Effect +(TDE) method has proposed firstly for unbiased learning by [13], which directly separates +the bias from biased predictions through the counterfactual methodologies on causal graphs; +CogTree [18] addressed the debiasing issue based on the coarse-to-fine structure of the rela- +tionships from the cognition viewpoint; Li et al. [19] improved the context modeling for tail +categories by designing the bipartite graph network and message propagation on resampled +objects and images. Lastly, the Predicate Probability Distribution based Loss (PPDL) [24] has +proposed to train the biased SGG models, which measure the semantic predicate representation +to re-balance the biased training loss. In this work, the Skew Class-balanced Re-weighting (SCR) +loss function is proposed for alleviating biased predicate predictions, leading to the most +generalized SGG models through the novel adaptive re-weighting learning scheme. +Re-Weighting based Unbiased SGG. Overall unbiased SGG models can be categorized into re- +balancing strategy of re-weighting [17,18,20,24] and re-sampling [19] and biased model-based +strategy [13,21,22]. For unbiased SGG modes, Tang et al. [13] first investigated the re-weighting +learning algorithm. However, they observed that the performances of the majority predicates +were drastically dropped, resulting in low recall scores while with high mean recall scores. This +shows the general tendency that there is a trade-off performance between majority predicates +and minority ones. To alleviate the trade-off issue, Yu et al. [18] proposed CogTree based on +the coarse-to-fine structure of the relationships from the cognition viewpoint. Recently, the +Predicate Probability Distribution (PPD) [24] re-balances the biased training loss according to +the similarity between the predicted probability distribution and the estimated one. However, +it has not yet correctly analyzed the trade-off performances between the majority and minority +classes in various SGG tasks. In this paper, we measure the sample skew score based on the +sample estimates for bias toward the majority classes to assign the sample weight correctly. The +sample skewness is computed as the Fisher-Pearson coefficient of skewness on its sample mean +value [43]. However, since the mean value tends to be biased toward the majority predicates, +we measure the sample skew score fairly on its target logit instead of its mean value. Based +on the sample skew score, the SCR assigns the sample weights adaptively - if there is no + +4 of 16 +man horse cat ... +man horse cat ... +(a) Object Model +Input & Label +Object Predictions +Predicate Predictions & Sample Estimates +Label: < man, riding, horse > +Sample Weight Estimates +(b) FREQ Model +(e) Sample Estimates +(c) Predicate Model +(d) Predicate Predictions +(f) Skew Measures +(g) Target Sample Weight +riding +near +on +... +riding +near +on +... +riding +near +on +... +riding +near +on +... +riding +near +on +... +riding +near +on +... +riding +near +on +... +riding +near +on +... +riding +near +on +... +riding +near +on +... +riding +near +on +... +riding +near +on +... +... +... +... +... +... +... +... +... +Figure 2. Skew Class-balanced Re-weighting (SCR) For Unbiased SGG models. The traditional SGG +model and (d) predicate prediction ( ˆR): (a) object detector outputs its predictions: man and hourse, (b) +FREQ embeds prior predicates and (c) predicate model outputs predicate predictions ( ˆRvis); Given man - +horse label predictions, our SCR estimates (e) a sample size of the possible predicate candidates through +FREQ (R freq or Remb ) and measures (f) the target label skew score and then calculates (g) the training +target sample weight for the adaptive re-weighted loss, and in (f) and (g), red lines around the target label +indicate predicate skew (Si +skew) and re-weight scores (Wi), respectively, where they have an approximately +inverse relationship. +bias, we assign fewer weights to the sample (majority). If it is biased to one side, we assign +larger weights to the samples (minority). Such that the SGG models with SCR show superior +performances and generality on the multiple SGG tasks. +2. Unbiased Scene Graph Generation +In this section, we discuss the general scene graph generation model of the object and +predicate predictions and depict the predicate sample estimates for measuring its skewness of +biased predictions. +2.1. Scene Graph Generations. +Given an image I, a scene graph model generates a graph G = (V, E), where V and E +are the sets of nodes and edges, respectively. Each node oi ∈ V is represented by a bounding +box vbbox ∈ R4 and a corresponding class label oy ∈ Cobj. Each edge ri,j ∈ E represents the +predicate between the subject node vi and the object node vj. The corresponding predicate +label is ry ∈ Crel. +2.1.1. Object Predictions. +Following [13], the node features are derived from the object detector. In particular, for +each bounding box vbbox +i +, the detector returns an RoI-Align feature xRoI +i +and an object label +embedding li. In general SGG model, the N number of node features are constructed by +vector concatenation X = {[xRoI +i +; li; bi]}N +i=1, where bi is the embedded box feature from the box +coordinate vbbox +i +. +˜X = f cobj(X) ∈ RN ×Dobj +(1) +where all f c∗ denote a fully connected layer for linear transformations or logits, and the object +feature dimension Dobj depends on the SGG model. The predicted object label of ˆO ∈ RN ×|Cobj| +is estimated by object logits f c( ˜X) as follows: +ˆO = f c( ˜X) +(2) + +man +horse +O5 of 16 +2.1.2. Predicate Predictions. +Predicate predictions can be made by employing multiple logits from visual and non- +visual features. We follow the sum over all outputs to generate the final predicate prediction. +The combined predicate logit ˆR is estimated based on the summation of the visual logits and +the non-visual logits as follows: +ˆR = ˆRvis ⊕ ˆR f req ⊕ ˆRemb +(3) +where ˆR ∈ RN (N −1)×|Crel|; ⊕ is an element-wise sum; ˆRvis is the predicate logits from visual +feature Fvis such as a Dvis dimensional union feature and a subject-object pair feature, which +also depends on the SGG model, +ˆRvis = f cvis(Fvis), Fvis ∈ RN (N −1)×Dvis, +ˆR f req = Sigmoid(R f req) ∈ RN (N −1)×|Crel|, +ˆRemb = f cemb(Lemb), Lemb ∈ RN (N −1)×400 +(4) +The FREQ [44] as a non-visual feature, R f req looks up the empirical distribution over rela- +tionships between subject ˆoi and object ˆoj as computed in the training set where ˆoi, ˆoj ∈ ˆO. +However, since FREQ does not consider any image representations when predicting predicates, +it tends to lead to biased predicate predictions due to its imbalanced predicate distribution. To +minimize the biased effects, we use the Sigmoid-activated FREQ predicate logits. In addition, +for acquiring the more smoothness of the empirical distribution, the concatenated subject-object +embedding ˆRemb is added to the predicate predictions where Lemb = {[li; lj]}N +i,j . +2.2. Sample Estimates +Non-visual predicate features tend to be more biased than visual features due to an +imbalanced training set. If the degree of biased predictions can be measured, we can leverage +its value to learn the SGG models without bias. According to [43], the degree of bias prediction +can generally be measured in a skew score. In general, if there is no bias, the skew score is close +to 0, and if it is biased to one side, the skew score is over either −1 or +1. +In SCR, we need to estimate how many predicate samples are biased to measure the +predicate skew scores. To approximate the biased sample numbers, we use two non-visual +prior predicate distributions of FREQ, ˆR f req and subject-object label embedding ˆRemb. Based +on the two predicate distributions, at first, we define the skew logit ˆRskew with the Sigmoid +activation function as follows: +ˆRskew = Sigmoid(R f req ⊕ ˆRemb) +(5) +To estimate the predicate sample weights properly, the SCR approximates skewness through the +predicate sample estimates ˆRskew in Eq. 5. We investigate the best predicate sample estimates +through several experiments with a combination of non-visual predictions as follows: +• +SCR of EMB: ˆRskew = Sigmoid( ˆRemb) +• +SCR of FREQ: ˆRskew = Sigmoid(R f req) +• +SCR of FREQ+EMB: ˆRskew = Sigmoid(R f req ⊕ ˆRemb) + +6 of 16 +Then, the predicate sample estimates are acquired as follows: +M = +N (N −1) +∑ +i +ˆri +(6) +where ˆri ∈ ˆRskew; the estimated my ∈ M is the number of yth predicate sample size and +my ∈ [0, N (N − 1)). +Moreover, the skew predicate ˆRskew serves to estimate the predicate candidates that a +subject-object pair can have fairly since the Sigmoid activation function suppresses the more +extensive biased predictions. For example, a Man-Horse pair may have predicates such as +riding, on, with, etc. The Sigmoid activation function amplifies the frequency of the minority +predicates while squeezing that of the majority ones, as shown in Fig.2 (b) and (e), which +is used to calculate Skew Class-Balanced Effective Number that we depict in the following +section. +3. Skew Class-balanced Re-weighting (SCR) +In this section, leveraged by the biased predictions deduced by the predicate sample +estimates, the Skew Class-balanced Re-weighting (SCR) performs sample weight estimates for +learning unbiased SGG models. +3.1. Skew Class-balanced Effective Number +The Skew Class-balanced Effective Number approximates the mini-batch class-balanced re- +weighting coefficients Emy based on the predicate skew logit ˆRskew in Eq. 5. The ith predicate +sample Ei +my is defined as follows as [45,46]: +Ei +my = 1 − β +my +i +(1 − βi) +(7) +where βi = (my − 1)/my; the effective number satisfies the following properties that Ei +my = 1 +if βi = 0 (my = 1); Ei +my → my as βi → 1 (my → ∞) such that βi controls how fast Ei +my grows +as the target predicate sample size my increases. +To estimate the ith predicate effective number Ei +my, we adaptively estimate the βi ∈ [0, 1) +by using the entropy Hi +skew and skew score function Si +skew of the ˆRskew as shown in Alg. 1: if +Si +skew > Sth then the βi ∈ [0, 1) that assigns more weights to the minority predicate samples +than the majority ones; otherwise, Si +skew ≤ Sth then βi = 0 that re-weights uniformly over the +entire class sample loss, i.e., conventional cross-entropy loss. The threshold Sth is determined +as follows: +Sth = ¯Sskew − δ +(8) +where ¯Sskew and δ = 0.7 are the mean value of Sskew and the hyper-parameter used respectively +in all experiments. +3.2. Skew Measures +In training, the predicate sample skewness depends on the predicate label. In other words, +according to [43], some majority predicate skew values tend to be greater than zero while +others tend to be zero or negative. This is an unfair skew measure, leading the SCR Alg. 1 to +more weights either in the majority predicates or in the minority ones. In this paper, to measure +the skew value of all target labels fairly, we use the target skew logit instead of the mean value. + +7 of 16 +0 +10 +20 +30 +40 +50 +Predicates +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +Softmax(Rskew) +Entorpy, Hskew = 0.984400 +Entorpy, Hskew = 0.996800 +Entorpy, Hskew = 0.998700 +Entorpy, Hskew = 0.999300 +Entorpy, Hskew = 0.999600 +Entorpy, Hskew = 0.999700 +Entorpy, Hskew = 0.999800 +Entorpy, Hskew = 0.999800 +Entorpy, Hskew = 0.999900 +(a) The Predicate Biased Logits, so f tmax( ˆRskew). +0 +10 +20 +30 +40 +50 +Predicates +1.000 +1.001 +1.002 +1.003 +1.004 +Weight, W +Hskew = 0.984400 +Hskew = 0.996800 +Hskew = 0.998700 +Hskew = 0.999300 +Hskew = 0.999600 +Hskew = 0.999700 +Hskew = 0.999800 +Hskew = 0.999800 +Hskew = 0.999900 +(b) The Predicate Weights, W = 1/Emy. +0.96 +0.97 +0.98 +0.99 +1.00 +Entropy, Hskew +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Skew, Sskew +Target Label, ry = 0 +Target Label, ry = 6 +Target Label, ry = 12 +Target Label, ry = 18 +Target Label, ry = 24 +Target Label, ry = 30 +Target Label, ry = 36 +Target Label, ry = 42 +Target Label, ry = 48 +(c) The Predicate Target Label-Wise Skew, Sskew +Figure 3. The Biased Predicate Weights: (a) an assumption that the predicate biased logits are represented +by its frequency M1/5 and that the degree of biased predictions is also modeled by its entropy Hskew, (b) +the discrete sample weights are assigned to the minority predicates according to the entropy Hskew and (c) +the skew in Eq. 9 measures not only the degree of true/false prediction but also biased predictions; the +true prediction has the negative skew Sskew < 0 while others do not. + +8 of 16 +Algorithm 1 Skew Class-balanced Effective Number +Require: Dataset D, SGG Model fθ +1: for t = 0, 1, 2, . . . , T do +2: +B ← Minibatch(D) +3: +ˆRskew ← Skew_Logits( ˆR f req, ˆRemb; B, fθ) +(Eq. 5) +4: +M ← Sample_Estimates( ˆRskew) +(Eq. 6) +5: +Sskew ← Skew( ˆRskew, R) +(Eq. 9) +6: +Hskew ← Entropy( ˆRskew) +(Eq. 10) +7: +if Sskew > Sth then +8: +β = 1.0 − Hskew +9: +else +10: +β = 0.0 +11: +end if +12: +Ei +my = 1−β +my +i +1−βi ; my ∈ M, i ∈ [0, N (N − 1)) +(Eq. 7) +13: end for +Therefore, the ith predicate sample skew Si +skew is firstly measured by the following equation +given the target label index y as follows: +Si +skew = +1 +|Crel| ∑ˆri,j∈ ˆRi +skew(ˆri,j − ˆri,y)3 +� +1 +|Crel| ∑ˆri,j∈ ˆRi +skew(ˆri,j − ˆri,y)2 +�3/2 +(9) +Then, we use the uniformness to determine the β. The uniformness provides confidence in +the predicate sample estimates. To calculate the uniformness of the ith predicate sample, we +estimate the entropy Hi +skew as follows: +Hi +skew = −λskew +∑ +ˆri,j∈ ˆRi +skew +p(ˆri,j) log|Crel| p(ˆri,j) +(10) +where the number of predicates |Crel| is used as the base of the logarithm, which Hi +skew ∈ [0, 1]; +the ˆri ∈ ˆRskew have uniform distributions when Hi +skew is close to 1; otherwise, the predicate +sample may have either skew distributions or correct distributions and the λskew = 0.06 is the +coefficient of Sskew. The following section depicts the relationship between skew and entropy +in detail. +3.3. Target Sample Weights +The interpretation of the relationship between skew and weight is depicted based on +biased predicate prediction as shown in Fig. 3. In Fig. 3a, to understand the sample weight +estimate from the predicate sample estimates, we assume that the predicate biased predictions +ˆRskew are given by the predicate sample frequency M1/5 (see Fig. 1a) and the predicate entropy +Hskew, i.e., the more frequent sample is the more biased prediction is; the more biased the +prediction is the less entropy is. The skew Sskew measures not only the degree of true/false +prediction but also biased predictions as shown in Fig. 3c. The following simple equation +summarizes the degree of true/false predictions: +� +Si +skew > 0 +if arg maxj ˆri,j ̸= ry +Si +skew < 0 +otherwise. +(11) + +9 of 16 +on +has +wearing +of +in +near +with +holding +behind +above +sitting on +wears +riding +under +in front of +standing on +at +carrying +attached to +walking on +for +over +laying on +looking at +hanging from +belonging to +parked on +eating +using +and +covering +part of +covered in +between +along +lying on +watching +on back of +to +walking in +mounted on +against +across +from +growing on +painted on +made of +playing +says +flying in +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +sg_transform+scr +sg_transform+cogtree +Figure 4. +The Recall@100 on PredCls: we compare the Skew Class-balanced Re-weight (SCR) of +FREQ+EMB with the Re-weighting method (CogTree), based on the SG-Transformer [18]. +where Si +skew ≈ −1 for the true prediction; Si +skew = 0 for the uniform prediction and Si +skew ≫ 0 +for the false and biased prediction. Moreover, the predicate target label-wise skew has more +discrete at high entropy. The final ith sample weight wi +y = 1/Ei +my is acquired by the criteria of +skew and entropy as shown in Fig. 3b. The minority predicates have larger weights than the +majority when Si +skew ≫ 0. +3.4. Learning with SCR +Except for object loss, all traditional SGG models are learned by the skew class-balanced +re-weighting cross-entropy loss functions. The conventional cross-entropy loss for the objects +is computed, given object predictions ˆO and the object ground-truth oi +y ∈ O as follows: +Lobj( ˆO, O) = +N +∑ +i +−γobj · wi +y log +� +exp(ˆoy) +∑ˆoj∈ ˆOi exp(ˆoj) +� +(12) +where wi +y = 1 s.t. |Cobj| = ∑ +|Cobj| +j +wi +j; γobj = 1/ ∑N +i wi +y for the mean cross-entropy loss. The +skew class-balanced cross-entropy loss computes the skew predicate-balanced cross-entropy, +based on the predicate predictions ˆR and the predicate ground-truth ri +y ∈ R: +Lrel( ˆR, R) = +N (N −1) +∑ +i +−γrel · wi +y log +� +exp(ˆry) +∑ˆrj∈ ˆRi exp(ˆrj) +� +(13) +where wi +y = +1 +Eimy = +1−βi +1−β +my +i +s.t. |Crel| = ∑|Crel| +j +wi +j; we set to γrel = 1/ ∑N (N −1) +i +wi +y for mean loss. +In summary, the total objective loss function for unbiased SGG learning can be formulated as +follows: +Ltotal = Lobj + Lrel +(14) +4. Experiments +The proposed SCR is evaluated with the traditional SGG models on the Visual Genome +benchmark datasets [26], and the performances of the SCR are compared with others in the +multiple SGG tasks. +4.1. Visual Genome +We used Visual Genome (VG) [26] dataset to train and evaluate our models, which is +composed of 108k images across 75k object categories and 37k predicate categories. We followed + +10 of 16 +the widely adopted VG split [1,12,44] containing the most frequent 150 object categories and 50 +predicate categories. The original split only has a training set (70%) and a test set (30%). We +followed [44] to sample a 5k validation set from the training set for parameter tuning. +4.2. Open Images +The Open Images dataset [27] is a large-scale dataset proposed by Google recently. Com- +pared with the Visual Genome dataset, it has a superior annotation quality for the scene graph +generation. In this work, we conduct experiments on Open Images V4&V6, following similar +data processing and evaluation protocols in [17,27,47]. The Open Images V4 is introduced as a +benchmark for scene graph generation by [47] and [17], which has 53, 953 and 3, 234 images +for the train and validation sets, 57 objects categories, and 9 predicate categories in total. The +Open Images V6 has 126, 368 images used for training, 1813, and 5322 images for validation +and testing, respectively, with 301 object categories and 31 predicate categories. This dataset +has a comparable amount of semantics categories with the VG. +4.3. Experiments Configurations +State-of-the-Art Comparisons. For fair comparisons, all the compared SGG models should use +the FREQ [44], which looks up the empirical distribution over relationships between subject +prediction ˆoi and object ones ˆoj. To evaluate the effectiveness of the SCR learning algorithm, +we follow the same experimental settings as the CogTree [18]. We also set the current state-of- +the-art SGG models as the baseline: MOTIFS[44], VCTree [11] and SG-Transformer [18] which +contains 3 O2O blocks and 2 R2O blocks with 12 attention heads, and Bipartite-Graph [23] +without resampling layers and compare the performance with the state-of-the-art debiasing +approach TDE [13], CogTree [18], PCPL [20], DLFE [21], BPL-SA [22], PPDL [24], and NICE +[25]. +Implementation. Following the previous works [13,18,23], the object detector is the pre-trained +Faster R-CNN [49] with ResNeXt-101-FPN [50]. In bi-level resampling[23], we also set the +repeat factor t = 0.07, instances drop rate γd = 0.7, and weight of fusion the entities features +ρ = −5. The α, β are initialized as 2.2 and 0.025, respectively. +4.4. Evaluations +Our SCR has the following two evaluations: +Relationship Retrieval (RR) contains three sub-task: (1) Predicate Classification (PredCls): +taking ground truth bounding boxes and labels as inputs, (2) Scene Graph Classification +(SGCls): using ground truth bounding boxes without labels, (3) Scene Graph Detection (SGDet): +detecting SGs from scratch. The conventional metric of RR is Recall@K (R@K), included in +this paper even though the biased prediction is reported by [51] for the performance of the +SCR. Moreover, to evaluate the general performances, we adopted mean Recall@ K (mR@K) +that retrieves each predicate separately and then averages R@K for all predicates. +Zero-Shot Relationship Retrieval (ZSRR). The Zero-Shot Recall@K was firstly evaluated on +the VG dataset in [13], which reports the R@K of those subject-predicate-object triplets that +have never been observed in the training set. ZSRR also has three sub-tasks as RR. +4.5. Quantitative Results +Visual Genome. The SCR is compared with others on the two evaluation tasks: RR and ZSRR, +which are the same as shown in Table. 1 and 3. The SCR achieves the best and second best +performances over the previous methods: TDE, PCPL, Cogtree, DLFE, BPL-SA PPDL, and +NICE, demonstrating its generality and effectiveness on the two measures of RR task. Moreover, +the SCR shows the best trade-off performances on the ZSRR task as shown in Fig. 3. + +11 of 16 +Table 1. The SGG performances of Relationship Retrieval on mean Recall@K and Recall@K. SCR† denotes +SCR of FREQ+EMB. Note the best and second best methods under each setting are marked according to +format. +PredCls +SGCls +SGDet +Model +mR@20/50/100 +R@20/50/100 +mR@20/50/100 +R@20/50/100 +mR@20/50/100 +R@20/50/100 +IMP+ [1] +-.-/ 9.8/10.5 +52.7/59.3/61.3 +-.-/ 5.8/ 6.0 +31.7/34.6/35.4 +-.-/ 3.8/ 4.8 +14.6/20.7/24.5 +FREQ [44] +8.3/13.0/16.0 +53.6/60.6/62.2 +5.1/ 7.2/ 8.5 +29.3/32.3/32.9 +4.5/ 6.1/ 7.1 +20.1/26.2/30.1 +KERN [16] +-.-/17.7/19.2 +-.-/65.8/67.6 +-.-/ 9.4/10.0 +-.-/36.7/37.4 +-.-/ 6.4/ 7.3 +-.-/27.1/29.8 +MOTIFS [44] +10.8/14.0/15.3 +58.5/65.2/67.1 +6.3/ 7.7/ 8.2 +32.9/35.8/36.5 +4.2/ 5.7/ 6.6 +21.4/27.2/30.3 +VCTree [11] +14.0/17.9/19.4 +60.1/66.4/68.1 +8.2/10.1/10.8 +35.2/38.1/38.8 +5.2/ 6.9/ 8.0 +22.0/27.9/31.3 +MSDN [48] +-.-/15.9/17.5 +-.-/64.6/66.6 +-.-/ 9.3/ 9.7 +-.-/38.4/39.8 +-.-/ 6.1/ 7.2 +-.-/31.9/36.6 +G-RCNN [7] +-.-/16.4/17.2 +-.-/64.8/66.7 +-.-/ 9.0/ 9.5 +-.-/38.5/37.0 +-.-/ 5.8/ 6.6 +-.-/29.7/32.8 +BGNN [23] +-.-/30.4/32.9 +-.-/59.2/61.3 +-.-/14.3/16.5 +-.-/37.4/38.5 +-.-/10.7/12.6 +-.-/31.0/35.8 +DT2-ACBS [14] +-.-/35.9/39.7 +-.-/23.3/25.6 +-.-/24.8/27.5 +-.-/16.2/17.6 +-.-/22.0/24.4 +-.-/15.0/16.3 +MOTIFS [44] +11.5/14.6/15.8 +59.5/66.0/67.9 +6.5/ 8.0/ 8.5 +35.8/39.1/39.9 +4.1/ 5.5/ 6.8 +25.1/32.1/36.9 ++ TDE [13] +18.5/25.5/29.1 +33.6/46.2/51.4 +9.8/13.1/14.9 +21.7/27.7/29.9 +5.8/ 8.2/ 9.8 +12.4/16.9/20.3 ++ PCPL [20] +-.-/24.3/26.1 +-.-/54.7/56.5 +-.-/12.0/12.7 +-.-/35.3/36.1 +-.-/10.7/12.6 +-.-/27.8/31.7 ++ CogTree [18] +20.9/26.4/29.0 +31.1/35.6/36.8 +12.1/14.9/16.1 +19.4/21.6/22.2 +7.9/10.4/11.8 +15.7/20.0/22.1 ++ DLFE [21] +22.1/26.9/28.8 +-.-/52.5/54.2 +12.8/15.2/15.9 +-.-/32.3/33.1 +8.6/11.7/13.8 +-.-/25.4/29.4 ++ BPL-SA [22] +24.8/29.7/31.7 +-.-/50.7/52.5 +14.0/16.5/17.5 +-.-/30.1/31.0 +10.7/13.5/15.6 +-.-/23.0/26.9 ++ PPDL [24] +-.-/32.2/33.3 +-.-/47.2/47.6 +-.-/17.5/18.2 +-.-/28.4/29.3 +-.-/11.4/13.5 +-.-/21.2/23.9 ++ NICE [25] +-.-/29.9/32.3 +-.-/55.1/57.2 +-.-/16.6/17.9 +-.-/33.1/34.0 +-.-/12.2/14.4 +-.-/27.8/31.8 ++ SCR† (ours) +25.9/31.5/33.6 +51.0/57.9/60.1 +14.2/17.1/18.2 +27.1/31.0/32.3 +9.6/13.5/15.9 +18.1/25.1/29.5 +VCTree [11] +11.7/14.9/16.1 +59.8/66.2/68.1 +6.2/ 7.5/ 7.9 +37.0/40.5/41.4 +4.2/ 5.7/ 6.9 +24.7/31.5/36.2 ++ TDE [13] +18.4/25.4/28.7 +36.2/47.2/51.6 +8.9/12.2/14.0 +19.9/25.4/27.9 +6.9/ 9.3/11.1 +14.0/19.4/23.2 ++ PCPL [20] +-.-/22.8/24.5 +-.-/56.9/58.7 +-.-/15.2/16.1 +-.-/40.6/41.7 +-.-/10.8/12.6 +-.-/26.6/30.3 ++ CogTree [18] +22.0/27.6/29.7 +39.0/44.0/45.4 +15.4/18.8/19.9 +27.8/30.9/31.7 +7.8/10.4/12.1 +14.0/18.2/20.4 ++ DLFE [21] +20.8/25.3/27.1 +-.-/51.8/53.5 +15.8/18.9/20.0 +-.-/33.5/34.6 +8.6/11.8/13.8 +-.-/22.7/26.3 ++ BPL-SA [22] +26.2/30.6/32.6 +-.-/50.0/51.8 +17.2/20.1/21.2 +-.-/34.0/35.0 +10.6/13.5/15.7 +-.-/21.7/25.5 ++ PPDL [24] +-.-/33.3/33.8 +-.-/47.6/48.0 +-.-/14.3/15.7 +-.-/32.1/33.0 +-.-/11.3/13.3 +-.-/20.1/22.9 ++ NICE [25] +-.-/30.7/33.0 +-.-/55.0/56.9 +-.-/19.9/21.3 +-.-/37.8/39.0 +-.-/11.9/14.1 +-.-/27.0/30.8 ++ SCR† (ours) +27.7/33.5/35.5 +49.7/56.4/58.3 +15.4/18.9/20.1 +26.7/30.6/31.9 +10.3/13.8/16.3 +18.1/25.0/29.4 +SG-Transformer [18] +14.8/19.2/20.5 +58.5/65.0/66.7 +8.9/11.6/12.6 +35.6/38.9/39.8 +5.6/ 7.7/ 9.0 +24.0/30.3/33.3 ++ CogTree[18] +22.9/28.4/31.0 +34.1/38.4/39.7 +13.0/15.7/16.7 +20.8/22.9/23.4 +7.9/11.1/12.7 +15.1/19.5/21.7 ++ SCR† (ours) +27.0/32.2/34.5 +45.3/52.7/55.0 +14.9/17.7/18.7 +25.1/28.9/30.2 +10.4 /13.4/15.0 +17.7/23.2/26.2 +Open Image V4 & V6. To show the effectiveness of SCR, we set BGNN as the baseline, as +shown in Table.2. On Open Images Dataset V4, SCR outperformed BGNN except for scorewtd +measurement. Mainly, SCR shows outstanding performance in terms of phrase evaluation. +Moreover, on Open Images Dataset V6, SCR outperformed all baselines such as BGNN, GPS- +Net, Etc., showing a good trade-off between mean recall@50 and recall@50. These results +proved that we could have a good trade-off performance in long-tailed predicated distributions +if properly assigning weights in training. +4.6. Ablation Study +We investigate the predicate-biased prediction and the best hyper-parameter settings of +the SCR loss function for the better generalized SGG models. +Predicate Bias. To estimate the predicate bias and assign the proper sample weights, we define +the predicate sample estimates ˆRskew of ˆR f req and ˆRemb in Eq. 5. To investigate the proper pred- +icate sample estimates, we examined the effectiveness of the predicate sample estimates-SCR +of EMB, SCR of FREQ, and SCR of FREQ+EMB with the fixed predicate predictions (Eq. 3) +as shown in Table. 4. In the experiments, the SCR of FREQ+EMB leads to more generalized + +12 of 16 +Table 2. The Performances of Open Images Dataset. ∗ denote results reproduced by Li et al. [23]. SCR† +denotes SCR of FREQ+EMB. +Dataset +Models +mR@50 +R@50 +wmAP +scorewtd +rel +phr +V4 +RelDN [47]∗ +70.40 +75.66 +36.13 +39.91 +45.21 +GPS-Net [17]∗ +69.50 +74.65 +35.02 +39.40 +44.70 +BGNN [23] +72.11 +75.46 +37.76 +41.70 +46.87 +BGNN+SCR† (ours) +72.20 +75.48 +38.64 +45.01 +45.01 +V6 +RelDN [47]∗ +33.98 +73.08 +32.16 +33.39 +40.84 +VCTree [11]∗ +33.91 +74.08 +34.16 +33.11 +40.21 +MOTIFS [44]∗ +32.68 +71.63 +29.91 +31.59 +38.93 +TDE [13]∗ +35.47 +69.30 +30.74 +32.80 +39.27 +GPS-Net [17]∗ +35.26 +74.81 +32.85 +33.98 +41.69 +BGNN [23] +40.45 +74.98 +33.51 +34.15 +42.06 +BGNN+SCR† (ours) +42.43 +75.21 +33.98 +35.13 +42.66 +Table 3. The SGG Performances of Zero-shot Relationship Retrieval on Recall@K. SCR† denotes SCR of +FREQ+EMB. The SGG models re-implemented under our codebase are denoted by the superscript∗. +Zero-Shot Relationship Retrieval +PredCls +SGCls +SGDet +Model +Method +R@50 +R@100 +R@50 +R@100 +R@50 +R@100 +MOTIFS [13] +baseline [13] +Reweight [13] +TDE [13] +CogTree [18]∗ +SCR† (ours) +10.9 +0.7 +14.4 +2.4 +18.0 +14.5 +0.9 +18.2 +4.0 +21.1 +2.2 +0.1 +3.4 +0.9 +5.1 +3.0 +0.1 +4.5 +1.5 +5.9 +0.1 +0.0 +2.3 +0.3 +2.4 +0.2 +0.0 +2.9 +0.6 +3.8 +VCTree [13] +Baseline [13] +TDE [13] +CogTree [18]∗ +SCR† (ours) +10.8 +14.3 +3.3 +17.6 +14.3 +17.6 +5.0 +20.4 +1.9 +3.2 +2.1 +4.5 +2.6 +4.0 +2.6 +5.2 +0.2 +2.6 +0.4 +2.5 +0.7 +3.2 +0.6 +3.5 +SG-Transformer [18]∗ +Baseline∗ +CogTree [18]∗ +SCR† (ours) +4.1 +5.2 +16.4 +6.3 +7.3 +19.6 +1.6 +2.3 +4.6 +2.3 +3.0 +5.3 +0.2 +0.3 +2.0 +0.5 +0.5 +3.2 +BGNN [23] +BGNN∗ +CogTree [18]∗ +SCR† (ours) +15.0 +13.4 +16.3 +18.0 +16.1 +19.5 +4.5 +5.0 +4.9 +5.4 +5.7 +5.9 +4.5 +0.5 +1.9 +5.3 +0.8 +3.0 +performances over others. In summary, the previous Re-Weight methods worsen the recall +performances, while the SCR does not. +Hyper-parameters. The hyper-parameters of SCR control the weight of the SGG loss. We +investigate the best hyper-parameter settings as shown in Table. 5. The best hyper-parameter +settings δ = 0.7 and λskew = 0.06 show the generalized performances of the SGG tasks. The +smaller λskew is, the higher mean Recall scores are, while the higher λskew is, the higher Recall +scores are, i.e., the λskew controls the trade-off between the majority predicates and the minority +ones since the smaller Hskew tend to assign more weights to the minority predicates in the SCR. +The δ = 0.7 shows the best proportion of Re-weighting the predicate samples. +4.7. Qualitative Examples +To demonstrate the effectiveness of the sample-wise SCR Re-weighting, we show the +comparison of Recall @100 on PredCls of all predicates based on the SG-Transformer [18] +as shown in Fig. 4. The SCR of FREQ+EMB achieves a significant performance gain on the +overall predicate categories. Moreover, the skew predicate ˆRskew serves to estimate not only +the predicate candidates that a subject-object pair can have but also the target skew Sskew which + +13 of 16 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +target skew, Si +skew +sample estimates, Rskew +backgrounds +on +has +wearing +of +in +near +with +holding +behind +above +sitting on +wears +riding +under +in front of +standing on +at +carrying +attached to +walking on +for +over +laying on +looking at +hanging from +belonging to +parked on +eating +using +and +covering +part of +covered in +between +along +lying on +watching +on back of +to +walking in +mounted on +against +across +from +growing on +painted on +made of +playing +says +flying in +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +predicates of woman-coat pair +predictions, Ri +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +target skew, Si +skew +sample estimates, Rskew +backgrounds +on +has +wearing +of +in +near +with +holding +behind +above +sitting on +wears +riding +under +in front of +standing on +at +carrying +attached to +walking on +for +over +laying on +looking at +hanging from +belonging to +parked on +eating +using +and +covering +part of +covered in +between +along +lying on +watching +on back of +to +walking in +mounted on +against +across +from +growing on +painted on +made of +playing +says +flying in +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +predicates of hill-snow pair +predictions, Ri +Figure 5. The Predicate Predictions and Sample Estimates: when one woman - coat pair (upper) have +possible predicates such as backgrounds, has, wearing, in, etc., we show predicate predictions ˆRi with its +target skew Si +skew deduced by its sample estimates ˆRi +skew. The larger ˆRi +skew ∈ ˆRi +skew tends to be a smaller +Si +skew, when compared to another hill - snow pair (lower). +determines the sample weight. Figure 5 includes a subject-object pair which have possible +predicates, its predicate prediction ˆRi, and the target skew Si +skew deduced by the predicate +sample estimate ˆRi +skew. The higher the sample estimate is, the lower the target skew is. +5. Conclusions +In this paper, the unbiased Scene Graph Generation (SGG) algorithm, referred to as Skew +Class-balanced Re-weighting (SCR), was proposed for considering the unbiased predicate +prediction caused by the long-tailed distribution. The prior works focus mainly on alleviating +the deteriorating performances of the minority predicate predictions, showing drastic dropping +recall scores, i.e., forgetting the majority predicate class. It has not yet properly analyzed the +trade-off performances between majority and minority predicates in the given SGG datasets. +In this paper, to address the issues leveraged by the skewness of biased predicate predictions, +firstly, the SCR estimated the predicate re-weighting coefficient and then re-weighted more +to the biased predicates for the better trading-off performances between the majority and the +minority predicates. Extensive experiments conducted on the standard Visual Genome dataset +and Open Image V4 & V6 showed the SCR’s effectiveness and generality with the traditional +SGG models. + +14 of 16 +Table 4. The Ablation Study for the Predicate Sample Estimates on Recall@100. The underbar represents +the predicate sample estimates for better generality. +Relationship Retrieval +PredCls +SGCls +SGDet +Model +Method +mRR +ZSRR +RR +mRR +ZSRR +RR +mRR +ZSRR +RR +VCTree +SCR of FREQ +SCR of EMB +SCR of FREQ+EMB +31.6 +33.3 +33.7 +22.2 +21.1 +20.1 +60.0 +60.1 +60.0 +16.9 +17.9 +18.0 +6.2 +5.9 +6.1 +35.4 +35.2 +33.3 +13.3 +15.4 +14.4 +3.5 +3.5 +3.5 +31.6 +30.0 +31.9 +Table 5. The Ablation Study for the SCR Hyper-Parameters on Recall@100. The underbar represents the +best trade-off performances between mRecall and Recall. SCR† denotes SCR of FREQ+EMB. +Relationship Retrieval +PredCls +SGCls +SGDet +Model +λskew +δ +mRR +RR +mRR +RR +mRR +RR +VCTree ++SCR† +0.03 +0.06 +0.08 +0.7 +0.7 +0.7 +35.5 +33.7 +32.8 +58.3 +60.0 +60.4 +20.1 +18.0 +17.2 +31.9 +33.3 +34.2 +16.3 +14.4 +13.5 +29.4 +31.9 +32.6 +0.06 +0.06 +0.06 +0.6 +0.7 +0.8 +33.5 +33.7 +33.3 +60.6 +60.0 +59.3 +17.8 +18.0 +17.6 +34.3 +33.3 +33.0 +14.2 +14.4 +14.0 +32.0 +31.9 +30.6 +References +1. +Xu, D.; Zhu, Y.; Choy, C.B.; Fei-Fei, L. Scene graph generation by iterative message passing. In Proceedings of the Proceedings of the +IEEE Conference on Computer Vision and Pattern Recognition, 2017, pp. 5410–5419. +2. +Dai, B.; Zhang, Y.; Lin, D. Detecting visual relationships with deep relational networks. In Proceedings of the Proceedings of the IEEE +Conference on Computer Vision and Pattern Recognition, 2017, pp. 3076–3086. +3. +Li, Y.; Ouyang, W.; Zhou, B.; Wang, K.; Wang, X. Scene graph generation from objects, phrases and region captions. 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Seeing through the human reporting bias: Visual classifiers from noisy +human-centric labels. In Proceedings of the Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, +pp. 2930–2939. + diff --git a/KdAyT4oBgHgl3EQff_ia/content/tmp_files/load_file.txt b/KdAyT4oBgHgl3EQff_ia/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..25b0b8b600f4fb336e8dae8097deee7abbd8f5a8 --- /dev/null +++ b/KdAyT4oBgHgl3EQff_ia/content/tmp_files/load_file.txt @@ -0,0 +1,2054 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf,len=2053 +page_content='Article Skew Class-balanced Re-weighting for Unbiased Scene Graph Generation Haeyong Kang and Chang D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Yoo* School of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' haeyong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='kang@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='kr Correspondence: cd_yoo@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='kr † Current address: Daejeon 34141, Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Abstract: An unbiased scene graph generation (SGG) algorithm referred to as Skew Class-balanced Re-weighting (SCR) is proposed for considering the unbiased predicate prediction caused by the long- tailed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The prior works focus mainly on alleviating the deteriorating performances of the minority predicate predictions, showing drastic dropping recall scores, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=', losing the majority predicate performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' It has not yet correctly analyzed the trade-off between majority and minority predicate performances in the limited SGG datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In this paper, to alleviate the issue, the Skew Class-balanced Re- weighting (SCR) loss function is considered for the unbiased SGG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Leveraged by the skewness of biased predicate predictions, the SCR estimates the target predicate weight coefficient and then re-weights more to the biased predicates for better trading-off between the majority predicates and the minority ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Extensive experiments conducted on the standard Visual Genome dataset and Open Image V4 & V6 show the performances and generality of the SCR with the traditional SGG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The source code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='com/ihaeyong/Unbiased-SGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Keywords: Scene Graph Generation (SGG), Skew Class-balanced Re-weighting (SCR), Predicate Sample Estimates, Skew Class-balanced Effective Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Introduction Scene Graph Generation (SGG) is receiving increased attention for improving image understanding [1–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The core building blocks of SGG are the objects in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' There can be diverse relationships among the objects [16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' All relationships can be represented as triplets subject, relation, object, which can be used for generating the scene graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The precise interpretation of an image depends on the core relations between the subject-object pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' For example, given an image of dog, woman, kite objects, the image can be interpreted in multiple ways such as ⟨dog, has, leg⟩, ⟨dog, near, woman⟩, and ⟨woman, holding, kite⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In general, however, the real-world large data sets often have shown long-tailed pred- icate distributions as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1a - the predicate proportion of the Visual Genome[1], containing 51 predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' One cause of long-tail distribution is the background predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The background predicates account for more than 80% in total predicates since it is not trivial to annotate all possible relationships, leading to possibly far more subject-object pairs without annotating ground truth than ones with ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Another reason is the visual event frequency, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The majority predicates more frequently occur than the minority ones given a visual subject-object pair in the real-world scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' For example, given a man-board pair, the predicate on is more frequently observed than holding and riding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' This long-tailed label distribution causes the trained model to tend to be biased toward the majority predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The SGG models [13,18–22] focus on increasing the minority predicate performances for the unbiased scene graph generation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=', increasing mean recall scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' How- ever, these methods lead to deteriorating performances of the majority predicates, drastically dropping recall scores, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=', losing the majority predicate performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='log(freq) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='the predicates of man-board pair ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='(b) The Predicate Distribution of the Man - Board Pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The Long-tailed Predicate Distribution: (a) The predicate proportion of the Visual Genome[1] which is possibly far more predicates of subject - object pairs without annotating ground-truth than others and (b) The majority predicates are more frequently observed than the minority ones given a subject - object pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Given a man - board pair, the possible predicates- backgrounds, on, holding, riding, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' represents a long-tail distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' been alleviated through resampling images and object instances [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' However, this method needs more computing power and time to train the model on the resampled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The current state-of-the-art models focus on re-balancing biased loss [24] or correcting noisy labels [25] to acquire an unbiased SGG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Nevertheless, the prior works have not yet adequately described and analyzed the trade-off performances between majority and minority predicates for learning SGG models based on the given imbalanced datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In this paper, the Skew Class-balanced Re-weighting (SCR) loss function is considered for alleviating the issues and acquiring the best trade-off performances in the multiple SGG tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Leveraged by the skewness of predicate predictions, the SCR estimates its weight coefficients and then reweights more to biased predicate samples to adaptively be unbiased SGG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The extensive experimental results show that the SCR loss function gives more generalized performances than priors in the multiple SGG tasks on the Visual Genome dataset[26] and the Open Images dataset [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Contributions of our SCR learning scheme to unbiased SGG models: Leveraged by the skewness of biased predicate predictions, the Skew Class-balanced Re- weighting (SCR) loss function is firstly proposed for the unbiased scene graph generation (SGG) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 3 of 16 The SCR is applied to the current state-of-the-art SGG models to show its effectiveness, leading to more generalized performances: the SCR outperforms the prior reweighted methods on both mean recall and recall measurements in the multiple SGG tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The Related Work section provides discussions on unbiased scene graph generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The unbiased SGG section presents scene graph generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In the Skew Class-balanced Re-weighting (SCR) section, the SCR loss function is depicted in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In the experimental section, the results of scene graph generation on the Visual Genome benchmark dataset are examined, along with an analysis and ablation study on the SCR with the current state-of-the-art SGG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Finally, we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Related Works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Unbiased Scene Graph Generation (SGG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Predicate distribution is much more long-tailed than object distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' For N objects and R predicates, the model has to address the fundamen- tal challenge of learning O(N2R) relations with few [28,28,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' To overcome the limited training dataset, the linguistic external knowledge [8,28,30] was used by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' [31], regularizing the deep neural network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' using linguistic knowledge, the probabilistic model has also alleviated the semantical ambiguity of visual relationships [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Furthermore, to alleviate the imbalanced relationship distribution, Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' [8] reformulated the conventional one-hot classification as a n-hot multiclass hierarchical recognition via a novel Intra-Hierarchical trees (IH-trees) for each label set in the triplet ⟨subject, predicate, object⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Recently, unbiased SGG [13,17–22,24,25,33–42] has drawn unprecedented interest for more generalized SGG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Occurrence-based Node Priority Sensitive (NPS)-loss [17] was used for balancing predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' the Total Direct Effect (TDE) method has proposed firstly for unbiased learning by [13], which directly separates the bias from biased predictions through the counterfactual methodologies on causal graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' CogTree [18] addressed the debiasing issue based on the coarse-to-fine structure of the rela- tionships from the cognition viewpoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' [19] improved the context modeling for tail categories by designing the bipartite graph network and message propagation on resampled objects and images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Lastly, the Predicate Probability Distribution based Loss (PPDL) [24] has proposed to train the biased SGG models, which measure the semantic predicate representation to re-balance the biased training loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In this work, the Skew Class-balanced Re-weighting (SCR) loss function is proposed for alleviating biased predicate predictions, leading to the most generalized SGG models through the novel adaptive re-weighting learning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Re-Weighting based Unbiased SGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Overall unbiased SGG models can be categorized into re- balancing strategy of re-weighting [17,18,20,24] and re-sampling [19] and biased model-based strategy [13,21,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' For unbiased SGG modes, Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' [13] first investigated the re-weighting learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' However, they observed that the performances of the majority predicates were drastically dropped, resulting in low recall scores while with high mean recall scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' This shows the general tendency that there is a trade-off performance between majority predicates and minority ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' To alleviate the trade-off issue, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' [18] proposed CogTree based on the coarse-to-fine structure of the relationships from the cognition viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Recently, the Predicate Probability Distribution (PPD) [24] re-balances the biased training loss according to the similarity between the predicted probability distribution and the estimated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' However, it has not yet correctly analyzed the trade-off performances between the majority and minority classes in various SGG tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In this paper, we measure the sample skew score based on the sample estimates for bias toward the majority classes to assign the sample weight correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The sample skewness is computed as the Fisher-Pearson coefficient of skewness on its sample mean value [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' However, since the mean value tends to be biased toward the majority predicates, we measure the sample skew score fairly on its target logit instead of its mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Based on the sample skew score, the SCR assigns the sample weights adaptively - if there is no 4 of 16 man horse cat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' man horse cat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' (a) Object Model Input & Label Object Predictions Predicate Predictions & Sample Estimates Label: < man, riding, horse > Sample Weight Estimates (b) FREQ Model (e) Sample Estimates (c) Predicate Model (d) Predicate Predictions (f) Skew Measures (g) Target Sample Weight riding near on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' riding near on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' riding near on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' riding near on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' riding near on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' riding near on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' riding near on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' riding near on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' riding near on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' riding near on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' riding near on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' riding near on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Skew Class-balanced Re-weighting (SCR) For Unbiased SGG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The traditional SGG model and (d) predicate prediction ( ˆR): (a) object detector outputs its predictions: man and hourse, (b) FREQ embeds prior predicates and (c) predicate model outputs predicate predictions ( ˆRvis);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Given man - horse label predictions, our SCR estimates (e) a sample size of the possible predicate candidates through FREQ (R freq or Remb ) and measures (f) the target label skew score and then calculates (g) the training target sample weight for the adaptive re-weighted loss, and in (f) and (g), red lines around the target label indicate predicate skew (Si skew) and re-weight scores (Wi), respectively, where they have an approximately inverse relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' bias, we assign fewer weights to the sample (majority).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' If it is biased to one side, we assign larger weights to the samples (minority).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Such that the SGG models with SCR show superior performances and generality on the multiple SGG tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Unbiased Scene Graph Generation In this section, we discuss the general scene graph generation model of the object and predicate predictions and depict the predicate sample estimates for measuring its skewness of biased predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Scene Graph Generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Given an image I, a scene graph model generates a graph G = (V, E), where V and E are the sets of nodes and edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Each node oi ∈ V is represented by a bounding box vbbox ∈ R4 and a corresponding class label oy ∈ Cobj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Each edge ri,j ∈ E represents the predicate between the subject node vi and the object node vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The corresponding predicate label is ry ∈ Crel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Object Predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Following [13], the node features are derived from the object detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In particular, for each bounding box vbbox i , the detector returns an RoI-Align feature xRoI i and an object label embedding li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In general SGG model, the N number of node features are constructed by vector concatenation X = {[xRoI i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' li;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' bi]}N i=1, where bi is the embedded box feature from the box coordinate vbbox i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' ˜X = f cobj(X) ∈ RN ×Dobj (1) where all f c∗ denote a fully connected layer for linear transformations or logits, and the object feature dimension Dobj depends on the SGG model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The predicted object label of ˆO ∈ RN ×|Cobj| is estimated by object logits f c( ˜X) as follows: ˆO = f c( ˜X) (2) man horse O5 of 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Predicate Predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Predicate predictions can be made by employing multiple logits from visual and non- visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' We follow the sum over all outputs to generate the final predicate prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The combined predicate logit ˆR is estimated based on the summation of the visual logits and the non-visual logits as follows: ˆR = ˆRvis ⊕ ˆR f req ⊕ ˆRemb (3) where ˆR ∈ RN (N −1)×|Crel|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' ⊕ is an element-wise sum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' ˆRvis is the predicate logits from visual feature Fvis such as a Dvis dimensional union feature and a subject-object pair feature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' which also depends on the SGG model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' ˆRvis = f cvis(Fvis),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Fvis ∈ RN (N −1)×Dvis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' ˆR f req = Sigmoid(R f req) ∈ RN (N −1)×|Crel|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' ˆRemb = f cemb(Lemb),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Lemb ∈ RN (N −1)×400 (4) The FREQ [44] as a non-visual feature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' R f req looks up the empirical distribution over rela- tionships between subject ˆoi and object ˆoj as computed in the training set where ˆoi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' ˆoj ∈ ˆO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' However, since FREQ does not consider any image representations when predicting predicates, it tends to lead to biased predicate predictions due to its imbalanced predicate distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' To minimize the biased effects, we use the Sigmoid-activated FREQ predicate logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In addition, for acquiring the more smoothness of the empirical distribution, the concatenated subject-object embedding ˆRemb is added to the predicate predictions where Lemb = {[li;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' lj]}N i,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Sample Estimates Non-visual predicate features tend to be more biased than visual features due to an imbalanced training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' If the degree of biased predictions can be measured, we can leverage its value to learn the SGG models without bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' According to [43], the degree of bias prediction can generally be measured in a skew score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In general, if there is no bias, the skew score is close to 0, and if it is biased to one side, the skew score is over either −1 or +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In SCR, we need to estimate how many predicate samples are biased to measure the predicate skew scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' To approximate the biased sample numbers, we use two non-visual prior predicate distributions of FREQ, ˆR f req and subject-object label embedding ˆRemb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Based on the two predicate distributions, at first, we define the skew logit ˆRskew with the Sigmoid activation function as follows: ˆRskew = Sigmoid(R f req ⊕ ˆRemb) (5) To estimate the predicate sample weights properly, the SCR approximates skewness through the predicate sample estimates ˆRskew in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' We investigate the best predicate sample estimates through several experiments with a combination of non-visual predictions as follows: SCR of EMB: ˆRskew = Sigmoid( ˆRemb) SCR of FREQ: ˆRskew = Sigmoid(R f req) SCR of FREQ+EMB: ˆRskew = Sigmoid(R f req ⊕ ˆRemb) 6 of 16 Then, the predicate sample estimates are acquired as follows: M = N (N −1) ∑ i ˆri (6) where ˆri ∈ ˆRskew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' the estimated my ∈ M is the number of yth predicate sample size and my ∈ [0, N (N − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Moreover, the skew predicate ˆRskew serves to estimate the predicate candidates that a subject-object pair can have fairly since the Sigmoid activation function suppresses the more extensive biased predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' For example, a Man-Horse pair may have predicates such as riding, on, with, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The Sigmoid activation function amplifies the frequency of the minority predicates while squeezing that of the majority ones, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 (b) and (e), which is used to calculate Skew Class-Balanced Effective Number that we depict in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Skew Class-balanced Re-weighting (SCR) In this section, leveraged by the biased predictions deduced by the predicate sample estimates, the Skew Class-balanced Re-weighting (SCR) performs sample weight estimates for learning unbiased SGG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Skew Class-balanced Effective Number The Skew Class-balanced Effective Number approximates the mini-batch class-balanced re- weighting coefficients Emy based on the predicate skew logit ˆRskew in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The ith predicate sample Ei my is defined as follows as [45,46]: Ei my = 1 − β my i (1 − βi) (7) where βi = (my − 1)/my;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' the effective number satisfies the following properties that Ei my = 1 if βi = 0 (my = 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Ei my → my as βi → 1 (my → ∞) such that βi controls how fast Ei my grows as the target predicate sample size my increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' To estimate the ith predicate effective number Ei my, we adaptively estimate the βi ∈ [0, 1) by using the entropy Hi skew and skew score function Si skew of the ˆRskew as shown in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 1: if Si skew > Sth then the βi ∈ [0, 1) that assigns more weights to the minority predicate samples than the majority ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' otherwise, Si skew ≤ Sth then βi = 0 that re-weights uniformly over the entire class sample loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=', conventional cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The threshold Sth is determined as follows: Sth = ¯Sskew − δ (8) where ¯Sskew and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='7 are the mean value of Sskew and the hyper-parameter used respectively in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Skew Measures In training, the predicate sample skewness depends on the predicate label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In other words, according to [43], some majority predicate skew values tend to be greater than zero while others tend to be zero or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' This is an unfair skew measure, leading the SCR Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 1 to more weights either in the majority predicates or in the minority ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In this paper, to measure the skew value of all target labels fairly, we use the target skew logit instead of the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 7 of 16 0 10 20 30 40 50 Predicates 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='07 Softmax(Rskew) Entorpy, Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='984400 Entorpy, Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='996800 Entorpy, Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='998700 Entorpy, Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='999300 Entorpy, Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='999600 Entorpy, Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='999700 Entorpy, Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='999800 Entorpy, Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='999800 Entorpy, Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='999900 (a) The Predicate Biased Logits, so f tmax( ˆRskew).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 0 10 20 30 40 50 Predicates 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='002 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='003 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='004 Weight, W Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='984400 Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='996800 Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='998700 Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='999300 Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='999600 Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='999700 Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='999800 Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='999800 Hskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='999900 (b) The Predicate Weights, W = 1/Emy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='00 Entropy, Hskew 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 Skew, Sskew Target Label, ry = 0 Target Label, ry = 6 Target Label, ry = 12 Target Label, ry = 18 Target Label, ry = 24 Target Label, ry = 30 Target Label, ry = 36 Target Label, ry = 42 Target Label, ry = 48 (c) The Predicate Target Label-Wise Skew, Sskew Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The Biased Predicate Weights: (a) an assumption that the predicate biased logits are represented by its frequency M1/5 and that the degree of biased predictions is also modeled by its entropy Hskew, (b) the discrete sample weights are assigned to the minority predicates according to the entropy Hskew and (c) the skew in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 9 measures not only the degree of true/false prediction but also biased predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' the true prediction has the negative skew Sskew < 0 while others do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 8 of 16 Algorithm 1 Skew Class-balanced Effective Number Require: Dataset D, SGG Model fθ 1: for t = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' , T do 2: B ← Minibatch(D) 3: ˆRskew ← Skew_Logits( ˆR f req, ˆRemb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' B, fθ) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 5) 4: M ← Sample_Estimates( ˆRskew) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 6) 5: Sskew ← Skew( ˆRskew, R) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 9) 6: Hskew ← Entropy( ˆRskew) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 10) 7: if Sskew > Sth then 8: β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 − Hskew 9: else 10: β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 11: end if 12: Ei my = 1−β my i 1−βi ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' my ∈ M, i ∈ [0, N (N − 1)) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 7) 13: end for Therefore, the ith predicate sample skew Si skew is firstly measured by the following equation given the target label index y as follows: Si skew = 1 |Crel| ∑ˆri,j∈ ˆRi skew(ˆri,j − ˆri,y)3 � 1 |Crel| ∑ˆri,j∈ ˆRi skew(ˆri,j − ˆri,y)2 �3/2 (9) Then, we use the uniformness to determine the β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The uniformness provides confidence in the predicate sample estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' To calculate the uniformness of the ith predicate sample, we estimate the entropy Hi skew as follows: Hi skew = −λskew ∑ ˆri,j∈ ˆRi skew p(ˆri,j) log|Crel| p(ˆri,j) (10) where the number of predicates |Crel| is used as the base of the logarithm, which Hi skew ∈ [0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' the ˆri ∈ ˆRskew have uniform distributions when Hi skew is close to 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' otherwise, the predicate sample may have either skew distributions or correct distributions and the λskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='06 is the coefficient of Sskew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The following section depicts the relationship between skew and entropy in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Target Sample Weights The interpretation of the relationship between skew and weight is depicted based on biased predicate prediction as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 3a, to understand the sample weight estimate from the predicate sample estimates, we assume that the predicate biased predictions ˆRskew are given by the predicate sample frequency M1/5 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 1a) and the predicate entropy Hskew, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=', the more frequent sample is the more biased prediction is;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' the more biased the prediction is the less entropy is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The skew Sskew measures not only the degree of true/false prediction but also biased predictions as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The following simple equation summarizes the degree of true/false predictions: � Si skew > 0 if arg maxj ˆri,j ̸= ry Si skew < 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' (11) 9 of 16 on has wearing of in near with holding behind above sitting on wears riding under in front of standing on at carrying attached to walking on for over laying on looking at hanging from belonging to parked on eating using and covering part of covered in between along lying on watching on back of to walking in mounted on against across from growing on painted on made of playing says flying in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 sg_transform+scr sg_transform+cogtree Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The Recall@100 on PredCls: we compare the Skew Class-balanced Re-weight (SCR) of FREQ+EMB with the Re-weighting method (CogTree), based on the SG-Transformer [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' where Si skew ≈ −1 for the true prediction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Si skew = 0 for the uniform prediction and Si skew ≫ 0 for the false and biased prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Moreover, the predicate target label-wise skew has more discrete at high entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The final ith sample weight wi y = 1/Ei my is acquired by the criteria of skew and entropy as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The minority predicates have larger weights than the majority when Si skew ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Learning with SCR Except for object loss, all traditional SGG models are learned by the skew class-balanced re-weighting cross-entropy loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The conventional cross-entropy loss for the objects is computed, given object predictions ˆO and the object ground-truth oi y ∈ O as follows: Lobj( ˆO, O) = N ∑ i −γobj · wi y log � exp(ˆoy) ∑ˆoj∈ ˆOi exp(ˆoj) � (12) where wi y = 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' |Cobj| = ∑ |Cobj| j wi j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' γobj = 1/ ∑N i wi y for the mean cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The skew class-balanced cross-entropy loss computes the skew predicate-balanced cross-entropy, based on the predicate predictions ˆR and the predicate ground-truth ri y ∈ R: Lrel( ˆR, R) = N (N −1) ∑ i −γrel · wi y log � exp(ˆry) ∑ˆrj∈ ˆRi exp(ˆrj) � (13) where wi y = 1 Eimy = 1−βi 1−β my i s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' |Crel| = ∑|Crel| j wi j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' we set to γrel = 1/ ∑N (N −1) i wi y for mean loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In summary, the total objective loss function for unbiased SGG learning can be formulated as follows: Ltotal = Lobj + Lrel (14) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Experiments The proposed SCR is evaluated with the traditional SGG models on the Visual Genome benchmark datasets [26], and the performances of the SCR are compared with others in the multiple SGG tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Visual Genome We used Visual Genome (VG) [26] dataset to train and evaluate our models, which is composed of 108k images across 75k object categories and 37k predicate categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' We followed 10 of 16 the widely adopted VG split [1,12,44] containing the most frequent 150 object categories and 50 predicate categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The original split only has a training set (70%) and a test set (30%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' We followed [44] to sample a 5k validation set from the training set for parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Open Images The Open Images dataset [27] is a large-scale dataset proposed by Google recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Com- pared with the Visual Genome dataset, it has a superior annotation quality for the scene graph generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In this work, we conduct experiments on Open Images V4&V6, following similar data processing and evaluation protocols in [17,27,47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The Open Images V4 is introduced as a benchmark for scene graph generation by [47] and [17], which has 53, 953 and 3, 234 images for the train and validation sets, 57 objects categories, and 9 predicate categories in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The Open Images V6 has 126, 368 images used for training, 1813, and 5322 images for validation and testing, respectively, with 301 object categories and 31 predicate categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' This dataset has a comparable amount of semantics categories with the VG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Experiments Configurations State-of-the-Art Comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' For fair comparisons, all the compared SGG models should use the FREQ [44], which looks up the empirical distribution over relationships between subject prediction ˆoi and object ones ˆoj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' To evaluate the effectiveness of the SCR learning algorithm, we follow the same experimental settings as the CogTree [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' We also set the current state-of- the-art SGG models as the baseline: MOTIFS[44], VCTree [11] and SG-Transformer [18] which contains 3 O2O blocks and 2 R2O blocks with 12 attention heads, and Bipartite-Graph [23] without resampling layers and compare the performance with the state-of-the-art debiasing approach TDE [13], CogTree [18], PCPL [20], DLFE [21], BPL-SA [22], PPDL [24], and NICE [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Following the previous works [13,18,23], the object detector is the pre-trained Faster R-CNN [49] with ResNeXt-101-FPN [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In bi-level resampling[23], we also set the repeat factor t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='07, instances drop rate γd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='7, and weight of fusion the entities features ρ = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The α, β are initialized as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='025, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Evaluations Our SCR has the following two evaluations: Relationship Retrieval (RR) contains three sub-task: (1) Predicate Classification (PredCls): taking ground truth bounding boxes and labels as inputs, (2) Scene Graph Classification (SGCls): using ground truth bounding boxes without labels, (3) Scene Graph Detection (SGDet): detecting SGs from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The conventional metric of RR is Recall@K (R@K), included in this paper even though the biased prediction is reported by [51] for the performance of the SCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Moreover, to evaluate the general performances, we adopted mean Recall@ K (mR@K) that retrieves each predicate separately and then averages R@K for all predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Zero-Shot Relationship Retrieval (ZSRR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The Zero-Shot Recall@K was firstly evaluated on the VG dataset in [13], which reports the R@K of those subject-predicate-object triplets that have never been observed in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' ZSRR also has three sub-tasks as RR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Quantitative Results Visual Genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The SCR is compared with others on the two evaluation tasks: RR and ZSRR, which are the same as shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The SCR achieves the best and second best performances over the previous methods: TDE, PCPL, Cogtree, DLFE, BPL-SA PPDL, and NICE, demonstrating its generality and effectiveness on the two measures of RR task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Moreover, the SCR shows the best trade-off performances on the ZSRR task as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 11 of 16 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The SGG performances of Relationship Retrieval on mean Recall@K and Recall@K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' SCR† denotes SCR of FREQ+EMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Note the best and second best methods under each setting are marked according to format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' PredCls SGCls SGDet Model mR@20/50/100 R@20/50/100 mR@20/50/100 R@20/50/100 mR@20/50/100 R@20/50/100 IMP+ [1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='-/ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='8/10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 /13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4/15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='7/23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2/26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 Open Image V4 & V6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' To show the effectiveness of SCR, we set BGNN as the baseline, as shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' On Open Images Dataset V4, SCR outperformed BGNN except for scorewtd measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Mainly, SCR shows outstanding performance in terms of phrase evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Moreover, on Open Images Dataset V6, SCR outperformed all baselines such as BGNN, GPS- Net, Etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=', showing a good trade-off between mean recall@50 and recall@50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' These results proved that we could have a good trade-off performance in long-tailed predicated distributions if properly assigning weights in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Ablation Study We investigate the predicate-biased prediction and the best hyper-parameter settings of the SCR loss function for the better generalized SGG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Predicate Bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' To estimate the predicate bias and assign the proper sample weights, we define the predicate sample estimates ˆRskew of ˆR f req and ˆRemb in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' To investigate the proper pred- icate sample estimates, we examined the effectiveness of the predicate sample estimates-SCR of EMB, SCR of FREQ, and SCR of FREQ+EMB with the fixed predicate predictions (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 3) as shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In the experiments, the SCR of FREQ+EMB leads to more generalized 12 of 16 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The Performances of Open Images Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' ∗ denote results reproduced by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' SCR† denotes SCR of FREQ+EMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Dataset Models mR@50 R@50 wmAP scorewtd rel phr V4 RelDN [47]∗ 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='40 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='66 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='13 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='91 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='21 GPS-Net [17]∗ 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='50 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='65 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='02 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='40 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='70 BGNN [23] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='11 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='46 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='76 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='70 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='87 BGNN+SCR† (ours) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='20 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='48 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='64 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='01 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='01 V6 RelDN [47]∗ 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='98 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='08 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='16 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='39 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='84 VCTree [11]∗ 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='91 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='08 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='16 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='11 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='21 MOTIFS [44]∗ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='68 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='63 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='91 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='59 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='93 TDE [13]∗ 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='47 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='30 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='74 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='80 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='27 GPS-Net [17]∗ 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='26 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='81 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='85 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='98 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='69 BGNN [23] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='45 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='98 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='51 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='15 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='06 BGNN+SCR† (ours) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='43 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='21 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='98 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='13 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='66 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The SGG Performances of Zero-shot Relationship Retrieval on Recall@K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' SCR† denotes SCR of FREQ+EMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The SGG models re-implemented under our codebase are denoted by the superscript∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Zero-Shot Relationship Retrieval PredCls SGCls SGDet Model Method R@50 R@100 R@50 R@100 R@50 R@100 MOTIFS [13] baseline [13] Reweight [13] TDE [13] CogTree [18]∗ SCR† (ours) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='8 VCTree [13] Baseline [13] TDE [13] CogTree [18]∗ SCR† (ours) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 SG-Transformer [18]∗ Baseline∗ CogTree [18]∗ SCR† (ours) 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 BGNN [23] BGNN∗ CogTree [18]∗ SCR† (ours) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='3 18.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 performances over others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In summary, the previous Re-Weight methods worsen the recall performances, while the SCR does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The hyper-parameters of SCR control the weight of the SGG loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' We investigate the best hyper-parameter settings as shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The best hyper-parameter settings δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='7 and λskew = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='06 show the generalized performances of the SGG tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The smaller λskew is, the higher mean Recall scores are, while the higher λskew is, the higher Recall scores are, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=', the λskew controls the trade-off between the majority predicates and the minority ones since the smaller Hskew tend to assign more weights to the minority predicates in the SCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='7 shows the best proportion of Re-weighting the predicate samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Qualitative Examples To demonstrate the effectiveness of the sample-wise SCR Re-weighting, we show the comparison of Recall @100 on PredCls of all predicates based on the SG-Transformer [18] as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The SCR of FREQ+EMB achieves a significant performance gain on the overall predicate categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Moreover, the skew predicate ˆRskew serves to estimate not only the predicate candidates that a subject-object pair can have but also the target skew Sskew which 13 of 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 target skew,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Si skew sample estimates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Rskew backgrounds on has wearing of in near with holding behind above sitting on wears riding under in front of standing on at carrying attached to walking on for over laying on looking at hanging from belonging to parked on eating using and covering part of covered in between along lying on watching on back of to walking in mounted on against across from growing on painted on made of playing says flying in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 predicates of woman-coat pair predictions, Ri 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 target skew,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Si skew sample estimates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Rskew backgrounds on has wearing of in near with holding behind above sitting on wears riding under in front of standing on at carrying attached to walking on for over laying on looking at hanging from belonging to parked on eating using and covering part of covered in between along lying on watching on back of to walking in mounted on against across from growing on painted on made of playing says flying in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 predicates of hill-snow pair predictions, Ri Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The Predicate Predictions and Sample Estimates: when one woman - coat pair (upper) have possible predicates such as backgrounds, has, wearing, in, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=', we show predicate predictions ˆRi with its target skew Si skew deduced by its sample estimates ˆRi skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The larger ˆRi skew ∈ ˆRi skew tends to be a smaller Si skew, when compared to another hill - snow pair (lower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' determines the sample weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Figure 5 includes a subject-object pair which have possible predicates, its predicate prediction ˆRi, and the target skew Si skew deduced by the predicate sample estimate ˆRi skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The higher the sample estimate is, the lower the target skew is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Conclusions In this paper, the unbiased Scene Graph Generation (SGG) algorithm, referred to as Skew Class-balanced Re-weighting (SCR), was proposed for considering the unbiased predicate prediction caused by the long-tailed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The prior works focus mainly on alleviating the deteriorating performances of the minority predicate predictions, showing drastic dropping recall scores, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=', forgetting the majority predicate class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' It has not yet properly analyzed the trade-off performances between majority and minority predicates in the given SGG datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' In this paper, to address the issues leveraged by the skewness of biased predicate predictions, firstly, the SCR estimated the predicate re-weighting coefficient and then re-weighted more to the biased predicates for the better trading-off performances between the majority and the minority predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Extensive experiments conducted on the standard Visual Genome dataset and Open Image V4 & V6 showed the SCR’s effectiveness and generality with the traditional SGG models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' 14 of 16 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The Ablation Study for the Predicate Sample Estimates on Recall@100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The underbar represents the predicate sample estimates for better generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Relationship Retrieval PredCls SGCls SGDet Model Method mRR ZSRR RR mRR ZSRR RR mRR ZSRR RR VCTree SCR of FREQ SCR of EMB SCR of FREQ+EMB 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='9 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='5 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='9 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The Ablation Study for the SCR Hyper-Parameters on Recall@100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' The underbar represents the best trade-off performances between mRecall and Recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' SCR† denotes SCR of FREQ+EMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Relationship Retrieval PredCls SGCls SGDet Model λskew δ mRR RR mRR RR mRR RR VCTree +SCR† 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='4 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='9 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content='6 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} +page_content=' Xu, D.' metadata={'source': 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+page_content=' 2930–2939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdAyT4oBgHgl3EQff_ia/content/2301.00351v1.pdf'} diff --git a/KdFLT4oBgHgl3EQfLC8K/vector_store/index.pkl b/KdFLT4oBgHgl3EQfLC8K/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..399024bedf37a34413a73d8546a3c4cb36e6097f --- /dev/null +++ b/KdFLT4oBgHgl3EQfLC8K/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0306c089714c4e0291a7829cbf6870ca47b5bfaa4184dca46ee346af23021f7d +size 222806 diff --git a/M9E0T4oBgHgl3EQfTABv/content/tmp_files/2301.02230v1.pdf.txt b/M9E0T4oBgHgl3EQfTABv/content/tmp_files/2301.02230v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a52b5e2e8f3c69ecc263166efa0b6e854e5f1e9 --- /dev/null +++ b/M9E0T4oBgHgl3EQfTABv/content/tmp_files/2301.02230v1.pdf.txt @@ -0,0 +1,610 @@ +Quantum Computing for the Wess–Zumino Model +Christopher Culver∗ and David Schaich +Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom +E-mail: C.Culver@liverpool.ac.uk, David.Schaich@liverpool.ac.uk +Future quantum computers will enable novel sign-problem-free studies of dynamical phenomena +in non-perturbative quantum field theories, including real-time evolution and spontaneous super- +symmetry breaking. We are investigating applications of quantum computing to low-dimensional +supersymmetric lattice systems that can serve as testbeds for existing and near-future quantum +devices. Here we present initial results for the N = 1 Wess–Zumino model in 1+1 dimensions, +building on our prior analyses of 0+1-dimensional supersymmetric quantum mechanics. In addi- +tion to exploring supersymmetry breaking using the variational quantum eigensolver, we consider +the prospects for real-time evolution. +The 38th International Symposium on Lattice Field Theory, LATTICE2022 +8–13 August 2022 +Bonn, Germany +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.02230v1 [hep-lat] 5 Jan 2023 + +Quantum Computing for the Wess–Zumino Model +Christopher Culver +1. +Introduction +Supersymmetry is an extension of Poincaré symmetry that has many important applications +throughout theoretical physics. These include potential extensions of the standard model, insight +into fundamental properties of quantum field theory (QFT), and holographic dualities with theories +of quantum gravity. Spontaneous symmetry breaking is an important topic in each of these realms. +In particular, dynamical supersymmetry breaking is a requirement of any experimentally viable +supersymmetric model of new physics, since experiments have not yet discovered superpartners of +the known particles of the standard model. Supersymmetry breaking is also a feature of simpler +QFTs, which we consider in this proceedings. +To non-perturbatively analyze supersymmetric QFTs, we employ lattice regularization. While +Monte Carlo importance sampling studies of supersymmetric lattice QFTs have been performed for +many years (see Refs. [1–3] for recent reviews), sign problems can prevent this approach from con- +sidering key dynamical phenomena including real-time evolution and spontaneous supersymmetry +breaking [2, 3]. Quantum computing in principle provides a novel means to study these phenomena +without introducing sign problems. +Existing and near-future quantum devices feature modest numbers (tens to hundreds) of qubits +with relatively high error rates, widely described as Noisy Intermediate-Scale Quantum (NISQ) +technology [4]. Lattice field theory studies employing such NISQ hardware are limited to small +systems and shallow circuit depths, leaving the calculations within the reach of classical diagonal- +ization. Even in the absence of quantum advantage, studies of these small systems are important to +explore, test, verify and refine quantum algorithms as hardware capabilities continue to improve [5]. +Here we investigate the N = 1 Wess–Zumino model in 1+1 dimensions, building on our prior +analyses of 0+1-dimensional supersymmetric quantum mechanics [6]. This is arguably the simplest +supersymmetric quantum field theory, and has previously been the subject of lattice investigations +from a variety of approaches. In addition to lattice calculations employing the traditional Lagrangian +formulation [7, 8], other studies also consider the continuous-time Hamiltonian formulation [9–12], +the fermion loop formulation [13], and tensor network formulations [14, 15]. See Ref. [1] for a +brief review. +Our current focus is on dynamical supersymmetry breaking in the 1+1d Wess–Zumino model +for specific prepotentials to be discussed below. +We will use the variational quantum eigen- +solver (VQE) to explore this. In the next section we begin by briefly summarizing the model, then +in Section 3 we review the quantum computing techniques we will apply. Considering two different +prepotentials, in Section 4 we present our initial results on dynamical supersymmetry breaking, and +also comment on prospects for real-time evolution. +2. +Wess–Zumino Model +The 1+1-dimensional N = 1 Wess–Zumino model involves a two-component fermionic field +𝜓 and a bosonic field 𝜙. It can be considered essentially a supersymmetric extension of 𝜙4 theory. +Following Refs. [9, 11], we construct the lattice Hamiltonian 𝐻 = 𝑄2 on the basis of the discretized +2 + +Quantum Computing for the Wess–Zumino Model +Christopher Culver +supercharge +𝑄 = 1 +√𝑎 +𝑁 +∑︁ +𝑛=1 +� +𝑝𝑛𝜓1,𝑛 − +� 𝜙𝑛+1 − 𝜙𝑛−1 +2 ++ 𝑎𝑉(𝜙𝑛) +� +𝜓2,𝑛 +� +, +(1) +for 𝑁 spatial sites separated by lattice spacing ‘𝑎’ (time remains continuous). Here 𝑉(𝜙𝑛) is an +arbitrary real ‘prepotential’ that depends on the bosonic field, and 𝑝𝑛 is the momentum conjugate +to 𝜙𝑛. Squaring this supercharge, we find the Hamiltonian +𝐻 = +∑︁ +𝑛 +� +𝑝2 +𝑛 +2𝑎 + 𝑎 +2 +� 𝜙𝑛+1 − 𝜙𝑛−1 +2𝑎 +�2 ++ 𝑎 +2𝑉(𝜙𝑛)2 + 𝑎𝑉(𝜙𝑛) 𝜙𝑛+1 − 𝜙𝑛−1 +2𝑎 ++(−1)𝑛𝑉 ′(𝜙𝑛) +� +𝜒† +𝑛𝜒𝑛 − 1 +2 +� ++ 1 +2𝑎 +� +𝜒† +𝑛𝜒𝑛+1 + 𝜒† +𝑛+1𝜒𝑛 +�� +, +(2) +where we have replaced the two fermion components 𝜓1,𝑛 and 𝜓2,𝑛 with creation and annihilation +operators 𝜒† +𝑛 and 𝜒𝑛 defined by +𝜓1,𝑛 = 1 − 𝑖(−1)𝑛 +2𝑖𝑛 +� +𝜒† +𝑛 + 𝑖𝜒𝑛 +� +𝜓2,𝑛 = 1 + 𝑖(−1)𝑛 +2𝑖𝑛 +� +𝜒† +𝑛 − 𝑖𝜒𝑛 +� +. +(3) +As mentioned in Section 1, we are interested in dynamical supersymmetry breaking in the +1+1d Wess–Zumino model, which depends on the prepotential 𝑉(𝜙𝑛). Considering polynomial +prepotentials of degree 𝑞, tree-level analyses suggest that supersymmetry should remain preserved +when 𝑞 is odd, but may break spontaneously for even 𝑞 [11]. In Sections 4.1 and 4.2 we will +consider 𝑞 = 1 and 𝑞 = 2, respectively, finding agreement with these expectations. +Of course there have been prior investigations of dynamical supersymmetry breaking both +for the 1+1d Wess–Zumino model [7–13] as well as for more complicated 1+1d systems including +super-Yang–Mills [16], super-QCD [17] and the supersymmetric Gross–Neveu–Yukawa model [18]. +A persistent challenge in such studies is the severe sign problem associated with spontaneous +supersymmetry breaking [2, 3], which has motivated the development of fermion loop [13], tensor +network [14] and conformal truncation [18] techniques. +We avoid sign problems by using the VQE to estimate the ground state energy, for now consid- +ering sufficiently small systems that the results can be checked through classical diagonalization. +Since 𝐻 = 𝑄2, the ground state energy 𝐸0 = ⟨Ω |𝐻| Ω⟩ = |𝑄 |Ω⟩|2 vanishes if and only if the ground +state is supersymmetric, 𝑄 |Ω⟩ = 0. Otherwise, if supersymmetry is spontaneously broken, the +ground state energy is strictly positive. While the classical computational costs of diagonalization +grow very rapidly as the system size increases, the VQE offers hope of efficient determination of +ground state energies using future quantum devices, as we now discuss. +3. +Quantum Computing +To perform computations on quantum devices, we map the bosonic and fermionic degrees of +freedom to qubit degrees of freedom. Qubits are physically realized as two state systems. This +allows for a straightforward mapping for the fermions via the Jordan–Wigner transformation, +𝜒† +𝑛 = 1 +2 (𝑋𝑛 − 𝑖𝑌𝑛) +𝜒𝑛 = 1 +2 (𝑋𝑛 + 𝑖𝑌𝑛) , +(4) +3 + +Quantum Computing for the Wess–Zumino Model +Christopher Culver +where 𝑋𝑛 and 𝑌𝑛 represent a Pauli gate acting on the 𝑛-th qubit. +The bosonic degrees of freedom have an infinite-dimensional Hilbert space at each lattice site +and need to be regulated. To do this we consider them in the harmonic oscillator basis and impose +a hard cutoff on the number Λ of allowed modes at each site. It is worth noting that this explicitly +breaks supersymmetry, which will only be exactly restored in the Λ → ∞ limit. The number of +qubits needed to define each 𝜙𝑛 truncated in this way is 𝑛𝑞 ≡ ⌈log2 Λ⌉. The raising and lowering +operators become +ˆ𝑎𝑛 = +Λ−2 +∑︁ +𝑙=0 +√ +𝑙 + 1 |𝑙⟩ ⟨𝑙 + 1| , +ˆ𝑎† +𝑛 = +Λ−2 +∑︁ +𝑙=0 +√ +𝑙 + 1 |𝑙 + 1⟩ ⟨𝑙| . +(5) +The introduction of this cutoff makes the bosonic Hilbert space finite and we can now perform the +mapping to qubits. +We follow the same steps as in Ref. [6], where more details can be found. Writing the state 𝑗 +in binary as 𝑗 = �𝑛𝑞−1 +𝑖=0 +𝑏𝑖2𝑖, we associate each digit with a qubit. Specific matrix elements can be +converted to their action on qubits using the relations +|0⟩ ⟨1| = 1 +2 (𝑋 + 𝑖𝑌) , +|1⟩ ⟨0| = 1 +2 (𝑋 − 𝑖𝑌) , +(6) +|0⟩ ⟨0| = 1 +2 (1 + 𝑍) , +|1⟩ ⟨1| = 1 +2 (1 − 𝑍) +(7) +and writing the full matrix element as a tensor product over all of the binary digits: +|𝑛⟩ ⟨𝑛′| = ⊗𝑛𝑞−1 +𝑖=0 +|𝑏𝑖⟩ +� +𝑏′ +𝑖 +�� . +(8) +This completes the mapping of all the degrees of freedom into qubits, which enables the application +of quantum algorithms of interest. +An important NISQ-era algorithm for determining whether or not supersymmetry is dynam- +ically broken is the VQE algorithm [19]. This algorithm outputs an upper bound on the lowest +eigenvalue of any matrix and by using the Hamiltonian as the target matrix we can investigate +whether or not the ground state energy is zero. Specifically we want to test whether or not, in +the Λ → ∞ limit, the VQE estimate of the energy tends towards zero or to a finite value. To run +the VQE algorithm, we first prepare some trial wavefunction for the ground state 𝜓 with tunable +parameters 𝜃𝑖. The energy of this trial state is computed with a quantum circuit and fed into a +classical optimization algorithm that adjusts the parameters 𝜃𝑖 in search of the minimum energy. +This hybrid classical–quantum algorithm will converge to some 𝐸var which is an upper bound for +the ground state energy, +𝐸0 ≤ 𝐸var = ⟨𝜓(𝜃𝑖) |𝐻| 𝜓(𝜃𝑖)⟩ +⟨𝜓(𝜃𝑖) |𝜓(𝜃𝑖)⟩ . +(9) +Another important algorithm, more relevant for the long-term prospects of quantum computing, +is the Suzuki–Trotter decomposition of the time-evolution operator, 𝑒𝑖𝐻𝑡. This gives us direct access +to the real-time evolution of a quantum state acting under the dynamics of our Hamiltonian. The +continuous time 𝑡 is broken down into 𝑁𝑡 steps of size 𝛿 = 𝑡/𝑁𝑡, +𝑒−𝑖𝐻𝑡 |𝜓⟩ = (exp [−𝑖𝐻𝛿])𝑁𝑡 |𝜓⟩ . +(10) +4 + +Quantum Computing for the Wess–Zumino Model +Christopher Culver +Noting that the Hamiltonian will be a sum of 𝑀 terms, a single Trotter step is +|𝜓(𝑡 + 𝛿)⟩ = exp +������ +−𝑖 +𝑀 +∑︁ +𝑗=1 +𝐻 𝑗𝛿 +������ +|𝜓(𝑡)⟩ . +For most physical applications 𝑀 > 1 and the Baker–Campbell–Hausdorff formula is used to +convert the above line into a product of exponentials of each of the 𝐻 𝑗. Unfortunately all of the +𝐻 𝑗 terms of the Hamiltonian do not commute in general, thus there will be a tradeoff introduced +between the number of gate operations to perform a single step, and the accuracy to which the final +state is obtained. This error in each step will then be compounded by the time discretization 𝛿. Of +course taking a larger number of smaller steps will reduce this error, but this comes at the cost of +more qubit operations which is not ideal in the NISQ era. We use the default implementation for +transpiling the Trotter circuit in Qiskit [20], based on the Hamiltonian in Eq. 2. See Ref. [21] for +further discussion of how Trotter decompositions can be optimized for quantum computing. +4. +Results +We now present results for the ground state energy of the Wess–Zumino model and the +entangling gate count for a single Trotter step of real-time evolution. The ground state energy +𝐸0 is important since supersymmetry is preserved if and only if 𝐸0 = 0. We consider a linear +prepotential 𝑉(𝜙) in Section 4.1 and a family of quadratic prepotentials in Section 4.2, in each case +using 𝑁 = 2–3 spatial lattice sites and several small values of the bosonic cutoff Λ ≤ 16. While we +can analyze any positive cutoff, in practice Λ should be a power of 2 to optimize the correspondence +between the bosonic degrees of freedom and the qubit degrees of freedom. Otherwise our approach +would leave part of the quantum computer’s Hilbert space unused. For each calculation we find +the ground state energy via classical diagonalization, and see how well this can be reproduced by a +modest number of VQE runs (minimum 100). +4.1 Linear prepotential +The linear prepotential is just +𝑉(𝜙𝑛) = 𝜙𝑛, +(11) +and is expected to preserve supersymmetry. This is the simplest nontrivial 𝜙-dependent prepotential, +and keeps the bosonic and fermionic fields from interacting with each other. +The ground state energy as a function of Λ from classical computations of the eigenvalues of +𝐻 on 2- and 3-site lattices is shown in Fig. 1. In the Λ → ∞ limit the ground state energy goes to +zero up to the precision of the solver, which confirms that supersymmetry is preserved as expected. +In Table 1a we compare the minimum energy obtained from 100 runs of the VQE against these +exact results from classical diagonalization. For the 2-site lattice with Λ ≤ 4 the VQE estimate of +the ground state energy behaves appropriately, and exponentially approaches zero. As an aside, the +negative energies shown in Table 1 for Λ = 2 reflect the significant supersymmetry breaking related +to such an extreme truncation. For larger values of the cutoff, and for the larger 3-site lattice, the +VQE struggles to converge to the correct order of magnitude. This provides us with immediate +5 + +Quantum Computing for the Wess–Zumino Model +Christopher Culver +2 +4 +6 +8 +10 +12 +14 +16 +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +100 +|E| +2 +4 +6 +8 +10 +12 +14 +16 +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +100 +|E| +Figure 1: Semi-log plots of the 1+1d Wess–Zumino ground state energy for 2-site (left) and 3-site (right) +lattices with the linear prepotential Eq. 11. As the bosonic cutoff Λ goes to infinity the ground state energy +goes to zero (up to numerical precision), consistent with the expected preservation of supersymmetry for this +prepotential. +Table 1: Ground state energies from classical diagonalization and the VQE for the linear and quadratic +prepotentials discussed in the text. The linear prepotential Eq. 11 is expected to preserve supersymmetry +and have a Λ → ∞ ground state energy of zero. Supersymmetry is expected to break dynamically for the +quadratic prepotential Eq. 12, so long as 𝑐 < 𝑐0 ≈ −0.5. +N Λ +Exact +VQE +2 2 +6.97e-03 +6.97e-03 +- +4 +3.22e-05 +6.61e-05 +- +8 +1.04e-09 +1.08e-01 +3 2 -9.97e-02 -1.28e+00 +- +4 +1.17e-04 +4.99e-01 +(a) Linear prepotential +N Λ +Exact +VQE +2 2 -4.87e-01 -9.11e-01 +- +4 +1.82e-01 +2.26e-01 +- +8 +1.31e-01 +7.49e-01 +3 2 -1.98e-01 -1.28e+00 +- +4 +3.02e-01 +5.08e-01 +(b) Quadratic prepotential, 𝑐 = −0.2 +N Λ +Exact +VQE +2 2 -4.87e-01 -9.11e-01 +- +4 +1.28e-01 -1.15e+00 +- +8 -1.74e-02 +6.89e-01 +3 2 -1.98e-01 -1.28e+00 +- +4 +2.47e-01 -1.10e+00 +(c) Quadratic prepotential, 𝑐 = −0.8 +targets we can use in work to improve the performance and reliability of VQE determinations of +the ground state energy. +In Table 2a we show the entangling CX gate counts for a single Trotter step of real-time +evolution for the Wess–Zumino model. As the bosonic cutoff and the number of sites increase, our +time-evolution circuits quickly exceed the capabilities of NISQ hardware. This motivates ongoing +work to optimize time-evolution schemes for the Wess–Zumino model, in order to minimize resource +requirements. +4.2 Quadratic prepotential +To introduce interactions between the bosons and fermions we consider the family of quadratic +prepotentials +𝑉(𝜙𝑛) = 𝑐 + 𝜙2 +𝑛, +(12) +with free parameter 𝑐, which was studied in Refs. [9, 11, 12]. This prepotential is expected to lead +to dynamical supersymmetry breaking for positive 𝑐, while supersymmetry should be preserved for +6 + +Quantum Computing for the Wess–Zumino Model +Christopher Culver +0.8 +0.6 +0.4 +0.2 +c +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +|E| += 8 += 12 += 16 +0.8 +0.6 +0.4 +0.2 +c +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +|E| += 8 += 12 += 16 +Figure 2: The 1+1d Wess–Zumino ground state energy for 2-site (left) and 3-site (right) lattices with various +values of the parameter 𝑐 in the quadratic prepotential of Eq. 12. Small horizontal offsets distinguish different +values of the cutoff Λ = 8, 12 and 16. Since non-zero values of 𝐸 correspond to supersymmetry breaking, the +results are consistent with the expected preservation of supersymmetry for sufficiently negative 𝑐 < 𝑐0 < 0 +and Λ → ∞. +sufficiently negative 𝑐 less than a critical value 𝑐0 < 0, found to be 𝑐0 ≈ −0.5 in Refs. [11, 12]. +We can investigate these expectations by computing the ground state energy for a range of +𝑐. Results from classical computations of the ground state energy as a function of 𝑐 are shown +in Fig. 2 for several values of Λ on 2- and 3-site lattices. We only show results from Λ ≥ 8 +since smaller Λ ≤ 4 produce wildly fluctuating results that are clearly not meaningful. For both +2- and 3-site lattices the ground state energy converges towards zero as 𝑐 → −∞, confirming that +supersymmetry is preserved for sufficiently negative 𝑐. For 𝑐 ≈ 0 little dependence on the cutoff is +visible for Λ ≥ 12. The results become more sensitive to Λ as 𝑐 becomes more negative, illustrating +the challenges that computations in this regime will face on near-term quantum hardware. +This is also reflected in the results for the VQE estimate of the ground state energy presented +in Table 1 for both 𝑐 = −0.2 and −0.8. In the first case of 𝑐 = −0.2, 100 runs of the VQE suffice to +recover the correct order of magnitude of the exact ground-state energy for the 2- and 3-site lattices +with Λ ≤ 8 we have considered so far. The VQE analysis becomes significantly more challenging +for 𝑐 = −0.8. Since these values of Λ are too small to be included in Fig. 2, for the time being +we are focusing our VQE improvement efforts on the linear prepotential discussed in the previous +subsection. +Finally, in Table 2 we again provide entangling gate counts for a single Trotter step. The results +are qualitatively similar to those for the linear prepotential. For Λ > 2, the interactions between the +bosons and fermions increase CX gate requirements by roughly a factor of 3. The mild dependence +on the value of 𝑐 is related to the realization of 𝑐-dependent terms in the Hamiltonian in terms of +the default basis gate set. +5. +Conclusion +We have presented initial results from our ongoing work using quantum computing to study +the Wess–Zumino model in 1+1 dimensions. We use a Hamiltonian lattice regularization of the +7 + +Quantum Computing for the Wess–Zumino Model +Christopher Culver +Table 2: Entangling CX gate counts for a single Trotter step of the time-evolution operator for the linear and +quadratic prepotentials discussed in the text. For each prepotential, we consider 2- and 3-site lattices, and +transpile the circuit for a few values of the bosonic cutoff Λ. +𝑁 +Λ +CX Gates +2 +2 +8 +- +4 +252 +- +8 +2556 +3 +2 +18 +- +4 +5728 +(a) Linear prepotential +𝑁 +Λ +CX Gates +2 +2 +14 +- +4 +754 +- +8 +7822 +3 +2 +30 +- +4 +2788 +(b) Quadratic prepotential, 𝑐 = −0.2 +𝑁 +Λ +CX Gates +2 +2 +14 +- +4 +718 +- +8 +7858 +3 +2 +30 +- +4 +2730 +(c) Quadratic prepotential, 𝑐 = −0.8 +theory to analyze spontaneous supersymmetry breaking for prepotentials with linear or quadratic +dependence on the bosonic field. The preservation or breaking of supersymmetry is determined by +the ground state energy, which we analyze with the VQE algorithm for systems with 2 or 3 spatial +sites and small bosonic cutoffs Λ ≤ 16 — small enough to check our results through classical +diagonalization. +Despite the small size of these systems, our results are consistent with the expected preservation +of supersymmetry for the linear prepotential, and show clear spontaneous supersymmetry breaking +for a quadratic prepotential with 𝑐 = −0.2. However, our VQE algorithm struggles to converge +for large values of the bosonic cutoff, motivating ongoing work to improve the performance and +reliability of this approach to exploring spontaneous supersymmetry breaking. +We also consider prospects for studying the real-time dynamics of the Wess–Zumino model, +based on a straightforward transpilation of the time-evolution operator corresponding to the Hamil- +tonian Eq. 2. Even for 2-site lattices, the number of gate operations in the resulting time-evolution +circuits exceed the capabilities of NISQ hardware for any reasonable cutoff Λ > 2. Here as well we +hope to improve this situation by more carefully considering time-evolution schemes in search of +optimal use of quantum resources. +Acknowledgments: We thank Johann Ostmeyer for discussions of Trotter schemes. This work +was supported by UK Research and Innovation Future Leader Fellowship MR/S015418/1 and STFC +grant ST/T000988/1. +References +[1] D. Kadoh, Recent progress in lattice supersymmetry: from lattice gauge theory to black +holes, Proc. Sci. LATTICE2015 (2016) 017 [1607.01170]. +[2] G. Bergner and S. Catterall, Supersymmetry on the lattice, Int. J. Mod. Phys. A 31 (2016) +1643005 [1603.04478]. +[3] D. Schaich, Lattice studies of supersymmetric gauge theories, Eur. Phys. J. Spec. Top. (2022) +[2208.03580]. +8 + +Quantum Computing for the Wess–Zumino Model +Christopher Culver +[4] J. 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Ostmeyer, Optimised Trotter Decompositions for Classical and Quantum Computing, +2211.02691. +10 + diff --git a/M9E0T4oBgHgl3EQfTABv/content/tmp_files/load_file.txt b/M9E0T4oBgHgl3EQfTABv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b913e0cf04de28ba87b11eca33f4093dc442e397 --- /dev/null +++ b/M9E0T4oBgHgl3EQfTABv/content/tmp_files/load_file.txt @@ -0,0 +1,353 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf,len=352 +page_content='Quantum Computing for the Wess–Zumino Model Christopher Culver∗ and David Schaich Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom E-mail: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='Culver@liverpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='uk, David.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='Schaich@liverpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='uk Future quantum computers will enable novel sign-problem-free studies of dynamical phenomena in non-perturbative quantum field theories, including real-time evolution and spontaneous super- symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' We are investigating applications of quantum computing to low-dimensional supersymmetric lattice systems that can serve as testbeds for existing and near-future quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Here we present initial results for the N = 1 Wess–Zumino model in 1+1 dimensions, building on our prior analyses of 0+1-dimensional supersymmetric quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' In addi- tion to exploring supersymmetry breaking using the variational quantum eigensolver, we consider the prospects for real-time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The 38th International Symposium on Lattice Field Theory, LATTICE2022 8–13 August 2022 Bonn, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='02230v1 [hep-lat] 5 Jan 2023 Quantum Computing for the Wess–Zumino Model Christopher Culver 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Introduction Supersymmetry is an extension of Poincaré symmetry that has many important applications throughout theoretical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' These include potential extensions of the standard model, insight into fundamental properties of quantum field theory (QFT), and holographic dualities with theories of quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Spontaneous symmetry breaking is an important topic in each of these realms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' In particular, dynamical supersymmetry breaking is a requirement of any experimentally viable supersymmetric model of new physics, since experiments have not yet discovered superpartners of the known particles of the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Supersymmetry breaking is also a feature of simpler QFTs, which we consider in this proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' To non-perturbatively analyze supersymmetric QFTs, we employ lattice regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' While Monte Carlo importance sampling studies of supersymmetric lattice QFTs have been performed for many years (see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' [1–3] for recent reviews), sign problems can prevent this approach from con- sidering key dynamical phenomena including real-time evolution and spontaneous supersymmetry breaking [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Quantum computing in principle provides a novel means to study these phenomena without introducing sign problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Existing and near-future quantum devices feature modest numbers (tens to hundreds) of qubits with relatively high error rates, widely described as Noisy Intermediate-Scale Quantum (NISQ) technology [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Lattice field theory studies employing such NISQ hardware are limited to small systems and shallow circuit depths, leaving the calculations within the reach of classical diagonal- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Even in the absence of quantum advantage, studies of these small systems are important to explore, test, verify and refine quantum algorithms as hardware capabilities continue to improve [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Here we investigate the N = 1 Wess–Zumino model in 1+1 dimensions, building on our prior analyses of 0+1-dimensional supersymmetric quantum mechanics [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' This is arguably the simplest supersymmetric quantum field theory, and has previously been the subject of lattice investigations from a variety of approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' In addition to lattice calculations employing the traditional Lagrangian formulation [7, 8], other studies also consider the continuous-time Hamiltonian formulation [9–12], the fermion loop formulation [13], and tensor network formulations [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' [1] for a brief review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Our current focus is on dynamical supersymmetry breaking in the 1+1d Wess–Zumino model for specific prepotentials to be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' We will use the variational quantum eigen- solver (VQE) to explore this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' In the next section we begin by briefly summarizing the model, then in Section 3 we review the quantum computing techniques we will apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Considering two different prepotentials, in Section 4 we present our initial results on dynamical supersymmetry breaking, and also comment on prospects for real-time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Wess–Zumino Model The 1+1-dimensional N = 1 Wess–Zumino model involves a two-component fermionic field 𝜓 and a bosonic field 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' It can be considered essentially a supersymmetric extension of 𝜙4 theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' [9, 11], we construct the lattice Hamiltonian 𝐻 = 𝑄2 on the basis of the discretized 2 Quantum Computing for the Wess–Zumino Model Christopher Culver supercharge 𝑄 = 1 √𝑎 𝑁 ∑︁ 𝑛=1 � 𝑝𝑛𝜓1,𝑛 − � 𝜙𝑛+1 − 𝜙𝑛−1 2 + 𝑎𝑉(𝜙𝑛) � 𝜓2,𝑛 � , (1) for 𝑁 spatial sites separated by lattice spacing ‘𝑎’ (time remains continuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Here 𝑉(𝜙𝑛) is an arbitrary real ‘prepotential’ that depends on the bosonic field, and 𝑝𝑛 is the momentum conjugate to 𝜙𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Squaring this supercharge, we find the Hamiltonian 𝐻 = ∑︁ 𝑛 � 𝑝2 𝑛 2𝑎 + 𝑎 2 � 𝜙𝑛+1 − 𝜙𝑛−1 2𝑎 �2 + 𝑎 2𝑉(𝜙𝑛)2 + 𝑎𝑉(𝜙𝑛) 𝜙𝑛+1 − 𝜙𝑛−1 2𝑎 +(−1)𝑛𝑉 ′(𝜙𝑛) � 𝜒† 𝑛𝜒𝑛 − 1 2 � + 1 2𝑎 � 𝜒† 𝑛𝜒𝑛+1 + 𝜒† 𝑛+1𝜒𝑛 �� , (2) where we have replaced the two fermion components 𝜓1,𝑛 and 𝜓2,𝑛 with creation and annihilation operators 𝜒† 𝑛 and 𝜒𝑛 defined by 𝜓1,𝑛 = 1 − 𝑖(−1)𝑛 2𝑖𝑛 � 𝜒† 𝑛 + 𝑖𝜒𝑛 � 𝜓2,𝑛 = 1 + 𝑖(−1)𝑛 2𝑖𝑛 � 𝜒† 𝑛 − 𝑖𝜒𝑛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' (3) As mentioned in Section 1, we are interested in dynamical supersymmetry breaking in the 1+1d Wess–Zumino model, which depends on the prepotential 𝑉(𝜙𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Considering polynomial prepotentials of degree 𝑞, tree-level analyses suggest that supersymmetry should remain preserved when 𝑞 is odd, but may break spontaneously for even 𝑞 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' In Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='2 we will consider 𝑞 = 1 and 𝑞 = 2, respectively, finding agreement with these expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Of course there have been prior investigations of dynamical supersymmetry breaking both for the 1+1d Wess–Zumino model [7–13] as well as for more complicated 1+1d systems including super-Yang–Mills [16], super-QCD [17] and the supersymmetric Gross–Neveu–Yukawa model [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' A persistent challenge in such studies is the severe sign problem associated with spontaneous supersymmetry breaking [2, 3], which has motivated the development of fermion loop [13], tensor network [14] and conformal truncation [18] techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' We avoid sign problems by using the VQE to estimate the ground state energy, for now consid- ering sufficiently small systems that the results can be checked through classical diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Since 𝐻 = 𝑄2, the ground state energy 𝐸0 = ⟨Ω |𝐻| Ω⟩ = |𝑄 |Ω⟩|2 vanishes if and only if the ground state is supersymmetric, 𝑄 |Ω⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Otherwise, if supersymmetry is spontaneously broken, the ground state energy is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' While the classical computational costs of diagonalization grow very rapidly as the system size increases, the VQE offers hope of efficient determination of ground state energies using future quantum devices, as we now discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Quantum Computing To perform computations on quantum devices, we map the bosonic and fermionic degrees of freedom to qubit degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Qubits are physically realized as two state systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' This allows for a straightforward mapping for the fermions via the Jordan–Wigner transformation, 𝜒† 𝑛 = 1 2 (𝑋𝑛 − 𝑖𝑌𝑛) 𝜒𝑛 = 1 2 (𝑋𝑛 + 𝑖𝑌𝑛) , (4) 3 Quantum Computing for the Wess–Zumino Model Christopher Culver where 𝑋𝑛 and 𝑌𝑛 represent a Pauli gate acting on the 𝑛-th qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The bosonic degrees of freedom have an infinite-dimensional Hilbert space at each lattice site and need to be regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' To do this we consider them in the harmonic oscillator basis and impose a hard cutoff on the number Λ of allowed modes at each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' It is worth noting that this explicitly breaks supersymmetry, which will only be exactly restored in the Λ → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The number of qubits needed to define each 𝜙𝑛 truncated in this way is 𝑛𝑞 ≡ ⌈log2 Λ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The raising and lowering operators become ˆ𝑎𝑛 = Λ−2 ∑︁ 𝑙=0 √ 𝑙 + 1 |𝑙⟩ ⟨𝑙 + 1| , ˆ𝑎† 𝑛 = Λ−2 ∑︁ 𝑙=0 √ 𝑙 + 1 |𝑙 + 1⟩ ⟨𝑙| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' (5) The introduction of this cutoff makes the bosonic Hilbert space finite and we can now perform the mapping to qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' We follow the same steps as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' [6], where more details can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Writing the state 𝑗 in binary as 𝑗 = �𝑛𝑞−1 𝑖=0 𝑏𝑖2𝑖, we associate each digit with a qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Specific matrix elements can be converted to their action on qubits using the relations |0⟩ ⟨1| = 1 2 (𝑋 + 𝑖𝑌) , |1⟩ ⟨0| = 1 2 (𝑋 − 𝑖𝑌) , (6) |0⟩ ⟨0| = 1 2 (1 + 𝑍) , |1⟩ ⟨1| = 1 2 (1 − 𝑍) (7) and writing the full matrix element as a tensor product over all of the binary digits: |𝑛⟩ ⟨𝑛′| = ⊗𝑛𝑞−1 𝑖=0 |𝑏𝑖⟩ � 𝑏′ 𝑖 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' (8) This completes the mapping of all the degrees of freedom into qubits, which enables the application of quantum algorithms of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' An important NISQ-era algorithm for determining whether or not supersymmetry is dynam- ically broken is the VQE algorithm [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' This algorithm outputs an upper bound on the lowest eigenvalue of any matrix and by using the Hamiltonian as the target matrix we can investigate whether or not the ground state energy is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Specifically we want to test whether or not, in the Λ → ∞ limit, the VQE estimate of the energy tends towards zero or to a finite value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' To run the VQE algorithm, we first prepare some trial wavefunction for the ground state 𝜓 with tunable parameters 𝜃𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The energy of this trial state is computed with a quantum circuit and fed into a classical optimization algorithm that adjusts the parameters 𝜃𝑖 in search of the minimum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' This hybrid classical–quantum algorithm will converge to some 𝐸var which is an upper bound for the ground state energy, 𝐸0 ≤ 𝐸var = ⟨𝜓(𝜃𝑖) |𝐻| 𝜓(𝜃𝑖)⟩ ⟨𝜓(𝜃𝑖) |𝜓(𝜃𝑖)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' (9) Another important algorithm, more relevant for the long-term prospects of quantum computing, is the Suzuki–Trotter decomposition of the time-evolution operator, 𝑒𝑖𝐻𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' This gives us direct access to the real-time evolution of a quantum state acting under the dynamics of our Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The continuous time 𝑡 is broken down into 𝑁𝑡 steps of size 𝛿 = 𝑡/𝑁𝑡, 𝑒−𝑖𝐻𝑡 |𝜓⟩ = (exp [−𝑖𝐻𝛿])𝑁𝑡 |𝜓⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' (10) 4 Quantum Computing for the Wess–Zumino Model Christopher Culver Noting that the Hamiltonian will be a sum of 𝑀 terms, a single Trotter step is |𝜓(𝑡 + 𝛿)⟩ = exp ������ −𝑖 𝑀 ∑︁ 𝑗=1 𝐻 𝑗𝛿 ������ |𝜓(𝑡)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' For most physical applications 𝑀 > 1 and the Baker–Campbell–Hausdorff formula is used to convert the above line into a product of exponentials of each of the 𝐻 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Unfortunately all of the 𝐻 𝑗 terms of the Hamiltonian do not commute in general, thus there will be a tradeoff introduced between the number of gate operations to perform a single step, and the accuracy to which the final state is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' This error in each step will then be compounded by the time discretization 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Of course taking a larger number of smaller steps will reduce this error, but this comes at the cost of more qubit operations which is not ideal in the NISQ era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' We use the default implementation for transpiling the Trotter circuit in Qiskit [20], based on the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' [21] for further discussion of how Trotter decompositions can be optimized for quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Results We now present results for the ground state energy of the Wess–Zumino model and the entangling gate count for a single Trotter step of real-time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The ground state energy 𝐸0 is important since supersymmetry is preserved if and only if 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' We consider a linear prepotential 𝑉(𝜙) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='1 and a family of quadratic prepotentials in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='2, in each case using 𝑁 = 2–3 spatial lattice sites and several small values of the bosonic cutoff Λ ≤ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' While we can analyze any positive cutoff, in practice Λ should be a power of 2 to optimize the correspondence between the bosonic degrees of freedom and the qubit degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Otherwise our approach would leave part of the quantum computer’s Hilbert space unused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' For each calculation we find the ground state energy via classical diagonalization, and see how well this can be reproduced by a modest number of VQE runs (minimum 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='1 Linear prepotential The linear prepotential is just 𝑉(𝜙𝑛) = 𝜙𝑛, (11) and is expected to preserve supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' This is the simplest nontrivial 𝜙-dependent prepotential, and keeps the bosonic and fermionic fields from interacting with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The ground state energy as a function of Λ from classical computations of the eigenvalues of 𝐻 on 2- and 3-site lattices is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' In the Λ → ∞ limit the ground state energy goes to zero up to the precision of the solver, which confirms that supersymmetry is preserved as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' In Table 1a we compare the minimum energy obtained from 100 runs of the VQE against these exact results from classical diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' For the 2-site lattice with Λ ≤ 4 the VQE estimate of the ground state energy behaves appropriately, and exponentially approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' As an aside, the negative energies shown in Table 1 for Λ = 2 reflect the significant supersymmetry breaking related to such an extreme truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' For larger values of the cutoff, and for the larger 3-site lattice, the VQE struggles to converge to the correct order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' This provides us with immediate 5 Quantum Computing for the Wess–Zumino Model Christopher Culver 2 4 6 8 10 12 14 16 10 10 10 8 10 6 10 4 10 2 100 |E| 2 4 6 8 10 12 14 16 10 10 10 8 10 6 10 4 10 2 100 |E| Figure 1: Semi-log plots of the 1+1d Wess–Zumino ground state energy for 2-site (left) and 3-site (right) lattices with the linear prepotential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' As the bosonic cutoff Λ goes to infinity the ground state energy goes to zero (up to numerical precision), consistent with the expected preservation of supersymmetry for this prepotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Table 1: Ground state energies from classical diagonalization and the VQE for the linear and quadratic prepotentials discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The linear prepotential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 11 is expected to preserve supersymmetry and have a Λ → ∞ ground state energy of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Supersymmetry is expected to break dynamically for the quadratic prepotential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 12, so long as 𝑐 < 𝑐0 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' N Λ Exact VQE 2 2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='97e-03 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='97e-03 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='22e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='61e-05 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='04e-09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='08e-01 3 2 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='97e-02 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='28e+00 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='17e-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='99e-01 (a) Linear prepotential N Λ Exact VQE 2 2 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='87e-01 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='11e-01 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='82e-01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='26e-01 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='31e-01 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='49e-01 3 2 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='98e-01 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='28e+00 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='02e-01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='08e-01 (b) Quadratic prepotential, 𝑐 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='2 N Λ Exact VQE 2 2 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='87e-01 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='11e-01 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='28e-01 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='15e+00 8 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='74e-02 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='89e-01 3 2 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='98e-01 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='28e+00 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='47e-01 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='10e+00 (c) Quadratic prepotential, 𝑐 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='8 targets we can use in work to improve the performance and reliability of VQE determinations of the ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' In Table 2a we show the entangling CX gate counts for a single Trotter step of real-time evolution for the Wess–Zumino model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' As the bosonic cutoff and the number of sites increase, our time-evolution circuits quickly exceed the capabilities of NISQ hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' This motivates ongoing work to optimize time-evolution schemes for the Wess–Zumino model, in order to minimize resource requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='2 Quadratic prepotential To introduce interactions between the bosons and fermions we consider the family of quadratic prepotentials 𝑉(𝜙𝑛) = 𝑐 + 𝜙2 𝑛, (12) with free parameter 𝑐, which was studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' [9, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' This prepotential is expected to lead to dynamical supersymmetry breaking for positive 𝑐, while supersymmetry should be preserved for 6 Quantum Computing for the Wess–Zumino Model Christopher Culver 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='2 c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='25 |E| = 8 = 12 = 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='2 c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='40 |E| = 8 = 12 = 16 Figure 2: The 1+1d Wess–Zumino ground state energy for 2-site (left) and 3-site (right) lattices with various values of the parameter 𝑐 in the quadratic prepotential of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Small horizontal offsets distinguish different values of the cutoff Λ = 8, 12 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Since non-zero values of 𝐸 correspond to supersymmetry breaking, the results are consistent with the expected preservation of supersymmetry for sufficiently negative 𝑐 < 𝑐0 < 0 and Λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' sufficiently negative 𝑐 less than a critical value 𝑐0 < 0, found to be 𝑐0 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='5 in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' We can investigate these expectations by computing the ground state energy for a range of 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Results from classical computations of the ground state energy as a function of 𝑐 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 2 for several values of Λ on 2- and 3-site lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' We only show results from Λ ≥ 8 since smaller Λ ≤ 4 produce wildly fluctuating results that are clearly not meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' For both 2- and 3-site lattices the ground state energy converges towards zero as 𝑐 → −∞, confirming that supersymmetry is preserved for sufficiently negative 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' For 𝑐 ≈ 0 little dependence on the cutoff is visible for Λ ≥ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The results become more sensitive to Λ as 𝑐 becomes more negative, illustrating the challenges that computations in this regime will face on near-term quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' This is also reflected in the results for the VQE estimate of the ground state energy presented in Table 1 for both 𝑐 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='2 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' In the first case of 𝑐 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='2, 100 runs of the VQE suffice to recover the correct order of magnitude of the exact ground-state energy for the 2- and 3-site lattices with Λ ≤ 8 we have considered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The VQE analysis becomes significantly more challenging for 𝑐 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Since these values of Λ are too small to be included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 2, for the time being we are focusing our VQE improvement efforts on the linear prepotential discussed in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Finally, in Table 2 we again provide entangling gate counts for a single Trotter step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The results are qualitatively similar to those for the linear prepotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' For Λ > 2, the interactions between the bosons and fermions increase CX gate requirements by roughly a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The mild dependence on the value of 𝑐 is related to the realization of 𝑐-dependent terms in the Hamiltonian in terms of the default basis gate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Conclusion We have presented initial results from our ongoing work using quantum computing to study the Wess–Zumino model in 1+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' We use a Hamiltonian lattice regularization of the 7 Quantum Computing for the Wess–Zumino Model Christopher Culver Table 2: Entangling CX gate counts for a single Trotter step of the time-evolution operator for the linear and quadratic prepotentials discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' For each prepotential, we consider 2- and 3-site lattices, and transpile the circuit for a few values of the bosonic cutoff Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 𝑁 Λ CX Gates 2 2 8 4 252 8 2556 3 2 18 4 5728 (a) Linear prepotential 𝑁 Λ CX Gates 2 2 14 4 754 8 7822 3 2 30 4 2788 (b) Quadratic prepotential, 𝑐 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='2 𝑁 Λ CX Gates 2 2 14 4 718 8 7858 3 2 30 4 2730 (c) Quadratic prepotential, 𝑐 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='8 theory to analyze spontaneous supersymmetry breaking for prepotentials with linear or quadratic dependence on the bosonic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' The preservation or breaking of supersymmetry is determined by the ground state energy, which we analyze with the VQE algorithm for systems with 2 or 3 spatial sites and small bosonic cutoffs Λ ≤ 16 — small enough to check our results through classical diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Despite the small size of these systems, our results are consistent with the expected preservation of supersymmetry for the linear prepotential, and show clear spontaneous supersymmetry breaking for a quadratic prepotential with 𝑐 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' However, our VQE algorithm struggles to converge for large values of the bosonic cutoff, motivating ongoing work to improve the performance and reliability of this approach to exploring spontaneous supersymmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' We also consider prospects for studying the real-time dynamics of the Wess–Zumino model, based on a straightforward transpilation of the time-evolution operator corresponding to the Hamil- tonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Even for 2-site lattices, the number of gate operations in the resulting time-evolution circuits exceed the capabilities of NISQ hardware for any reasonable cutoff Λ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} +page_content=' Here as well we hope to improve this situation by more carefully 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9E0T4oBgHgl3EQfTABv/content/2301.02230v1.pdf'} diff --git a/MtFJT4oBgHgl3EQfzS0r/content/tmp_files/2301.11642v1.pdf.txt b/MtFJT4oBgHgl3EQfzS0r/content/tmp_files/2301.11642v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..679ca978001fc69a3f9265e0a2d5916934feff72 --- /dev/null +++ b/MtFJT4oBgHgl3EQfzS0r/content/tmp_files/2301.11642v1.pdf.txt @@ -0,0 +1,1168 @@ +A NUMERICAL METHOD FOR A NONLOCAL FORM OF +RICHARDS’ EQUATION BASED ON PERIDYNAMIC THEORY +MARCO BERARDI, FABIO V. DIFONZO, AND SABRINA F. PELLEGRINO +Abstract. Forecasting water content dynamics in heterogeneous porous media has +significant interest in hydrological applications; in particular, the treatment of infil- +tration when in presence of cracks and fractures can be accomplished resorting to +peridynamic theory, which allows a proper modeling of non localities in space. In this +framework, we make use of Chebyshev transform on the diffusive component of the +equation and then we integrate forward in time using an explicit method. We prove +that the proposed spectral numerical scheme provides a solution converging to the +unique solution in some appropriate Sobolev space. We finally exemplify on several +different soils, also considering a sink term representing the root water uptake. +1. Introduction +Environmental protection and related sustainability management policies demand a +thorough understanding of complex coupling between hydrology, soil sciences, ecology, +agronomy, atmospheric sciences, calling for deeper mathematical modeling and numeri- +cal methods able to deal with the multiphysics processes involved in these environmental +phenomena. In particular, flow processes in unsaturated media have to be studied for +a better understanding of the whole water cycle; a correct managing of irrigation needs +relies, for instance, on robust numerical solvers for unsaturated flows with root water +uptake (see for instance, [21, 44]), or it is the basis for forecasting contaminant transport +in the vadose zone (see for instance [50]). Classical local advection-diffusion equations +in porous media often fail to describe accurately such complex phenomena. +The idea of incorporating non-local behaviors in standard unsaturated flow models is +gaining interest in recent times. Besides non-localities in space, which are the focus of +this paper, also non-local effects in time can be considered, that generally account for +memory terms in the advection-diffusion equations: in some pioneering works in the +early ’60s [43]) it had been already noticed that diffusivity depends not only on water +content, but also explicitly on time, and this argument has been then extended also to +hydraulic conductivity (see [23]; later on a model, in which derivative of water content +on time is fractional, has been first proposed in [41] and then generalized in [28]. A +memory component has been observed also when modeling water stress in the root wa- +ter uptake: the experimental evidence of such ”ecological memory” of plant roots has +been noticed, for instance, in [48, 13] and has been recently formalized in [6]. +2020 Mathematics Subject Classification. 65M70, 42B30. +Key words and phrases. Richards’ equation, Peridynamic, Nonlocal Model, Spectral Numerical +Method. +1 +arXiv:2301.11642v1 [math.NA] 27 Jan 2023 + +2 +M. BERARDI, F.V. DIFONZO, S.F. PELLEGRINO +When dealing with spatial discontinuities or significant heterogeneities, classical local +formulations of flow and transport phenomena present severe limitations; for instance, in +some cases, standard unsaturated flow models can not forecast correctly water dynam- +ics; as reported in [39], when modeling fast infiltration processes (for instance infiltration +after a heavy rainfall event), ”first arrival time at the groundwater [...] are often under- +predicted” because of preferential flow paths. These preferential flows can be ascribed +to non-equilibrium of water pressure at a local scale. As a matter of facts, there is an +experimental evidence that pore structures in natural soils dynamically change due to +alternating swelling and shrinkage processes (see for instance [17]): this phenomenon +can be described by a dual permeability approach, by which the bulk porous medium +consists of two dynamic interacting pore domains: (i) the fracture (from shrinkage) +pore domain and (ii) the aggregate (interparticles plus structural pores), respectively +(see [18]): in practice, two different unsaturated flow equations are considered in each +part of this domain. Analogously, in the context of solute transport, the solute exchange +between mobile and immobile water has been modeled by a delay term in [36], and, in +a computational framework, this approach has been implemented in [38]. +More in detail, multirate mass transfer is modeled assuming advection-diffusion on +the fast mobile continuum and only diffusion in the slow immobile continuum: after +solving analytically the diffusion model, the consequent fast domain model results non- +local in time ([14]). In this dual-continuum framework, the pioneering work [39] shows +that the linearization of the nonlinear diffusion equation, governing capillary flow in the +slow continuum, ensures a good description of the averaged water content dynamics in +the slow domain: therefore, they derive a non-local Richards’ equation in the mobile +domain, endowed with a memory kernel encoding mass transfer dynamics of the slow +domains. +From the viewpoint of applications, in this context, dessication cracks impact the effi- +ciency of irrigation and provokes a fast leakage of nutrients and contaminants below the +root zone into the groundwater. Even worst is the effect of such cracks into an earth +dam, since it may lead to the failure of the dam itself. +On the other hand, several efforts have been accomplished towards a comprehensive +modeling and efficient numerical solvers for such nonlocal problems. In [49] the cou- +pling of peridynamic formulation of chemical transport with water flow is proposed in +the unsaturated context, and an implicit numerical solver is implemented, and tested +over different case studies, in order to show the ability of the model to recognize dis- +continuities and heterogeneities, including stationary cracks, propagating cracks, and +randomly distributed permeable and impermeable inclusions. In [19] authors discuss +how a single continuum model can properly catch the contributions from all the flow +paths only if the control volume (i.e. the computational cell) is much larger than the +longest connections between pores: therefore a non-local model is proposed therein, +showing that if the longest connections are much smaller than the size of the control +volume, these models converge to Darcy’s law. A significant work has be presented +in [40], in which the peridynamic theory is employed for simulating hydraulic fracture +propagation in saturated porous media, and it is coupled with FEM for forecasting +fluid flow therein. In this paper we aim at introducing a tailored numerical method for +the corresponding peridynamic model of Richards’ equation describing the unsaturated + +PERIDYNAMIC PINN +3 +flow; for the sake of clarity we should say that peridynamic theory was introduced by +Silling in [46] as a nonlocal version of elasticity theory, for modeling long-range interac- +tions occurring in real materials, ruling several phenomena like fractures, instabilities +and cracks. In general, peridynamic models consist of an integro-differential equation +not involving spatial derivatives and describe the motion of a material body subjected to +external loading conditions. The theory prescribes the existence of a domain influence, +called horizon, which represents a measure of the nonlocality of the model and defines +the range of interactions between material particles. +In this framework, the remaining of the paper is structured as follows. In Section +2 an introduction to nonlocal framework and a peridynamic formulation of Richards’ +equation is given, with all the necessary assumptions to justify the current setting. +Then, in Section 3, we propose a numerical method to integrate forward in time a semi- +discretized version of the equation, leveraging spectral theory and Chebyshev transform +properties to prove convergence results of the discretized solution to the exact one. The +implementation of Chebyshev collocation method provides a good accuracy and does +not require to impose periodic boundary conditions. Finally, in Section 4 we exemplify +on different soils with several type of Dirichlet boundary conditions to support our +findings. +1.1. A short overview on Richards’ equation. It is well known that Richards’ +equation is a mass conservation law in terms of the volumetric moisture content θ and +of the soil matric head hm defined on some compact domain Ω ⊆ R3, coupled with the +Buckingham-Darcy’s law for the description on the flux: +∂θ +∂t (x, t) = −∇q(x, t) + S(x, θ), +x ∈ Ω +q(x, t) = −K(hm)∇(hm − z), +where z is the elevation component of the space variable x, θ represents the volumetric +water content, K is the so called hydraulic conductivity and S(x, θ) is a source or sink +term describing, for instance, the root water uptake. Thus, Richards’ equation reads as +(1.1) +∂θ +∂t (x, t) − ∇ (K(hm)∇(hm − z)) = S(x, θ), +x ∈ Ω, t ∈ [0, T], +endowed with suitable initial and boundary conditions. +With the hypothesis that air pressure in the pores is constant, Richards’ equation +assumes that matric head at a given location is in equilibrium and that there exists a +bijective function relating θ with hm, called water retention curve (see [39]), which is +generally defined according to empirical functions. Moreover, for Richards’ equation +to be well posed, K must be smooth enough to guarantee existence and uniqueness of +solutions, also in case of heterogeneous soils with smooth boundary (see [8] and refer- +ences therein). In particular, hereafter and through the whole paper, K and hm will be +assumed to be locally Lipschitz on their respective domains. +However, in case of desiccation cracks or anisotropic soils could affect well-posedness of +Richards’ equation (1.1) and prevent existence of any solution. An alternative approach +has been proposed in [26], where theory of elasticity for solid mechanics has been ap- +plied to unsaturated, heterogeneous, anisotropic soils. In this case, though, the flow +density function depends on the position, matric head or moisture content, instead of + +4 +M. BERARDI, F.V. DIFONZO, S.F. PELLEGRINO +the relative distance and relative displacement [46, 47]. +The numerous numerical issues arising when solving Richards’ equation in a compu- +tational framework rely mainly in its nature of highly nonlinear degenerate elliptic +parabolic PDE. Here we just mention some significant references for the main numer- +ical problems arising in Richards’ equation. For instance, since implicit methods are +generally used for time integration, the arising nonlinear problems have been studied +with different methods, such as Newton’s (e.g. [15, 10], Picard ([16]), L-Scheme or its +variants ([42, 37]). Even richer is the literature on spatial discretization techniques, for +which we refer to [2, 30, 35, 29] and references therein. As regards numerical integration +over layered discontinuous geological formations, a domain decomposition approach is +followed in [45], while a transversal method of lines is adopted in [5, 9]. +In this paper, we are looking at the 1D version of (1.1) equipped with initial and Dirich- +let boundary conditions, in which diffusion evolves exclusively along the depth, so that +Ω = [0, Z] for some Z > 0, and the forcing term S only depends on z ∈ [0, Z]. Thus, +one considers +(1.2) +∂θ +∂t (z, t) − ∂ +∂z +� +K(hm) ∂ +∂z (hm − z) +� += S(z), +z ∈ [0, Z], t ∈ [0, T]. +2. Peridynamic Model: assumptions and derivation +Let us consider a compact domain Ω with smooth boundary and let us define +(2.1) +Bδ(z) := {z′ ∈ Ω : ∥z′ − z∥ ≤ δ}, +the horizon of z of radius δ > 0. +We assume that moisture dynamics at z is only +affected by pairwise interaction with z′ ∈ Bδ(z); points outside the horizon of z do not +contribute to any dynamics therein. +The model is built on the concept of peripipes. Given any z ∈ Ω, we assume that for +each z′ ∈ Bδ(z) there exists a fictitious pipe, called peripipe, connecting every z to z′. +We assume that the following requirements hold for any peripipe (see [25]): +(1) Moisture is stored at the endpoints z, x′ of a peripipe, and zero moisture content +is located along a peripipe; +(2) moisture flows in the direction of the peripipe and no transversal flux crosses its +boundaries +(3) a peripipe is purely resistive, it has zero reactance and its response is propor- +tional to H(z) − H(z′); +(4) a peripipe has uniform conductivity; +(5) peripipe conductivity is function of medium conductivity at its endpoints; +(6) the length of a peripipe is ∥z − z′∥; +(7) peripipe response may also depend on the its length. +Following [26] and requirements above, we assume that the rate of volumetric mois- +ture flow from a point z′ to a point z per unit volume of z and per unit volume of z′ is +given by +(2.2) +J(z, z′, t) = C(z, z′)(H(z′, t) − H(z, t)), + +PERIDYNAMIC PINN +5 +where C(z, z′) is the peridynamic hydraulic conductance density and H(z, t) is the total +hydraulic potential, defined as +H(z, t) = hm(z, t) + z. +Hereafter, for the sake of readability, we omit time dependence, unless required by the +context. +The peripipe conductance depends on the peridynamic hydraulic conductivity κ(z′, z), +which is an intrinsic material property (related to the classical hydraulic conductivity +K), in the following way: +(2.3) +C(z, z′) = κ(z′, z) +∥z − z′∥, +where +(2.4) +κ(z, z′) := Kϕ(z − z′). +The function ϕ(z − z′) is the so-called influence function, representing a convolution +kernel relating the horizon (2.1) with the nature of boundary conditions assigned to +(1.2). The shape of such an even function and the way to select it turns out to be +crucial, as we will see in Section 2.1. +Therefore, the changes of moisture stored at z and at z′, mediated by the peripipe +zz′, are given by +∆Vm(z, z′) = κ(z′, z)H(z′) − H(z) +∥z − z′∥ +dVz′dVz, +∆Vm(z′, z) = κ(z, z′)H(z) − H(z′) +∥z′ − z∥ +dVzdV ′ +z. +As an immediate consequence it must hold κ(z, z′) = κ(z′, z). +In case of inhomogeneous soils in unsaturated regime, above relations could be leveraged +to define a peridynamic conductivity density by setting +(2.5) +κ(z, z′) := κ(z) + κ(z′) +2 +, +where κ(z) ≡ κ(z, 0), as proposed in [49, 26]. +Now, since the change over time of volumetric moisture content due to z′ at time t, on +the account of (2.2), is given by +∂θ +∂t (z|z′, t) = J(z, z′), +from which +∂θ +∂t (z, t) = +� +Bδ(z) +J(z, z′) dVz′ + S(z). + +6 +M. BERARDI, F.V. DIFONZO, S.F. PELLEGRINO +Thus, using (2.3) and with peridynamic conductivity given by (2.5), our model (1.2), +endowed with Dirichlet boundary conditions, reads +∂θ +∂t = +� +Bδ(z) +ϕ(z′ − z) +∥z′ − z∥ +K(z) + K(z′) +2 +[H(z′) − H(z)] dVz′ + S(z), +(2.6a) +θ(z, 0) = θ0(z), +z ∈ [0, Z], +(2.6b) +θ(0, t) = θ0(t), +t ∈ [0, T], +(2.6c) +θ(Z, t) = θZ(t), +t ∈ [0, T]. +(2.6d) +2.1. Selection of the influence function. Usually (see, e.g. [49, 11, 26]) ϕ(z) in (2.4) +represents a convolution kernel, which can be chosen as a uniform influence function +ϕ(z) := +� +2 +δ, +∥z∥ ≤ δ, +0, +∥z∥ > δ, +or as a linear influence function +ϕ(z) := +� +1 − ∥z∥ +δ , +∥z∥ ≤ δ, +0, +∥z∥ > δ. +However, since such kernels would suggest the model to weigh more those cells where +they are nonzero, and since our boundary conditions would typically be of Dirichlet type, +we propose to consider a distributed influence function (see Figure 1), concentrated on +the domain boundary, of the form +(2.7) +ϕ(z) := +� ∥z∥−1+δ +δ +, +∥z∥ ≥ 1 − δ, +0, +∥z∥ < 1 − δ. +In so doing, we are suggesting the model to averaging out not just what happens in +the middle of the dynamics, but rather the behavior around each point of the spatial +domain. In all our experiments, presented in Section 4, uniform and linear influence +functions do not make our proposed numerical method converge, resulting in instabilities +and blow-ups after a relatively small amount of time integration; on the other hand, and +as presented below, using (2.7) guarantees stability and convergence, plus a reasonable +shape of the numerical solutions. +3. Numerical Method +The nonlocal Richards’ equation (2.6) can be discretized in space by using Cheby- +shev polynomials. This approach is typically used when the integral operator can be +expressed in terms of convolution products [34, 33, 32]. Moreover, the choice of such +kind of polynomials allows us to overcome the limitation of imposing periodic boundary +condition, which is necessary when dealing with Fourier trigonometric polynomials. +The proposed technique consists in looking for an approximation of θ(x, t) in the +form of a finite linear combination of Chebyshev polynomials of the first kind. To do so, +we can assume the spatial domain to be [−1, 1], as we can benefit of the orthogonality +properties of the polynomials. However, a more general interval can be used as spatial +domain by linearity. Moreover, for time integration we use the explicit Euler method, +as in [49]. + +PERIDYNAMIC PINN +7 +−1 −0.8−0.6−0.4−0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +z +ϕ(z) +Figure 1. Distributed influence function defined in (1) with δ = 0.15. +In this section we briefly make a review on Chebyshev polynomials, then we derive the +semidiscrete model of (2.6) and finally prove the convergence of the proposed method. +3.1. Basic overview on Chebyshev polynomials. Chebyshev polynomials of the +first kind, Tk(z) are defined by +Tk(z) = cos (k arccos(z)) , +z ∈ [−1, 1], k ∈ N, +and are orthogonal with respect to the weight function w(z) := +�√ +1 − z2 +�−1 +. +These polynomials are commonly used in the context of spectral approximation be- +cause they satisfy an interpolation property: given an integer N, any sufficiently smooth +function u defined on [−1, 1] can be expanded as an (N + 1)-term linear combination +of polynomials uN given by +(3.1) +uN(z) := +N +� +k=0 +�uk Tk(z), +where �uk are the coefficients of the expansion and approximate the Chebyshev coeffi- +cients +�uk = +2 +πck +� 1 +−1 +u(x)Tk(z)w(x) dz, +with +ck = +� +2 +k = 0 +1 +k ̸= 0. + +8 +M. BERARDI, F.V. DIFONZO, S.F. PELLEGRINO +The explicit expression of �uk depends on the choice of the grid points used to discretize +[−1, 1]. In particular, if we choose the Gauss-Lobatto collocation points +(3.2) +zh := cos +�hπ +N +� +, +h = 0, . . . , N, +the expression of �uk is +(3.3) +�uk = 1 +γk +N +� +h=0 +u(xh) Tk(zh)wh, +where γk is a normalization constant defined by +(3.4) +γk := +� +π +k = 0, N +π +2 +k = 1, . . . , N − 1 +and +(3.5) +wh := +� +π +2N +h = 0, N +π +N +h = 1, . . . , N − 1. +Equation (3.1) represents the inverse discrete Chebyshev transform, while the co- +efficients �uk in (3.3) correspond to the discrete Chebyshev transform. +They can be +efficiently computed using the Fast Fourier Transform. Additionally, they fulfill the +same properties of the Fourier transform. In particular, we can rewrite a convolution +product in the physic space as a multiplication of the Chebyshev transform of each +factor in the frequency space. +The following result shows the rate of convergence of the Chebyshev approximation. +Theorem 3.1 (see [12]). For any 0 ≤ µ ≤ 2 and u ∈ L2([−1, 1]), there exists a positive +constant C independent on N, such that +(3.6) +��u − uN�� +L2([−1,1]) ≤ +C +N2−µ ∥u∥L2([−1,1]) . +In the next section, to lighten the notation, we denote the Chebyshev transform by +F and the inverse Chebyshev transform by F−1. +3.2. Chebyshev semi-discrete collocation method for the nonlocal Richards’ +equation. In what follows, we develop a spectral approximation of (2.6) by using the +Chebyshev transform. We fix N > 0 and assume Ω = [−1, 1]. We can discretize the +spatial domain by the Guass-Lobatto points zh, h = 0, . . . , N defined in (3.2). +If we set +Λ(z) := K(z)H(z), +ϕ(z) := ϕ(z) +∥z∥ , +β := +� 1 +−1 +ϕ(z) dz, + +PERIDYNAMIC PINN +9 +then, since from distributed influence function definition (2.7) it follows that β = +2 +� +1 + 1−δ +δ ln(1 − δ) +� +, we can rewrite model (2.6a) as +∂θ +∂t = +� +Bδ(z) +ϕ(z′ − z) +∥z′ − z∥ +K(z′) + K(z) +2 +[H(z′) − H(z)] dVz′ + S(z), += 1 +2 [(ϕ ∗ Λ) (z) + K(z) (ϕ ∗ H) (z) − H(z) (ϕ ∗ K) (z) − βΛ(z)] + S(z). +(3.7) +Thus, the right hand side of (3.7) can be computed by means of the finite discrete +Chebyshev transform. Indeed, we have +(ϕ ∗ Λ) (z) = F−1 (F (ϕ) F (Λ)) (z), +(3.8) +(ϕ ∗ H) (z) = F−1 (F (ϕ) F (H)) (z), +(3.9) +(ϕ ∗ K) (z) = F−1 (F (ϕ) F (K)) (z). +(3.10) +So, at each collocation point zh, the semi-discretization of the model reads +∂θ +∂t (zh, t) = 1 +2 +� +F−1 (F (ϕ) F (Λ)) (zh) + K(zh) F−1 (F (ϕ) F (H)) (zh) +� +− 1 +2 +� +H(zh) F−1 (F (ϕ) F (K)) (zh) + βΛ(zh) +� ++ S(zh) +(3.11) +The function Λ is defined as the product between the conductivity K and the hy- +draulic potential H: therefore, to compute its Chebyshev transform, we first need to +compute a product. The computational cost to obtain this term could be efficiently +reduced by observing that the Chebyshev coefficients of Λ can be obtained from the +Chebyshev coefficients of H and K. +Indeed, the following result holds (see [3]). +Theorem 3.2. Let N ∈ N. If H and K are approximated by a finite series of Chebyshev +polynomials HN and KN, respectively, given by +HN(z) = +N +� +j=0 +�HkTk(z), +KN(z) = +N +� +j=0 +�KjTj(z), +then the product Λ(z) = H(z)K(z) can be approximated by the following 2N + 1 com- +bination of Chebyshev polynomials +ΛN(z) = +2N +� +j=0 +�ΛjTj(z), +where the coefficients �Λj are given by +2�Λj = +� +� +� +� +� +2 �H0 �K0 + �N +ℓ=1 �Hℓ �Kℓ, +j = 0 +�j +ℓ=0 �Hj−ℓ �Kℓ + �N−j +ℓ=0 �Hj+ℓ �Kℓ + �N +ℓ=j �Hℓ−j �Kℓ, +j = 1, . . . , N +�N +ℓ=j−N �Hj−ℓ �Kℓ, +j = N + 1, . . . , 2N. +The application of Theorem 3.2 implies that the first term in the right hand side +of (3.11) is discretized by 2N + 1 mesh points. Therefore, to maintain the consistency +of the scheme, the discretization of the remaining terms on the right hand side of (3.11) +is accomplished by considering 2N + 1 Gauss-Lobatto collocation points. + +10 +M. BERARDI, F.V. DIFONZO, S.F. PELLEGRINO +3.3. Convergence of the semi-discrete scheme. We prove the convergence of the +spectral semi-discrete method in a suitable weighted Hilbert space. Throughout this +section, C denotes a generic constant. +We consider the weighted Lebesgue space +L2 +w([−1, 1]) = +� +u ∈ L2 : +� 1 +−1 +u2(z)w(z)dz < +∞ +� +equipped with the inner product and the norm respectively +(u, v)w = +� 1 +−1 +u(z)v(z)w(z) dz, +∥u∥2 +w = (u, u)w, +where w(z) = +�√ +1 − z2 +�−1 +. +For any s ≥ 0, we set +Hs +w([−1, 1]) = +� +u ∈ L2 +w([−1, 1]) | ∥u∥s,w < +∞ +� +, +where +∥u∥2 +s,w = +� +|α≤s| +∥Dαu∥2 +w . +Let SN be the space of Chebyshev polynomials of degree N, +SN := span {Th(x) | 0 ≤ h ≤ N} ⊂ L2 +w([−1, 1]), +and PN : L2 +w([−1, 1]) → SN be an orthogonal projection operator +PNu(x) := +N +� +h=0 +�uhTh(x)wh, +for wh defined in (3.5), such that for any u ∈ L2 +w([−1, 1]), the following equality holds +(3.12) +(u − PNu, ϕ)w = 0, +for every ϕ ∈ SN. +The projection operator PN commutes with derivatives in the distributional sense: +∂q +t PNu = PN∂q +t u, +q ∈ N, q ≥ 1, +where, as usual, ∂t := ∂ +∂t. +Letting s ≥ 1, we denote by Xs := C0 (0, T; Hs +w([−1, 1])) the space of all continuous +functions in the weighted Sobolev space Hs +w([−1, 1]), with norm +∥u∥2 +Xs := max +t∈[0,T] ∥u(·, t)∥2 +s,w , +for any T > 0. We denote by L the nonlocal integral operator of (2.6), namely +(3.13) +L (θ) := +� +Bδ(z) +ϕ(z′ − z) +∥z′ − z∥ +K(z) + K(z′) +2 +[H(z′) − H(z)] dVz′. +Then, the semi-discrete spectral scheme for (2.6) can be rewritten as +∂θN +∂t += PNL(θN) + PNS(z), +(3.14) +θN(z, 0) = PNθ0(z), +(3.15) + +PERIDYNAMIC PINN +11 +where θN(z, t) ∈ SN for every 0 ≤ t ≤ T. +To obtain the convergence of the semi-discrete scheme, we need of the following +lemma. +Lemma 3.3 ([12, Theorem 3.1]). For any real 0 ≤ µ ≤ s, there exists a positive constant +C such that +(3.16) +∥u − PNu∥Hµ +s,wµ([−1,1]) ≤ +C +Ns−µ ∥θ∥Hsw([−1,1]) , +for every θ ∈ Hs +w([−1, 1]). +Recalling that K and hm are locally Lipschitz in their respective domains, we can +prove the following theorem. +Theorem 3.4. Let s ≥ 1 and θ(z, t) ∈ Xs be the solution to the initial-boundary-valued +problem (2.6) and θN(z, t) be the solution to the semi-discrete scheme (3.14)-(3.15). +Then, there exists a positive constant C, independent on N, such that +(3.17) +��θ − θN�� +X1 ≤ C(T) +� 1 +N +�s−1 +∥u∥Xs , +for any initial data θ0 ∈ Hs +w([−1, 1]) and for any T > 0. +Proof. Let s ≥ 1. Using the triangular inequality, we have +(3.18) +��θ − θN�� +X1 ≤ ∥θ − PNθ∥X1 + +��PNθ − θN�� +X1 . +Lemma 3.3 implies +∥(θ − PNθ)(·, t)∥H1w([−1,1]) ≤ +C +Ns−1 ∥θ(·, t)∥Hsw([−1,1]) . +Therefore, +(3.19) +∥θ − PNθ∥X1 ≤ +C +Ns−1 ∥θ∥Xs . +Subtracting (3.14) from (2.6) and taking the weighted inner product with PNθ − θN ∈ +SN, we have +0 = +� 1 +−1 +� +∂tθ(z, t) − ∂tθN(z, t) +� � +PNθ(z, t) − θN(z, t) +� +w(z) dz +� +�� +� +=:I1 +− +� 1 +−1 +� +L(θ(z, t)) − PNL(θN(z, t)) +� � +PNθ(z, t) − θN(z, t) +� +w(z) dz +� +�� +� +=:I2 +− +� 1 +−1 +(S(z, t) − PNS(z, t)) +� +PNθ(z, t) − θN(z, t) +� +w(z) dz +� +�� +� +=:I3 +. +(3.20) +The orthogonal condition (3.12) implies that +� 1 +−1 +(∂tθ(z, t) − PN∂tθ(z, t)) +� +PNθ(z, t) − θN(z, t) +� +w(z) dz = 0, + +12 +M. BERARDI, F.V. DIFONZO, S.F. PELLEGRINO +and +� 1 +−1 +(S(z) − PNS(z)) +� +PNθ(z, t) − θN(z, t) +� +w(z) dz = 0. +Thus, +I1 = +� 1 +−1 +(∂tθ(z, t) − PN∂tθ(z, t)) +� +PNθ(z, t) − θN(z, t) +� +w(z) dz ++ +� 1 +−1 +� +PN∂tθ(z, t) − θN +t (z, t) +� � +PNθ(z, t) − θN(z, t) +� +w(z) dz += 1 +2 +d +dt +��(PNθ − θN)(·, t) +��2 +H1w([−1,1]) . +(3.21) +Since it is straightforward to see that I3 = 0, we can focus on I2. From (3.12) it follows +that +� 1 +−1 +� +L(θN(z, t)) − PNL(θN(z, t)) +� � +PNθ(z, t) − θN(z, t) +� +w(z) dz = 0, +and so, due to the locally Lipschitzianity of K and hm and from Cauchy and triangular +inequalities, we obtain +I2 = +� 1 +−1 +� +L(θ(z, t)) − L(θN(z, t)) +� � +PNθ(z, t) − θN(z, t) +� +w(z) dz +≤ C +��� +θ − θN� +(·, t) +��2 +H1w([−1,1]) + C +��(PNθ − θN)(·, t) +��2 +H1w([−1,1]) +≤ 2C ∥(θ − PNθ) (·, t)∥2 +H1w([−1,1]) + 3C +��(PNθ − θN)(·, t) +��2 +H1w([−1,1]) . +(3.22) +Plugging (3.21) and (3.22) in (3.20), we have +(3.23) +d +dt +��(PNθ − θN)(·, t) +��2 +H1w([−1,1]) ≤ 4C ∥(θ − PNθ)(·, t)∥2 +H1w([−1,1])+6C +��(PNθ − θN)(·, t) +��2 +H1w([−1,1]) . +Since +��(PNθ − θN)(·, 0) +�� +H1w([−1,1]) = 0, Lemma 3.3 and Gronwall’s inequality imply +that +��(PNθ − θN)(·, t) +��2 +H1([−1,1]) ≤ 4C +� t +0 +e6C(t−τ) ∥(θ − PNθ)(·, τ)∥2 +H1w([−1,1]) dτ +≤ C(T) +N2s−2 +� t +0 +∥θ(·, τ)∥2 +H1w([−1,1]) dτ. +Hence, +(3.24) +��PNθ − θN�� +X1 ≤ C(T) +Ns−1 ∥θ∥Xs . +Finally, using (3.19) and (3.24) in (3.18), we complete the proof. +□ +4. Numerical Simulations +In this section we test our proposed method on different soils, possibly with a sink +forcing term, representing the water uptake due to root systems. +Van Genuchten - +Mualem constitutive relations are considered in the following numerical simulations. + +PERIDYNAMIC PINN +13 +Figure 2. Numerical solution relative to Example 4.1. +Example 4.1. Drawing from [4], we consider a soil with the following parameters: +θr = 0.075, θS = 0.287, α = 0.036, n = 1.56, KS = 0.94e − 3 cm/s. +We added a sink term S = −700 s−1 and parameter δ = 0.15 in (2.1). We set our +initial and boundary conditions as follows +θ(0, t) = 0.2234 +� +1 − t +T +� ++ 0.181 t +T , t ∈ [0, T], +θ(Z, t) = 0.1368 +� +1 − t +T +� ++ 0.1174 t +T , t ∈ [0, T], +θ(z, 0) = 0.2234 − +� +1 − z +Z +� 0.0848 +2 +, z ∈ [0, Z]. +Here Z = 30 cm, T = 60 s; moreover, we used ∆t = 0.06 s and ∆x = 0.3 cm. Results +are shown in Figure 2. +Example 4.2. As already considered by [24], we select a Glendale clay loam, charac- +terized by the following parameters +θr = 0.1060, θS = 0.4686, α = 0.0104, n = 1.3954, KS = 1.5162e − 4 cm/s. +We put neither sink nor source on this simulation. Peridynamic parameter is δ = 0.15 +in (2.1). Our boundary conditions are constant with values +θ(0, t) = 0.2, t ∈ [0, T], +θ(Z, t) = 0.3, t ∈ [0, T], + +0 +5 +10 +N 15 +t= 0 s +t= 7 s +t=13 s +20 +t=20 s +t=27 s +t=33 s +t=40 s +25 +t=47 s +t=53 s +t=60 s +30 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +014 +M. BERARDI, F.V. DIFONZO, S.F. PELLEGRINO +Figure 3. Numerical solution relative to Example 4.2. +while initial condition follows a nonlinear profile of the form +θ(z, 0) = −0.05z3 + 0.25, z ∈ [0, Z]. +We select Z = 70 cm, T = 2400 s; moreover, we used ∆t = 2.4 s and ∆x = 0.3 cm. +The resulting water content profiles are shown in Figure 3. +Example 4.3. As in [24], we consider a Berino loamy fine sand, with parameters +θr = 0.0286, θS = 0.3658, α = 0.0280, n = 2.2390, KS = 0.0063 cm/s. +We added a sink term S = −100 s−1 and parameter δ = 0.15 in (2.1). We set our +initial and boundary conditions as follows +θ(0, t) = 0.3 +� +1 − t +T +� ++ 0.29 t +T , t ∈ [0, T], +θ(Z, t) = 0.2, t ∈ [0, T], +while initial condition has a nonlinear profile +θ(z, 0) = 0.05z3 + 0.25, z ∈ [0, Z]. +We select Z = 70 cm, T = 600 s; moreover, we used ∆t = 0.06 s and ∆x = 0.3 cm. +Results are shown in Figure 4. +5. Conclusions and future works +Starting from an appropriate and physically based rewriting of Richards’ equation +using non-locality theory, we propose to compute its numerical solution using a semi- +discretized time forward scheme based on Chebyshev transform. We prove that such + +0 +t= 0 s +t=267 s +10 +t=533 s +t=800 s +t=1067 s +20 +t=1333 s +t=1600 s +t=1867 s +30 +t=2133 s +t=2400 s +N +40 +50 +60 +70 +0.18 +0.2 +0.22 +0.24 +0.26 +0.28 +0.3 +0PERIDYNAMIC PINN +15 +Figure 4. Numerical solution relative to Example 4.3. +approach converges in suitable Sobolev spaces, providing a theoretical background for +further extensions of the present work to higher dimensional domains. We also propose +a new kind of convolutional kernel, or influence function, in order to manage the peridy- +namic behavior of the proposed model. Such an influence function distributes its effect +on the domain so to correctly catch boundary conditions. In fact, we experienced major +benefits from this choice, as numerical convergence turns out to be robust with respect +to time, and compared to results coming from classical choices of influence functions. +To testify our theoretical analysis, we have performed several experiments, on a wide +range on soils, in MATLAB. We have considered different Dirichlet boundary conditions +and linear and non-linear initial conditions and show that, with suitable discretization +step-sizes, our method is reliable and accurate. +The present work suggests several possible directions for future and already ongoing +research studies. For instance, it would be of interest applying Eulerian-Lagrangian +methods (e.g. +[1]) in the proposed peridynamic framework for Richards’ equation, +while the idea of introducing a basic control approach on the boundary conditions, +as in [7], could be adapted as well. +Another development would consider non-local +terms in time, so to resort to numerical solvers coming from specific tools in fractional +differential calculus (see [22, 20]), or, even more promisingly, by spectral methods in 2D +(see [33, 27, 31]). + +0 +10 +20 +30 +N +t= 0 s +40 +t=67 s +t=133 s +t=200 s +50 +t=267 s +t=333 s +t=400 s +60 +t=467 s +t=533 s +t=600 s +70 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +0.26 +0.28 +0.3 +016 +M. BERARDI, F.V. DIFONZO, S.F. PELLEGRINO +Acknowledgments +All authors are member of the INdAM Research group GNCS. MB and FVD also +acknowledge the partial support of the 2022 project “Modelli di evoluzione non lo- +cali: analisi, trattamento numerico e algoritmi” funded by GNCS-INdAM. SFP ac- +knowledges the partial support of “Finanziamento giovani ricercatori 2022” funded by +GNCS-INdAM. MB acknowledges the partial support of the CNR project “MENTOR”. +FVD has been supported by REFIN Project, grant number 812E4967 funded by Re- +gione Puglia. SFP has been supported by REFIN Project, grant number D1AB726C +funded by Regione Puglia. +References +[1] E. Abreu, R. De la cruz, J.C. Juajibioy, and W. Lambert. Lagrangian-Eulerian Approach for +Nonlocal Conservation Laws. Journal of Dynamics and Differential Equations, 2022. +[2] T. Arbogast, M.F. Wheeler, and N.Y. Zhang. A Nonlinear Mixed Finite Element Method for a +Degenerate Parabolic Equation Arising in Flow in Porous Media. SIAM Journal on Numerical +Analysis, 33(4):1669—-1687, 1996. +[3] G¨unter Baszenski and Manfred Tasche. Fast polynomial multiplication and convolutions related to +the discrete cosine transform. Linear Algebra and Its Applications, 252(1-3):1 – 25, 1997. +[4] M. Berardi, F. Difonzo, F. 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Reformulation of elasticity theory for discontinuities and long-range forces. Journal of +the Mechanics and Physics of Solids, 48(1):175–209, 2000. +[47] S.A. Silling and R.B. Lehoucq. Peridynamic Theory of Solid Mechanics. In Hassan Aref and Erik +van der Giessen, editors, Advances in Applied Mechanics, volume 44 of Advances in Applied Me- +chanics, pages 73–168. Elsevier, 2010. +[48] Xun Wu, Qiang Zuo, Jianchu Shi, Lichun Wang, Xuzhang Xue, and Alon Ben-Gal. Introducing +water stress hysteresis to the Feddes empirical macroscopic root water uptake model. Agricultural +Water Management, 240:106293, 2020. +[49] Huaxiang Yan, Majid Sedighi, and Andrey P. Jivkov. Peridynamics modelling of coupled water +flow and chemical transport in unsaturated porous media. Journal of Hydrology, 591:125648, 2020. +[50] Jiˇr´ı ˇSim˚unek and Martinus Th. van Genuchten. Contaminant Transport in the Unsaturated Zone: +Theory and Modeling. In J.H. Cushman and D.M Tartakovsky, editors, The Handbook of Ground- +water Engineering, chapter 8, pages 266–290. CRC Press, Boca Raton, 2016. +Istituto di Ricerca sulle Acque, Consiglio Nazionale delle Ricerche, Via F. de Blasio +5, 70132 Bari, Italy +Email address: marco.berardi@ba.irsa.cnr.it +Dipartimento di Matematica, Universit`a degli Studi di Bari Aldo Moro, Via E. Orabona +4, 70125 Bari, Italy +Email address: fabio.difonzo@uniba.it +Dipartimento di Management, Finanza e Tecnologia, Universit`a LUM Giuseppe Degen- +naro, S.S. 100 Km 18, 70010 Casamassima,Italy +Email address: pellegrino@lum.it + diff --git a/MtFJT4oBgHgl3EQfzS0r/content/tmp_files/load_file.txt b/MtFJT4oBgHgl3EQfzS0r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a8ffc1e619a4a80d19009dc0f9a0fd3394d72cc --- /dev/null +++ b/MtFJT4oBgHgl3EQfzS0r/content/tmp_files/load_file.txt @@ -0,0 +1,765 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf,len=764 +page_content='A NUMERICAL METHOD FOR A NONLOCAL FORM OF RICHARDS’ EQUATION BASED ON PERIDYNAMIC THEORY MARCO BERARDI, FABIO V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' DIFONZO, AND SABRINA F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' PELLEGRINO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Forecasting water content dynamics in heterogeneous porous media has significant interest in hydrological applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' in particular, the treatment of infil- tration when in presence of cracks and fractures can be accomplished resorting to peridynamic theory, which allows a proper modeling of non localities in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In this framework, we make use of Chebyshev transform on the diffusive component of the equation and then we integrate forward in time using an explicit method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We prove that the proposed spectral numerical scheme provides a solution converging to the unique solution in some appropriate Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We finally exemplify on several different soils, also considering a sink term representing the root water uptake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Introduction Environmental protection and related sustainability management policies demand a thorough understanding of complex coupling between hydrology, soil sciences, ecology, agronomy, atmospheric sciences, calling for deeper mathematical modeling and numeri- cal methods able to deal with the multiphysics processes involved in these environmental phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In particular, flow processes in unsaturated media have to be studied for a better understanding of the whole water cycle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' a correct managing of irrigation needs relies, for instance, on robust numerical solvers for unsaturated flows with root water uptake (see for instance, [21, 44]), or it is the basis for forecasting contaminant transport in the vadose zone (see for instance [50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Classical local advection-diffusion equations in porous media often fail to describe accurately such complex phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The idea of incorporating non-local behaviors in standard unsaturated flow models is gaining interest in recent times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Besides non-localities in space, which are the focus of this paper, also non-local effects in time can be considered, that generally account for memory terms in the advection-diffusion equations: in some pioneering works in the early ’60s [43]) it had been already noticed that diffusivity depends not only on water content, but also explicitly on time, and this argument has been then extended also to hydraulic conductivity (see [23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' later on a model, in which derivative of water content on time is fractional, has been first proposed in [41] and then generalized in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' A memory component has been observed also when modeling water stress in the root wa- ter uptake: the experimental evidence of such ”ecological memory” of plant roots has been noticed, for instance, in [48, 13] and has been recently formalized in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 65M70, 42B30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Richards’ equation, Peridynamic, Nonlocal Model, Spectral Numerical Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='11642v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='NA] 27 Jan 2023 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' BERARDI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' DIFONZO, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' PELLEGRINO When dealing with spatial discontinuities or significant heterogeneities, classical local formulations of flow and transport phenomena present severe limitations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' for instance, in some cases, standard unsaturated flow models can not forecast correctly water dynam- ics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' as reported in [39], when modeling fast infiltration processes (for instance infiltration after a heavy rainfall event), ”first arrival time at the groundwater [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='] are often under- predicted” because of preferential flow paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' These preferential flows can be ascribed to non-equilibrium of water pressure at a local scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' As a matter of facts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' there is an experimental evidence that pore structures in natural soils dynamically change due to alternating swelling and shrinkage processes (see for instance [17]): this phenomenon can be described by a dual permeability approach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' by which the bulk porous medium consists of two dynamic interacting pore domains: (i) the fracture (from shrinkage) pore domain and (ii) the aggregate (interparticles plus structural pores),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' respectively (see [18]): in practice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' two different unsaturated flow equations are considered in each part of this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Analogously, in the context of solute transport, the solute exchange between mobile and immobile water has been modeled by a delay term in [36], and, in a computational framework, this approach has been implemented in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' More in detail, multirate mass transfer is modeled assuming advection-diffusion on the fast mobile continuum and only diffusion in the slow immobile continuum: after solving analytically the diffusion model, the consequent fast domain model results non- local in time ([14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In this dual-continuum framework, the pioneering work [39] shows that the linearization of the nonlinear diffusion equation, governing capillary flow in the slow continuum, ensures a good description of the averaged water content dynamics in the slow domain: therefore, they derive a non-local Richards’ equation in the mobile domain, endowed with a memory kernel encoding mass transfer dynamics of the slow domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' From the viewpoint of applications, in this context, dessication cracks impact the effi- ciency of irrigation and provokes a fast leakage of nutrients and contaminants below the root zone into the groundwater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Even worst is the effect of such cracks into an earth dam, since it may lead to the failure of the dam itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' On the other hand, several efforts have been accomplished towards a comprehensive modeling and efficient numerical solvers for such nonlocal problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In [49] the cou- pling of peridynamic formulation of chemical transport with water flow is proposed in the unsaturated context, and an implicit numerical solver is implemented, and tested over different case studies, in order to show the ability of the model to recognize dis- continuities and heterogeneities, including stationary cracks, propagating cracks, and randomly distributed permeable and impermeable inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In [19] authors discuss how a single continuum model can properly catch the contributions from all the flow paths only if the control volume (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' the computational cell) is much larger than the longest connections between pores: therefore a non-local model is proposed therein, showing that if the longest connections are much smaller than the size of the control volume, these models converge to Darcy’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' A significant work has be presented in [40], in which the peridynamic theory is employed for simulating hydraulic fracture propagation in saturated porous media, and it is coupled with FEM for forecasting fluid flow therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In this paper we aim at introducing a tailored numerical method for the corresponding peridynamic model of Richards’ equation describing the unsaturated PERIDYNAMIC PINN 3 flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' for the sake of clarity we should say that peridynamic theory was introduced by Silling in [46] as a nonlocal version of elasticity theory, for modeling long-range interac- tions occurring in real materials, ruling several phenomena like fractures, instabilities and cracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In general, peridynamic models consist of an integro-differential equation not involving spatial derivatives and describe the motion of a material body subjected to external loading conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The theory prescribes the existence of a domain influence, called horizon, which represents a measure of the nonlocality of the model and defines the range of interactions between material particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In this framework, the remaining of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In Section 2 an introduction to nonlocal framework and a peridynamic formulation of Richards’ equation is given, with all the necessary assumptions to justify the current setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Then, in Section 3, we propose a numerical method to integrate forward in time a semi- discretized version of the equation, leveraging spectral theory and Chebyshev transform properties to prove convergence results of the discretized solution to the exact one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The implementation of Chebyshev collocation method provides a good accuracy and does not require to impose periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Finally, in Section 4 we exemplify on different soils with several type of Dirichlet boundary conditions to support our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' A short overview on Richards’ equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' It is well known that Richards’ equation is a mass conservation law in terms of the volumetric moisture content θ and of the soil matric head hm defined on some compact domain Ω ⊆ R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' coupled with the Buckingham-Darcy’s law for the description on the flux: ∂θ ∂t (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' t) = −∇q(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' t) + S(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' x ∈ Ω q(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' t) = −K(hm)∇(hm − z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' where z is the elevation component of the space variable x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' θ represents the volumetric water content,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' K is the so called hydraulic conductivity and S(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' θ) is a source or sink term describing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' for instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' the root water uptake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Thus, Richards’ equation reads as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1) ∂θ ∂t (x, t) − ∇ (K(hm)∇(hm − z)) = S(x, θ), x ∈ Ω, t ∈ [0, T], endowed with suitable initial and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' With the hypothesis that air pressure in the pores is constant, Richards’ equation assumes that matric head at a given location is in equilibrium and that there exists a bijective function relating θ with hm, called water retention curve (see [39]), which is generally defined according to empirical functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Moreover, for Richards’ equation to be well posed, K must be smooth enough to guarantee existence and uniqueness of solutions, also in case of heterogeneous soils with smooth boundary (see [8] and refer- ences therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In particular, hereafter and through the whole paper, K and hm will be assumed to be locally Lipschitz on their respective domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' However, in case of desiccation cracks or anisotropic soils could affect well-posedness of Richards’ equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1) and prevent existence of any solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' An alternative approach has been proposed in [26], where theory of elasticity for solid mechanics has been ap- plied to unsaturated, heterogeneous, anisotropic soils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In this case, though, the flow density function depends on the position, matric head or moisture content, instead of 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' BERARDI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' DIFONZO, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' PELLEGRINO the relative distance and relative displacement [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The numerous numerical issues arising when solving Richards’ equation in a compu- tational framework rely mainly in its nature of highly nonlinear degenerate elliptic parabolic PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Here we just mention some significant references for the main numer- ical problems arising in Richards’ equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' For instance, since implicit methods are generally used for time integration, the arising nonlinear problems have been studied with different methods, such as Newton’s (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' [15, 10], Picard ([16]), L-Scheme or its variants ([42, 37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Even richer is the literature on spatial discretization techniques, for which we refer to [2, 30, 35, 29] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' As regards numerical integration over layered discontinuous geological formations, a domain decomposition approach is followed in [45], while a transversal method of lines is adopted in [5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In this paper, we are looking at the 1D version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1) equipped with initial and Dirich- let boundary conditions, in which diffusion evolves exclusively along the depth, so that Ω = [0, Z] for some Z > 0, and the forcing term S only depends on z ∈ [0, Z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Thus, one considers (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2) ∂θ ∂t (z, t) − ∂ ∂z � K(hm) ∂ ∂z (hm − z) � = S(z), z ∈ [0, Z], t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Peridynamic Model: assumptions and derivation Let us consider a compact domain Ω with smooth boundary and let us define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1) Bδ(z) := {z′ ∈ Ω : ∥z′ − z∥ ≤ δ}, the horizon of z of radius δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We assume that moisture dynamics at z is only affected by pairwise interaction with z′ ∈ Bδ(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' points outside the horizon of z do not contribute to any dynamics therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The model is built on the concept of peripipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Given any z ∈ Ω, we assume that for each z′ ∈ Bδ(z) there exists a fictitious pipe, called peripipe, connecting every z to z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We assume that the following requirements hold for any peripipe (see [25]): (1) Moisture is stored at the endpoints z, x′ of a peripipe, and zero moisture content is located along a peripipe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' (2) moisture flows in the direction of the peripipe and no transversal flux crosses its boundaries (3) a peripipe is purely resistive, it has zero reactance and its response is propor- tional to H(z) − H(z′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' (4) a peripipe has uniform conductivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' (5) peripipe conductivity is function of medium conductivity at its endpoints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' (6) the length of a peripipe is ∥z − z′∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' (7) peripipe response may also depend on the its length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Following [26] and requirements above, we assume that the rate of volumetric mois- ture flow from a point z′ to a point z per unit volume of z and per unit volume of z′ is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2) J(z, z′, t) = C(z, z′)(H(z′, t) − H(z, t)), PERIDYNAMIC PINN 5 where C(z, z′) is the peridynamic hydraulic conductance density and H(z, t) is the total hydraulic potential, defined as H(z, t) = hm(z, t) + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Hereafter, for the sake of readability, we omit time dependence, unless required by the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The peripipe conductance depends on the peridynamic hydraulic conductivity κ(z′, z), which is an intrinsic material property (related to the classical hydraulic conductivity K), in the following way: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3) C(z, z′) = κ(z′, z) ∥z − z′∥, where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='4) κ(z, z′) := Kϕ(z − z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The function ϕ(z − z′) is the so-called influence function, representing a convolution kernel relating the horizon (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1) with the nature of boundary conditions assigned to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The shape of such an even function and the way to select it turns out to be crucial, as we will see in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Therefore, the changes of moisture stored at z and at z′, mediated by the peripipe zz′, are given by ∆Vm(z, z′) = κ(z′, z)H(z′) − H(z) ∥z − z′∥ dVz′dVz, ∆Vm(z′, z) = κ(z, z′)H(z) − H(z′) ∥z′ − z∥ dVzdV ′ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' As an immediate consequence it must hold κ(z, z′) = κ(z′, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In case of inhomogeneous soils in unsaturated regime, above relations could be leveraged to define a peridynamic conductivity density by setting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='5) κ(z, z′) := κ(z) + κ(z′) 2 , where κ(z) ≡ κ(z, 0), as proposed in [49, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Now, since the change over time of volumetric moisture content due to z′ at time t, on the account of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2), is given by ∂θ ∂t (z|z′, t) = J(z, z′), from which ∂θ ∂t (z, t) = � Bδ(z) J(z, z′) dVz′ + S(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' BERARDI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' DIFONZO, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' PELLEGRINO Thus, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3) and with peridynamic conductivity given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='5), our model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2), endowed with Dirichlet boundary conditions, reads ∂θ ∂t = � Bδ(z) ϕ(z′ − z) ∥z′ − z∥ K(z) + K(z′) 2 [H(z′) − H(z)] dVz′ + S(z), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6a) θ(z, 0) = θ0(z), z ∈ [0, Z], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6b) θ(0, t) = θ0(t), t ∈ [0, T], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6c) θ(Z, t) = θZ(t), t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6d) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Selection of the influence function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Usually (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' [49, 11, 26]) ϕ(z) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='4) represents a convolution kernel, which can be chosen as a uniform influence function ϕ(z) := � 2 δ, ∥z∥ ≤ δ, 0, ∥z∥ > δ, or as a linear influence function ϕ(z) := � 1 − ∥z∥ δ , ∥z∥ ≤ δ, 0, ∥z∥ > δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' However, since such kernels would suggest the model to weigh more those cells where they are nonzero, and since our boundary conditions would typically be of Dirichlet type, we propose to consider a distributed influence function (see Figure 1), concentrated on the domain boundary, of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='7) ϕ(z) := � ∥z∥−1+δ δ , ∥z∥ ≥ 1 − δ, 0, ∥z∥ < 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In so doing, we are suggesting the model to averaging out not just what happens in the middle of the dynamics, but rather the behavior around each point of the spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In all our experiments, presented in Section 4, uniform and linear influence functions do not make our proposed numerical method converge, resulting in instabilities and blow-ups after a relatively small amount of time integration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' on the other hand, and as presented below, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='7) guarantees stability and convergence, plus a reasonable shape of the numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Numerical Method The nonlocal Richards’ equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6) can be discretized in space by using Cheby- shev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' This approach is typically used when the integral operator can be expressed in terms of convolution products [34, 33, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Moreover, the choice of such kind of polynomials allows us to overcome the limitation of imposing periodic boundary condition, which is necessary when dealing with Fourier trigonometric polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The proposed technique consists in looking for an approximation of θ(x, t) in the form of a finite linear combination of Chebyshev polynomials of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' To do so, we can assume the spatial domain to be [−1, 1], as we can benefit of the orthogonality properties of the polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' However, a more general interval can be used as spatial domain by linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Moreover, for time integration we use the explicit Euler method, as in [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' PERIDYNAMIC PINN 7 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='8−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='4−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='8 1 z ϕ(z) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Distributed influence function defined in (1) with δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In this section we briefly make a review on Chebyshev polynomials, then we derive the semidiscrete model of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6) and finally prove the convergence of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Basic overview on Chebyshev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Chebyshev polynomials of the first kind, Tk(z) are defined by Tk(z) = cos (k arccos(z)) , z ∈ [−1, 1], k ∈ N, and are orthogonal with respect to the weight function w(z) := �√ 1 − z2 �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' These polynomials are commonly used in the context of spectral approximation be- cause they satisfy an interpolation property: given an integer N, any sufficiently smooth function u defined on [−1, 1] can be expanded as an (N + 1)-term linear combination of polynomials uN given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1) uN(z) := N � k=0 �uk Tk(z), where �uk are the coefficients of the expansion and approximate the Chebyshev coeffi- cients �uk = 2 πck � 1 −1 u(x)Tk(z)w(x) dz, with ck = � 2 k = 0 1 k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' BERARDI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' DIFONZO, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' PELLEGRINO The explicit expression of �uk depends on the choice of the grid points used to discretize [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In particular, if we choose the Gauss-Lobatto collocation points (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2) zh := cos �hπ N � , h = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' , N, the expression of �uk is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3) �uk = 1 γk N � h=0 u(xh) Tk(zh)wh, where γk is a normalization constant defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='4) γk := � π k = 0, N π 2 k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' , N − 1 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='5) wh := � π 2N h = 0, N π N h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1) represents the inverse discrete Chebyshev transform, while the co- efficients �uk in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3) correspond to the discrete Chebyshev transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' They can be efficiently computed using the Fast Fourier Transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Additionally, they fulfill the same properties of the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In particular, we can rewrite a convolution product in the physic space as a multiplication of the Chebyshev transform of each factor in the frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The following result shows the rate of convergence of the Chebyshev approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1 (see [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' For any 0 ≤ µ ≤ 2 and u ∈ L2([−1, 1]), there exists a positive constant C independent on N, such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6) ��u − uN�� L2([−1,1]) ≤ C N2−µ ∥u∥L2([−1,1]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In the next section, to lighten the notation, we denote the Chebyshev transform by F and the inverse Chebyshev transform by F−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Chebyshev semi-discrete collocation method for the nonlocal Richards’ equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In what follows, we develop a spectral approximation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6) by using the Chebyshev transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We fix N > 0 and assume Ω = [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We can discretize the spatial domain by the Guass-Lobatto points zh, h = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' , N defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' If we set Λ(z) := K(z)H(z), ϕ(z) := ϕ(z) ∥z∥ , β := � 1 −1 ϕ(z) dz, PERIDYNAMIC PINN 9 then, since from distributed influence function definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='7) it follows that β = 2 � 1 + 1−δ δ ln(1 − δ) � , we can rewrite model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6a) as ∂θ ∂t = � Bδ(z) ϕ(z′ − z) ∥z′ − z∥ K(z′) + K(z) 2 [H(z′) − H(z)] dVz′ + S(z), = 1 2 [(ϕ ∗ Λ) (z) + K(z) (ϕ ∗ H) (z) − H(z) (ϕ ∗ K) (z) − βΛ(z)] + S(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='7) Thus, the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='7) can be computed by means of the finite discrete Chebyshev transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Indeed, we have (ϕ ∗ Λ) (z) = F−1 (F (ϕ) F (Λ)) (z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='8) (ϕ ∗ H) (z) = F−1 (F (ϕ) F (H)) (z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='9) (ϕ ∗ K) (z) = F−1 (F (ϕ) F (K)) (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='10) So, at each collocation point zh, the semi-discretization of the model reads ∂θ ∂t (zh, t) = 1 2 � F−1 (F (ϕ) F (Λ)) (zh) + K(zh) F−1 (F (ϕ) F (H)) (zh) � − 1 2 � H(zh) F−1 (F (ϕ) F (K)) (zh) + βΛ(zh) � + S(zh) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='11) The function Λ is defined as the product between the conductivity K and the hy- draulic potential H: therefore, to compute its Chebyshev transform, we first need to compute a product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The computational cost to obtain this term could be efficiently reduced by observing that the Chebyshev coefficients of Λ can be obtained from the Chebyshev coefficients of H and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Indeed, the following result holds (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Let N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' If H and K are approximated by a finite series of Chebyshev polynomials HN and KN, respectively, given by HN(z) = N � j=0 �HkTk(z), KN(z) = N � j=0 �KjTj(z), then the product Λ(z) = H(z)K(z) can be approximated by the following 2N + 1 com- bination of Chebyshev polynomials ΛN(z) = 2N � j=0 �ΛjTj(z), where the coefficients �Λj are given by 2�Λj = � � � � � 2 �H0 �K0 + �N ℓ=1 �Hℓ �Kℓ, j = 0 �j ℓ=0 �Hj−ℓ �Kℓ + �N−j ℓ=0 �Hj+ℓ �Kℓ + �N ℓ=j �Hℓ−j �Kℓ, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' , N �N ℓ=j−N �Hj−ℓ �Kℓ, j = N + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' , 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The application of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2 implies that the first term in the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='11) is discretized by 2N + 1 mesh points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Therefore, to maintain the consistency of the scheme, the discretization of the remaining terms on the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='11) is accomplished by considering 2N + 1 Gauss-Lobatto collocation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' BERARDI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' DIFONZO, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' PELLEGRINO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Convergence of the semi-discrete scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We prove the convergence of the spectral semi-discrete method in a suitable weighted Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Throughout this section, C denotes a generic constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We consider the weighted Lebesgue space L2 w([−1, 1]) = � u ∈ L2 : � 1 −1 u2(z)w(z)dz < +∞ � equipped with the inner product and the norm respectively (u, v)w = � 1 −1 u(z)v(z)w(z) dz, ∥u∥2 w = (u, u)w, where w(z) = �√ 1 − z2 �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' For any s ≥ 0, we set Hs w([−1, 1]) = � u ∈ L2 w([−1, 1]) | ∥u∥s,w < +∞ � , where ∥u∥2 s,w = � |α≤s| ∥Dαu∥2 w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Let SN be the space of Chebyshev polynomials of degree N, SN := span {Th(x) | 0 ≤ h ≤ N} ⊂ L2 w([−1, 1]), and PN : L2 w([−1, 1]) → SN be an orthogonal projection operator PNu(x) := N � h=0 �uhTh(x)wh, for wh defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='5), such that for any u ∈ L2 w([−1, 1]), the following equality holds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='12) (u − PNu, ϕ)w = 0, for every ϕ ∈ SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The projection operator PN commutes with derivatives in the distributional sense: ∂q t PNu = PN∂q t u, q ∈ N, q ≥ 1, where, as usual, ∂t := ∂ ∂t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Letting s ≥ 1, we denote by Xs := C0 (0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Hs w([−1, 1])) the space of all continuous functions in the weighted Sobolev space Hs w([−1, 1]), with norm ∥u∥2 Xs := max t∈[0,T] ∥u(·, t)∥2 s,w , for any T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We denote by L the nonlocal integral operator of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6), namely (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='13) L (θ) := � Bδ(z) ϕ(z′ − z) ∥z′ − z∥ K(z) + K(z′) 2 [H(z′) − H(z)] dVz′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Then, the semi-discrete spectral scheme for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6) can be rewritten as ∂θN ∂t = PNL(θN) + PNS(z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='14) θN(z, 0) = PNθ0(z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='15) PERIDYNAMIC PINN 11 where θN(z, t) ∈ SN for every 0 ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' To obtain the convergence of the semi-discrete scheme, we need of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3 ([12, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' For any real 0 ≤ µ ≤ s, there exists a positive constant C such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='16) ∥u − PNu∥Hµ s,wµ([−1,1]) ≤ C Ns−µ ∥θ∥Hsw([−1,1]) , for every θ ∈ Hs w([−1, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Recalling that K and hm are locally Lipschitz in their respective domains, we can prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Let s ≥ 1 and θ(z, t) ∈ Xs be the solution to the initial-boundary-valued problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6) and θN(z, t) be the solution to the semi-discrete scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='14)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Then, there exists a positive constant C, independent on N, such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='17) ��θ − θN�� X1 ≤ C(T) � 1 N �s−1 ∥u∥Xs , for any initial data θ0 ∈ Hs w([−1, 1]) and for any T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Let s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Using the triangular inequality, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='18) ��θ − θN�� X1 ≤ ∥θ − PNθ∥X1 + ��PNθ − θN�� X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3 implies ∥(θ − PNθ)(·, t)∥H1w([−1,1]) ≤ C Ns−1 ∥θ(·, t)∥Hsw([−1,1]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Therefore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='19) ∥θ − PNθ∥X1 ≤ C Ns−1 ∥θ∥Xs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Subtracting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='14) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='6) and taking the weighted inner product with PNθ − θN ∈ SN, we have 0 = � 1 −1 � ∂tθ(z, t) − ∂tθN(z, t) � � PNθ(z, t) − θN(z, t) � w(z) dz � �� � =:I1 − � 1 −1 � L(θ(z, t)) − PNL(θN(z, t)) � � PNθ(z, t) − θN(z, t) � w(z) dz � �� � =:I2 − � 1 −1 (S(z, t) − PNS(z, t)) � PNθ(z, t) − θN(z, t) � w(z) dz � �� � =:I3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='20) The orthogonal condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='12) implies that � 1 −1 (∂tθ(z, t) − PN∂tθ(z, t)) � PNθ(z, t) − θN(z, t) � w(z) dz = 0, 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' BERARDI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' DIFONZO, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' PELLEGRINO and � 1 −1 (S(z) − PNS(z)) � PNθ(z, t) − θN(z, t) � w(z) dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Thus, I1 = � 1 −1 (∂tθ(z, t) − PN∂tθ(z, t)) � PNθ(z, t) − θN(z, t) � w(z) dz + � 1 −1 � PN∂tθ(z, t) − θN t (z, t) � � PNθ(z, t) − θN(z, t) � w(z) dz = 1 2 d dt ��(PNθ − θN)(·, t) ��2 H1w([−1,1]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='21) Since it is straightforward to see that I3 = 0, we can focus on I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='12) it follows that � 1 −1 � L(θN(z, t)) − PNL(θN(z, t)) � � PNθ(z, t) − θN(z, t) � w(z) dz = 0, and so, due to the locally Lipschitzianity of K and hm and from Cauchy and triangular inequalities, we obtain I2 = � 1 −1 � L(θ(z, t)) − L(θN(z, t)) � � PNθ(z, t) − θN(z, t) � w(z) dz ≤ C ��� θ − θN� (·, t) ��2 H1w([−1,1]) + C ��(PNθ − θN)(·, t) ��2 H1w([−1,1]) ≤ 2C ∥(θ − PNθ) (·, t)∥2 H1w([−1,1]) + 3C ��(PNθ − θN)(·, t) ��2 H1w([−1,1]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='22) Plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='21) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='22) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='20), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='23) d dt ��(PNθ − θN)(·, t) ��2 H1w([−1,1]) ≤ 4C ∥(θ − PNθ)(·, t)∥2 H1w([−1,1])+6C ��(PNθ − θN)(·, t) ��2 H1w([−1,1]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Since ��(PNθ − θN)(·, 0) �� H1w([−1,1]) = 0, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3 and Gronwall’s inequality imply that ��(PNθ − θN)(·, t) ��2 H1([−1,1]) ≤ 4C � t 0 e6C(t−τ) ∥(θ − PNθ)(·, τ)∥2 H1w([−1,1]) dτ ≤ C(T) N2s−2 � t 0 ∥θ(·, τ)∥2 H1w([−1,1]) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='24) ��PNθ − θN�� X1 ≤ C(T) Ns−1 ∥θ∥Xs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Finally, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='24) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='18), we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Numerical Simulations In this section we test our proposed method on different soils, possibly with a sink forcing term, representing the water uptake due to root systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Van Genuchten - Mualem constitutive relations are considered in the following numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' PERIDYNAMIC PINN 13 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Numerical solution relative to Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Drawing from [4], we consider a soil with the following parameters: θr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='075, θS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='287, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='036, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='56, KS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='94e − 3 cm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We added a sink term S = −700 s−1 and parameter δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='15 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We set our initial and boundary conditions as follows θ(0, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2234 � 1 − t T � + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='181 t T , t ∈ [0, T], θ(Z, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1368 � 1 − t T � + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1174 t T , t ∈ [0, T], θ(z, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2234 − � 1 − z Z � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='0848 2 , z ∈ [0, Z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Here Z = 30 cm, T = 60 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' moreover, we used ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='06 s and ∆x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Results are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' As already considered by [24], we select a Glendale clay loam, charac- terized by the following parameters θr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1060, θS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='4686, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='0104, n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3954, KS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='5162e − 4 cm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We put neither sink nor source on this simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Peridynamic parameter is δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='15 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Our boundary conditions are constant with values θ(0, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2, t ∈ [0, T], θ(Z, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3, t ∈ [0, T], 0 5 10 N 15 t= 0 s t= 7 s t=13 s 20 t=20 s t=27 s t=33 s t=40 s 25 t=47 s t=53 s t=60 s 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='24 014 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' BERARDI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' DIFONZO, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' PELLEGRINO Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Numerical solution relative to Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' while initial condition follows a nonlinear profile of the form θ(z, 0) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='05z3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='25, z ∈ [0, Z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We select Z = 70 cm, T = 2400 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' moreover, we used ∆t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='4 s and ∆x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The resulting water content profiles are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' As in [24], we consider a Berino loamy fine sand, with parameters θr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='0286, θS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3658, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='0280, n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2390, KS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='0063 cm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We added a sink term S = −100 s−1 and parameter δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='15 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We set our initial and boundary conditions as follows θ(0, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3 � 1 − t T � + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='29 t T , t ∈ [0, T], θ(Z, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2, t ∈ [0, T], while initial condition has a nonlinear profile θ(z, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='05z3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='25, z ∈ [0, Z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We select Z = 70 cm, T = 600 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' moreover, we used ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='06 s and ∆x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Results are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Conclusions and future works Starting from an appropriate and physically based rewriting of Richards’ equation using non-locality theory, we propose to compute its numerical solution using a semi- discretized time forward scheme based on Chebyshev transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We prove that such 0 t= 0 s t=267 s 10 t=533 s t=800 s t=1067 s 20 t=1333 s t=1600 s t=1867 s 30 t=2133 s t=2400 s N 40 50 60 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3 0PERIDYNAMIC PINN 15 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Numerical solution relative to Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' approach converges in suitable Sobolev spaces, providing a theoretical background for further extensions of the present work to higher dimensional domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We also propose a new kind of convolutional kernel, or influence function, in order to manage the peridy- namic behavior of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Such an influence function distributes its effect on the domain so to correctly catch boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' In fact, we experienced major benefits from this choice, as numerical convergence turns out to be robust with respect to time, and compared to results coming from classical choices of influence functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' To testify our theoretical analysis, we have performed several experiments, on a wide range on soils, in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' We have considered different Dirichlet boundary conditions and linear and non-linear initial conditions and show that, with suitable discretization step-sizes, our method is reliable and accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' The present work suggests several possible directions for future and already ongoing research studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' For instance, it would be of interest applying Eulerian-Lagrangian methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' [1]) in the proposed peridynamic framework for Richards’ equation, while the idea of introducing a basic control approach on the boundary conditions, as in [7], could be adapted as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Another development would consider non-local terms in time, so to resort to numerical solvers coming from specific tools in fractional differential calculus (see [22, 20]), or, even more promisingly, by spectral methods in 2D (see [33, 27, 31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 0 10 20 30 N t= 0 s 40 t=67 s t=133 s t=200 s 50 t=267 s t=333 s t=400 s 60 t=467 s t=533 s t=600 s 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='3 016 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' BERARDI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' DIFONZO, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' PELLEGRINO Acknowledgments All authors are member of the INdAM Research group GNCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' MB and FVD also acknowledge the partial support of the 2022 project “Modelli di evoluzione non lo- cali: analisi, trattamento numerico e algoritmi” funded by GNCS-INdAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' SFP ac- knowledges the partial support of “Finanziamento giovani ricercatori 2022” funded by GNCS-INdAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' MB acknowledges the partial support of the CNR project “MENTOR”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' FVD has been supported by REFIN Project, grant number 812E4967 funded by Re- gione Puglia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' SFP has been supported by REFIN Project, grant number D1AB726C funded by Regione Puglia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Abreu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' De la cruz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Juajibioy, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Lambert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Lagrangian-Eulerian Approach for Nonlocal Conservation Laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Journal of Dynamics and Differential Equations, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Arbogast, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Wheeler, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' A Nonlinear Mixed Finite Element Method for a Degenerate Parabolic Equation Arising in Flow in Porous Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' SIAM Journal on Numerical Analysis, 33(4):1669—-1687, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' [3] G¨unter Baszenski and Manfred Tasche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Fast polynomial multiplication and convolutions related to the discrete cosine transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Linear Algebra and Its Applications, 252(1-3):1 – 25, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Berardi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Difonzo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Notarnicola, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Vurro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' A transversal method of lines for the numerical modeling of vertical infiltration into the vadose zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Applied Numerical Mathematics, 135:264 – 275, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Berardi, F.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='it Dipartimento di Matematica, Universit`a degli Studi di Bari Aldo Moro, Via E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' Orabona 4, 70125 Bari, Italy Email address: fabio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='difonzo@uniba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='it Dipartimento di Management, Finanza e Tecnologia, Universit`a LUM Giuseppe Degen- naro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content=' 100 Km 18, 70010 Casamassima,Italy Email address: pellegrino@lum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} +page_content='it' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf'} diff --git a/NNAyT4oBgHgl3EQfgvgA/content/tmp_files/2301.00362v1.pdf.txt b/NNAyT4oBgHgl3EQfgvgA/content/tmp_files/2301.00362v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..881055b7d617c9e14d3b5469eb7ba47811f2bfc2 --- /dev/null +++ b/NNAyT4oBgHgl3EQfgvgA/content/tmp_files/2301.00362v1.pdf.txt @@ -0,0 +1,1512 @@ +1 +Goal-guided Transformer-enabled Reinforcement +Learning for Efficient Autonomous Navigation +Wenhui Huang, Student Member, IEEE, Yanxin Zhou, Xiangkun He, Member, IEEE, and Chen Lv, Senior +Member, IEEE +Abstract—Despite some successful applications of goal-driven +navigation, +existing +deep +reinforcement +learning-based +ap- +proaches notoriously suffers from poor data efficiency issue. One +of the reasons is that the goal information is decoupled from +the perception module and directly introduced as a condition +of decision-making, resulting in the goal-irrelevant features of +the scene representation playing an adversary role during the +learning process. In light of this, we present a novel Goal-guided +Transformer-enabled reinforcement learning (GTRL) approach +by considering the physical goal states as an input of the scene +encoder for guiding the scene representation to couple with the +goal information and realizing efficient autonomous navigation. +More specifically, we propose a novel variant of the Vision Trans- +former as the backbone of the perception system, namely Goal- +guided Transformer (GoT), and pre-train it with expert priors to +boost the data efficiency. Subsequently, a reinforcement learning +algorithm is instantiated for the decision-making system, taking +the goal-oriented scene representation from the GoT as the input +and generating decision commands. As a result, our approach +motivates the scene representation to concentrate mainly on goal- +relevant features, which substantially enhances the data efficiency +of the DRL learning process, leading to superior navigation per- +formance. Both simulation and real-world experimental results +manifest the superiority of our approach in terms of data effi- +ciency, performance, robustness, and sim-to-real generalization, +compared with other state-of-art baselines. Demonstration videos +are available at https://youtu.be/93LGlGvaN0c. +Index Terms—Autonomous navigation, deep reinforcement +learning, goal guidance, transformer, learning efficiency. +I. INTRODUCTION +R +EINFORCEMENT Learning (RL) algorithms have sig- +nificantly contributed to a wide range of domains over +the past years, including but not limited to autonomous driving +[1, 2], unmanned ground vehicle (UGV) navigation [3, 4], +and computer games [5]. With the representative capability +of handling high-dimensional states, recent RL algorithms, +e.g., deep Q-learning (DQN) [6, 7], Soft Actor-Critic (SAC) +[8, 9], are increasingly adopted by the robotics community to +address decision-making problems, especially for autonomous +navigation. +Conventional autonomous navigation methods that rely on +prior knowledge of maps have been well-studied thanks to the +Simultaneous Localization and Mapping (SLAM) technique +[10]. In reality, however, such an approach significantly de- +pends on the map’s precision and might even fail in an un- +W. Huang, Y. Zhou, X. He, and C. Lv are with the School of Mechan- +ical and Aerospace Engineering, Nanyang Technological University, Sin- +gapore, 639798. (E-mail: wenhui001@e.ntu.edu.sg, yanxin001@e.ntu.edu.sg, +xiangkun.he@ntu.edu.sg, lyuchen@ntu.edu.sg) +Corresponding author: Chen Lv. (E-mail: lyuchen@ntu.edu.sg) +known environment. Therefore, developing a simple mapless +navigation strategy directly utilizing sensor input, such as laser +scan [11, 12] or visual images [13, 14] is an emerging field +that is garnering significant attention in current UGV research. +Especially having the advantage of visual fidelity, depth image- +based autonomous navigation has been intensively studied by +several works [15, 16]. Similarly, segmentation images are +often employed in mapless end-to-end navigation as well due +to their powerful representative capability [17, 18]. In order +to approach a target position, the approaches mentioned above +train their models in a goal-conditional learning manner [19], +directly concatenating the physical goal information (i.e., goal +location in polar coordinate) with the latent states from the +perception system (i.e., convolutional neural network) and feed +into subsequent networks. Despite various degrees of success, +these methods decouple the goal information from the scene +representation, leading to poor data efficiency. For instance, +latent states from the goal information-less scene encoder may +include certain mismatched features that are unnecessary for +reaching a goal position and thus play an adverse role during +the RL training process. +Self-attention-based approaches, especially Transformers +[20], have become the dominant model of choice in the +natural language processing field. Motivated by adapting a +standard Transformer architecture to images with the fewest +modifications, a variant named Vision Transformer (ViT) [21] +that can deal with image input is proposed in the computer +vision community and has been applied to various domains, +such as robotic manipulation [22] and autonomous driving +[23]. However, there has yet to be an existing work that +develops ViT-enabled DRL algorithms to UGV for realizing +mapless autonomous navigation, especially for goal-driven +tasks. +In +light +of +this, +we +present +a +novel +Goal-guided +Transformer-enabled reinforcement learning (GTRL) approach +by considering the physical goal states as an input of the scene +encoder for guiding the scene representation to couple with the +goal information and achieving efficient autonomous naviga- +tion. To realize a ViT architecture that treats both physical +and visual states as the input, we propose a novel ViT variant +with minimal modifications, which we call Goal-guided Trans- +former (GoT) for the rest of the paper, as the backbone of our +perception system. Then, we instantiate a GoT-enabled actor- +critic algorithm, namely GoT-SAC, for the decision-making +system, receiving the goal-oriented scene representation from +the perception system and generating decision commands +for the UGV. To boost the data efficiency, we pre-train the +arXiv:2301.00362v1 [cs.RO] 1 Jan 2023 + +2 +GoT with expert priors and then learn the decision-making +with the subsequent RL process. As a result, our method +makes the scene representation more interpretable in terms +of reaching the goal information, which is confirmed through +qualitative and quantitative evaluations. Most importantly, such +an approach motivates the scene representation to concentrate +mainly on goal-relavent features, which substantially enhances +the data efficiency of the DRL learning process, leading +to superior navigation performance. Therefore, the proposed +approach is an efficient DRL-based autonomous navigation +method for UGV from the goal-driven task perspective. We +summarize the main contributions of this paper as follows: +1) A novel and Transformer architecture-based DRL ap- +proach, Goal-guided Transformer-enabled reinforcement +learning (GTRL), is realized to achieve an efficient goal- +driven autonomous navigation for the UGV. +2) A novel Transformer architecture is proposed, which +we call Goal-guided Transformer (GoT) in this paper, +through minimal modifications of ViT to handle the +multimodal input: physical goal states and visual states. +Most importantly, the GoT enables the scene representa- +tion to concentrate mainly on the goal-relavent features, +significantly enhancing the data efficiency of the DRL +learning process from the goal-driven task perspective. +3) As for the practical contribution, we instantiated a +concrete GoT-enabled DRL algorithm for the proposed +method and validated it both in simulation and the +physical world. The experimental results demonstrate +the clear superiority of the proposed approach in data +efficiency and performance compared with other state- +of-art baselines. Moreover, the investigation of goal- +driven navigation in the unknown environment confirms +our approach’s robustness and sim-to-real transferability. +II. RELATED WORKS +Due to the powerful representative capability to high dimen- +sional states and superior data efficiency, DRL algorithms are +gaining increasing attention among the robot community, es- +pecially for autonomous navigation of the UGV. Several works +that infer the decision and control commands from laser scans +have been proposed thanks to their robust transfer performance +from the simulation to the real world. For instance, Zhelo +et al. [24] trains the Asynchronous Advantage Actor-Critic +(A3C) algorithm with intrinsic reward signals measured by +curiosity to achieve mapless navigation. In [25], the steering +angle is discretized into seven actions and trained together +with the forward commands by the Advantage Actor-Critic +(A2C) algorithm, and the trained model is applied to real- +world obstacle avoidance. Similarly, Tai et al. [26] presents +goal-driven mapless navigation based on an asynchronous +Deep Deterministic Policy Gradient (DDPG) algorithm and +successfully generalizes the learned model to the physical +environment. In order to reduce the training time, Pfeiffer et al. +[3] proposes to pre-train the Constrained Policy Optimization +(CPO) algorithm with IL and then continuously train it in the +RL manner. +Nevertheless, laser scans cannot provide sufficient infor- +mation to describe the environment in some cases [14], and +thus scholars turn their attention to visual sensors-based ap- +proaches. In [16], a DDPG algorithm that considers depth +images as input is employed to train the control policy by +switching the different controllers. Similarly, work from Tai +et al. [27] utilizes the same depth-based information but +combines Behavior Cloning (BC) and Generative Adversarial +Imitation Learning (GAIL) to demonstrate the enhanced per- +formance of the social force-driven path planning. Mousavian +et al. [17] presents a deep learning model consisting of the +Convolutional Networks (ConvNets) and Long Short-Term +Memory (LSTM) network to make a decision among seven +commands based on semantic segmentation images in real- +time. However, existing works mainly learn goal-driven tasks +in a goal-conditional learning manner which is mentioned +in [19]. Though this work focuses on conditional imitation +learning (CIL), the logic behind utilizing goal information +is similar to the DRL-based methods, treating the physical +goal information as a condition of the decision-making and +directly concatenating to the latent states provided by the +perception system. On the contrary, we consider the physical +goal information as an input of the scene encoder, rather +than a condition, to extract matching features w.r.t. the goal- +driven autonomous navigation and improve the DRL learning +efficiency. +ViT-based architecture has been a dominant choice not +only for CV tasks but also for robotic research to achieve +better scene representation and analysis. In [22], the ViT is +utilized as one of the encoders to measure the stability of their +manipulation approach. Similarly, Godoy et al. [28] proposes +a temporal multi-channel ViT to classify the hand motions +for achieving better control of the bionic hands. To learn a +more effective global context of the scene, Kargar and Kyrki +[23] presents a perception utilizing ViT instead of ConvNets +architecture, handling with the birds-eye-view (BEV) images. +The simulation results indicate that such an encoder can +identify the significant surrounding cars for the ego car to learn +a safe and effective policy in complex environments. Another +ViT-based work related to Vehicle-to-Everything (V2X) is +presented in [29]. They propose a robust cooperative per- +ception framework by means of building a holistic attention +model, effectively integrating information across road users. +Despite various degrees of success in the above domains, to +our best knowledge, no current work develops ViT-enabled +DRL algorithms for UGV’s mapless autonomous navigation, +especially for goal-driven tasks. +III. PRELIMINARIES +A. Reinforcement Learning +The objective of goal-driven autonomous navigation is that +infers the linear and angular velocity of the UGV from +the input states, including images and goal information. We +consider such a task as a standard Markov decision process +(MDP) formulated by a tuple < 풮, 풜, 풫, ℛ >, where 풮 is +a set of states denoting the possible condition of the agent +and environment, 풜 represents action space, 풫 models the +transition of the environment, and ℛ is the reward function +evaluating the future overall payoff. At each time step t, the RL + +3 +agent percepts the state 푠푡 ∈ 풮 and executes an action 푎푡 ∈ 풜, +receiving an immediate reward 푟푡 = ℛ(푠푡, 푎푡) : 풮 × 풜 → R, +as well as next state 푠푡+1 ∈ 풮 based on the transition proba- +bility 풫(푠푡+1|푠푡, 푎푡) : 풮 × 풜 → [0, 1]. Usually, the RL agent +selects an action based on a policy 푎푡 ∼ 휋(·|푠푡) : 풮 → 풜, +which represents a probability distribution denoting the belief +that the agent holds about its decision at each time step. The +target of the RL agent is to maximize the discounted total +return along the future from an initial state 푠, i.e., 푉휋(푠), +denoted as: +푉휋(푠) = +E +푠푡∼풫[ +푇 +� +푡=0 +훾푡 · 푟푡], +(1) +where 푉휋 is called value function and 훾 is the discounting +factor constrained by 0 < 훾 ≤ 1. Similarly, the state-value +function 푄휋 based on the state 푠푡 and the action 푎푡 at time +step t is defined as: +푄휋(푠푡, 푎푡) = 푟푡 + 훾 · +E +푠푡+1∼풫[푉휋(푠푡+1)] += 푟푡 + 훾 · +E +푠푡+1∼풫, 푎푡+1∼휋[푄휋(푠푡+1, 푎푡+1)]. +(2) +In the actor-critic method, an optimal policy 휋∗ can be +obtained by maximizing the overall future payoff for all states +along one trajectory. Additionally, an entropy term can be +augmented to the objective to prevent the policy from trapping +in the local optima in the early stage [30]: +푚푎푥 +휋 +E +푠푡∼풫,푎푡∼휋[ +푇 +� +푡=0 +훾푡(푟푡 + 훼ℋ(휋(·|푠푡)))] +(3) +B. Behavior Cloning +As one technique of imitation learning (IL) method, be- +havior cloning (BC) aims at directly mimicking the decision +policy given a set of state-action pairs 풟 = {< 푠푖, 푎푖 >}푁 +푖=1, +where N represents the number of samples. Therefore, it is +a supervised learning problem by minimizing the statistical +distance between action 푎푖 and parameterized function approx- +imator F(푠푖; 휃): +푚푖푛푖푚푖푧푒 +휃 +� +푖 +ℒ(F(푠푖; 휓), 푎푖) +(4) +where ℒ indicates loss function. Usually, we assume the action +푎E is directly from a human expert which means the Eq. 4 can +be reformulated as: +푚푖푛푖푚푖푧푒 +휃 +� +푖 +ℒ(F(푠푖; 휓), 푎E +푖 ) +(5) +C. Vision Transformer +The main idea behind ViT is splitting the images into +patches and mapping them into linear embeddings in the same +way the standard Transformer architecture treats tokens in +natural language processing (NLP). Given an input image +푥 ∈ R퐻×푊×퐶, the ViT first reshapes it into a sequence of +symbol representation (푥1, 푥2, ...푥푛), where (퐻, 푊, 퐶) are the +resolution and channel dimension of the input image 푥 and +푥푛 ∈ R푁×(푃2·퐶) is a representation of flattened 2D patches with +Goal-guided +Transformer +Latent +Features +Goal-oriented +Scene +Representation +Soft +Actor-Critic +Input +Goal State +RGB Images +Interactive +Environment +State +Transition +Goal-guided Transformer-enabled +Reinforcement Learning (GTRL) +Visual Attention Flow +RGB Image +Multi-Head Self-Attention +Qualitative Analysis +Quantitative Analysis +Goal-oriented Scene Representation +Gini +Coefficient +Shannon-Wiener +Index +Metrics +Fig. 1: Overall framework of the proposed approach. +the resolution 푃. Therefore, the total number of 2D patches +can be calculated as follows: +푁 = 퐻 · 푊 +푃2 +(6) +Then, the input of the ViT encoder can be obtained by +augmenting the position embeddings E푝표푠 ∈ R(푁+1)×퐷 to D- +dimensional flattened 2D patches: +푧0 = [푥0; LP(푥1); LP(푥2); · · · ; LP(푥푛)] + E푝표푠 +(7) +where LP represents linear projection, and 푥0 ∈ R1×퐷 is +an extra learnable embedding called class token. By feeding +the embedded patches into the classic Transformer encoder, +we can get multi-head self-attention (MSA) through the self- +attention (SA) mechanism: +푀푆퐴(푄, 퐾, 푉) = LP([ATT1(푄, 퐾, 푉); ATT2(푄, 퐾, 푉); · · · ; ATT푘(푄, 퐾, 푉)]) +(8) +where 푘 denotes k-th head and ATT indicates self-attention +(SA) mechanism. As demonstrated in [20], we compute SA +through the query Q, keys K, and values V: +ATT(푄, 퐾, 푉) = 푠표 푓 푡푚푎푥(푄퐾푇 +� +푑푘 +)푉 +[푄, 퐾, 푉] = LP(푧) +(9) +where z represents a set of embedded patches and 푑푘 is a +scaling factor. +IV. METHODOLOGY +A. Framework +Realizing goal-driven autonomous navigation requires the +DRL-based approach to understand and analyze the goal +information. One possible solution is to treat the parameterized +goal states as an input rather than a condition, feeding it +together with the visual input, such as raw RGB images, to +enhance the capability of the scene representation. Specifically, +we learn the goal-oriented scene representation through a novel + +4 +Human +Decision +Visual State +Input +Tokens +· 0 +· 1 +· 2 +· 3 +· 4 +· 5 +· 6 +· 7 +· 8 +· 9 +Embedded +Tokens +Linear Projection of Patch-Tokens +Goal-guided Transformer Encoder +Layer Normalization +MLP +Layer Normalization +Fully Connected Layer +Multi-head Attention +Multi-head Attention +Multi-head Attention +Multi-head Attention +Perception +Imitation +Learning +Expert +Priors +Decision +Decision Making +Goal +Information +Physical State +MLP +Hierarchical +Architecture +Human +Expert +Fig. 2: Goal-guided Transformer Architecture and Pre-train with Expert Priors. +Transformer-based architecture that considers multimodal (i.e., +physical goal states and visual states) input as a sequence +of continuous representations. In light of this, we term the +backbone of our perception system Goal-guided Transformer +(GoT). Once the goal-oriented latent features are extracted, +we motivate the SAC algorithm to learn the decision policy +for approaching the goal position by interacting with the +environment. Therefore, the two main ingredients, GoT and +Transformer architecture-based SAC algorithm, complete our +approach that we term Goal-guided Transformer-enabled rein- +forcement learning (GTRL). +The overall framework of our approach is depicted in Fig. 1. +In our case, the input consists of two parts, i.e., goal position +in polar coordinates and raw fisheye RGB images stacked +over four frames. In the first stage, they are flattened into the +same dimension and fed into the GoT encoder. Then, these +embedded patches are encoded to goal-relevant latent features +through the MSA and provided to the subsequent decision- +making system. Finally, the GoT-SAC algorithm makes a +decision according to the goal-relevant latent features, and +the UGV executes the decision command to trigger the state +transition of the environment. After the algorithm converges, +we qualitatively (visual attention flow maps) and quantitatively +(Gini coefficient and Shannon-Wiener Index) evaluate the +trained model in terms of the SA mechanism to analyze and +interpret the significance of the goal-oriented scene represen- +tation (Section V). +B. Goal-guided Transformer +In order to deal with the multimodality of the input, i.e., +the goal states and visual images, we propose a novel variant +of the ViT that we term GoT in this paper. In model design, +We construct the architecture of the GoT by the minimum +modification of ViT for the purpose of a simple setup. Specif- +ically, inspired by BERT [31], we define a special goal token +풢 ∈ R1×퐷 that is mapped from input goal states 푠푔표푎푙 ∈ R1×2 +through a multilayer perception (MLP) network: +풢 = MLP(푠푔표푎푙) +(10) +Therefore, the embeddings of GoT can be formulated as: +푧 +′ +0 = E푖푛푝푢푡(CONCAT(LP(푠), 풢)) +푧0 = 푧 +′ +0 + +E푝표푠 +(11) +where CONCAT represents tensor concatenate operation. By +feeding the embeddings to the GoT encoder: +푧 +′ +푙 = MSA(LN(푧푙−1)) + 푧푙−1 +푧푙 = FC(LN(MLP(푧 +′ +푙) + 푧 +′ +푙)) +(12) +where l indicates the l-th block. We decide the depth of GoT +as two blocks in this work, and hence, the latent features can +be obtained from the output of the second block, denoted as: +ℎ = 푮풐푻(CONCAT(LP(푠), 풢); 휑) +(13) +where 휑 represents parameters of the GoT. +Figure 2 illustrates an overview of the GoT architecture and +pre-train process. As the figure shows, the input consists of +two modalities: goal information as the physical state and raw +RGB images as the visual state. The physical state is fed into +MLP network and encoded as feature patches while the visual +state is decomposed to eight by eight small image patches +(we illustrate this process with three by three image patches +in the figure due to limited space). Therefore, we can obtain + +Latent Features +Goal-relevant5 +Fig. 3: RGB images from the fisheye camera stacked for most recent four frames. The upside pair of figures show the raw +RGB images, whereas those on the downside illustrate pixel-level Gaussian noise-augmented images after preprocessing. +complete input tokens by integrating both kinds of patches. +Furthermore, we add position embeddings for each input token +and fix the one encoded from goal information to the first +position in particular. As for the GoT encoder, it consists of +an MSA block, the MLP, the fully connected layer (FC), the +layer normalization operation [32], and the residual connec- +tions [33]. Considering the limited computational power and +lightweight design, we employ two blocks of the encoder with +only four heads per block. Having the latent features from +perception and the subsequent decision system, we are able to +perform deep imitation learning through expert demonstration +data to pre-train the GoT, boosting the subsequential learning +efficiency. In a standard IL, in terms of the goal-driven end- +to-end navigation problem, the function approximator depends +on the environment state 푠푖 and goal state 푠{푔표푎푙, 푖}: +푚푖푛푖푚푖푧푒 +휓, 휓푠 +� +푖 +ℒ(F(F푠(푠푖; 휓푠), 푠{푔표푎푙, 푖}; 휓), 푎E +푖 ) +(14) +where 휓 is the parameters of the function approximator. In +our proposed approach, however, the goal state is no longer +a condition but an input. Thus, the objective of goal-oriented +imitation learning becomes: +푚푖푛푖푚푖푧푒 +휓 +� +푖 +ℒ(F(CONCAT(LP(푠푖), 풢푖); 휓), 푎E +푖 ) +(15) +In our case, such a design is essential since we aim to +guide the scene representation to couple with the physical goal +information so that the perception can extract goal-relevant and +rational features to promote the data efficiency of the subse- +quent goal-driven decision process. To clearly demonstrate the +point, we visualize goal-oriented scene representation through +visual attention flow maps [34] and quantitatively evaluate +the reliability of our approach in section V. Additionally, this +design allows us to generalize the Transformer architecture to +the multimodal input while keeping the original characteristics. +C. Goal-guided Transformer-enabled Reinforcement Learning +As mentioned in section IV-A, the input of GTRL consists +of two ingredients: visual states and goal states in polar +coordinates. In this work, we employ 160 × 120 raw RGB +images from a fisheye camera and stack the four most recent +frames. Additionally, we augment a pixel-level noise to the +input images to learn a more robust and transferable decision +policy for the sim-to-real experiments. Figure 3 demonstrates +the difference between the original images and our input. The +upside pair of four figures show the most recent four raw +RGB images from the fisheye camera, whereas those on the +downside illustrate Gaussian noise-augmented images that are +utilized for training our algorithm. As for the goal state 푠푔표푎푙, +we provide it in a 2-D dimensional manner with the relative +distance and heading error. Specifically, we define the first +dimension of the goal state as the normalized relative distance +and compute it as: +푑푡 = min( +∥푝<푥,푦> +푡 +− 푞<푥,푦>∥2 +휆 +, 1.0) +(16) +where 푝<푥,푦> +푡 +denotes the real-time position of the UGV, +푞<푥,푦> indicates an arbitrary location of the goal point, ∥ · ∥2 +represents euclidean norm operation, and 휆 is a constant +normalizer that maps the relative distance in the range of [0, +1]. Correspondingly, we associate the second dimension of the +goal state as the heading error between UGV’s orientation and +the directional vector points to the goal position: +Δ휑푡 = atan� +(푞<푦> − 푝<푦> +푡 +), (푞<푥> − 푝<푥> +푡 +)� +− 휓푡 +(17) +where 휓푡 represents the heading angle of the UGV. Similar to +the relative distance, we normalize the heading error as: +Δ휑푡 = +����� +����� +Δ휑푡−2휋 +휋 +, +푖 푓 Δ휑푡 > 휋 +Δ휑푡+2휋 +휋 +, +푖 푓 Δ휑푡 < −휋 +Δ휑푡 +휋 , +표푡ℎ푒푟푤푖푠푒 +(18) + +6 +Algorithm 1 Goal-guided Transformer-enabled Reinforcement +Learning (GTRL) +Initialize Goal-guided Transformer (GoT) network with pre- +trained parameters: 휑∗. +Initialize actor and critic network parameters: 휙, 휃. +Initialize entropy parameters: 훼. +Initialize batch size N and replay buffer 풟 ← ∅. +Assign target parameters: 휃푡푎푟푔푒푡 ← 휃. +for episode=1 to E do +Initialize the environment state: 푠푡 ∼ 퐸푛푣 +Initialize the goal state: 푠{푔표푎푙, 푡} ∼ 퐸푛푣 +for step=1 to S do +Map goal token: 풢푡 = MLP(푠{푔표푎푙, 푡}) +Scene Representation: +ℎ푡 ← 푮풐푻(CONCAT(LP(푠푡), 풢푡); 휑∗) +Sample an action: 푎푡 ← 휋휙(푎푡|ℎ푡) +Interact with the environment: +푟푡, 푠푡+1, 푠{푔표푎푙, 푡+1} ∼ 퐸푛푣 +Store the transition: +풟 ← 풟 ∪ (푠푡, 푠{푔표푎푙, 푡}, 푎푡, 푟푡, 푠푡+1, 푠{푔표푎푙, 푡+1}) +If time to update critic then +Sample a batch of the data: +(푠푖 +푡, 푠푖 +{푔표푎푙, 푡}, 푎푖 +푡, 푟푖 +푡, 푠푖 +푡+1, 푠푖 +{푔표푎푙, 푡+1}) +푁 +푖=1 ∼ 풟 +Compute critic (MBSE) loss: ℒ(휃) +Update parameters of critic network. +end if +If time to update actor then +Sample a batch of the data: +(푠푖 +푡, 푠푖 +{푔표푎푙, 푡}, 푎푖 +푡, 푟푖 +푡, 푠푖 +푡+1, 푠푖 +{푔표푎푙, 푡+1}) +푁 +푖=1 ∼ 풟 +Compute actor loss: ℒ(휙) +Update parameters of actor network. +If automatic tune is True then +Update temperature parameter 훼. +end if +end if +If time to update target network then +Update target network: 휃푡푎푟푔푒푡 ← 휃. +end if +end for +end for +Receiving the above-mentioned input, the GTRL outputs +decision commands 푎푡 = [푣푡, 휔푡], i.e., linear velocity 푣푡 ∈ +[0, 1] and angular velocity 휔푡 ∈ [− 휋 +2 , 휋 +2 ], and delivers them +to the UGV through the Robot Operating System (ROS). +The target of autonomous navigation is to demonstrate +a goal-driven decision and collision-free path planning for +reaching the goal position. Therefore, we carefully design +the reward function in combination with the continuous and +sparse reward to boost the converge efficiency of the GTRL. +More specifically, the overall payoff consists of four individual +ingredients as follows: +푟푡 = 푟ℎ + 푟푎 + 푟푔 + 푟푐 +(19) +where 푟ℎ denotes heuristic reward, 푟푎 represents action reward, +푟푔 indicates reward for arriving the goal position, and 푟푐 is the +(a) Gazebo Environment. +(b) UGV. +Fig. 4: Laboratory environment and UGV model. +collision penalty. The heuristic reward is designed to motivate +the UGV to move toward the goal position: +푟ℎ = 휂ℎ × (∥푝<푥,푦> +푡−1 +− 푞<푥,푦>∥2 − ∥푝<푥,푦> +푡 +− 푞<푥,푦>∥2) +(20) +where 휂ℎ is a constant weight. Similarly, we design the action +reward to drive the UGV to approach the goal position as +soon as possible but with the minimum number of steering +operations: +푟푎 = 푣푡 − 휂푎 × abs(휔푡) +(21) +where abs is an absolute value operation. Last but not least, +two sparse rewards, i.e., the goal reach reward and the collision +penalty, are designed as follows: +푟푔 = +� +100, +푖 푓 푑푡 <= 휉 +0, +표푡ℎ푒푟푤푖푠푒 +푟푐 = +� +−100, +푖 푓 푐표푙푙푖푠푖표푛 +0, +표푡ℎ푒푟푤푖푠푒 +(22) +where 휉 represents a constant margin w.r.t. the goal position. +Subsequently, given the extracted latent features ℎ푡 from +GoT at a specific timestep t, the SAC algorithm learns the de- +cision policy 휋(푎푡|ℎ푡) based on the reward function mentioned +above. One common technique widely utilized in the SAC +algorithm is double Q-networks to tackle the over-estimation +issue. Hence, the parameters of the critic network of GoT- +SAC are updated by minimizing the mean bellman-squared +error (MBSE) loss function: +ℒ(휃푖) = +E +ℎ푡∼풫,푎푡∼휋∥푄휋 +휃푖(ℎ푡, 푎푡) − (푟푡 + 훾 · ˆ푄휋)∥2 +(23) +where ˆ푄휋 is the state-action value of the next step from double +target Q-networks and calculated by: +ˆ푄휋 = +E +ℎ푡+1∼풫,푎푡+1∼휋 +� +min +푖=1,2푄휋 +휃푡푎푟푔푒푡 +푖 +(ℎ푡+1, 푎푡+1) − 훼 · 푙표푔휋(푎푡+1|ℎ푡+1) +� +(24) +where 훼 is a temperature parameter that trades off between +the stochasticity of the optimal policy and the state-action +value. Accordingly, the actor network updates its parameters +by maximizing the soft state-action function: +ℒ(휙) = +E +ℎ푡∼풫,푎푡∼휋 +� +min +푖=1,2푄휋 +휃푖(ℎ푡, 푎푡) − 훼 · 푙표푔휋휙(푎푡|ℎ푡) +� +(25) +The detailed implementation of our approach is provided in +Algorithm 1. + +T7 +0 +100 +200 +300 +400 +500 +300 +200 +100 +0 +100 +200 +GoT-SAC w/ Pre-Train +GoT-SAC w/o Pre-Train +ViT-SAC (Goal Conditional) +ConvNet-SAC (Goal Conditional) +Reward +Episode +Fig. 5: Convergence curve comparison. The red dotted line +and solid lines represent the average rewards of our algorithms +and baselines per episode, while the shaded areas depict the +variances over five runs. +V. EXPERIMENTS +A. Baseline Algorithms +To benchmark the proposed GTRL method for trustworthy +end-to-end autonomous navigation, we employ state-of-the-art +RL and IL algorithms as baselines to compare the qualitative +and quantitative performance both in simulation and the real +world. +1) ConvNet-SAC [8]: A state-of-art off-policy DRL algo- +rithm that employs ConvNets as its scene representation +encoder. We augment the physical goal state to the latent +features encoded from ConvNets in a goal-conditional +manner. +2) ViT-SAC: This baseline is derived from a state-of- +art ViT-based DRL algorithm called ViT-DQN [23], +which employs ViT-DINO as the backbone of the DQN +encoder. Without losing the original vital characteristics, +we replace the DQN with SAC to fit the end-to-end +navigation demand and call it ViT-SAC in the rest of +the paper. +3) MultiModal CIL [35]: A state-of-art conditional IL +(CIL) algorithm that considers the human command or +goal vector as a condition in the learning process. We +select the command-input method among two architec- +tures proposed in the original work to fit the goal-driven +autonomous navigation task. +4) MoveBase Planner: A conventional planner widely uti- +lized in UGV for goal-driven autonomous navigation. To +be fair enough, we turn off the global map while keep- +ing an eight-by-eight local map for real-time obstacle +avoidance. +In addition, we employ vanilla GoT-SAC to learn the policy +from scratch without any expert priors during the reinforce- +ment training process to validate our proposed algorithm’s data +efficiency thoroughly. +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +ViT-SAC + GoT-SAC +ConvNet-SAC +w/ Pre-Train +Success Rate + GoT-SAC +w/o Pre-Train +Fig. 6: Success Rate Boxplot. The black-solid line and ”star” +located at the box body denote the median and average, while +the hollow circles represent the outliers. +B. Simulation Assessment +All algorithms are trained on a computer equipped with an +Intel Core i7-10700 CPU, 64 GB of RAM, and an NVIDIA +GTX 1660 SUPER graphics card. A high-fidelity autonomous +navigation simulator, Gazebo, is employed to build the realistic +laboratory environment and the UGV model for goal-driven +mapless navigation, shown in Fig. 4. We train the instantiated +algorithm for 500 episodes with a maximum of 200 steps for +each. An episode ends when the goal position is reached, +a collision occurs, or the UGV runs out of maximum step +numbers. To well generalize the DRL-based policy and achieve +better sim-to-real transferability, we not only augment a pixel- +level Gaussian noise to the RGB image but also vary the +initial location and goal position for each episode. Though +the proposed algorithm only needs one fisheye camera for +autonomous navigation, we also set a laser sensor in the sim- +ulation to detect the collision (4(b)). Furthermore, we employ +the Robot Operating System (ROS) open-source platform to +communicate with Gazebo and derive the goal information +through subscribing to odometry messages. +Figure 5 illustrates the learning curves of GoT-SAC and +all the DRL-based baselines. We run each algorithm with five +different random seeds to measure statistics and evaluate the +robustness. Specifically, the red dotted line and solid lines rep- +resent the average rewards of our algorithms and baselines per +episode, while the shaded areas depict the variances over five +runs. As the figure shows, both versions of GoT-SAC achieve +higher reward levels with relatively lower variances than those +of goal-conditional DRL-based algorithms, which indicates +the significance of the goal-oriented scene representation. +Moreover, both GoT-SAC models exhibit a faster convergence +and enhance the training efficiency by over 129% and 86% +compared with the ViT-SAC model. It should be noticed that +though the convergence pace of ConvNet-SAC at the early +stage is slightly faster due to its relatively small number of +parameters, the average episode return is much lower than our +proposed algorithm. To evaluate the performance, we validate + +8 +(a) Scenario I. +(b) Scenario II. +(c) Scenario III. +Fig. 7: Attention Flow Visualization. The left pair of diagrams for each subfigure shows the original RGB image and goal +information, while the right side diagram depicts the revised RGB image masked by the attention flow. A red square highlights +the queried image patch, and the attention level is represented through a color transition from blue (low) to red (high). a) +Scenario I: query for 59th image patch occupied with drivable space, b) Scenario II: query for 60th image patch occupied with +drivable space, c) Scenario III: query for 34th image patch occupied with obstacles. +TABLE I: Quantitative statistics of the self-attention mechanism behavior for the three goal-driven tasks. +Model +Episode I +Episode II +Episode III +Gini Coefficient +Shannon-Wiener Index +Gini Coefficient +Shannon-Wiener Index +Gini Coefficient +Shannon-Wiener Index +GoT-SAC +0.927 +0.896 +0.848 +1.133 +0.901 +0.984 +ViT-SAC +0.802 +1.613 +0.616 +1.807 +0.695 +1.545 +all the trained policies with 20 random seeds and run for 50 +episodes for each seed. The success rate, which is obtained +as the number of goal-reached episodes divided by the total +runs, is employed as the metric for the evaluation. From Fig. +6, we can observe the dominant success rate and superior +robustness of the proposed algorithm compared with other +baselines regardless of varying a wide range of random seeds. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Normal +Success Rate +Attention +Perturbation +Perturbation +w/ 10% Attention +w/ Avg. Attention +Fig. 8: Success Rate Comparison in terms of the different +attention levels. The black-solid line and ”star” located at the +box body denote the median and average, while the hollow +circles represent the outliers. +C. Attention Visualization and Evaluation +Besides the superior efficiency and performance, the GTRL +approach also possesses a significant advantage in model in- +terpretability thanks to the goal-oriented scene representation. +To analyze the rationale behind fast convergence and excellent +performance of our algorithm, we extract the attention from +the GoT encoder w.r.t. randomly sampled RGB images and +visualize it in Fig. 7. As the figure shows, the left pair of +diagrams for each subfigure shows the original RGB image +and goal information, while the right side diagram depicts +the visual attention flow map [34] masked by the extracted +attention. The queried image patch is highlighted by a red +square, and the attention level is represented through a color +transition from blue (low) to red (high). In Scenario I (Fig. +7(a)), the UGV is facing the oncoming T intersection, and +the goal position locates on the left side behind the office +chair and table. From the visual attention flow map, we can +observe that the overall attention generates a visual path by +mainly focusing on goal-oriented image patches. It should be +noticed that the orientation of such a visual path obviously +towards the goal position though the right turn is also feasible +in this scenario, which proves that the scene representation +successfully couples with the goal information. Similarly, a +clear goal-driven visual path is shown in Scenario II (Fig. +7(b)). In Fig. 7(c), different from the previous two scenarios, +we query for the image patch that occupies an obstacle (office +chair) instead of drivable space. We surprisingly find that the +attention highlights most of the adjacent obstacles, evidently +pointing out the undrivable regions. Therefore, we qualitatively +verify that our approach can provide a clear explanation of how +the UGV analyzes the scene and arrives at the destination with + +9 +Xavier NX +Stereo Camera +Fisheye Camera +9-axis IMU +(a) SCOUTMINI. +(b) Robotics Research Center. +Fig. 9: The real UGV platform and sim-to-real experiment +environment. a) SCOUTMINI: an omnidirectional steering +mobile robot from Agilex. b) Robotics research center: an +indoor laboratory space in Nanyang Technological University. +a collision-free path. +Furthermore, we quantitatively evaluate and compare our +approach with the ViT-SAC (goal-conditional) model to sup- +port the above conclusion. Specifically, we run each model +with three random episodes and measure the statistical char- +acteristics by averaging the whole frame. Two automated unsu- +pervised metrics: Gini coefficient for measuring the evenness +of the attention weights distribution [36] and Shannon-Wiener +index for evaluating the concentration of the attention [37] +are reported in Table I. Both metrics clearly reveal that the +attention of the GoT-SAC model is sparser and tends to be +more concentrated on task-related image patches, proving that +better interpretability is achieved through goal-oriented scene +representation. +Last but not least, we also investigate the impact of the +significant attention through perturbation-based method [38] +to observe how modifications of critical attention affect the +navigation task performance. In light of this, we measure +the success rate of the GoT-SAC model over another twenty +random seeds with fifty episodes for each by dynamically +replacing the essential attention (weights higher than 0.995) +with a moving average and 10% of the original value. The +boxplot illustrated in Figure 8 shows the overall result. We +can observe that the performance of the GoT-SAC model, the +one that decreases the significant attention to 10%, degrades +catastrophically (62.5%) in terms of the success rate. Though +the success rate of the model employing the average perturba- +tion method is slightly higher than the previous one, it is still +clearly lower than the normal GoT-SAC model, indicating the +significance of the attention learned by our approach. +D. Sim-To-Real Assessment +In addition to evaluating the feasibility and performance +of the algorithm in a virtual simulation environment, we also +expect to apply our approach in real-world navigation tasks. In +terms of the UGV, we use the omnidirectional steering mobile +platform from Agilex called SCOUTMINI. The SCOUTMINI +equips with an edge computing platform NVIDIA Jetson +Xavier, a ZED2i stereo camera, an inertial measurement unit +(IMU), and a fisheye camera with an ultra-wide FOV of 220 +degrees (Fig. 9(a)). Regarding the software, we deliver the +goal information and raw RGB images to the UGV through +the ROS. Then, the GoT-SAC sends the real-time decision +inference to the UGV chassis via CAN communication to +realize motion control. In this real-world experiment, all the +algorithms are applied at the Robotics Research Center at +Nanyang Technological University to complete a loop naviga- +tion task, as shown in Fig. 9(b). More specifically, we design +four destinations that motivate the UGV to reach one by one +with a small break after each arrival and finally return to the +vicinity of the starting point. This experiment aims to test +the algorithm’s ability to avoid static obstacles and quickly +navigate to given goal positions. +Figure 10 illustrates the qualitative measurement of per- +formance for each algorithm. We plot both trajectories from +the UGV and the human with two different colors, blue for +ground truth and red for human engagement, respectively. As +shown in the figure, the GoT-SAC policy performs smooth +and collision-free navigation, while the other three algorithms +(ConvNet-SAC, MultiModal CIL, and MoveBase) all need +human engagement to arrive at the destinations. Surprisingly, +we find that ViT-SAC policy also demonstrates an equally +excellent performance despite the low average success rate +during the evaluation in the simulation environment. It is +reasonable since we select the best model for each algorithm +for sim-to-real transfer assessment. It may also indicate the +significance of the self-attention mechanism for goal-driven +autonomous navigation. +To compare the performance and robustness of the policies +in a deeper sight, we employ six statistical metrics for each +goal-driven task: the average and variance of traveling dis- +tance, average and variance of navigation time, success rate, +and engagement number. Especially the successful arrival is +determined if the UGV reaches each goal position within one +minute, and we actively engage the UGV control once the col- +lision is likely to happen. A detailed quantitative measurement +is reported in the Table. II. It is clear that the GoT-SAC model +demonstrates dominant performance and robustness from all +the domains compared with other baselines, including ViT- +SAC and MoveBase planner. The performance of ViT-SAC +is also comparably excellent, besides the longest navigated +distance for the fourth destination and high average time for +the third goal position. The worst performance is provided +by the MultiModal CIL model, whose success rate is only +20% for reaching the third goal position. We can observe a +similar performance from the statistical result of the ConvNet- +SAC model in terms of the number of engagements, which is +15 in total. As for the MoveBase planner, the performance +highly depends on the local cost-map quality, especially in +the turning cases (the second and fourth goal position). For +instance, the local cost-map occurs false detection frequently +due to limited field of view and occlusion from the obstacles, +leading to improper path planning. Overall, both quantitative +and qualitative results in this sim-to-real experiment highlight +the superiority of the proposed algorithm compared with +other baselines, including against state-of-art leaning-based +approaches and the classic UGV navigation method. +Additionally, our approach is tested in an unknown envi- +ronment to validate the generalization capability thoroughly. + +10 +(a) GoT-SAC. +(b) ViT-SAC. +(c) ConvNet-SAC. +(d) MultiModal CIL. +(e) MoveBase. +Fig. 10: Qualitative measurement of proposed algorithm and baselines. The solid blue line depicts the ground truth of the +trajectory, while the solid red line represents the human-engaged path. +TABLE II: Quantitative performance of proposed algorithm compared with baselines. +Approach +Goal Position +Avg. Dist. +Var. Dist. +Avg. Time. +Var. Time +Success Rate +Engage Number +GoT-SAC +1st +6.044 +± 0.049 +14.048 +± 0.282 +100% +0 +2nd +5.747 +± 0.110 +11.987 +± 0.368 +100% +0 +3rd +6.171 +± 0.036 +12.821 +± 0.076 +100% +0 +4th +6.360 +± 0.113 +13.902 +± 1.287 +100% +0 +ViT-SAC +1st +5.958 +± 0.036 +15.274 +± 0.394 +100% +0 +2nd +6.506 +± 0.189 +15.432 +± 0.741 +100% +0 +3rd +6.226 +± 0.015 +22.502 +± 1.601 +100% +0 +4th +7.274 +± 0.210 +16.449 +± 0.722 +100% +0 +ConvNet-SAC +1st +6.150 +± 0.294 +17.918 +± 2.563 +100% +4 +2nd +6.780 +± 0.252 +20.735 +± 1.519 +100% +5 +3rd +6.322 +± 0.245 +18.757 +± 2.683 +100% +6 +4th +5.971 +± 0.123 +13.550 +± 0.414 +100% +0 +MultiModal CIL +1st +5.852 +± 0.021 +13.458 +± 0.180 +100% +0 +2nd +9.868 +± 0.480 +57.263 +± 6.026 +60% +5 +3rd +6.666 +± 0.134 +75.121 +± 12.019 +20% +5 +4th +6.913 +± 0.217 +28.893 +± 4.475 +100% +0 +MoveBase +1st +5.822 +± 0.063 +12.180 +± 0.088 +100% +0 +2nd +7.070 +± 0.529 +23.446 +± 9.388 +100% +4 +3rd +6.092 +± 0.141 +14.010 +± 3.122 +100% +1 +4th +6.510 +± 0.517 +24.098 +± 5.442 +100% +5 +Due to the unstable connection and limitation of hardware, +we select an unseen office environment rather than an outdoor +space. It is worthwhile to test the generalization and trans- +ferability of the proposed algorithm in such an environment +since it has a number of corner cases to be addressed, such +as planning a collision-free path in narrow corridors, handling +unseen obstacles (in terms of shape and color), and performing +U-turn operation in order to reach the goal position. Figure 11 +demonstrates the details of the experiment, where the yellow +circle labels the initial and goal positions, and the performed +trajectory is highlighted with the solid blue line. In particular, +the UGV has to pass a narrow corridor to reach the first two +destinations and perform 90-degree-turn and U-turn operations +to arrive last two goal positions. Nevertheless, the GoT-SAC +model can still approach all five goals without collision or +engagement, indicating the excellent generalization capability +and transferability of our approach. +VI. CONCLUSION +This paper presents a Transformer-enabled DRL approach, +namely GTRL, to realize efficient goal-driven autonomous +navigation. Specifically, we first propose a novel Transformer- +based architecture called Goal-guided Transformer (GoT) for +the perception to consider the goal information as an input +of the scene representation rather than a condition. For the +purpose of boosting data efficiency, deep imitation learning +is employed to pre-train the GoT. Then, a GoT-enabled soft +actor-critic algorithm (GoT-SAC) is instantiated to train the de- +cision policy based on the goal-oriented scene representation. +As a result, our approach motivates the scene representation to + +2341213424132314241311 +1 +2 +3 +4 +5 +0 +0 → 1 +1 → 2 +2 → 3 +3 → 4 +4 → 5 +Real-time Pose +Goal Pose +Fig. 11: Office Environment. 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Attentional bottleneck: +Towards an interpretable deep driving network. +In +Proceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition Workshops, pages 322– +323, 2020. +[38] ´Eloi Zablocki, H´edi Ben-Younes, Patrick P´erez, and +Matthieu Cord. +Explainability of deep vision-based +autonomous driving systems: Review and challenges. +International Journal of Computer Vision, pages 1–28, +2022. + diff --git a/NNAyT4oBgHgl3EQfgvgA/content/tmp_files/load_file.txt b/NNAyT4oBgHgl3EQfgvgA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d018253a7325b878670886b02021f7e08fc6177e --- /dev/null +++ b/NNAyT4oBgHgl3EQfgvgA/content/tmp_files/load_file.txt @@ -0,0 +1,692 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf,len=691 +page_content='1 Goal-guided Transformer-enabled Reinforcement Learning for Efficient Autonomous Navigation Wenhui Huang, Student Member, IEEE, Yanxin Zhou, Xiangkun He, Member, IEEE, and Chen Lv, Senior Member, IEEE Abstract—Despite some successful applications of goal-driven navigation, existing deep reinforcement learning-based ap- proaches notoriously suffers from poor data efficiency issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' One of the reasons is that the goal information is decoupled from the perception module and directly introduced as a condition of decision-making, resulting in the goal-irrelevant features of the scene representation playing an adversary role during the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In light of this, we present a novel Goal-guided Transformer-enabled reinforcement learning (GTRL) approach by considering the physical goal states as an input of the scene encoder for guiding the scene representation to couple with the goal information and realizing efficient autonomous navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' More specifically, we propose a novel variant of the Vision Trans- former as the backbone of the perception system, namely Goal- guided Transformer (GoT), and pre-train it with expert priors to boost the data efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Subsequently, a reinforcement learning algorithm is instantiated for the decision-making system, taking the goal-oriented scene representation from the GoT as the input and generating decision commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' As a result, our approach motivates the scene representation to concentrate mainly on goal- relevant features, which substantially enhances the data efficiency of the DRL learning process, leading to superior navigation per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Both simulation and real-world experimental results manifest the superiority of our approach in terms of data effi- ciency, performance, robustness, and sim-to-real generalization, compared with other state-of-art baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Demonstration videos are available at https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='be/93LGlGvaN0c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Index Terms—Autonomous navigation, deep reinforcement learning, goal guidance, transformer, learning efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' INTRODUCTION R EINFORCEMENT Learning (RL) algorithms have sig- nificantly contributed to a wide range of domains over the past years, including but not limited to autonomous driving [1, 2], unmanned ground vehicle (UGV) navigation [3, 4], and computer games [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' With the representative capability of handling high-dimensional states, recent RL algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=', deep Q-learning (DQN) [6, 7], Soft Actor-Critic (SAC) [8, 9], are increasingly adopted by the robotics community to address decision-making problems, especially for autonomous navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Conventional autonomous navigation methods that rely on prior knowledge of maps have been well-studied thanks to the Simultaneous Localization and Mapping (SLAM) technique [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In reality, however, such an approach significantly de- pends on the map’s precision and might even fail in an un- W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' He, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Lv are with the School of Mechan- ical and Aerospace Engineering, Nanyang Technological University, Sin- gapore, 639798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' (E-mail: wenhui001@e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='sg, yanxin001@e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='sg, xiangkun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='he@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='sg, lyuchen@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='sg) Corresponding author: Chen Lv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' (E-mail: lyuchen@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='sg) known environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Therefore, developing a simple mapless navigation strategy directly utilizing sensor input, such as laser scan [11, 12] or visual images [13, 14] is an emerging field that is garnering significant attention in current UGV research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Especially having the advantage of visual fidelity, depth image- based autonomous navigation has been intensively studied by several works [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Similarly, segmentation images are often employed in mapless end-to-end navigation as well due to their powerful representative capability [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In order to approach a target position, the approaches mentioned above train their models in a goal-conditional learning manner [19], directly concatenating the physical goal information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=', goal location in polar coordinate) with the latent states from the perception system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=', convolutional neural network) and feed into subsequent networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Despite various degrees of success, these methods decouple the goal information from the scene representation, leading to poor data efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' For instance, latent states from the goal information-less scene encoder may include certain mismatched features that are unnecessary for reaching a goal position and thus play an adverse role during the RL training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Self-attention-based approaches, especially Transformers [20], have become the dominant model of choice in the natural language processing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Motivated by adapting a standard Transformer architecture to images with the fewest modifications, a variant named Vision Transformer (ViT) [21] that can deal with image input is proposed in the computer vision community and has been applied to various domains, such as robotic manipulation [22] and autonomous driving [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' However, there has yet to be an existing work that develops ViT-enabled DRL algorithms to UGV for realizing mapless autonomous navigation, especially for goal-driven tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In light of this, we present a novel Goal-guided Transformer-enabled reinforcement learning (GTRL) approach by considering the physical goal states as an input of the scene encoder for guiding the scene representation to couple with the goal information and achieving efficient autonomous naviga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' To realize a ViT architecture that treats both physical and visual states as the input, we propose a novel ViT variant with minimal modifications, which we call Goal-guided Trans- former (GoT) for the rest of the paper, as the backbone of our perception system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Then, we instantiate a GoT-enabled actor- critic algorithm, namely GoT-SAC, for the decision-making system, receiving the goal-oriented scene representation from the perception system and generating decision commands for the UGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' To boost the data efficiency, we pre-train the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='00362v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='RO] 1 Jan 2023 2 GoT with expert priors and then learn the decision-making with the subsequent RL process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' As a result, our method makes the scene representation more interpretable in terms of reaching the goal information, which is confirmed through qualitative and quantitative evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Most importantly, such an approach motivates the scene representation to concentrate mainly on goal-relavent features, which substantially enhances the data efficiency of the DRL learning process, leading to superior navigation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Therefore, the proposed approach is an efficient DRL-based autonomous navigation method for UGV from the goal-driven task perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' We summarize the main contributions of this paper as follows: 1) A novel and Transformer architecture-based DRL ap- proach, Goal-guided Transformer-enabled reinforcement learning (GTRL), is realized to achieve an efficient goal- driven autonomous navigation for the UGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 2) A novel Transformer architecture is proposed, which we call Goal-guided Transformer (GoT) in this paper, through minimal modifications of ViT to handle the multimodal input: physical goal states and visual states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Most importantly, the GoT enables the scene representa- tion to concentrate mainly on the goal-relavent features, significantly enhancing the data efficiency of the DRL learning process from the goal-driven task perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 3) As for the practical contribution, we instantiated a concrete GoT-enabled DRL algorithm for the proposed method and validated it both in simulation and the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The experimental results demonstrate the clear superiority of the proposed approach in data efficiency and performance compared with other state- of-art baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Moreover, the investigation of goal- driven navigation in the unknown environment confirms our approach’s robustness and sim-to-real transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' RELATED WORKS Due to the powerful representative capability to high dimen- sional states and superior data efficiency, DRL algorithms are gaining increasing attention among the robot community, es- pecially for autonomous navigation of the UGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Several works that infer the decision and control commands from laser scans have been proposed thanks to their robust transfer performance from the simulation to the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' For instance, Zhelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' [24] trains the Asynchronous Advantage Actor-Critic (A3C) algorithm with intrinsic reward signals measured by curiosity to achieve mapless navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In [25], the steering angle is discretized into seven actions and trained together with the forward commands by the Advantage Actor-Critic (A2C) algorithm, and the trained model is applied to real- world obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Similarly, Tai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' [26] presents goal-driven mapless navigation based on an asynchronous Deep Deterministic Policy Gradient (DDPG) algorithm and successfully generalizes the learned model to the physical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In order to reduce the training time, Pfeiffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' [3] proposes to pre-train the Constrained Policy Optimization (CPO) algorithm with IL and then continuously train it in the RL manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Nevertheless, laser scans cannot provide sufficient infor- mation to describe the environment in some cases [14], and thus scholars turn their attention to visual sensors-based ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In [16], a DDPG algorithm that considers depth images as input is employed to train the control policy by switching the different controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Similarly, work from Tai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' [27] utilizes the same depth-based information but combines Behavior Cloning (BC) and Generative Adversarial Imitation Learning (GAIL) to demonstrate the enhanced per- formance of the social force-driven path planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Mousavian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' [17] presents a deep learning model consisting of the Convolutional Networks (ConvNets) and Long Short-Term Memory (LSTM) network to make a decision among seven commands based on semantic segmentation images in real- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' However, existing works mainly learn goal-driven tasks in a goal-conditional learning manner which is mentioned in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Though this work focuses on conditional imitation learning (CIL), the logic behind utilizing goal information is similar to the DRL-based methods, treating the physical goal information as a condition of the decision-making and directly concatenating to the latent states provided by the perception system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' On the contrary, we consider the physical goal information as an input of the scene encoder, rather than a condition, to extract matching features w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' the goal- driven autonomous navigation and improve the DRL learning efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' ViT-based architecture has been a dominant choice not only for CV tasks but also for robotic research to achieve better scene representation and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In [22], the ViT is utilized as one of the encoders to measure the stability of their manipulation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Similarly, Godoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' [28] proposes a temporal multi-channel ViT to classify the hand motions for achieving better control of the bionic hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' To learn a more effective global context of the scene, Kargar and Kyrki [23] presents a perception utilizing ViT instead of ConvNets architecture, handling with the birds-eye-view (BEV) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The simulation results indicate that such an encoder can identify the significant surrounding cars for the ego car to learn a safe and effective policy in complex environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Another ViT-based work related to Vehicle-to-Everything (V2X) is presented in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' They propose a robust cooperative per- ception framework by means of building a holistic attention model, effectively integrating information across road users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Despite various degrees of success in the above domains, to our best knowledge, no current work develops ViT-enabled DRL algorithms for UGV’s mapless autonomous navigation, especially for goal-driven tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' PRELIMINARIES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Reinforcement Learning The objective of goal-driven autonomous navigation is that infers the linear and angular velocity of the UGV from the input states, including images and goal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' We consider such a task as a standard Markov decision process (MDP) formulated by a tuple < 풮, 풜, 풫, ℛ >, where 풮 is a set of states denoting the possible condition of the agent and environment, 풜 represents action space, 풫 models the transition of the environment, and ℛ is the reward function evaluating the future overall payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' At each time step t, the RL 3 agent percepts the state 푠푡 ∈ 풮 and executes an action 푎푡 ∈ 풜, receiving an immediate reward 푟푡 = ℛ(푠푡, 푎푡) : 풮 × 풜 → R, as well as next state 푠푡+1 ∈ 풮 based on the transition proba- bility 풫(푠푡+1|푠푡, 푎푡) : 풮 × 풜 → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Usually, the RL agent selects an action based on a policy 푎푡 ∼ 휋(·|푠푡) : 풮 → 풜, which represents a probability distribution denoting the belief that the agent holds about its decision at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The target of the RL agent is to maximize the discounted total return along the future from an initial state 푠, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=', 푉휋(푠), denoted as: 푉휋(푠) = E 푠푡∼풫[ 푇 � 푡=0 훾푡 · 푟푡], (1) where 푉휋 is called value function and 훾 is the discounting factor constrained by 0 < 훾 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Similarly, the state-value function 푄휋 based on the state 푠푡 and the action 푎푡 at time step t is defined as: 푄휋(푠푡, 푎푡) = 푟푡 + 훾 · E 푠푡+1∼풫[푉휋(푠푡+1)] = 푟푡 + 훾 · E 푠푡+1∼풫, 푎푡+1∼휋[푄휋(푠푡+1, 푎푡+1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' (2) In the actor-critic method, an optimal policy 휋∗ can be obtained by maximizing the overall future payoff for all states along one trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Additionally, an entropy term can be augmented to the objective to prevent the policy from trapping in the local optima in the early stage [30]: 푚푎푥 휋 E 푠푡∼풫,푎푡∼휋[ 푇 � 푡=0 훾푡(푟푡 + 훼ℋ(휋(·|푠푡)))] (3) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Behavior Cloning As one technique of imitation learning (IL) method, be- havior cloning (BC) aims at directly mimicking the decision policy given a set of state-action pairs 풟 = {< 푠푖, 푎푖 >}푁 푖=1, where N represents the number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Therefore, it is a supervised learning problem by minimizing the statistical distance between action 푎푖 and parameterized function approx- imator F(푠푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 휃): 푚푖푛푖푚푖푧푒 휃 � 푖 ℒ(F(푠푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 휓), 푎푖) (4) where ℒ indicates loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Usually, we assume the action 푎E is directly from a human expert which means the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 4 can be reformulated as: 푚푖푛푖푚푖푧푒 휃 � 푖 ℒ(F(푠푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 휓), 푎E 푖 ) (5) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Vision Transformer The main idea behind ViT is splitting the images into patches and mapping them into linear embeddings in the same way the standard Transformer architecture treats tokens in natural language processing (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Given an input image 푥 ∈ R퐻×푊×퐶, the ViT first reshapes it into a sequence of symbol representation (푥1, 푥2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='푥푛),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' where (퐻,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 푊,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 퐶) are the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='resolution and channel dimension of the input image 푥 and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='푥푛 ∈ R푁×(푃2·퐶) is a representation of flattened 2D patches with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Goal-guided ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Transformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Latent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Goal-oriented ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Scene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Soft ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Actor-Critic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Goal State ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='RGB Images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Interactive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Environment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='State ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Transition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Goal-guided Transformer-enabled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Reinforcement Learning (GTRL) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Visual Attention Flow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='RGB Image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Multi-Head Self-Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Qualitative Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Quantitative Analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Goal-oriented Scene Representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Gini ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Shannon-Wiener ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Metrics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 1: Overall framework of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' the resolution 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Therefore, the total number of 2D patches can be calculated as follows: 푁 = 퐻 · 푊 푃2 (6) Then, the input of the ViT encoder can be obtained by augmenting the position embeddings E푝표푠 ∈ R(푁+1)×퐷 to D- dimensional flattened 2D patches: 푧0 = [푥0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' LP(푥1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' LP(푥2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' LP(푥푛)] + E푝표푠 (7) where LP represents linear projection, and 푥0 ∈ R1×퐷 is an extra learnable embedding called class token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' By feeding the embedded patches into the classic Transformer encoder, we can get multi-head self-attention (MSA) through the self- attention (SA) mechanism: 푀푆퐴(푄, 퐾, 푉) = LP([ATT1(푄, 퐾, 푉);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' ATT2(푄, 퐾, 푉);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' ATT푘(푄, 퐾, 푉)]) (8) where 푘 denotes k-th head and ATT indicates self-attention (SA) mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' As demonstrated in [20], we compute SA through the query Q, keys K, and values V: ATT(푄, 퐾, 푉) = 푠표 푓 푡푚푎푥(푄퐾푇 � 푑푘 )푉 [푄, 퐾, 푉] = LP(푧) (9) where z represents a set of embedded patches and 푑푘 is a scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Framework Realizing goal-driven autonomous navigation requires the DRL-based approach to understand and analyze the goal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' One possible solution is to treat the parameterized goal states as an input rather than a condition, feeding it together with the visual input, such as raw RGB images, to enhance the capability of the scene representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='we learn the goal-oriented scene representation through a novel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Human ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Decision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Visual State ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Tokens ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Embedded ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Tokens ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Linear Projection of Patch-Tokens ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Goal-guided Transformer Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Layer Normalization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Layer Normalization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Fully Connected Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Multi-head Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Multi-head Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Multi-head Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Multi-head Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Perception ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Imitation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Expert ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Priors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Decision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Decision Making ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Goal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Physical State ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Hierarchical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Architecture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Human ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Expert ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 2: Goal-guided Transformer Architecture and Pre-train with Expert Priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Transformer-based architecture that considers multimodal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=', physical goal states and visual states) input as a sequence of continuous representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In light of this, we term the backbone of our perception system Goal-guided Transformer (GoT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Once the goal-oriented latent features are extracted, we motivate the SAC algorithm to learn the decision policy for approaching the goal position by interacting with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Therefore, the two main ingredients, GoT and Transformer architecture-based SAC algorithm, complete our approach that we term Goal-guided Transformer-enabled rein- forcement learning (GTRL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The overall framework of our approach is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In our case, the input consists of two parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=', goal position in polar coordinates and raw fisheye RGB images stacked over four frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In the first stage, they are flattened into the same dimension and fed into the GoT encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Then, these embedded patches are encoded to goal-relevant latent features through the MSA and provided to the subsequent decision- making system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Finally, the GoT-SAC algorithm makes a decision according to the goal-relevant latent features, and the UGV executes the decision command to trigger the state transition of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' After the algorithm converges, we qualitatively (visual attention flow maps) and quantitatively (Gini coefficient and Shannon-Wiener Index) evaluate the trained model in terms of the SA mechanism to analyze and interpret the significance of the goal-oriented scene represen- tation (Section V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Goal-guided Transformer In order to deal with the multimodality of the input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=', the goal states and visual images, we propose a novel variant of the ViT that we term GoT in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In model design, We construct the architecture of the GoT by the minimum modification of ViT for the purpose of a simple setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Specif- ically, inspired by BERT [31], we define a special goal token 풢 ∈ R1×퐷 that is mapped from input goal states 푠푔표푎푙 ∈ R1×2 through a multilayer perception (MLP) network: 풢 = MLP(푠푔표푎푙) (10) Therefore, the embeddings of GoT can be formulated as: 푧 ′ 0 = E푖푛푝푢푡(CONCAT(LP(푠), 풢)) 푧0 = 푧 ′ 0 + +E푝표푠 (11) where CONCAT represents tensor concatenate operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' By feeding the embeddings to the GoT encoder: 푧 ′ 푙 = MSA(LN(푧푙−1)) + 푧푙−1 푧푙 = FC(LN(MLP(푧 ′ 푙) + 푧 ′ 푙)) (12) where l indicates the l-th block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' We decide the depth of GoT as two blocks in this work, and hence, the latent features can be obtained from the output of the second block, denoted as: ℎ = 푮풐푻(CONCAT(LP(푠), 풢);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 휑) (13) where 휑 represents parameters of the GoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Figure 2 illustrates an overview of the GoT architecture and pre-train process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' As the figure shows, the input consists of two modalities: goal information as the physical state and raw RGB images as the visual state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The physical state is fed into MLP network and encoded as feature patches while the visual state is decomposed to eight by eight small image patches (we illustrate this process with three by three image patches in the figure due to limited space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Therefore, we can obtain Latent Features Goal-relevant5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 3: RGB images from the fisheye camera stacked for most recent four frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The upside pair of figures show the raw RGB images, whereas those on the downside illustrate pixel-level Gaussian noise-augmented images after preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' complete input tokens by integrating both kinds of patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Furthermore, we add position embeddings for each input token and fix the one encoded from goal information to the first position in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' As for the GoT encoder, it consists of an MSA block, the MLP, the fully connected layer (FC), the layer normalization operation [32], and the residual connec- tions [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Considering the limited computational power and lightweight design, we employ two blocks of the encoder with only four heads per block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Having the latent features from perception and the subsequent decision system, we are able to perform deep imitation learning through expert demonstration data to pre-train the GoT, boosting the subsequential learning efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In a standard IL, in terms of the goal-driven end- to-end navigation problem, the function approximator depends on the environment state 푠푖 and goal state 푠{푔표푎푙, 푖}: 푚푖푛푖푚푖푧푒 휓, 휓푠 � 푖 ℒ(F(F푠(푠푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 휓푠), 푠{푔표푎푙, 푖};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 휓), 푎E 푖 ) (14) where 휓 is the parameters of the function approximator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In our proposed approach, however, the goal state is no longer a condition but an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Thus, the objective of goal-oriented imitation learning becomes: 푚푖푛푖푚푖푧푒 휓 � 푖 ℒ(F(CONCAT(LP(푠푖), 풢푖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 휓), 푎E 푖 ) (15) In our case, such a design is essential since we aim to guide the scene representation to couple with the physical goal information so that the perception can extract goal-relevant and rational features to promote the data efficiency of the subse- quent goal-driven decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' To clearly demonstrate the point, we visualize goal-oriented scene representation through visual attention flow maps [34] and quantitatively evaluate the reliability of our approach in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Additionally, this design allows us to generalize the Transformer architecture to the multimodal input while keeping the original characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Goal-guided Transformer-enabled Reinforcement Learning As mentioned in section IV-A, the input of GTRL consists of two ingredients: visual states and goal states in polar coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In this work, we employ 160 × 120 raw RGB images from a fisheye camera and stack the four most recent frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Additionally, we augment a pixel-level noise to the input images to learn a more robust and transferable decision policy for the sim-to-real experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Figure 3 demonstrates the difference between the original images and our input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The upside pair of four figures show the most recent four raw RGB images from the fisheye camera, whereas those on the downside illustrate Gaussian noise-augmented images that are utilized for training our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' As for the goal state 푠푔표푎푙, we provide it in a 2-D dimensional manner with the relative distance and heading error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Specifically, we define the first dimension of the goal state as the normalized relative distance and compute it as: 푑푡 = min( ∥푝<푥,푦> 푡 − 푞<푥,푦>∥2 휆 , 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='0) (16) where 푝<푥,푦> 푡 denotes the real-time position of the UGV, 푞<푥,푦> indicates an arbitrary location of the goal point, ∥ · ∥2 represents euclidean norm operation, and 휆 is a constant normalizer that maps the relative distance in the range of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Correspondingly, we associate the second dimension of the goal state as the heading error between UGV’s orientation and the directional vector points to the goal position: Δ휑푡 = atan� (푞<푦> − 푝<푦> 푡 ), (푞<푥> − 푝<푥> 푡 )� − 휓푡 (17) where 휓푡 represents the heading angle of the UGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Similar to the relative distance, we normalize the heading error as: Δ휑푡 = ����� ����� Δ휑푡−2휋 휋 , 푖 푓 Δ휑푡 > 휋 Δ휑푡+2휋 휋 , 푖 푓 Δ휑푡 < −휋 Δ휑푡 휋 , 표푡ℎ푒푟푤푖푠푒 (18) 6 Algorithm 1 Goal-guided Transformer-enabled Reinforcement Learning (GTRL) Initialize Goal-guided Transformer (GoT) network with pre- trained parameters: 휑∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Initialize actor and critic network parameters: 휙, 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Initialize entropy parameters: 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Initialize batch size N and replay buffer 풟 ← ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Assign target parameters: 휃푡푎푟푔푒푡 ← 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' for episode=1 to E do Initialize the environment state: 푠푡 ∼ 퐸푛푣 Initialize the goal state: 푠{푔표푎푙, 푡} ∼ 퐸푛푣 for step=1 to S do Map goal token: 풢푡 = MLP(푠{푔표푎푙, 푡}) Scene Representation: ℎ푡 ← 푮풐푻(CONCAT(LP(푠푡), 풢푡);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 휑∗) Sample an action: 푎푡 ← 휋휙(푎푡|ℎ푡) Interact with the environment: 푟푡, 푠푡+1, 푠{푔표푎푙, 푡+1} ∼ 퐸푛푣 Store the transition: 풟 ← 풟 ∪ (푠푡, 푠{푔표푎푙, 푡}, 푎푡, 푟푡, 푠푡+1, 푠{푔표푎푙, 푡+1}) If time to update critic then Sample a batch of the data: (푠푖 푡, 푠푖 {푔표푎푙, 푡}, 푎푖 푡, 푟푖 푡, 푠푖 푡+1, 푠푖 {푔표푎푙, 푡+1}) 푁 푖=1 ∼ 풟 Compute critic (MBSE) loss: ℒ(휃) Update parameters of critic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' end if If time to update actor then Sample a batch of the data: (푠푖 푡, 푠푖 {푔표푎푙, 푡}, 푎푖 푡, 푟푖 푡, 푠푖 푡+1, 푠푖 {푔표푎푙, 푡+1}) 푁 푖=1 ∼ 풟 Compute actor loss: ℒ(휙) Update parameters of actor network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' If automatic tune is True then Update temperature parameter 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' end if end if If time to update target network then Update target network: 휃푡푎푟푔푒푡 ← 휃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' end if end for end for Receiving the above-mentioned input, the GTRL outputs decision commands 푎푡 = [푣푡, 휔푡], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=', linear velocity 푣푡 ∈ [0, 1] and angular velocity 휔푡 ∈ [− 휋 2 , 휋 2 ], and delivers them to the UGV through the Robot Operating System (ROS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The target of autonomous navigation is to demonstrate a goal-driven decision and collision-free path planning for reaching the goal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Therefore, we carefully design the reward function in combination with the continuous and sparse reward to boost the converge efficiency of the GTRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' More specifically, the overall payoff consists of four individual ingredients as follows: 푟푡 = 푟ℎ + 푟푎 + 푟푔 + 푟푐 (19) where 푟ℎ denotes heuristic reward, 푟푎 represents action reward, 푟푔 indicates reward for arriving the goal position, and 푟푐 is the (a) Gazebo Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' (b) UGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 4: Laboratory environment and UGV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' collision penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The heuristic reward is designed to motivate the UGV to move toward the goal position: 푟ℎ = 휂ℎ × (∥푝<푥,푦> 푡−1 − 푞<푥,푦>∥2 − ∥푝<푥,푦> 푡 − 푞<푥,푦>∥2) (20) where 휂ℎ is a constant weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Similarly, we design the action reward to drive the UGV to approach the goal position as soon as possible but with the minimum number of steering operations: 푟푎 = 푣푡 − 휂푎 × abs(휔푡) (21) where abs is an absolute value operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Last but not least, two sparse rewards, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=', the goal reach reward and the collision penalty, are designed as follows: 푟푔 = � 100, 푖 푓 푑푡 <= 휉 0, 표푡ℎ푒푟푤푖푠푒 푟푐 = � −100, 푖 푓 푐표푙푙푖푠푖표푛 0, 표푡ℎ푒푟푤푖푠푒 (22) where 휉 represents a constant margin w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' the goal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Subsequently, given the extracted latent features ℎ푡 from GoT at a specific timestep t, the SAC algorithm learns the de- cision policy 휋(푎푡|ℎ푡) based on the reward function mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' One common technique widely utilized in the SAC algorithm is double Q-networks to tackle the over-estimation issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' the parameters of the critic network of GoT- SAC are updated by minimizing the mean bellman-squared error (MBSE) loss function: ℒ(휃푖) = E ℎ푡∼풫,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='푎푡∼휋∥푄휋 휃푖(ℎ푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 푎푡) − (푟푡 + 훾 · ˆ푄휋)∥2 (23) where ˆ푄휋 is the state-action value of the next step from double target Q-networks and calculated by: ˆ푄휋 = E ℎ푡+1∼풫,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='푎푡+1∼휋 � min 푖=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='2푄휋 휃푡푎푟푔푒푡 푖 (ℎ푡+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 푎푡+1) − 훼 · 푙표푔휋(푎푡+1|ℎ푡+1) � (24) where 훼 is a temperature parameter that trades off between the stochasticity of the optimal policy and the state-action value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Accordingly, the actor network updates its parameters by maximizing the soft state-action function: ℒ(휙) = E ℎ푡∼풫,푎푡∼휋 � min 푖=1,2푄휋 휃푖(ℎ푡, 푎푡) − 훼 · 푙표푔휋휙(푎푡|ℎ푡) � (25) The detailed implementation of our approach is provided in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' T7 0 100 200 300 400 500 300 200 100 0 100 200 GoT-SAC w/ Pre-Train GoT-SAC w/o Pre-Train ViT-SAC (Goal Conditional) ConvNet-SAC (Goal Conditional) Reward Episode Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 5: Convergence curve comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The red dotted line and solid lines represent the average rewards of our algorithms and baselines per episode, while the shaded areas depict the variances over five runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Baseline Algorithms To benchmark the proposed GTRL method for trustworthy end-to-end autonomous navigation, we employ state-of-the-art RL and IL algorithms as baselines to compare the qualitative and quantitative performance both in simulation and the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 1) ConvNet-SAC [8]: A state-of-art off-policy DRL algo- rithm that employs ConvNets as its scene representation encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' We augment the physical goal state to the latent features encoded from ConvNets in a goal-conditional manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 2) ViT-SAC: This baseline is derived from a state-of- art ViT-based DRL algorithm called ViT-DQN [23], which employs ViT-DINO as the backbone of the DQN encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Without losing the original vital characteristics, we replace the DQN with SAC to fit the end-to-end navigation demand and call it ViT-SAC in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 3) MultiModal CIL [35]: A state-of-art conditional IL (CIL) algorithm that considers the human command or goal vector as a condition in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' We select the command-input method among two architec- tures proposed in the original work to fit the goal-driven autonomous navigation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 4) MoveBase Planner: A conventional planner widely uti- lized in UGV for goal-driven autonomous navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' To be fair enough, we turn off the global map while keep- ing an eight-by-eight local map for real-time obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In addition, we employ vanilla GoT-SAC to learn the policy from scratch without any expert priors during the reinforce- ment training process to validate our proposed algorithm’s data efficiency thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='0 ViT-SAC GoT-SAC ConvNet-SAC w/ Pre-Train Success Rate GoT-SAC w/o Pre-Train Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 6: Success Rate Boxplot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The black-solid line and ”star” located at the box body denote the median and average, while the hollow circles represent the outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Simulation Assessment All algorithms are trained on a computer equipped with an Intel Core i7-10700 CPU, 64 GB of RAM, and an NVIDIA GTX 1660 SUPER graphics card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' A high-fidelity autonomous navigation simulator, Gazebo, is employed to build the realistic laboratory environment and the UGV model for goal-driven mapless navigation, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' We train the instantiated algorithm for 500 episodes with a maximum of 200 steps for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' An episode ends when the goal position is reached, a collision occurs, or the UGV runs out of maximum step numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' To well generalize the DRL-based policy and achieve better sim-to-real transferability, we not only augment a pixel- level Gaussian noise to the RGB image but also vary the initial location and goal position for each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Though the proposed algorithm only needs one fisheye camera for autonomous navigation, we also set a laser sensor in the sim- ulation to detect the collision (4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Furthermore, we employ the Robot Operating System (ROS) open-source platform to communicate with Gazebo and derive the goal information through subscribing to odometry messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Figure 5 illustrates the learning curves of GoT-SAC and all the DRL-based baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' We run each algorithm with five different random seeds to measure statistics and evaluate the robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Specifically, the red dotted line and solid lines rep- resent the average rewards of our algorithms and baselines per episode, while the shaded areas depict the variances over five runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' As the figure shows, both versions of GoT-SAC achieve higher reward levels with relatively lower variances than those of goal-conditional DRL-based algorithms, which indicates the significance of the goal-oriented scene representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Moreover, both GoT-SAC models exhibit a faster convergence and enhance the training efficiency by over 129% and 86% compared with the ViT-SAC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' It should be noticed that though the convergence pace of ConvNet-SAC at the early stage is slightly faster due to its relatively small number of parameters, the average episode return is much lower than our proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' To evaluate the performance, we validate 8 (a) Scenario I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' (b) Scenario II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' (c) Scenario III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 7: Attention Flow Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The left pair of diagrams for each subfigure shows the original RGB image and goal information, while the right side diagram depicts the revised RGB image masked by the attention flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' A red square highlights the queried image patch, and the attention level is represented through a color transition from blue (low) to red (high).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' a) Scenario I: query for 59th image patch occupied with drivable space, b) Scenario II: query for 60th image patch occupied with drivable space, c) Scenario III: query for 34th image patch occupied with obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' TABLE I: Quantitative statistics of the self-attention mechanism behavior for the three goal-driven tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Model Episode I Episode II Episode III Gini Coefficient Shannon-Wiener Index Gini Coefficient Shannon-Wiener Index Gini Coefficient Shannon-Wiener Index GoT-SAC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='927 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='848 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='984 ViT-SAC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='802 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='616 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='695 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='545 all the trained policies with 20 random seeds and run for 50 episodes for each seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The success rate, which is obtained as the number of goal-reached episodes divided by the total runs, is employed as the metric for the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 6, we can observe the dominant success rate and superior robustness of the proposed algorithm compared with other baselines regardless of varying a wide range of random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='0 Normal Success Rate Attention Perturbation Perturbation w/ 10% Attention w/ Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Attention Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 8: Success Rate Comparison in terms of the different attention levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The black-solid line and ”star” located at the box body denote the median and average, while the hollow circles represent the outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Attention Visualization and Evaluation Besides the superior efficiency and performance, the GTRL approach also possesses a significant advantage in model in- terpretability thanks to the goal-oriented scene representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' To analyze the rationale behind fast convergence and excellent performance of our algorithm, we extract the attention from the GoT encoder w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' randomly sampled RGB images and visualize it in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' As the figure shows, the left pair of diagrams for each subfigure shows the original RGB image and goal information, while the right side diagram depicts the visual attention flow map [34] masked by the extracted attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The queried image patch is highlighted by a red square, and the attention level is represented through a color transition from blue (low) to red (high).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In Scenario I (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 7(a)), the UGV is facing the oncoming T intersection, and the goal position locates on the left side behind the office chair and table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' From the visual attention flow map, we can observe that the overall attention generates a visual path by mainly focusing on goal-oriented image patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' It should be noticed that the orientation of such a visual path obviously towards the goal position though the right turn is also feasible in this scenario, which proves that the scene representation successfully couples with the goal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Similarly, a clear goal-driven visual path is shown in Scenario II (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 7(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 7(c), different from the previous two scenarios, we query for the image patch that occupies an obstacle (office chair) instead of drivable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' We surprisingly find that the attention highlights most of the adjacent obstacles, evidently pointing out the undrivable regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Therefore, we qualitatively verify that our approach can provide a clear explanation of how the UGV analyzes the scene and arrives at the destination with 9 Xavier NX Stereo Camera Fisheye Camera 9-axis IMU (a) SCOUTMINI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' (b) Robotics Research Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 9: The real UGV platform and sim-to-real experiment environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' a) SCOUTMINI: an omnidirectional steering mobile robot from Agilex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' b) Robotics research center: an indoor laboratory space in Nanyang Technological University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' a collision-free path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Furthermore, we quantitatively evaluate and compare our approach with the ViT-SAC (goal-conditional) model to sup- port the above conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Specifically, we run each model with three random episodes and measure the statistical char- acteristics by averaging the whole frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Two automated unsu- pervised metrics: Gini coefficient for measuring the evenness of the attention weights distribution [36] and Shannon-Wiener index for evaluating the concentration of the attention [37] are reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Both metrics clearly reveal that the attention of the GoT-SAC model is sparser and tends to be more concentrated on task-related image patches, proving that better interpretability is achieved through goal-oriented scene representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Last but not least, we also investigate the impact of the significant attention through perturbation-based method [38] to observe how modifications of critical attention affect the navigation task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In light of this, we measure the success rate of the GoT-SAC model over another twenty random seeds with fifty episodes for each by dynamically replacing the essential attention (weights higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='995) with a moving average and 10% of the original value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The boxplot illustrated in Figure 8 shows the overall result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' We can observe that the performance of the GoT-SAC model, the one that decreases the significant attention to 10%, degrades catastrophically (62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='5%) in terms of the success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Though the success rate of the model employing the average perturba- tion method is slightly higher than the previous one, it is still clearly lower than the normal GoT-SAC model, indicating the significance of the attention learned by our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Sim-To-Real Assessment In addition to evaluating the feasibility and performance of the algorithm in a virtual simulation environment, we also expect to apply our approach in real-world navigation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In terms of the UGV, we use the omnidirectional steering mobile platform from Agilex called SCOUTMINI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The SCOUTMINI equips with an edge computing platform NVIDIA Jetson Xavier, a ZED2i stereo camera, an inertial measurement unit (IMU), and a fisheye camera with an ultra-wide FOV of 220 degrees (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 9(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Regarding the software, we deliver the goal information and raw RGB images to the UGV through the ROS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Then, the GoT-SAC sends the real-time decision inference to the UGV chassis via CAN communication to realize motion control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In this real-world experiment, all the algorithms are applied at the Robotics Research Center at Nanyang Technological University to complete a loop naviga- tion task, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 9(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' More specifically, we design four destinations that motivate the UGV to reach one by one with a small break after each arrival and finally return to the vicinity of the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' This experiment aims to test the algorithm’s ability to avoid static obstacles and quickly navigate to given goal positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Figure 10 illustrates the qualitative measurement of per- formance for each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' We plot both trajectories from the UGV and the human with two different colors, blue for ground truth and red for human engagement, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' As shown in the figure, the GoT-SAC policy performs smooth and collision-free navigation, while the other three algorithms (ConvNet-SAC, MultiModal CIL, and MoveBase) all need human engagement to arrive at the destinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Surprisingly, we find that ViT-SAC policy also demonstrates an equally excellent performance despite the low average success rate during the evaluation in the simulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' It is reasonable since we select the best model for each algorithm for sim-to-real transfer assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' It may also indicate the significance of the self-attention mechanism for goal-driven autonomous navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' To compare the performance and robustness of the policies in a deeper sight, we employ six statistical metrics for each goal-driven task: the average and variance of traveling dis- tance, average and variance of navigation time, success rate, and engagement number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Especially the successful arrival is determined if the UGV reaches each goal position within one minute, and we actively engage the UGV control once the col- lision is likely to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' A detailed quantitative measurement is reported in the Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' It is clear that the GoT-SAC model demonstrates dominant performance and robustness from all the domains compared with other baselines, including ViT- SAC and MoveBase planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The performance of ViT-SAC is also comparably excellent, besides the longest navigated distance for the fourth destination and high average time for the third goal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The worst performance is provided by the MultiModal CIL model, whose success rate is only 20% for reaching the third goal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' We can observe a similar performance from the statistical result of the ConvNet- SAC model in terms of the number of engagements, which is 15 in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' As for the MoveBase planner, the performance highly depends on the local cost-map quality, especially in the turning cases (the second and fourth goal position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' For instance, the local cost-map occurs false detection frequently due to limited field of view and occlusion from the obstacles, leading to improper path planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Overall, both quantitative and qualitative results in this sim-to-real experiment highlight the superiority of the proposed algorithm compared with other baselines, including against state-of-art leaning-based approaches and the classic UGV navigation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Additionally, our approach is tested in an unknown envi- ronment to validate the generalization capability thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 10 (a) GoT-SAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' (b) ViT-SAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' (c) ConvNet-SAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' (d) MultiModal CIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' (e) MoveBase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 10: Qualitative measurement of proposed algorithm and baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The solid blue line depicts the ground truth of the trajectory, while the solid red line represents the human-engaged path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' TABLE II: Quantitative performance of proposed algorithm compared with baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Approach Goal Position Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Time Success Rate Engage Number GoT-SAC 1st 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='044 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='049 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='048 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='282 100% 0 2nd 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='747 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='110 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='987 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='368 100% 0 3rd 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='171 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='036 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='821 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='076 100% 0 4th 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='360 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='113 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='902 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='287 100% 0 ViT-SAC 1st 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='958 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='036 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='274 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='394 100% 0 2nd 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='506 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='189 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='432 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='741 100% 0 3rd 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='226 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='015 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='502 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='601 100% 0 4th 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='274 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='210 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='449 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='722 100% 0 ConvNet-SAC 1st 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='150 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='294 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='918 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='563 100% 4 2nd 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='780 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='252 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='735 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='519 100% 5 3rd 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='322 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='245 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='757 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='683 100% 6 4th 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='971 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='123 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='550 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='414 100% 0 MultiModal CIL 1st 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='852 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='021 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='458 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='180 100% 0 2nd 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='868 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='480 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='263 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='026 60% 5 3rd 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='666 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='134 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='121 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='019 20% 5 4th 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='913 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='217 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='893 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='475 100% 0 MoveBase 1st 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='822 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='063 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='180 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='088 100% 0 2nd 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='070 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='529 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='446 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='388 100% 4 3rd 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='092 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='141 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='010 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='122 100% 1 4th 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='510 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='517 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='098 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content='442 100% 5 Due to the unstable connection and limitation of hardware, we select an unseen office environment rather than an outdoor space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' It is worthwhile to test the generalization and trans- ferability of the proposed algorithm in such an environment since it has a number of corner cases to be addressed, such as planning a collision-free path in narrow corridors, handling unseen obstacles (in terms of shape and color), and performing U-turn operation in order to reach the goal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Figure 11 demonstrates the details of the experiment, where the yellow circle labels the initial and goal positions, and the performed trajectory is highlighted with the solid blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In particular, the UGV has to pass a narrow corridor to reach the first two destinations and perform 90-degree-turn and U-turn operations to arrive last two goal positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Nevertheless, the GoT-SAC model can still approach all five goals without collision or engagement, indicating the excellent generalization capability and transferability of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' CONCLUSION This paper presents a Transformer-enabled DRL approach, namely GTRL, to realize efficient goal-driven autonomous navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Specifically, we first propose a novel Transformer- based architecture called Goal-guided Transformer (GoT) for the perception to consider the goal information as an input of the scene representation rather than a condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' For the purpose of boosting data efficiency, deep imitation learning is employed to pre-train the GoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Then, a GoT-enabled soft actor-critic algorithm (GoT-SAC) is instantiated to train the de- cision policy based on the goal-oriented scene representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' As a result, our approach motivates the scene representation to 2341213424132314241311 1 2 3 4 5 0 0 → 1 1 → 2 2 → 3 3 → 4 4 → 5 Real-time Pose Goal Pose Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' 11: Office Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' The initial and goal positions are labeled by yellow circles, and the performed trajectory is highlighted with the solid blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' concentrate mainly on goal-relavent features, which substan- tially enhances the data efficiency of the DRL learning process, leading to superior navigation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Both simulative and sim-to-real transfer experiments confirm our approach’s superiority in data efficiency, performance, robustness, and sim-to-real generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' In the future, we would like to upgrade the hardware and current algorithm to deal with the dynamic objects in the outdoor environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' REFERENCES [1] Mofan Zhou, Yang Yu, and Xiaobo Qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Development of an efficient driving strategy for connected and automated vehicles at signalized intersections: A reinforcement learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' IEEE Transactions on Intelligent Transportation Systems, 21(1):433–443, 2019.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' Explainability of deep vision-based autonomous driving systems: Review and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} +page_content=' International Journal of Computer Vision, pages 1–28, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNAyT4oBgHgl3EQfgvgA/content/2301.00362v1.pdf'} diff --git a/OtFKT4oBgHgl3EQffy6V/content/tmp_files/2301.11831v1.pdf.txt b/OtFKT4oBgHgl3EQffy6V/content/tmp_files/2301.11831v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4350730aa0eaa6594cbd3d9b52306a2ff8d81411 --- /dev/null +++ b/OtFKT4oBgHgl3EQffy6V/content/tmp_files/2301.11831v1.pdf.txt @@ -0,0 +1,803 @@ +Data Volume-aware Computation Task Scheduling +for Smart Grid Data Analytic Applications +Binquan Guo∗, Hongyan Li∗, Ye Yan†, Zhou Zhang†, and Peng Wang∗ +∗State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an P. R. China +†Tianjin Artificial Intelligence Innovation Center (TAIIC), Tianjin, P. R. China +Email:bqguo@stu.xidian.edu.cn, hyli@xidian.edu.cnyanye1971@sohu.com, zt.sy1986@163.com, pengwangclz@163.com +Abstract—Emerging smart grid applications analyze large +amounts of data collected from millions of meters and systems +to facilitate distributed monitoring and real-time control tasks. +However, current parallel data processing systems are designed +for common applications, unaware of the massive volume of +the collected data, causing long data transfer delay during the +computation and slow response time of smart grid systems. +A promising direction to reduce delay is to jointly schedule +computation tasks and data transfers. We identify that the +smart grid data analytic jobs require the intermediate data +among different computation stages to be transmitted orderly +to avoid network congestion. This new feature prevents current +scheduling algorithms from being efficient. In this work, an +integrated computing and communication task scheduling scheme +is proposed. The mathematical formulation of smart grid data +analytic jobs scheduling problem is given, which is unsolvable +by existing optimization methods due to the strongly coupled +constraints. Several techniques are combined to linearize it for +adapting the Branch and Cut method. Based on the topological +information in the job graph, the Topology Aware Branch and +Cut method is further proposed to speed up searching for optimal +solutions. Numerical results demonstrate the effectiveness of the +proposed method. +Index Terms—Smart grid applications, data analytics, task +scheduling, job completion time, branch and cut, disjunctive +formulation. +I. INTRODUCTION +The smart grid uses smart meters, sensors to collect data, +and adopts information technologies to make smart decisions +to fulfill the demand and supply of modern electrical power +[1]. To facilitate distributed monitoring and real-time control +tasks [2], a huge amount of raw data are collected in real +time from smart meters and sensors deployed in different +geographical areas, and uploaded periodically on a hourly, +daily or monthly basis (depending on the customers and pur- +poses) to computing systems for data analysis [3]. According +to [4], a smart grid system with 2 million customers will +generate about 22 Gigabytes of data each day. To efficiently +handle such massive data, various computing and communi- +cation optimization technologies are proposed to improve the +performance, such as fog computing for energy consumption +scheduling [5], software define networking in smart grid for +resilient communications [6]. Moreover, many standardization +This work is supported by the Natural Science Foundation of China +(61931017). The corresponding author is Hongyan Li. +The source code is publicly available at https://github.com/wilixx/ICCTS. +activities supported by governments and stakeholders are mak- +ing continuous efforts to makes use of advanced information, +control, and communications technologies in smart grids sys- +tems to save energy, reduce cost and increase reliability and +transparency [7]. +Traditionally, the typical data processing architectures like +MapReduce [8], Pregel [9] and Spark [10], designed for gen- +eral applications, usually partition input data over a number of +parallel machines, such that a data analytic job is decomposed +into multiple tasks. Before generating the final results, the +partial results between the adjacent stages of computation +need to be exchanged through the network during the job +execution. These systems developed for common purposes, +focus on data partitioning and computing, and rarely optimize +data transmission performance. With the rapid growth and +accumulation of the data volume in smart grid, the data transfer +time has become an increasingly significant bottleneck in the +performance of data analytic jobs. Firstly, when tasks with +precedence constraints are scheduled on different machines, +the data transfer time will be increased. Secondly, when a +large number of data transfers are performed at the same time, +the competition with occur, leading additional delays. The +increased job execution time will greatly affect the response +time, which will impact the rapid decision-making ability of +the smart grid systems. Therefore, optimizing data transfers is +important for minimizing the job completion time, with the +aim of enabling rapid decision-making for smart grid data +analytic applications. +With respect to minimizing the job completion time, tradi- +tional works designed for common applications focused on +either computation task placement (e.g., [11]) or network +flow scheduling (e.g., [12]). However, the separation between +scheduling computation and communication tasks results in +inefficient job processing performance especially when the +data volume is huge. To overcome this, some researchers +recently began to break the barrier and attempted to coordinate +the computation and communication tasks. The authors in [13] +designed the Symbiosis framework to co-locate computation- +intensive and network-intensive tasks, where computation +tasks can utilize the idle computing resource during the +transmission of other network-intensive tasks to reduce the +completion time. In the Firebird framework proposed in [14], +computation tasks are placed based on machines’ available +arXiv:2301.11831v1 [cs.DC] 27 Jan 2023 + +bandwidths to avoid network contention. In [15], the authors +proposed to place computation tasks according to the predicted +flow transfer time under given network conditions. In [16], the +authors considered jointly optimizing the reducer placement +and bandwidth scheduling to minimize the coflow completion +time. Those works usually aimed to reduce the completion +time of either computation or communication tasks rather than +optimize the whole job. In [17], the authors considered the +joint computation and communication task scheduling problem +from the job perspective, and proposed heuristic scheduling +algorithms. In [18], the authors investigated the problem of +joint computation and communication task scheduling with +bandwidth augmentation, where mathematical optimization +method was designed to solve it optimally. However, both of +their methods are designed for general data center scenarios, +which cannot be directly applied in smart grid systems. +Besides, the mathematical model of the integrated computation +and communication task scheduling problem, especially for the +smart grid applications, is still missing. Indeed, the inherent +causal relationship between the computation task placement +and data transfer condition (i.e., data transfer may or may +not be necessary depending on the placement of computation +tasks) in data analytic jobs will greatly increase the complexity +of the integrated scheduling problem. +In this work, an efficient Integrated Computing and Com- +munication Task Scheduling (ICCTS) scheme for smart grid +applications is proposed. At first, by exploring the causal +relationship, we construct the Completion Time Minimization +oriented Integrated Computation and Communication Task +Scheduling Problem (CTM-ICCTSP) to mathematically model +the data analytic job scheduling problem, which is not solvable +by exiting optimization tools due to a large number of com- +plicated coupled non-linearity constraints. Second, the general +flow concept is defined to represent the internal or external +data transfer between adjacent computation tasks. Based on +the general flow concept, the auxiliary virtual channel is +introduced to linearize CTM-ICCSP, so that it can be directly +solved by the Branch and Cut (B&C) method. Then, to reduce +the searching space, we utilize the topology information in the +job graph model and design a Topology Aware Branch and Cut +(TABC) method to effectively speed up searching for optimal +solutions. Finally, numerical results validate the necessity of +optimizing the data transfers as well as the effectiveness of +the proposed ICCTS method. +II. SYSTEM MODEL +In this work, we model the smart grid data analytic jobs +as periodic jobs, which are executed on a hourly or daily or +monthly basis depending on their purposes, and their detailed +knowledges can be profiled from historical logs. Each job +is represented by a Dual Weight Directed Acyclic Graph +(DWDAG) G = {V, E, P, Q, R}. V is the computation task +set, and vj ∈ V denotes the j-th computation task. The +execution of task vj lasts for pj time slots, and P = {pj|1 ≤ +j ≤ |V|} is the execution time set for computation tasks. E +is the dual-weighted edge set, and edge euv ∈ E represents +� +� +� +� +� +� +��� +��� +��� +��� +��� +��� +���������������� +��������������� +j +� +� +jp +� +� +� +v +uv +u +e +e +r +q +uv +e +Fig. 1. A smart grid data analytic job example with six computation tasks +and eight possible data transfers (i.e., network flows). +the dependency between the computation task u and v, which +means the execution of task v requires the data from u. +The two weights of a edge separately represent the internal +and external data transfer time corresponding to the different +placement of precedence-constrained computation tasks. If the +pair of precedence-constrained computation tasks, denoted by +u and v, are placed in the same machine, the intermediate +data on edge euv are transmitted internally, and the network +flow will not occur; otherwise flow fuv on edge euv will +be transmitted externally through the communication channel +between machines accommodating u and v. The corresponding +internal and external transfer times are separately denoted by +reuv ∈ R and qeuv ∈ Q. +Fig. 1 illustrates a DWDAG example with six computation +tasks and eight possible network flows. For a job, a set of +available machines are reserved to process its data, denoted +by M = {αi|1 ≤ i ≤ M}, while the network resource +shared among these machines is modeled by a set of individual +communication channels with the same bandwidth, denoted +by N += {βk|1 ≤ k ≤ N}. The time axis is cut into +multiple identical slots, and the slot index set is denoted as +T += {τ|1 ≤ τ ≤ Tmax}. We assume that each computa- +tion task’s processing time and each data’s transfer time are +predetermined, and computation tasks and network flows are +scheduled non-preemptively, i.e., once started, their processing +or transmission cannot be interrupted. +III. INTEGRATED COMPUTATION AND COMMUNICATION +TASK SCHEDULING PROBLEM +In this section, the smart grid data analytic job scheduling +problem is formulated as a Completion Time Minimization +oriented Integrated Computation and Communication Task +Scheduling Problem (CTM-ICCTSP). The computation task +set and possible network flow set are denoted by ˆ +J and F, +respectively. For single job scheduling, we have ˆ +J = V and +F = {fuv|euv ∈ E}. The external flow set ˆF ⊆ F only +contains the flows transferred via the external physical links +between machines. +The binary computation task placement decision variable is +denoted by Xjiτ. If task vj begins to be executed at time τ +in machine αi, Xjiτ = 1; otherwise Xjiτ = 0. Since each task + +can be executed only once, we have +� +1≤i≤M +� +τ∈T +Xjiτ = 1, ∀vj ∈ ˆ +J . +(1) +Similarly, the binary network flow scheduling decision variable +Yfkτ = 1 means flow f begins to transmit at time τ in +communication channel βk, and thus we have +� +1≤k≤N +� +τ∈T +Yfkτ = 1, ∀f ∈ ˆF. +(2) +The start time of computation task vj and flow f, separately +denoted by sM +j +and sN +f , are calculated as follows. +sM +j +∆= +� +τ∈T +� +1≤i≤M +τXjiτ, +sN +f +∆= +� +τ∈T +� +1≤k≤N +τYfkτ. +The optimization objective is to minimize the job com- +pletion time, i.e., the maximum completion time among all +computation and communication tasks: +min +Xjiτ ,Yfkτ Cmax = max +� +sM +j ++ pj +� +. +The number of the machine to place vj is denoted by +mj +∆= +� +1≤i≤M +� +τ∈T +i · Xjiτ, +and the number of communication channel to send f is +denoted by +nf +∆= +� +1≤k≤N +� +τ∈T +k · Yfkτ. +In the smart grid data analytic job scheduling problem, the +following inherent constraints must be carefully taken into +consideration. +Computing resource constraints: The computing resource +constraints include (3) and (4). Constraint (4) means that the +execution times of computation tasks in the same machine will +not overlap, which is a typical disjunctive constraint. +sM +j +≥ 0, ∀vj ∈ ˆ +J +(3) +sM +j +pj ≤ sM +j′ or sM +j′ +pj′ ≤ sM +j , ∀vj ̸= vj′, mj = mj′ (4) +Causality constraints: The occurrence of flow fuv depends +on the placement of its precedence-constrained tasks u and v. +Thus the external flow set ˆF can be rewritten in the causal +relationship based formulation as +ˆF = {fuv|fuv ∈ F, mu ̸= mv}. +(5) +Precedence constraints: The precedence relationship exists +between two adjacent upstream and downstream computation +tasks. A task starts to be executed after all of its precedent +tasks and upstream necessary flows end. Depending on where +precedence-constrained tasks are placed in the same machine, +two cases should be considered. +Case 1: If task u and v are placed on the same machine, +the data between u and v is transferred within the machine, +and the start time of the downstream task v should be after +u’s end time adding the internal transfer time, denoted by +sM +u + pu + reuv ≤ sM +v , ∀fuv ∈ F − ˆF. +(6) +Case 2: If task u and v are placed on different machines, +the flow between u and v is transferred between the machines. +Thus the start time of fuv should be after u’s end time, while +v’s start time should be after the end time of fuv, denoted by +(7) and (8). +sM +u + pu ≤ sN +fuv, ∀fuv ∈ ˆF +(7) +sN +fuv + qeuv ≤ sM +v , ∀fuv ∈ ˆF +(8) +Communication resource constraints: The Communication +resource constraints include (9) and (10). Constraint (10) +means that the flow transmissions in the same network channel +will not overlap, which is also a typical disjunctive constraint. +sN +f > 0, ∀f ∈ ˆF +(9) +sN +f + qf′ ≤ sN +f′ or sN +f′ + qf′ ≤ sN +f , ∀f, f ′ ∈ ˆF, f ̸= f ′, nf = nf′ +(10) +Finally, the resulting CTM-ICCTSP can be expressed as +follows. +P1 : min Cmax +s.t. (1) − (10) +IV. LINEARIZED REFORMULATION AND TOPOLOGY +AWARE BRANCH AND CUT METHOD +A. General flow concept and auxiliary virtual channel +Due to constraints (4), (5), (6), (7), (8) and (10), CTM- +ICCTSP is a non-linear problem. Constrains (4) and (10) can +be transferred into linear constraint formulations by the Big- +M and Convex Hull reformulation methods in disjunctive pro- +gramming. However, constraint (5) is not so easy to handle. To +eliminate the volatility of ˆF in (5), we define the general flow +concept. A general flow may be an internal or external flow +transferred between two precedence-constrained tasks. Only +the external flows compete for network resource. To construct +a unified flow scheduling framework compatible with the two +types of flows, we introduce the auxiliary virtual channel, +which is contention-free for all internal flow transfers. Thus the +external flows are transferred via physical network links while +the internal flows are handled by the auxiliary virtual channel. +By introducing this auxiliary virtual channel, the uncertainty of +external flows can be eliminated. The auxiliary virtual channel +is denoted by ˆk, and thus the communication resource set is +N ∪ ˆk. +A general flow f ∈ F must be placed on either real com- +munication channels or the auxiliary virtual channel, which +separately corresponds to the external or internal transfer. +Since each general flow must be transferred, we have +� +τ∈T +n +� +k∈N ∪ˆk +Yfkτ = 1, ∀f ∈ F. +(11) + +B. Linearization of computation and communication disjunc- +tive constraints +By introducing the general flow and auxiliary virtual chan- +nel, the scheduling entity ˆF in P1 can be replaced by the +general flow set F, which eliminates the uncertainty of the the +scheduling entity. Therefore, we can continue to adopt the re- +formulation methods in disjunctive programming [19] to trans- +form P1 into an equivalent Integer Linear Programming (ILP) +problem. To linearize constraint (4), we introduce two types of +auxiliary variables to describe the placement of computation +tasks. Binary variables ψjj′i ∈ {0, 1} indicate whether two +computation tasks are placed in the same machine, where +vj, vj′ ∈ � +J , vj ̸= vj′, 1 ≤ i ≤ M. If computation tasks vj and +vj′ are both placed on the i-th machine, ψjj′i = 1; otherwise +ψjj′i = 0. To construct ψjj′i, we have +0 ≤ +� +τ∈T +Xjiτ + +� +τ∈T +Xj′iτ − 2 · ψjj′i ≤ 1, +∀vj, vj′ ∈ ˆ +J , vj ̸= vj′, 1 ≤ i ≤ M. +(12) +Binary precedence indicator variables σjj′ ∈ {0, 1} rep- +resent the precedence relationship between two computation +tasks. If task vj starts no later than task vj′, σjj′ = 1; +otherwise σjj′ = 0. These constraints can be guaranteed by +sM +j′ − sM +j +≤ Tmax · σjj′ − ϵ(1 − σjj′), +∀vj, vj′ ∈ ˆ +J , vj ̸= vj′. +(13) +where ϵ ∈ (0, 1) is a small enough constant commonly used +in the logical formulation of integer programming. With ψjj′i +and σjj′, the disjunctive constraint (4) can be linearized as +sM +j ++ pj − sM +j′ ≤ Tmax · (2 − σjj′ − +� +1≤i≤M +ψjj′i), +∀vj, vj′ ∈ ˆ +J , vj ̸= vj′. +(14) +Constraint (14) guarantees that task vj′ must start after the +completion of task vj when they are placed in the same +machine, i.e., σjj′ = 1 and � +1≤i≤M ψjj′i = 1 hold simulta- +neously. +Similarly, to linearize constraint (10), auxiliary binary vari- +ables χff′k and φff′ are introduced. χff′k indicates whether two +flows are both placed in the k-th communication channel, +while precedence indicator variable φff′ represents the prece- +dence relationship between two flows. The constraints of χff’k +and φff’ as well as the reformulation of constraint (10) are +shown as follows. +0 ≤ +� +τ∈T +Yfkτ + +� +τ∈T +Yf ′kτ − 2 · χff′k ≤ 1, +∀f, f ′ ∈ F, f ̸= f ′ +(15) +sN +f − sN +f ′ ≤ Tmax · φff′ − ϵ(1 − φff′), +∀f, f ′ ∈ F, f ̸= f ′ +(16) +sN +f + qf − sN +f ′ ≤ Tmax · (2 − φff − +� +1≤k≤N +χff′k), +∀f, f ′ ∈ F, f ̸= f ′ +(17) +C. Linearization of causality and precedence constraints be- +tween computation and communication tasks +After the separate linearizations of computation and com- +munication disjunctive constraints, we continue to reformulate +the causality and precedence constraints (5), (6), (7) and (8) +to integrate the scheduling of computation and communication +tasks. +With the introduced auxiliary variable ψjj′i, the causality +relationship between computation task placement and network +flow occurrence in constraint (6) can be rewritten by +� +1≤i≤M +ψuvi = +� +τ∈T +Yfuv,ˆk,τ, ∀fuv ∈ F. +(18) +With the general flow concept, the two cases in the previous +precedence constraints can be represented in a unified form. +For each pair of precedence-constrained tasks u and v con- +nected by general flow fuv, the start time of fuv must be after +the end time of u, i.e., +sM +u + pu ≤ sN ∪ˆk +fuv +, ∀fuv ∈ F. +(19) +If flow fuv is transferred internally, � +τ∈T Yfuv,ˆk,τ = 1; +otherwise, the answer is zero. No matter fuv is transferred +externally or internally, the start time of task v must be after +the end time of flow fuv, denoted by +sN ∪ˆk +fuv ++ ru +� +τ∈T +Yfuv,ˆk,τ + qfuv(1 − +� +τ∈T +Yfuv,ˆk,τ) ≤ sM +v . (20) +Finally, CTM-ICCTSP can be linearized as P2. +P2 :min Cmax +s.t. (1), (3), (11) − (20), +Cmax ≥ sM +j ++ pj, ∀vj ∈ ˆ +J , +Cmax ≥ sN ∪ˆk +f ++ qef(1 − +� +τ∈T +Yfuv,ˆk,τ), ∀f ∈ F, +Cmax ≥ sN ∪ˆk +f ++ ref +� +τ∈T +Yfuv,ˆk,τ, ∀f ∈ F. +D. Topology-aware Branch and Cut (TABC) Algorithm +Problem P2 is an ILP problem and can be solved by +the classic B&C algorithm. Since the searching iterations of +B&C may vary dramatically in different problem instances, its +running time may be unacceptable in some cases. To avoid +this situation, we take the topological relationship among +computation tasks and network flows in the DWDAG into +consideration and propose an efficient Topology Aware Branch +and Cut (TABC) Algorithm based on the following strategies. +1) Branch with the chain precedence constraints: +The +branching process can be carried out from enumerating the +precedence indicator matrices, Θ = (σjj′) or Φ = (φff′). +Based on the precedence relationships among computation and +communication tasks, some searching branches can be directly +pruned, and thus the searching space can be greatly reduced. +The chain precedence constraints include the precedence con- +straints along successive computation task and flow chains. +For example, in computation tasks if σjj′ = σjj′′ = 1, then + +σjj′′ = 1. For network flows, if φff′ = φf′f′′ = 1, then φff′′ = 1. +In general,if fuv ∈ F, then σuv = 1; if fuv, fvv’ ∈ F, then +φfuvfvv’ = 1. +2) Update task interval constraints: In each job’s DWDAG, +the earliest and latest start time of each computation task +or network flow can be inferred according to both the job’s +incumbent upper and lower bounds and the processing and +transfer times of the other computation tasks and flows along +its longest branch. A graph theory-based method to obtain the +longest branch of a directed acyclic graph can be found in [18], +which is applicable to this work and very easy to implement. +Therefore, the searching tree can be further pruned according +to the reduced interval constraint for each task or flow. +3) Utilize the symmetry of solution space: One symmetry +feature is from the homogeneity of physical machines and +communication channels. If we switch the indexes of two +scheduled machines or communication channels in a feasible +solution, the result is an equivalent solution. Setting each task’s +machine affinity value in advance can eliminate a large number +of symmetric solutions. The other symmetry feature comes +from the symmetry of nodes or edges in the job graph, thus +different priorities are added to the equivalent computation +tasks or network flows to reduce redundant searching. +Algorithm 1 Topology Aware Branch and Cut Algorithm +Input: Job G = {V, E, P, Q, R}, resource set {M} and {N}. +Output: Optimal solution S∗. +1: Initialization: +2: Calculate a solution S using heuristics. +3: Set LB = sum(P ) +|M| +, UB = min(Cmax(S), sum(P)). +4: Set the initial precedence matrices with chain precedence +constraints in G, the initial interval constraints, and the +affinities and priorities of tasks/flows. +5: Repeat +6: Solve P2 using B&C for an incumbent solution. +7: If new LB or UB is obtained, recalculate and update +interval constraints to the current active node in P2. +8: Until Optimal solution S∗ found. +9: return S∗. +The TABC procedures are shown in Algorithm 1. The +three pruning strategies are all adopted to efficiently reduce +the iterations. Due to the inherent unreliability and instability +properties of B&C in solving ILP problems, the performance +of the proposed TABC may also vary in different cases. Proper +termination conditions can be set to avoid too long running +time. Though TABC may terminate before producing a global +optimal solution, the incumbent solution can still be better +than most heuristics. In addition, since the variables in P2 are +binary, TABC can be efficiently implemented by splitting P2 +into multiple parallelly executed sub-problems. Thus the time +complexity can be greatly reduced. +V. SIMULATION RESULTS +To evaluate the performance of the proposed scheme, we +conduct simulations over the synthetic smart grid data analytic +jobs. As in [18], the computation task processing times and +the data transfer times are randomly and uniformly chosen +from [1, 100] and [1, 50], respectively, which is to mimic the +different data volumes by normalizing the maximum time to +100. The larger the transfer time corresponds to the larger the +data volume transfered among the adjacent tasks. +In Fig. 2, we compare the average normalized makespans +(i.e., job completion times) of six scheduling schemes with +different machine numbers and one network channel. The +Random Scheduling scheme places the computation tasks ran- +domly, while the List Scheduling scheme is from [20]. Both of +them only considered the placement of computation tasks. The +Partition Scheduling, Generalized List (G-List) Scheduling and +G-List-Master Scheduling schemes are from [17]. For each +scheduling scheme, we generated 3000 job cases each with +ten computation tasks, calculated the normalized makespans +of these jobs, and averaged them. The normalized makespan is +the ratio of the makespan obtained by one scheduling scheme +and the upper bound makespan when the job’s computation +tasks were all placed on a single machine. For the Random +Scheduling and the List Scheduling schemes, their average +normalized makespans increase with the machine numbers due +to the ignorance of data transfer optimization. For the other +four scheduling schemes, their average normalized makespans +decrease with the increase of machine numbers since the +computation and communication tasks are jointly scheduled. +The proposed Integrated Computing and Communication Task +Scheduling (ICCTS) scheme obtains the lowest average nor- +malized makespans among all the six scheduling schemes. +Since the linearization reformulation keeps the optimality +of the original CTM-ICCTSP problem, ICCTS can obtain +the optimal solution and act as an important benchmark for +heuristic schemes. It can be observed that when the machines +(computing resources) are sufficient, ICCTS can averagely +reduce the job completion time by up to 5%. +1 +2 +3 +4 +5 +6 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +Average normalized makespan +Number of machines +Random Scheduling +List Scheduling +Partition Scheduling +G-List-Master Scheduling +G-List Scheduling +ICCTS +Fig. 2. Average normalized makespans of different scheduling schemes versus +the number of machines. +In Fig. 3, we compare the efficiency of B&C and TABC +in the ICCTS scheme, and the simplex iteration number + +is adopted as the algorithm efficiency metric. Though their +average simplex iterations both increases exponentially with +the computation task number, TABC can significantly reduce +the iterations due to the pruning rules from DWDAG. +4 +5 +6 +7 +8 +9 +10 +11 +12 +10 +0 +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +7 +Average simplex iterations +Number of computing tasks +B&C +TABC +Fig. 3. +Average simplex iterations of B&C and TABC versus the number of +computing tasks. +VI. CONCLUSION +In this work, an integrated computation and communication +task scheduling scheme for smart grid data analytic appli- +cations is proposed. The mathematical formulation and the +corresponding constraint linearization of the job scheduling +problem were introduced, and an efficient Topology Aware +Branch and Cut method was designed to improve the searching +speed for the optimal solutions. Numerical results confirmed +the necessity of considering data volume and validity of the +proposed integrated scheduling scheme. +REFERENCES +[1] Q. Li, Y. 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Rayward-Smith, “Uet scheduling with unit interprocessor com- +munication delays,” Discrete Applied Mathematics, vol. 18, no. 1, pp. +55–71, 1987. + diff --git a/OtFKT4oBgHgl3EQffy6V/content/tmp_files/load_file.txt b/OtFKT4oBgHgl3EQffy6V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0a8f0368b827fd203e72f62d98664ec5ef719b8e --- /dev/null +++ b/OtFKT4oBgHgl3EQffy6V/content/tmp_files/load_file.txt @@ -0,0 +1,415 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf,len=414 +page_content='Data Volume-aware Computation Task Scheduling for Smart Grid Data Analytic Applications Binquan Guo∗, Hongyan Li∗, Ye Yan†, Zhou Zhang†, and Peng Wang∗ ∗State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' China †Tianjin Artificial Intelligence Innovation Center (TAIIC), Tianjin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' China Email:bqguo@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='cn, hyli@xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='cnyanye1971@sohu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='com, zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='sy1986@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='com, pengwangclz@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='com Abstract—Emerging smart grid applications analyze large amounts of data collected from millions of meters and systems to facilitate distributed monitoring and real-time control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' However, current parallel data processing systems are designed for common applications, unaware of the massive volume of the collected data, causing long data transfer delay during the computation and slow response time of smart grid systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' A promising direction to reduce delay is to jointly schedule computation tasks and data transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' We identify that the smart grid data analytic jobs require the intermediate data among different computation stages to be transmitted orderly to avoid network congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' This new feature prevents current scheduling algorithms from being efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' In this work, an integrated computing and communication task scheduling scheme is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The mathematical formulation of smart grid data analytic jobs scheduling problem is given, which is unsolvable by existing optimization methods due to the strongly coupled constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Several techniques are combined to linearize it for adapting the Branch and Cut method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Based on the topological information in the job graph, the Topology Aware Branch and Cut method is further proposed to speed up searching for optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Numerical results demonstrate the effectiveness of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Index Terms—Smart grid applications, data analytics, task scheduling, job completion time, branch and cut, disjunctive formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' INTRODUCTION The smart grid uses smart meters, sensors to collect data, and adopts information technologies to make smart decisions to fulfill the demand and supply of modern electrical power [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' To facilitate distributed monitoring and real-time control tasks [2], a huge amount of raw data are collected in real time from smart meters and sensors deployed in different geographical areas, and uploaded periodically on a hourly, daily or monthly basis (depending on the customers and pur- poses) to computing systems for data analysis [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' According to [4], a smart grid system with 2 million customers will generate about 22 Gigabytes of data each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' To efficiently handle such massive data, various computing and communi- cation optimization technologies are proposed to improve the performance, such as fog computing for energy consumption scheduling [5], software define networking in smart grid for resilient communications [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Moreover, many standardization This work is supported by the Natural Science Foundation of China (61931017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The corresponding author is Hongyan Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The source code is publicly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='com/wilixx/ICCTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' activities supported by governments and stakeholders are mak- ing continuous efforts to makes use of advanced information, control, and communications technologies in smart grids sys- tems to save energy, reduce cost and increase reliability and transparency [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Traditionally, the typical data processing architectures like MapReduce [8], Pregel [9] and Spark [10], designed for gen- eral applications, usually partition input data over a number of parallel machines, such that a data analytic job is decomposed into multiple tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Before generating the final results, the partial results between the adjacent stages of computation need to be exchanged through the network during the job execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' These systems developed for common purposes, focus on data partitioning and computing, and rarely optimize data transmission performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' With the rapid growth and accumulation of the data volume in smart grid, the data transfer time has become an increasingly significant bottleneck in the performance of data analytic jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Firstly, when tasks with precedence constraints are scheduled on different machines, the data transfer time will be increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Secondly, when a large number of data transfers are performed at the same time, the competition with occur, leading additional delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The increased job execution time will greatly affect the response time, which will impact the rapid decision-making ability of the smart grid systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Therefore, optimizing data transfers is important for minimizing the job completion time, with the aim of enabling rapid decision-making for smart grid data analytic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' With respect to minimizing the job completion time, tradi- tional works designed for common applications focused on either computation task placement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=', [11]) or network flow scheduling (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=', [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' However, the separation between scheduling computation and communication tasks results in inefficient job processing performance especially when the data volume is huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' To overcome this, some researchers recently began to break the barrier and attempted to coordinate the computation and communication tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The authors in [13] designed the Symbiosis framework to co-locate computation- intensive and network-intensive tasks, where computation tasks can utilize the idle computing resource during the transmission of other network-intensive tasks to reduce the completion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' In the Firebird framework proposed in [14], computation tasks are placed based on machines’ available arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='11831v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='DC] 27 Jan 2023 bandwidths to avoid network contention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' In [15], the authors proposed to place computation tasks according to the predicted flow transfer time under given network conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' In [16], the authors considered jointly optimizing the reducer placement and bandwidth scheduling to minimize the coflow completion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Those works usually aimed to reduce the completion time of either computation or communication tasks rather than optimize the whole job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' In [17], the authors considered the joint computation and communication task scheduling problem from the job perspective, and proposed heuristic scheduling algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' In [18], the authors investigated the problem of joint computation and communication task scheduling with bandwidth augmentation, where mathematical optimization method was designed to solve it optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' However, both of their methods are designed for general data center scenarios, which cannot be directly applied in smart grid systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Besides, the mathematical model of the integrated computation and communication task scheduling problem, especially for the smart grid applications, is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Indeed, the inherent causal relationship between the computation task placement and data transfer condition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=', data transfer may or may not be necessary depending on the placement of computation tasks) in data analytic jobs will greatly increase the complexity of the integrated scheduling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' In this work, an efficient Integrated Computing and Com- munication Task Scheduling (ICCTS) scheme for smart grid applications is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' At first, by exploring the causal relationship, we construct the Completion Time Minimization oriented Integrated Computation and Communication Task Scheduling Problem (CTM-ICCTSP) to mathematically model the data analytic job scheduling problem, which is not solvable by exiting optimization tools due to a large number of com- plicated coupled non-linearity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Second, the general flow concept is defined to represent the internal or external data transfer between adjacent computation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Based on the general flow concept, the auxiliary virtual channel is introduced to linearize CTM-ICCSP, so that it can be directly solved by the Branch and Cut (B&C) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Then, to reduce the searching space, we utilize the topology information in the job graph model and design a Topology Aware Branch and Cut (TABC) method to effectively speed up searching for optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Finally, numerical results validate the necessity of optimizing the data transfers as well as the effectiveness of the proposed ICCTS method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' SYSTEM MODEL In this work, we model the smart grid data analytic jobs as periodic jobs, which are executed on a hourly or daily or monthly basis depending on their purposes, and their detailed knowledges can be profiled from historical logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Each job is represented by a Dual Weight Directed Acyclic Graph (DWDAG) G = {V, E, P, Q, R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' V is the computation task set, and vj ∈ V denotes the j-th computation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The execution of task vj lasts for pj time slots, and P = {pj|1 ≤ j ≤ |V|} is the execution time set for computation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' E is the dual-weighted edge set, and edge euv ∈ E represents � � � � � � ��� ��� ��� ��� ��� ��� ���������������� ��������������� j � � jp � � � v uv u e e r q uv e Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' A smart grid data analytic job example with six computation tasks and eight possible data transfers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=', network flows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' the dependency between the computation task u and v, which means the execution of task v requires the data from u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The two weights of a edge separately represent the internal and external data transfer time corresponding to the different placement of precedence-constrained computation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' If the pair of precedence-constrained computation tasks, denoted by u and v, are placed in the same machine, the intermediate data on edge euv are transmitted internally, and the network flow will not occur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' otherwise flow fuv on edge euv will be transmitted externally through the communication channel between machines accommodating u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The corresponding internal and external transfer times are separately denoted by reuv ∈ R and qeuv ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 1 illustrates a DWDAG example with six computation tasks and eight possible network flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' For a job, a set of available machines are reserved to process its data, denoted by M = {αi|1 ≤ i ≤ M}, while the network resource shared among these machines is modeled by a set of individual communication channels with the same bandwidth, denoted by N = {βk|1 ≤ k ≤ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The time axis is cut into multiple identical slots, and the slot index set is denoted as T = {τ|1 ≤ τ ≤ Tmax}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' We assume that each computa- tion task’s processing time and each data’s transfer time are predetermined, and computation tasks and network flows are scheduled non-preemptively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=', once started, their processing or transmission cannot be interrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' INTEGRATED COMPUTATION AND COMMUNICATION TASK SCHEDULING PROBLEM In this section, the smart grid data analytic job scheduling problem is formulated as a Completion Time Minimization oriented Integrated Computation and Communication Task Scheduling Problem (CTM-ICCTSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The computation task set and possible network flow set are denoted by ˆ J and F, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' For single job scheduling, we have ˆ J = V and F = {fuv|euv ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The external flow set ˆF ⊆ F only contains the flows transferred via the external physical links between machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The binary computation task placement decision variable is denoted by Xjiτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' If task vj begins to be executed at time τ in machine αi, Xjiτ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' otherwise Xjiτ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Since each task can be executed only once, we have � 1≤i≤M � τ∈T Xjiτ = 1, ∀vj ∈ ˆ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (1) Similarly, the binary network flow scheduling decision variable Yfkτ = 1 means flow f begins to transmit at time τ in communication channel βk, and thus we have � 1≤k≤N � τ∈T Yfkτ = 1, ∀f ∈ ˆF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (2) The start time of computation task vj and flow f, separately denoted by sM j and sN f , are calculated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' sM j ∆= � τ∈T � 1≤i≤M τXjiτ, sN f ∆= � τ∈T � 1≤k≤N τYfkτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The optimization objective is to minimize the job com- pletion time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=', the maximum completion time among all computation and communication tasks: min Xjiτ ,Yfkτ Cmax = max � sM j + pj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The number of the machine to place vj is denoted by mj ∆= � 1≤i≤M � τ∈T i · Xjiτ, and the number of communication channel to send f is denoted by nf ∆= � 1≤k≤N � τ∈T k · Yfkτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' In the smart grid data analytic job scheduling problem, the following inherent constraints must be carefully taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Computing resource constraints: The computing resource constraints include (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Constraint (4) means that the execution times of computation tasks in the same machine will not overlap, which is a typical disjunctive constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' sM j ≥ 0, ∀vj ∈ ˆ J (3) sM j +pj ≤ sM j′ or sM j′ +pj′ ≤ sM j , ∀vj ̸= vj′, mj = mj′ (4) Causality constraints: The occurrence of flow fuv depends on the placement of its precedence-constrained tasks u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Thus the external flow set ˆF can be rewritten in the causal relationship based formulation as ˆF = {fuv|fuv ∈ F, mu ̸= mv}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (5) Precedence constraints: The precedence relationship exists between two adjacent upstream and downstream computation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' A task starts to be executed after all of its precedent tasks and upstream necessary flows end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Depending on where precedence-constrained tasks are placed in the same machine, two cases should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Case 1: If task u and v are placed on the same machine, the data between u and v is transferred within the machine, and the start time of the downstream task v should be after u’s end time adding the internal transfer time, denoted by sM u + pu + reuv ≤ sM v , ∀fuv ∈ F − ˆF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (6) Case 2: If task u and v are placed on different machines, the flow between u and v is transferred between the machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Thus the start time of fuv should be after u’s end time, while v’s start time should be after the end time of fuv, denoted by (7) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' sM u + pu ≤ sN fuv, ∀fuv ∈ ˆF (7) sN fuv + qeuv ≤ sM v , ∀fuv ∈ ˆF (8) Communication resource constraints: The Communication resource constraints include (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Constraint (10) means that the flow transmissions in the same network channel will not overlap, which is also a typical disjunctive constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' sN f > 0, ∀f ∈ ˆF (9) sN f + qf′ ≤ sN f′ or sN f′ + qf′ ≤ sN f , ∀f, f ′ ∈ ˆF, f ̸= f ′, nf = nf′ (10) Finally, the resulting CTM-ICCTSP can be expressed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' P1 : min Cmax s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (1) − (10) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' LINEARIZED REFORMULATION AND TOPOLOGY AWARE BRANCH AND CUT METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' General flow concept and auxiliary virtual channel Due to constraints (4), (5), (6), (7), (8) and (10), CTM- ICCTSP is a non-linear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Constrains (4) and (10) can be transferred into linear constraint formulations by the Big- M and Convex Hull reformulation methods in disjunctive pro- gramming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' However, constraint (5) is not so easy to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' To eliminate the volatility of ˆF in (5), we define the general flow concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' A general flow may be an internal or external flow transferred between two precedence-constrained tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Only the external flows compete for network resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' To construct a unified flow scheduling framework compatible with the two types of flows, we introduce the auxiliary virtual channel, which is contention-free for all internal flow transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Thus the external flows are transferred via physical network links while the internal flows are handled by the auxiliary virtual channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' By introducing this auxiliary virtual channel, the uncertainty of external flows can be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The auxiliary virtual channel is denoted by ˆk, and thus the communication resource set is N ∪ ˆk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' A general flow f ∈ F must be placed on either real com- munication channels or the auxiliary virtual channel, which separately corresponds to the external or internal transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Since each general flow must be transferred, we have � τ∈T n � k∈N ∪ˆk Yfkτ = 1, ∀f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (11) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Linearization of computation and communication disjunc- tive constraints By introducing the general flow and auxiliary virtual chan- nel, the scheduling entity ˆF in P1 can be replaced by the general flow set F, which eliminates the uncertainty of the the scheduling entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Therefore, we can continue to adopt the re- formulation methods in disjunctive programming [19] to trans- form P1 into an equivalent Integer Linear Programming (ILP) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' To linearize constraint (4), we introduce two types of auxiliary variables to describe the placement of computation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Binary variables ψjj′i ∈ {0, 1} indicate whether two computation tasks are placed in the same machine, where vj, vj′ ∈ � J , vj ̸= vj′, 1 ≤ i ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' If computation tasks vj and vj′ are both placed on the i-th machine, ψjj′i = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' otherwise ψjj′i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' To construct ψjj′i, we have 0 ≤ � τ∈T Xjiτ + � τ∈T Xj′iτ − 2 · ψjj′i ≤ 1, ∀vj, vj′ ∈ ˆ J , vj ̸= vj′, 1 ≤ i ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (12) Binary precedence indicator variables σjj′ ∈ {0, 1} rep- resent the precedence relationship between two computation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' If task vj starts no later than task vj′, σjj′ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' otherwise σjj′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' These constraints can be guaranteed by sM j′ − sM j ≤ Tmax · σjj′ − ϵ(1 − σjj′), ∀vj, vj′ ∈ ˆ J , vj ̸= vj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (13) where ϵ ∈ (0, 1) is a small enough constant commonly used in the logical formulation of integer programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' With ψjj′i and σjj′, the disjunctive constraint (4) can be linearized as sM j + pj − sM j′ ≤ Tmax · (2 − σjj′ − � 1≤i≤M ψjj′i), ∀vj, vj′ ∈ ˆ J , vj ̸= vj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (14) Constraint (14) guarantees that task vj′ must start after the completion of task vj when they are placed in the same machine, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=', σjj′ = 1 and � 1≤i≤M ψjj′i = 1 hold simulta- neously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Similarly, to linearize constraint (10), auxiliary binary vari- ables χff′k and φff′ are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' χff′k indicates whether two flows are both placed in the k-th communication channel, while precedence indicator variable φff′ represents the prece- dence relationship between two flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The constraints of χff’k and φff’ as well as the reformulation of constraint (10) are shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 0 ≤ � τ∈T Yfkτ + � τ∈T Yf ′kτ − 2 · χff′k ≤ 1, ∀f, f ′ ∈ F, f ̸= f ′ (15) sN f − sN f ′ ≤ Tmax · φff′ − ϵ(1 − φff′), ∀f, f ′ ∈ F, f ̸= f ′ (16) sN f + qf − sN f ′ ≤ Tmax · (2 − φff − � 1≤k≤N χff′k), ∀f, f ′ ∈ F, f ̸= f ′ (17) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Linearization of causality and precedence constraints be- tween computation and communication tasks After the separate linearizations of computation and com- munication disjunctive constraints, we continue to reformulate the causality and precedence constraints (5), (6), (7) and (8) to integrate the scheduling of computation and communication tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' With the introduced auxiliary variable ψjj′i, the causality relationship between computation task placement and network flow occurrence in constraint (6) can be rewritten by � 1≤i≤M ψuvi = � τ∈T Yfuv,ˆk,τ, ∀fuv ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (18) With the general flow concept, the two cases in the previous precedence constraints can be represented in a unified form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' For each pair of precedence-constrained tasks u and v con- nected by general flow fuv, the start time of fuv must be after the end time of u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=', sM u + pu ≤ sN ∪ˆk fuv , ∀fuv ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (19) If flow fuv is transferred internally, � τ∈T Yfuv,ˆk,τ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' otherwise, the answer is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' No matter fuv is transferred externally or internally, the start time of task v must be after the end time of flow fuv, denoted by sN ∪ˆk fuv + ru � τ∈T Yfuv,ˆk,τ + qfuv(1 − � τ∈T Yfuv,ˆk,τ) ≤ sM v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (20) Finally, CTM-ICCTSP can be linearized as P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' P2 :min Cmax s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' (1), (3), (11) − (20), Cmax ≥ sM j + pj, ∀vj ∈ ˆ J , Cmax ≥ sN ∪ˆk f + qef(1 − � τ∈T Yfuv,ˆk,τ), ∀f ∈ F, Cmax ≥ sN ∪ˆk f + ref � τ∈T Yfuv,ˆk,τ, ∀f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Topology-aware Branch and Cut (TABC) Algorithm Problem P2 is an ILP problem and can be solved by the classic B&C algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Since the searching iterations of B&C may vary dramatically in different problem instances, its running time may be unacceptable in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' To avoid this situation, we take the topological relationship among computation tasks and network flows in the DWDAG into consideration and propose an efficient Topology Aware Branch and Cut (TABC) Algorithm based on the following strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 1) Branch with the chain precedence constraints: The branching process can be carried out from enumerating the precedence indicator matrices, Θ = (σjj′) or Φ = (φff′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Based on the precedence relationships among computation and communication tasks, some searching branches can be directly pruned, and thus the searching space can be greatly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The chain precedence constraints include the precedence con- straints along successive computation task and flow chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' For example, in computation tasks if σjj′ = σjj′′ = 1, then σjj′′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' For network flows, if φff′ = φf′f′′ = 1, then φff′′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' In general,if fuv ∈ F, then σuv = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' if fuv, fvv’ ∈ F, then φfuvfvv’ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 2) Update task interval constraints: In each job’s DWDAG, the earliest and latest start time of each computation task or network flow can be inferred according to both the job’s incumbent upper and lower bounds and the processing and transfer times of the other computation tasks and flows along its longest branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' A graph theory-based method to obtain the longest branch of a directed acyclic graph can be found in [18], which is applicable to this work and very easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Therefore, the searching tree can be further pruned according to the reduced interval constraint for each task or flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 3) Utilize the symmetry of solution space: One symmetry feature is from the homogeneity of physical machines and communication channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' If we switch the indexes of two scheduled machines or communication channels in a feasible solution, the result is an equivalent solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Setting each task’s machine affinity value in advance can eliminate a large number of symmetric solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The other symmetry feature comes from the symmetry of nodes or edges in the job graph, thus different priorities are added to the equivalent computation tasks or network flows to reduce redundant searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Algorithm 1 Topology Aware Branch and Cut Algorithm Input: Job G = {V, E, P, Q, R}, resource set {M} and {N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Output: Optimal solution S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 1: Initialization: 2: Calculate a solution S using heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 3: Set LB = sum(P ) |M| , UB = min(Cmax(S), sum(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 4: Set the initial precedence matrices with chain precedence constraints in G, the initial interval constraints, and the affinities and priorities of tasks/flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 5: Repeat 6: Solve P2 using B&C for an incumbent solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 7: If new LB or UB is obtained, recalculate and update interval constraints to the current active node in P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 8: Until Optimal solution S∗ found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 9: return S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The TABC procedures are shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The three pruning strategies are all adopted to efficiently reduce the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Due to the inherent unreliability and instability properties of B&C in solving ILP problems, the performance of the proposed TABC may also vary in different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Proper termination conditions can be set to avoid too long running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Though TABC may terminate before producing a global optimal solution, the incumbent solution can still be better than most heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' In addition, since the variables in P2 are binary, TABC can be efficiently implemented by splitting P2 into multiple parallelly executed sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Thus the time complexity can be greatly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' SIMULATION RESULTS To evaluate the performance of the proposed scheme, we conduct simulations over the synthetic smart grid data analytic jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' As in [18], the computation task processing times and the data transfer times are randomly and uniformly chosen from [1, 100] and [1, 50], respectively, which is to mimic the different data volumes by normalizing the maximum time to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The larger the transfer time corresponds to the larger the data volume transfered among the adjacent tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 2, we compare the average normalized makespans (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=', job completion times) of six scheduling schemes with different machine numbers and one network channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The Random Scheduling scheme places the computation tasks ran- domly, while the List Scheduling scheme is from [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Both of them only considered the placement of computation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The Partition Scheduling, Generalized List (G-List) Scheduling and G-List-Master Scheduling schemes are from [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' For each scheduling scheme, we generated 3000 job cases each with ten computation tasks, calculated the normalized makespans of these jobs, and averaged them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The normalized makespan is the ratio of the makespan obtained by one scheduling scheme and the upper bound makespan when the job’s computation tasks were all placed on a single machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' For the Random Scheduling and the List Scheduling schemes, their average normalized makespans increase with the machine numbers due to the ignorance of data transfer optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' For the other four scheduling schemes, their average normalized makespans decrease with the increase of machine numbers since the computation and communication tasks are jointly scheduled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The proposed Integrated Computing and Communication Task Scheduling (ICCTS) scheme obtains the lowest average nor- malized makespans among all the six scheduling schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Since the linearization reformulation keeps the optimality of the original CTM-ICCTSP problem, ICCTS can obtain the optimal solution and act as an important benchmark for heuristic schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' It can be observed that when the machines (computing resources) are sufficient, ICCTS can averagely reduce the job completion time by up to 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content='2 Average normalized makespan Number of machines Random Scheduling List Scheduling Partition Scheduling G-List-Master Scheduling G-List Scheduling ICCTS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Average normalized makespans of different scheduling schemes versus the number of machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 3, we compare the efficiency of B&C and TABC in the ICCTS scheme, and the simplex iteration number is adopted as the algorithm efficiency metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Though their average simplex iterations both increases exponentially with the computation task number, TABC can significantly reduce the iterations due to the pruning rules from DWDAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 4 5 6 7 8 9 10 11 12 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 Average simplex iterations Number of computing tasks B&C TABC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Average simplex iterations of B&C and TABC versus the number of computing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' CONCLUSION In this work, an integrated computation and communication task scheduling scheme for smart grid data analytic appli- cations is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' The mathematical formulation and the corresponding constraint linearization of the job scheduling problem were introduced, and an efficient Topology Aware Branch and Cut method was designed to improve the searching speed for the optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' Numerical results confirmed the necessity of considering data volume and validity of the proposed integrated scheduling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFKT4oBgHgl3EQffy6V/content/2301.11831v1.pdf'} +page_content=' REFERENCES [1] Q.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' † and Hussain Gohar1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' ‡ 1Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' University of Szczecin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Wielkopolska 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 70-451 Szczecin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Poland 2National Centre for Nuclear Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Andrzeja Sołtana 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 05-400 Otwock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Poland 3Copernicus Center for Interdisciplinary Studies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Szczepa´nska 1/5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 31-011 Kraków,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Poland (Dated: January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 2023) The effect of the generalized uncertainty principle (GUP) on nonextensive thermodynamics ap- plied to black holes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' as well as the sparsity of radiation at different temperatures associated with each nonextensive entropy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We examine the Rényi, Tsallis-Cirto, Kaniadakis, Sharma Mittal, and Barrow entropies, temperatures, and heat capacities and show that, in each case, due to GUP corrections, the temperature and entropy have finite values, implying that the final state of the black hole is a remnant at the end of the evaporation process and that the sparsity of the radiation at each temperature depends on the mass of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We also find that GUP reduces the value of the sparsity parameter for each case as compared to the sparsity parameter at Hawking temperature, which is always constant throughout the evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' INTRODUCTION Black holes emit radiation due to the Hawking evap- oration process, and therefore, there is an established concept of Hawking temperature [1] and Bekenstein entropy [2] connected with the black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The black hole evaporation process operates within the purview of quantum field theory, and one of its more in- triguing aspects may be that it appears to indicate a non- unitary evolution, which gives rise to the well-known is- sue of the information loss paradox [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In this regard, black holes behave like thermodynamic objects, and the laws of black hole thermodynamics [6–10] are analogous to the conventional thermodynamic laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The thermo- dynamics of black holes have been extensively studied and used in a variety of cosmological and gravitational applications [11–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Entropy measures how difficult it is for an outside ob- server to get information about the underlying structure of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This is a clear reflection of the macro- scopic features that result from the quantum statisti- cal mechanics that govern the behavior of quantum mi- crostates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the case of black holes, there is no defi- nition of Bekenstein entropy in quantum statistical me- chanics and it only relies on Hawking’s area theorem [21], therefore, it would be required to have a complete theory of quantum gravity in order to fully comprehend the origin of this entropy and the nature of microstates in the case of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Therefore, we rely on the defi- nition of Bekenstein entropy for black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the case of a Schwarzschild black hole with mass M, the Hawk- ing temperature TH and Bekenstein entropy SB are given by [1, 2] TH = ¯hκ 2πkBc , SB = kBc3A 4G¯h , (1) ∗ ilim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='cimdiker@phd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='usz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='pl † mariusz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='dabrowski@usz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='pl ‡ hussain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='gohar@usz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='pl where ¯h, G, kB, and c are the reduced Planck constant, the Newton gravitational constant, the Boltzmann con- stant, and the speed of light, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The area A of the event horizon is defined as A = 4πr2 h in the above equation (1), where rh = 2GM/c2 is the Schwarzschild radius and κ = c4/4πGM is the surface gravity defined on the event horizon of the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The core assumption of Gibbs thermodynamics and statistical mechanics is that entropy is extensive and ad- ditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Nonextensive statistical mechanics, such as Tsal- lis nonextensive statistical mechanics [22–31], is the out- come of removing this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The assumption of the extensive nature of entropy is connected to ig- noring the long-range forces between thermodynamic sub-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Since the size of the system exceeds the range of the interaction between the system’s compo- nents, Gibbs thermodynamics ignores these long-range forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Because of this, the total entropy of a composite system equals the sum of the entropies of the individ- ual subsystems and entropy grows with the size of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' However, long-range forces are important in various unique thermodynamic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For instance, if we think of a black hole as a (3 + 1) dimensional ob- ject, it is vital to note that Bekenstein entropy scales with the area and is thus regarded as a nonextensive quan- tity [32–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Furthermore, because of the area scaling, Bekenstein entropy is nonadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Therefore, Gibbs thermodynamics or statistical mechanics may not be the appropriate choice for studying the thermodynamics of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In order to understand the nonextensive and nonadditive nature of Bekenstein entropy, several extensions [22, 39–44] of standard Gibbs thermodynam- ics have been applied to black holes and cosmologi- cal horizons [45–70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' One of the main proposals is the Tsallis-Cirto’s black hole entropy definition [32], which makes the black entropy extensive and compatible with the Legendre structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Rényi entropy [39], being a mea- sure of entanglement, is another definition of entropy applied to black holes and cosmological horizons which is nonextensive, but additive (by assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' There arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='00609v1 [gr-qc] 2 Jan 2023 2 have been some other nonextensive forms of entropy suggested such as the Sharma-Mittal entropy [40, 41] as a generalization of Rényi entropy, the Kaniadakis en- tropy [42] which takes inspiration from Lorentz group transformations and the Barrow entropy [44] which is based on a hypothetical fractal structure of black hole horizon as a result of quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Due to the prevalence of quantum gravity effects, it is anticipated that the semiclassical technique would fail during the last phases of Hawking evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' There is currently no satisfactory theory of quantum gravity that enables us to completely explain that regime, despite the development of several quite diverse proposals [71–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Investigating the phenomenological consequences of an underlying theory of quantum gravity is one technique to explore the quantum gravity effects at those scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The generalized uncertainty principle (GUP) [76–79] is one approach that has the benefit of being sufficiently generic to be compatible with several quantum gravity theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The Bekenstein entropy and Hawking temper- ature of a black hole in its last phases of evaporation are modified within this framework [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Because of these modifications, black holes do not entirely evapo- rate during the evaporation process, and the final state of the black hole is a remnant of the order of Planck mass Sparsity [80–91] is an important feature of Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It is defined as the average time between the emission of successive quanta over the timescales set by the energies of the emitted quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It was shown that Hawking radiation is very sparse during the black hole evaporation process [84], which is one of the key char- acteristics that distinguish it from black-body radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' However, it has been found that when GUP corrections are incorporated [87–89], the sparsity decreases toward the late stages of evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' When nonextensivity is considered in the context of Rényi temperature [90], the Rényi radiation is initially not sparse, but as evaporation progresses, it begins to become sparse and eventually approaches the case of Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In this paper, we are interested in exploring the GUP modifications to the nonextensive entropies and corre- sponding thermodynamic quantities in Rényi, Tsallis- Cirto, Sharma-Mittal, Kaniadakis, and Barrow nonex- tensive statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Furthermore, the sparsity of the radia- tion is analyzed at different temperatures corresponding to different nonextensive entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The following is the outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' II, we introduce the notion of GUP and apply it to the case of standard thermodynamic black hole quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' III, we introduce nonextensive entropies and accompa- nying nonextensive thermodynamic quantities, as well as GUP modifications to nonextensive black hole ther- modynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' IV, we summarize and discuss our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' GUP AND BLACK HOLE THERMODYNAMICS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Generalized Uncertainty Principle One common aspect of several quantum gravity the- ories is that they all predict a minimum measurable length [77, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For example, the notion of minimal length is defined in string theory as the string length [72, 93], in loop quantum gravity [74] it is the expec- tation value of the length operator, and this notion can also be developed by the phenomenological aspects coming from black hole physics [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Because of the ap- pearance of a minimum length at the Planck scale in var- ious quantum gravity approaches, it has been proposed that the Heisenberg Uncertainty Principle (HUP) ∆x0∆p ≥ ¯h, or ∆x0 ∼ ¯h ∆p (2) where ∆x0 and ∆p are position and momentum uncer- tainties can be modified when gravitational interaction is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The simplest argument for the modifica- tion of HUP within the framework of Newtonian theory is that there is a gravitational acceleration⃗a of an electron due to photon of mass E/c2 [73], where E is the pho- ton energy and r is the photon-electron distance, which reads ⃗a = ¨⃗r = − G(E/c2) r2 ⃗r r, (3) and the interaction takes place in a characteristic region of length L ∼ r and in characteristic time t ∼ L/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Then, the velocity acquired by an electron ∆v is ∆v ∼ GE c2r2 L c , (4) and the (extra due to gravity) distance ∆x1 it is shifted reads ∆x1 ∼ GE c2r2 L2 c2 ∼ G∆p c3 = c∆p 4Fmax = l2 p ∆p ¯h , (5) where lp = √ G¯h/c3 is the Planck length, and Fmax = c4/4G is the maximum force [94–97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Extra uncertainty (5) adds to the standard HUP uncertainty of position ∆x0 as in (2) giving ∆x = ∆x0 + ∆x1 ∼ ¯h ∆p + l2 p ∆p ¯h , (6) leading to the generalized uncertainty principle (GUP) ∆x∆p ≥ ¯h � 1 + l2 p ¯h2 (∆p)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (7) Taking an algebraic point of view, GUP can be derived from the deformed commutation relation between the 3 position operator ˆx and the momentum operator ˆp such that [ ˆx, ˆp] = i¯h f ( ˆp), (8) where f ( ˆp) is a general function of momentum operator ˆp and there exist different proposed functions for f ( ˆp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In order to make the function f ( ˆp) compatible with (7), following the literature, we choose f ( ˆp) = 1 + α l2 p ¯h2 ˆp2, (9) where we the introduce GUP parameter α – a dimen- sionless parameter predicted to be an order of unity, but in reality bounded by different experiments and obser- vations to be much larger than that [98–102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' By intro- ducing α, the equation (10), now, reads as ∆x∆p ≥ ¯h � 1 + α l2 p ¯h2 (∆p)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (10) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' GUP Modified Hawking Temperature and Bekenstein Entropy An interesting application of (10) to black hole physics is the modification to the Hawking temperature, which can be derived by solving it for ∆p, which gives ∆p = ∆x ¯h αl2p � �1 ± � 1 − αl2p (∆x)2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (11) We consider the + sign in (11), following the discus- sion in [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Considering the minimum position uncer- tainty near the event horizon of the Schwarzschild black hole as ∆x = 2lp = 4GM/c2, where lp is taken as the Schwarzschild radius rh, the GUP modified Hawking temperature TGUP reads TGUP = m2 pc2 8πkBM � ��� 4 2 + � 4 − α m2p M2 � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (12) By introducing a correction term due to GUP, K(α, M), TGUP can be written in terms of TH and K, such that TGUP = TH(M)K(α, M), (13) where the GUP correction term is defined as K(α, M) = 4 2 + � 4 − α m2p M2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (14) This provides us with a more compact form of TGUP, which will be used in the next sections for GUP mod- ifications to the thermodynamic quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Using the Clausius relation, the GUP modified Bekenstein entropy SGUP in terms of SB and the correction term K(α, M) can be written as SGUP = SB K − απkB 2 ln � 4M m0K � , (15) where m0 is a dimensionful constant of unit mass, which is introduced in order to make the logarithm dimension- less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Note that in the limit α → 0, the correction term K goes to one, and hence TGUP and SGUP reduce to TH and SB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The plots of (12) and (15) are given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It is important to mention that all the plots in the pa- per, unless explicitly stated, are given in natural units ¯h = c = G = 1 and also with the GUP parameter α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' TH TGUP,α=1 TGUP,α=-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 M T Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Temperature vs mass for the Hawking temperature TH and the GUP corrected temperature with positive and neg- ative values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Threshold with positive α for mass lies at the remnant mass M2r = (α/4)m2p (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' formula (16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' SB SGUP,α=1 SGUPα=-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0 10 20 30 40 50 M S Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Entropy vs mass for the Hawking temperature and GUP corrected temperatures with positive and negative val- ues of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The threshold for mass lies at the remnant mass given by M2r = (α/4)m2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It is interesting to note that, for real physical situa- tions, the equation (14) gives a bound on the mass which reads: M2 ≥ αm2 p/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This means that for positive values of α, the black hole evaporation stops when the mass of the black hole reaches some critical value of mass Mr = √αmp 2 = 2lp √α c2 Fmax, (16) which is called the black hole remnant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Therefore, we can say that the final state of the black hole evapora- tion is a remnant having the mass Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In fact, without 4 a well-defined quantum gravity theory, we cannot pre- dict what happens if the mass of a black hole is smaller than this critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the critical mass value Mr, the formulas (12) and (15) for TGUP and SGUP, give the temperature Tr and the entropy Sr for the remnant as [90] Tr = mpc2 2πkB √α, Sr = παkB 2 � 1 − ln �√αmp m0 �� , (17) provided that α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For α < 0 in (14), we have a smooth correction function defined for all black hole mass val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In this case, the black hole continues to radiate slowly and yields an infinite lifetime [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' When M ap- proaches zero, interestingly, the temperature is still fi- nite, and for this case, in [103], it is referred to as a rem- nant with zero rest mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' GUP Modified Heat Capacity In order to investigate the GUP modifications to the heat capacity of a black hole with mass M, we use the definition of heat capacity C, which reads C = −S′2(M) S′′(M) , (18) where S is the black hole entropy and prime and dou- ble prime denote the first and second derivative with respect to the mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the case of Schwarzschild black hole, we have (denoting C as CSc) CSc = −8πkB M2 m2p , (19) and we can see that it is negative for all mass values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This means that the Schwarzschild black hole is thermo- dynamically unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In order to introduce GUP cor- rections, we introduce the quantity βGUP = 1 kBTGUP , (20) which after using (12) gives S′ GUP(M) kBc2 = βGUP = β K , (21) where β = 1/kBTH is the inverse Hawking tempera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Differentiating βGUP once more, and using equa- tions (18) and (21), we obtain the GUP modified heat capacity CGUP, which can be written as (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 3) CGUP = CSc �2 − K K2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (22) This means that the GUP corrections still yield a nega- tive heat capacity for M > Mr, and when the black hole mass approaches the critical mass Mr, we have K = 2 and interestingly, we get the zero heat capacity for the remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In such a case, from the thermodynamic point of view, a small amount of heat would then increase the temperature of the remnant by an infinite amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' CSc CGUP,α=1 CGUP,α=-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 30 25 20 15 10 5 0 M C Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Specific heat capacity of the Hawking radiation for GUP corrected black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For positive α, there is a remnant with zero heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' GUP Modified Sparsity of Hawking Radiation One of the most important aspects of Hawking radia- tion is that it is extremely sparse as compared to black- body radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The sparsity can be defined by using the parameter η [84, 87, 90], η = C g � λ2 t Ae f f � , (23) where C is a dimensionless constant associated with dif- ferent physical cases [84], g is the spin degeneracy fac- tor of the particle, λt = 2π¯hc/kBT is the thermal wave- length in terms of the temperature T and Ae f f = 27A/4 [80, 84] is the effective area with A being the horizon area for the case of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the Schwarzschild black hole, one can find the thermal wavelength λt by taking T = TH = 1/kBβ as λt = 2π¯hc kBTH = 2π¯hcβ, (24) and the sparsity parameter for the Hawking radiation reads [84] ηH = 64π3 27 ≈ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='38, (25) which is constant and is much greater than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Note that for standard black body radiation, the value of η is less than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This implies that the sparsity param- eter clearly differentiates the Hawking radiation from blackbody radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' One can obtain the GUP effects on the sparsity by replacing the Hawking temperature with the GUP corrected temperature TGUP given by (12) [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' However, it is assumed that GUP also modifies the black hole horizon area [87, 90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Thus, it is logical to take the effective area that GUP modifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In fact, the GUP mod- ifications to A can be derived from the equation (15) by writing it as SGUP = kBc3AGUP 4¯hG , (26) 5 ηH ηGUP,α=1 ηGUP,α=-1 0 1 2 3 4 0 20 40 60 80 100 M η Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Sparsity of Hawking vs GUP corrected black holes in natural units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For positive values of α, we observe that sparsity decreases when a black hole is near the final evaporation state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' where the GUP modified area AGUP reads AGUP = A K − απl2 p ln � 16A A0K2 � , (27) and A0 = 16πm2 0G2/c4 is a constant having the dimen- sion of area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Note that in [90], corrections are only in the first order of α, while in the above equation (27) the area is corrected to all orders in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Now, the GUP modified sparsity can be found by replacing T by TGUP and A by AGUP in (23), which now reads ηGUP = ηH K2 � A AGUP � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (28) Interestingly, GUP modified sparsity ηGUP, depends on the mass of the black hole and the GUP parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the negative values of α, the sparsity parameter in- creases as M goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the positive values of α, the sparsity parameter decreases below the values of sparsity for the Hawking radiation until it reaches the critical mass Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 4, we can see that the GUP corrected sparsity is not a constant and it increases first before M approaches Mr for α > 0 and then it decreases to finite value when M approaches to Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the case of α < 0, first, it decreases, and then it goes to plus in- finity when M approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It is due to the fact that A/AGUP > 1 for α > 0 and ηH/K2 turns back the spar- sity from a maximum value to a constant value, which is less than ηH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Therefore, we can clearly see the effects of GUP on sparsity due to TGUP and AGUP as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Similarly, A/AGUP < 1 for α < o and K goes to zero when M approaches zero, therefore, sparsity de- creases first, and then it goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Note that in [89], the GUP corrected area is not taken into account, therefore, there is no bump in the sparsity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' GUP AND NONEXTENSIVE BLACK HOLE THERMODYNAMICS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Tsallis Nonextensive Entropy Entropy plays a significant role in Gibbs thermody- namics or statistical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It is extensive and adheres to the additive composition rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' However, Gibbs statistical mechanics ignores long-range forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' On the other hand, there are some physical systems for which Gibbs thermodynamics cannot be the appropri- ate choice to apply [24] since they are subject to long- range forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Important examples are the some self- gravitating systems such as black holes, since for them long-range forces play significant role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For that reason Constantino Tsallis in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' [22, 24] generalized the con- ventional Gibbs entropy for nonextensive systems in or- der to encompass and address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Tsallis entropy ST was one of the earliest proposals to extend Gibbs en- tropy and the suggested new form of it reads ST = −kB ∑ i [p(i)]q lnq p(i), (29) where p(i) is the probability distribution defined on a set of microstates Ω, with the parameter q determining the degree of nonextensivity, and we consider it positive to ensure the concavity of Sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The q-logarithmic function lnq p is given by lnq p = p1−q − 1 1 − q , (30) where, in the limit q → 1, Tsallis entropy Sq given by (29), reduces to Gibbs entropy SG SG = −kB ∑ i p(i) ln p(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (31) In fact, the Tsallis entropy (29) satisfies quite general, nonadditive composition rule of the following form ST 12 = ST 1 + ST 2 + λ kB ST 1ST 2, (32) for a composite system ”12”, made up of two subsys- tems ”1” and ”2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In above equation, we have defined a new nonextensivity parameter λ = 1 − q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Rényi Entropy The Rényi entropy [39], a measure of entanglement in quantum information that is additive and preserves event independence, is another important generaliza- tion of the Gibbs-Shannon entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It is defined as SR = kB ln ∑i pq(i) 1 − q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (33) 6 It is important that SR can be written in terms of ST by using the formal logarithm approach [30], and both en- tropies are related as follows SR = kB λ ln[1 + λ kB ST ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (34) It is interesting to mention here that SR is the equilib- rium entropy which corresponds to an equilibrium tem- perature TR defined from the equilibrium condition by maximizing the Tsallis entropy (32), which is given by [53] TR = (1 + λ kB ST ) 1 kBβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (35) Here, kBβ = ∂ST /∂U, where U is the internal energy of the nonextensive system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Rényi black hole Entropy and Temperature For the case of a Schwarzschild black hole, assuming that the Bekenstein entropy SB is just the Tsallis entropy ST , and replacing internal energy U with the mass of the black hole M in equations (34) and (35), the Rényi entropy can be defined on the horizon of a black hole as [33–37] SR = kB λ ln[1 + λ kB SB], (36) and the associated Rényi temperature reads TR = (1 + λ kB SB)TH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (37) Furthermore, we can write down the GUP corrected Rényi entropy using GUP corrected Bekenstein entropy as follows [90] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 5) SRgup = kB λ ln � 1 + λ kB (SGUP) � , (38) and corresponding GUP modified Rényi temperature TRgup can be written as (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 6) TRgup = � 1 + λ kB (SGUP) � KTH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (39) The Rényi entropy increases logarithmically (for 0 < λ < 1), whereas the Bekenstein entropy (λ → 0) in- creases quadratically, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Furthermore, for the GUP corrections, the Rényi black holes do not completely evaporate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' rather, evaporation stops at the critical mass Mr, leaving a remnant with finite entropy and temperature as the Rényi black hole’s final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Using (37) and (39), we can write the inverse Rényi temperature parameters, βR and βRgup, which will fur- ther be used in calculating the heat capacities, such that kBβR = S′ B(M)/c2 1 + λ kB SB = kBβ 1 + λ kB SB , (40) λ=0 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 λ=1 λ=0 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 λ=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0 2 4 6 8 10 M SR Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Rényi entropy SR of a black hole vs its mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines represent GUP corrected cases, λ → 0 limit is the Bekenstein-Hawking case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' λ=0 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 λ=1 λ=0 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 λ=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 M TR Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Rényi temperature TR of a black hole vs its mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines represent GUP corrected cases, λ → 0 limit is the Bekenstein-Hawking case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' and the GUP-corrected inverse Rényi temperature reads kBβRgup = S′ GUP(M)/c2 1 + λ kB SGUP = kBβGUP 1 + λ kB SGUP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (41) One may determine the characteristic length scale LR for λ [49, 50, 52], which reveals the impact of nonexten- sive parameter λ in SR and SRgup, and in TR and TRgup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' As a result, it can be concluded that below this charac- teristic length scale LR, the Rényi temperature behaves like TH, and that above LR, the nonextensive effects in- crease and TR grows linearly with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The precise value for the length scale is found in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Heat Capacity for the Rényi black hole In order to investigate the thermodynamic stability of Rényi black holes, we define the heat capacity CR of the Rényi black hole as CR = −S′2 R(M) S′′ R(M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (42) 7 Inserting (40) and (41) into (42), the heat capacity for the non-GUP case reads CR = CSc 1 + λ kB SB + λ kB CSc , (43) and for the GUP case, we have CRgup = CGUP 1 + λ kB SGUP + λ kB CGUP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (44) We plot the heat capacity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (7), where we can λ=0 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 λ=1 λ=0 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 λ=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 10 5 0 5 10 M CR Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Heat capacity CR of a Rényi black hole vs its mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines represent GUP corrected cases, λ → 0 limit is the Bekenstein-Hawking case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' see that L differentiates two regions for non-GUP and GUP cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In order to understand the behavior of CR in both regions, we find LR in terms of λ from the singular points of equation (43) for the case Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We find, for the non-GUP case λ = − kB [SB + CSc] = m2 p 4πM2 , (45) and for the GUP case, we have λ = − kB [SGUP + CGUP] (46) ≈ m2 p 4πM2 + 3αm4 p 64πM4 + αm4 p log � 4M mp � 32πM4 by ignoring the higher order terms in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This means that for the non-GUP case, we define the mass scale Mc = mp 2 √ πλ , (47) which differentiates the two regions and can be further used to define the characteristic length scale LR, which can be written as LR = 2lp √ πλ, (48) where we have defined LR = GMc/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the GUP case, we would expect the characteristic length scale LRgup ≈ LR + α f (λ) by using equation (47), where f is a function of the nonextensivity parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' However, we can not solve it exactly, and it again shows the effects of α and λ for the values of M greater than the GUP cor- rected mass scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Interestingly, for the non-GUP case, the heat capacity is positive for the values greater than this scale, and below this scale, black holes have neg- ative heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This means that black holes with higher masses than Mc are thermodynamically stable and with masses lower than Mc, they are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Note that, if we exclude quantum gravity effects, LR should be greater than lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This puts a numerical constraint on the nonextensive parameter λ > 1/4π and this can also be derived by considering Mc > mp by excluding the quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In [49, 50, 52], the authors de- rived this constraint as λ > 1/π because they consid- ered LR = 2GMc/c2 as characteristic length scale for λ, where the extra 2 in LR is motivated by Schwarzschild radius rh = 2GM/c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We believe that the proper way to introduce the length or mass scale for λ should be irre- spective of the definition which is motivated by rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Sparsity of the Rényi Radiation In order to calculate the sparsity of Rényi radiation, we replace T with TR in (23), and so the sparsity param- eter ηR reads ηR = ηH [1 + λ kB SB]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (49) Replacing T with TRgup and using GUP modified area AGUP in equation (23), the GUP modified sparsity pa- rameter ηRgup reads ηRgup = ηGUP [1 + λ kB SGUP]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (50) From (49), we conclude that the sparsity parameter ηR λ=0 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 λ=1 λ=0 λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 λ=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0 20 40 60 80 M ηR Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Sparsity ηR of a Rényi blackhole vs its mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines represent GUP corrected cases, λ → 0 limit is the Bekenstein-Hawking case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' depends on both the mass of the black hole and the nonextensivity parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (8), we can easily see that the radiation is not sparse initially and then, at the final stages of the evaporation, the sparsity 8 grows, reaching the value of ηH, when M approaches to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the GUP case, initially, the behavior of spar- sity is similar to the non-GUP case, however, when M approaches Mr, it has a finite value which is much less than the sparsity of Hawking radiation for the non-GUP and GUP cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Again, we can see the bump before M reaches Mr, which is due to the effect of GUP correc- tions to the Rényi temperature and GUP corrections to the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Tsallis-Cirto Black Hole Entropy Tsallis-Cirto black hole entropy [32] is based on key principles of Gibbs thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' First, the entropy must be extensive and additive, and second, the entropy and associated temperature for a thermodynamic sys- tem must satisfy the Legendre structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the case of black holes, if we rely on the definition of Beken- stein entropy, then black holes are considered to be two-dimensional thermodynamic objects since Beken- stein entropy scales with area and Bekenstein entropy and Hawking temperature fulfill the Legendre struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' However, if we consider a black hole as a (3 + 1) dimensional thermodynamic object, then the Bekenstein entropy is thought to be nonextensive due to its area scaling and also because it follows a nonadditive com- position rule S12 = S1 + S2 + 2√S1 √S2 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' [90]), whereas Gibbs statistical mechanics or thermodynam- ics is based on the extensive and additive properties of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This indicates that Bekenstein entropy vio- lates a key principle of classical Gibbs thermodynamics and that new definitions of entropy and temperature for black holes are required in order to comply with the fun- damental principles of thermodynamics in the case of (3 + 1)-dimensional black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Therefore, Tsallis and Cirto proposed the following entropy definition [32, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Sδ kB = �SB kB �δ , (51) where δ > 0 is a real parameter and it follows the com- position rule for a composite thermodynamic system, which is given by Sδ12 = kB ��Sδ1 kB �1/δ + �Sδ2 kB �1/δ�δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (52) In this context, the SB is additive, and Sδ is nonadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For δ = 3/2, Sδ is proportional to the volume for the case of the Schwarzschild black hole, and so it is an ex- tensive quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The corresponding Tsallis-Cirto tem- perature can be written by using the Clausius relation [53] Tδ = TH δ �SB kB �1−δ , (53) and it scales with 1/M2 for δ = 3/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=', Tδ ∝ 1/M2, for the case of Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' GUP corrections to the Tsallis-Cirto black hole entropy can be obtained by the GUP corrected Bekenstein entropy SGUP given by (15) into (51), which results in Sδgup kB = �SGUP kB �δ , (54) and the corresponding GUP-modified Tsallis-Cirto tem- perature can be derived from the Clausius relation, giv- ing Tδgup = TGUP δ �SGUP kB �1−δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (55) From the Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (9) and (10), it shows that the evap- δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='7 δ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='7 δ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0 5 10 15 20 M Sδ Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Tsallis-Cirto entropy ST of a black hole vs its mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines represent GUP-corrected cases in this figure oration process stops at the critical value Mr for the Tsallis-Cirto case when GUP corrections are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This means that the final state of the black hole for the Tsallis-Cirto case is also a remnant with finite entropy and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Generally, for the non-GUP case, the parameter δ plays a significant role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For δ > 1/2, the Tsallis-Cirto entropy behaves similarly to Bekenstein en- tropy and increases exponentially with mass, whereas for δ < 1/2, it increases with mass sub-linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For δ = 1/2, the entropy depends linearly on mass, and in this case, Tsallis-Cirto temperature becomes constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Furthermore, the behavior of the Tsallis temperature is similar to the Hawking temperature for δ > 1/2 while for δ < 1/2, the behavior is completely different for the non-GUP case and, interestingly, it behaves like Rényi temperature for the GUP-corrected case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Note that, un- like λ parameter of the Rényi entropy, δ is not associated with the length scale for the non-GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' On the other hand, introducing GUP corrections to Tsallis-Cirto en- tropy, one can define a characteristic length scale for δ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Heat Capacity for Tsallis-Cirto black holes Following the previous subsection, the heat capacity for the Tsallis-Cirto case can be written in terms of Csc, 9 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='7 δ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='7 δ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='6 M Tδ Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Temperature Tδ vs the mass M for Tsallis-Cirto black hole entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines correspond to a GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' and SB Cδ = CSc � SB SB − (δ − 1)CSc � , (56) where for the Schwarzschild black hole, we have CSc = −2SB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For δ = 1/2, we have infinite heat capacity for all masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For δ < 1/2, we have positive heat capac- ity values and negative heat capacity for δ > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This means that black holes are thermodynamically stable for δ < 1/2, and unstable for δ > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the GUP correc- δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='7 δ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='4 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='7 δ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 20 10 0 10 20 M Cδ Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Heat Capacity Cδ for Tsallis-Cirto black hole en- tropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines correspond to a GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' tions, we can write the GUP-corrected heat capacity as Cδgup = CGUP � SGUP SGUP − (δ − 1)CGUP � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (57) Note that from equations (15) and (22), we have −2SGUP ̸= CGUP, therefore, we can find an associated characteristic length scale Lδgup for the δ parameter, for which, we have two regions, which corresponds to pos- itive and negative values of GUP corrected heat capac- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The length scale Lδgup can be found by using the singular points of the above equation (57) for δ, which is given by δ = SGUP CGUP + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (58) One could solve the above equation (58) for mass M, which gives Lδgup as a function of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' However, it is ana- lytically not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' One may use the perturbative ap- proach to solve the equation for M and define the corre- sponding length scale or mass scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' From the Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (9) and (11), for δ < 1/2, and below Lδgup, the GUP cor- rected Tsallis-Cirto entropy behaves like SR and it gives positive GUP modified heat capacity for the GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For values δ > 1/2, Lδgup does not exist as (58) yields imaginary numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Thus, it gives negative heat capac- ity, implying that GUP-corrected Tsallis black holes are thermodynamically stable for δ < 1/2, and unstable for δ > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Sparsity of the Tsallis-Cirto Radiation By following the previous subsection, and using the Tsallis-Cirto temperature, we can write the sparsity pa- rameter ηδ for Tsallis-Cirto radiation as ηδ = ηHδ2 �SB kB �2δ−2 , (59) and the GUP-corrected sparsity ηδgup, by using (23) and (55), it can be written as ηδgup = ηGUPδ2 �SGUP kB �2δ−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (60) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (12) depicts the sparsity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' mass relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='8 δ=1 δ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='1 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='8 δ=1 δ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0 50 100 150 200 M ηδ Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Sparsity ηδ for Tsallis-Cirto black hole entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines correspond to a GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' the Tsallis-Cirto temperature, the sparsity scales with M4δ−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Again, the value of δ, significantly changes the behavior of the sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It should be noted that the spar- sity parameter is now affected by mass as well as δ and the GUP-parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In the non-GUP case, ηδ = ηH for δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' When δ > 1, the value of ηδ is initially very high and approaches zero at the end of the black hole evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This means that, initially, the Tsallis-Cirto radiation is highly sparse, and during the final stages of evaporation, it is not sparse at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In this way, for δ < 1, Tsallis-Cirto radiation is initially not sparse, but at the end of the evaporation, it is extremely sparse with the sparsity parameter infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the GUP case, initially, the behavior is the same as for the non-GUP case, but 10 when the mass approaches the order of Planck mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=', the remnant mass Mr, the sparsity parameter de- creases to some finite values for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Note that all these finite values of sparsity parameters are less than the standard sparsity parameter ηH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Sharma-Mittal Entropy Sharma-Mittal (SM) is an entropic form [40, 104] that generalizes the Rényi and Tsallis entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It is defined as SSM = 1 R � � � W ∑ i=1 p1−λ i � R λ − 1 � � (61) where R is another free parameter that is introduced in SM entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Under the equiprobability condition of the states [69], the above equation (61) reduces to SSM = kB R � (1 + λ kB ST)R/λ − 1 � , (62) where R → λ limit yields the Tsallis entropy, and R → 0 yields Rényi entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The Sharma-Mittal entropy obeys the same general nonextensive composition rule (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Assuming that the Bekenstein entropy SB is the same as the Tsallis entropy ST , we can write SSM for the case of a Schwarzschild black hole as SSM = kB R � (1 + λ kB SB)R/λ − 1 � , (63) and replacing SGUP with ST in equation (62), the GUP corrected SM entropy SSMgup reads as SSMgup = kB R � (1 + λ kB SGUP)R/λ − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (64) The corresponding temperatures can be found by using the Clausius relation, as TSM = TH(1 + λ kB SB)1− R λ , (65) and the GUP corrected SM temperature TSMgup reads as TSMgup = TGUP(1 + λ kB SGUP)1− R λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (66) We can now define the inverse temperature parameters for GUP and non-GUP cases by using the above equa- tions (65) and (66), which are given, for the non-GUP case, as βSM = S′ SM kBc2 = β(1 + λ kB SB) R λ −1, (67) and for the GUP case, as βSMgup = S′ SMgup kBc2 = βGUP(1 + λ kB SGUP) R λ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (68) R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='2 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='6 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='9 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='2 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='6 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0 10 20 30 40 50 M SSM Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Plot of the Sharma-Mittal entropy for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines correspond to a GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Since SM entropy is the generalization of the Tsallis and Rényi entropy, the behavior of the temperature and the entropy are similar to that of SB and SR and TH and TR for different values of Sharma-Mittal parameter R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Also, the black hole does not evaporate in this case as well, and the evaporation process stops at Mr, leaving the fi- nal state of the black hole as a remnant having finite en- tropy and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The plots of SM entropy and temperature are given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='2 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='6 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='9 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='2 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='6 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='30 M TSM Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Sharma-Mittal temperature for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines correspond to a GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Heat Capacity for Sharma-Mittal Black Holes By following the previous subsections, we can calcu- late the heat capacity CSM for the SM black holes as CSM = CSc(1 + λ kB SB) R λ (1 + λ kB SB) − λ kB CSc � R λ − 1 � , (69) and for the GUP SM black holes case, it reads as CSMgup = CGUP(1 + λ kB SGUP) R λ (1 + λ kB SGUP) − λ kB CGUP � R λ − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (70) The plots of (69) and (70) are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Similarly as for the Rényi case, we define the characteristic length scale LSM in terms of λ and R by employing the singular 11 point of CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the non-GUP case, we have such a singular point for λ = RCSc − kB CSc + SB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (71) From (71), we can easily define the following character- istic relation by solving it for M, which reads LSM = 2lp � π(λ − 2R), (72) where LSM = GMc/c2, and the mass scale Mc is defined as Mc = mp 2 � π(λ − 2R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (73) Similarly, one can define LSMgup for the GUP case by using the following singular point at λ = RCGUP − kB CGUP + SGUP , (74) and solve it for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Since the analytic solution is not pos- sible, one could use a perturbative approach to find the GUP corrections to LSM up to the first order in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Note that R → 0 limit yields the LR for the Rényi case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For λ − 2R > 0 and M > Mc, the heat capacity is positive for both non-GUP and GUP cases, and for M < Mc, the heat capacity is negative for both non-GUP and GUP cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='2 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='6 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='9 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='2 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='6 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 30 20 10 0 10 20 30 M CSM Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Heat capacity CSM for Sharma-Mittal entropy for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines correspond to a GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Sparsity of the Sharma-Mittal Radiation The sparsity parameter ηSM can be derived by apply- ing the Sharma-Mittal temperature to (23), and reads ηSM = ηH(1 + λ kB SB)2( R λ −1), (75) and for the GUP case, substituting equations (66) and (27) in (23), the GUP modified sparsity parameter for the Sharma-Mittal radiation reads as ηSMgup = ηGUP(1 + λ kB SGUP)2( R λ −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (76) R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='45 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='6 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='45 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='6 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='3 R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0 100 200 300 400 500 600 M ηSM Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Sparsity for Sharma-Mittal entropy for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines correspond to a GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The plots of the sparsity for SM (75) and SM GUP (76) cases are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The behavior of the sparsity parameter again depends on the Sharma-Mittal param- eter R in addition to the nonextensive parameter λ and also the GUP parameter α in the case of GUP corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the values of λ and R, which satisfy the inequality λ + 2R > 0, the sparsity of the Sharma-Mittal radiation behaves like the sparsity of the Rényi radiation for both non-GUP and GUP cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This means that, initially, the Sharma-Mittal radiation is not sparse, and at the end of the evaporation, its value approaches the value of Hawking’s case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=', ηH, for the non-GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the GUP case, when M approaches Mr, the Sharma-Mittal sparsity parameter approaches some finite value, which is less than ηH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For λ > R, initially, the Sharma-Mittal sparsity parameter is higher than ηH and its value ex- actly approaches ηH at the end of the evaporation, while for the case of GUP, it approaches to some finite value less than ηH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It is interesting to note that, for α > 0, the GUP modified sparsity parameter is always less than the standard Hawking case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Kaniadakis Entropy Kaniadakis entropy [42, 70] is a type of nonextensive entropy that results from the Lorentz transformation of special relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It is a single parameter deformation of Gibbs entropy in which The standard Gibbs entropy is generalized to the relativistic regime with the help of a new parameter K that is connected to the dimensionless rest energy of the various parts of a multibody relativis- tic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The Kaniadakis entropy SK is defined as SK = kB logK Ω (77) where logK(Ω) = ΩK − Ω−K 2K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (78) Considering SB = kB ln Ω, which means that the num- ber of microstates Ω for a black hole is proportional to 12 eSB/kB, the above equation (77) can be written in the fol- lowing form SK = kB K sinh � K SB kB � , (79) where we have used equation (78) for the sinh x function and used the relation Ω = eSB/kB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Replacing SB with SGUP, the GUP modified Kaniadakis entropy SKGUP reads as SKGUP = kB K sinh � K SGUP kB � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (80) Note that, in the limit K → 0, SK reduces to Gibbs en- K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='1 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='9 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='1 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0 20 40 60 80 100 M SK Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Kaniadakis Entropy SK vs mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines correspond to a GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' tropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (17), one can see the characteristic form of sine hyperbolic (sinh) function for different small val- ues of K which shows the similar behaviour like the Bekenstein entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' As expected, for the GUP case, black holes do not evaporate completely and the final state of the black hole is a remnant like for the case of standard GUP modified Bekenstein-Hawking case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Fur- thermore, as K increases, the entropy increases sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' By using the Clausius relation, the corresponding Kani- adakis black black hole temperature TK reads as TK = TH sech � K SB kB � , (81) and the GUP modified Kaniadakis temperature TKGUP can be written as TKgup = TGUP sech � K SGUP kB � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (82) By using (81) and (82), one can write the following in- verse temperature parameters βK as follows kBβK = kBβ cosh � K SB kB � , (83) and for the GUP case, βKGUP reads kBβKgup = kBβGUP cosh � K SGUP kB � , (84) K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='1 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='9 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='1 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='20 M TK Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Kaniadakis temprature TK vs mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines correspond to a GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' which can further be used to find the heat capacities for Kaniadiakis black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (18) shows that Ka- niadakis temperature behaves as Hawking temperature with a slight change depending on the parameter K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the GUP case, it stops at some finite value, when M ap- proaches to Mr during the final stages of the black hole evaporation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Heat capacity for Kaniadakis Black Holes The heat capacities for Kaniadakis entropy can be calculated by following the previous subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the non-GUP case, the heat capacity CK for Kaniadakis black hole reads as CK = CSc cosh2[K SB kB ] cosh[K SB kB ] − CSc sinh[K SB kB ] , (85) and for the GUP modified heat capacity, CKgup, it can written as CKgup = CGUP cosh2[K SGUP kB ] cosh[K SGUP kB ] − CGUP sinh[K SGUP kB ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (86) From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (19), one can easily notice the negative heat K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='1 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='9 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='1 δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 100 80 60 40 20 0 M CK Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Kaniadakis heat capacity CK vs mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines correspond to a GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' capacities for all values of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This means that Kani- adakis black holes are thermodynamically unstable for all M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 13 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='1 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='7 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='1 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='5 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='4 0 50 100 150 200 250 300 M ηK Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Sparsity ηK for Kaniadakis radiation vs mass M of Kaniadakis black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Dashed lines correspond to a GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Sparsity of the Kaniadakis Radiation The sparsity parameter ηK for the Kaniadakis radia- tion can be derived by applying (81) into (23), and reads ηK = ηH cosh2 � K SB kB � , (87) and for the GUP modified sparsity parameter ηKGUP, we apply (82) and (27) into (23), to obtain ηKGUP = ηGUP cosh2 � K SGUP kB � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (88) From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (20), the sparsity parameter for the Kani- adakis case is always high from the beginning of the evaporation process as compared to the standard Beken- stein Hawking case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' However, for the non-GUP case, ηK approaches to the value of ηH at the end of the evapo- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the GUP case, again, it approaches to some finite value of sparsity when M approaches Mr, which is always less than the sparsity parameter ηH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Further- more, we see that increasing value of K directly results in sparser Kaniadakis radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Barrow entropy Barrow entropy [44] is an entropic form that has no statistical roots, but is closely tied to black hole hori- zon geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It is proposed to replace the smooth black hole horizon with a fractal of spheres known as a sphereflake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This structure is distinguished by its frac- tal dimension d f , where 3 ≥ d f ≥ 2, and results in an effective horizon area of r+d f , where r+ is the horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' As a result, in this scenario, the horizon area is modified, yielding Barrow entropy as below SBarrow SBarrow = kB � A Ap �1+ ∆ 2 (89) where A is the horizon area, Ap is the Planck area, and ∆ is the parameter directly tied to the fractal dimension d f through ∆ = d f − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In this form, ∆ can take values between 0 and 1, and ∆ → 1 limit yields maximally frac- tal structure, where the horizon area effectively behaves like a 3−dimensional volume, while ∆ → 0 limit yields the well-known Bekenstein area law where no fractal- ization occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Although Barrow entropy offers a dif- ferent picture in the geometrical sense, in its essence, it has the same form as Tsallis-Cirto entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We can see that they are equivalent by making the following parametrization in Tsallis-Cirto entropy [105] δ → 1 + ∆ 2 (90) Thus, qualitatively, both entropic forms yield the same temperatures and heat capacities as a function of black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Similarly, the Tsallis-Cirto entropy limit ∆ = 1 (δ = 3/2 for Sδ) yields an extensive, but still nonaddi- tive entropy for black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' SUMMARY AND DISCUSSION We have investigated the nonextensive thermody- namics of black holes, the impact of the generalized uncertainty principle on nonextensive thermodynamics quantities, and the sparsity and GUP-modified sparsity of the radiation in the nonextensive scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We have found that all nonextensive black hole entropies and as- sociated temperatures have finite values at the end of the black hole evaporation process due to GUP modifi- cations, indicating the existence of a remnant at the end of the evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This means that black holes do not evaporate fully in the nonextensive setup as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We have also investigated the sparsity parameter in each nonextensive configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Despite the fact that the be- havior of the sparsity parameter varies for each nonex- tensive scenario, GUP consistently lowers the radiation sparsity in all circumstances toward the end of the evap- oration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Even though multiple nonextensive scenarios have the same temperatures and entropic pro- files, we have demonstrated that the sparsity parameter can be used to distinguish between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We have introduced GUP and GUP-corrected thermo- dynamic parameters and have revised otherwise well- known GUP corrected quantities to a better form in which the two crucial limits - the extensivity limit for λ → 0 and the HUP limit for α → 0 - are easily iden- tified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Even though GUP corrections on Rényi entropy in black hole thermodynamics have been researched in the literature, we presented a full discussion of it in or- der to help readers distinguish between various sorts of nonextensive scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Additionally, we have provided non-perturbative results for each quantity, with a focus on the Rényi sparsity parameter, which rises (as shown by the "bump" in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' (8)) before the value of the rem- nant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This is because it is assumed that the area can change as a result of the GUP-modified Bekenstein entropy, which is explicitly shown in (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This indi- 14 cates that AGUP as well as TGUP have an impact on the sparsity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Furthermore, we have introduced black hole mass scale Mc = mp/2 √ πλ for the nonexten- sive parameter λ for the Rényi black hole quantities and we defined corresponding characteristic length for λ in terms of Mc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' LR = GMc/c2 = 2lp √ πλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We have shown that, for M > Mc, the heat capacity is positive and hence black holes in Rényi scenario are thermody- namically stable, while for M < Mc, the heat capacity is negative and SR and TR behave like Bekenstein entropy SB and Hawking temperature TH, hence unstable black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Similarly, we have also analyzed the thermodynamic black hole quantities associated with Tsallis-Cirto black hole entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Particularly, we have focused on GUP corrections and the sparsity of the Tsallis-Cirto radia- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We have shown that, when GUP corrections are included, Tsallis-Cirto entropy and associated temper- ature have a finite value, and this proves that the fi- nal state of the black hole is also a remnant with finite entropy and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It is interesting to note that the Tsallis-Cirto parameter δ plays a significant role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We have found that, for δ > 1/2, Tsallis-Cirto entropy and temperature behave similarly to Bekenstein entropy and Hawking temperature, and hence have negative heat ca- pacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the GUP case, Tsallis-Cirto temperature be- haves like Rényi temperature and has positive heat ca- pacity for δ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This means that, in this framework, we must have δ < 1/2 for thermodynamic stability of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In this way, we have shown that the Tsallis- Cirto sparsity parameter is very high during the start of the evaporation for δ > 1, but it approaches zero at the the end of the black hole evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' On the contrary, for δ < 1, we have shown that the Tsallis-Cirto radi- ation is not sparse during the start of the evaporation, but at the end of the evaporation, the sparsity parame- ter becomes infinite and hence shows the highly sparse Tsallis-Cirto radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' The behavior of the GUP case is initially the same as that of the non-GUP case, but as the mass approaches the order of Planck mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=', Mr, the Tsallis-Cirto sparsity parameter for each case reduces to some finite values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It should be noted that all of these fi- nite sparsity parameter values are less than the sparsity parameter ηH for the standard Hawking case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We have also shown that the behavior of the tempera- ture and the entropy for the Sharma-Mittal case is com- parable to that of SB and SR and TH and TR for differ- ent values of the Sharma-Mittal parameter R since the Sharma-Mittal entropy is the extension of the Tsallis and Rényi entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Also, in this instance, the black hole does not evaporate, and the evaporation process stops at Mr, leaving the black hole in its ultimate state as a remnant of mass Mr with finite entropy and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We have analysed the sparsity of the Sharma-Mittal radia- tion and compared it with the standard Hawking case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We have found that the sparsity of the Sharma-Mittal ra- diation behaves similarly to the Rényi radiation in both non-GUP and GUP instances for values of λ and R that fulfill the condition λ − 2R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' This indicates that the Sharma-Mittal radiation is initially not sparse and that by the end of the evaporation, its value approaches that of Hawking’s scenario, or ηH, for the non-GUP case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' When M approaches Mr for the GUP case, the Sharma- Mittal sparsity parameter approaches a finite value that is smaller than ηH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the case, R > λ, we have shown that the Sharma-Mittal sparsity parameter is initially larger than ηH and its value exactly approaches ηH by the end of the evaporation whereas for the case of GUP, it approaches a finite value that is smaller than ηH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It is noteworthy to notice that, for α > 0, the GUP modified sparsity parameter is always lower than the standard Hawking case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Moreover, we have also introduced the characteristic mass scale, Mc = mp/2 � π(λ − 2R), for the Sharma-Mittal scenario and also, defined the corre- sponding characteristic length scale LSM = GMc/c2 = 2lp � π(λ − 2R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We have shown that, for M > Mc with λ − 2R > 0, the black holes are thermodynamically sta- ble in the Sharma-Mittal scenario for both GUP and non- GUP cases, while for M < Mc, black holes are thermo- dynamically unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' We have also examined the Kaniadakis thermody- namic black hole quantities, and the results demonstrate that, with a little variation depending on the parame- ter K, Kaniadakis entropy and temperature behave sim- ilarly to Bekenstein entropy and Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In the case of the GUP, both quantities reach a finite value as black hole mass approaches Mr during the late stages of the black hole evaporation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' It results in negative heat capacity for all values of K, indicating that Kaniadakis black holes are thermodynamically unstable for all values of black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Furthermore, in con- trast to the typical Hawking example, the sparsity pa- rameter for the Kaniadakis instance is consistently high from the beginning of the evaporation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' For the non-GUP example, however, ηK approaches the value of ηH at the end of the evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In the GUP situation, it approaches some finite value of sparsity when M ap- proaches Mr, which is always smaller than the sparsity parameter ηH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Additionally, it is clear that a rise in the value of K causes the Kaniadakis radiation to become sparser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' Finally, our short look onto the Barrow entropy has proven its equivalence (though in a restricted range of parameters) to the Tsallis-Cirto entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' In view of that, all the discussion of termodynamical quantities for Bar- row entropy should be the same as for Tsallis-Cirto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' ACKNOWLEDGMENTS The work of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdAyT4oBgHgl3EQfuPmB/content/2301.00609v1.pdf'} +page_content='D.' metadata={'source': 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a/TNFAT4oBgHgl3EQf2R4K/content/tmp_files/2301.08713v1.pdf.txt b/TNFAT4oBgHgl3EQf2R4K/content/tmp_files/2301.08713v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..85f9342dcc324669208872b1c758ae541027b668 --- /dev/null +++ b/TNFAT4oBgHgl3EQf2R4K/content/tmp_files/2301.08713v1.pdf.txt @@ -0,0 +1,1190 @@ +Massively Parallel Genetic Optimization through +Asynchronous Propagation of Populations +Oskar Taubert[0000−0002−3707−499X], Marie Weiel[0000−0001−9648−4385], +Daniel Coquelin[0000−0001−8552−5153], Anis Farshian[0000−0002−9888−0653], +Charlotte Debus[0000−0002−7156−2022], Alexander Schug[0000−0002−0534−502X], +Achim Streit[0000−0002−5065−469X], and Markus G¨otz[0000−0002−2233−1041] +Steinbuch Centre for Computing (SCC), Karlsruhe Institute of Technology (KIT), +76344 Eggenstein-Leopoldshafen, Germany, markus.goetz@kit.edu +Abstract. We present Propulate, an evolutionary optimization algo- +rithm and software package for global optimization and in particular hy- +perparameter search. For efficient use of HPC resources, Propulate omits +the synchronization after each generation as done in conventional genetic +algorithms. Instead, it steers the search with the complete population +present at time of breeding new individuals. We provide an MPI-based +implementation of our algorithm, which features variants of selection, +mutation, crossover, and migration and is easy to extend with custom +functionality. We compare Propulate to the established optimization tool +Optuna. We find that Propulate is up to three orders of magnitude faster +without sacrificing solution accuracy, demonstrating the efficiency and +efficacy of our lazy synchronization approach. Code and documentation +are available at https://github.com/Helmholtz-AI-Energy/propulate. +Keywords: Genetic Optimization · AI · Parallelization · Evolutionary Algo- +rithm +1 +Introduction +Machine learning (ML) algorithms are heavily used in almost every area of +human life today, from medical diagnosis and critical infrastructure to trans- +portation and food production. Almost all ML algorithms have non-learnable +hyperparameters (HPs) that influence the training and in particular their pre- +dictive capacity. As evaluating a set of HPs involves at least a partial train- +ing, state-free approaches to HP optimization (HPO), like grid and random +search, often go beyond available compute resources [15]. To explore the high- +dimensional HP spaces efficiently, information from previous evaluations must +be leveraged to guide the search. Such state-dependent strategies minimize the +number of evaluations to find a useful model, reducing search times and thus the +energy consumption of the computation. Bayesian and bio-inspired optimizers +are the most popular of these AutoML approaches. Among the latter, genetic +algorithms (GAs) are versatile metaheuristics inspired by natural evolution. To +arXiv:2301.08713v1 [cs.NE] 20 Jan 2023 + +2 +O. Taubert et al. +solve a search-for-solutions problem, a population of candidate solutions (or in- +dividuals) is evolved in an iterative interplay of selection and variation [23,30]. +Although reaching the global optimum is not guaranteed, GAs often find near- +optimal solutions with less computational effort than classical optimizers [9,8]. +They have become popular for various optimization problems, including HPO +for ML and neural architecture search (NAS) [14]. +To take full advantage of the increasingly bigger models and datasets, de- +signing scalable algorithms for high performance computing (HPC) has become +a must [40]. While Bayesian optimization is inherently serial, the structure of +GAs renders them suitable for parallelization [34]: Since all candidates in each +iteration are independent, they can be evaluated in parallel. To breed the next +generation, however, the previous one has to be completed. As the computational +expenses for evaluating different candidates vary, synchronizing the parallel evo- +lutionary process affects the scalability by introducing a substantial bottleneck. +Approaches to reducing the overall communication in parallel GAs like the island +model (IM) [34] do not address the underlying synchronization problem. +To solve the issues arising from explicit synchronization, we introduce Prop- +ulate, a massively parallel genetic optimizer with asynchronous propagation of +populations and migration. Unlike classical GAs, Propulate maintains a contin- +uous population of already evaluated individuals with a softened notion of the +typically strictly separated, discrete generations. Our contributions include: +– A novel parallel genetic algorithm based on a fully asynchronous island model +with independently processing workers, allowing to parallelize the optimiza- +tion process and distribute the internal evaluation of the objective function. +– Massive parallelism by asynchronous propagation of continuous populations +and migration. +– A prototypical implementation in Python using extremely efficient commu- +nication via the message passing interface (MPI). +– Optimal use of parallel hardware by minimizing idle times in HPC systems. +We use Propulate to optimize various benchmark functions and the HPs of a deep +neural network on a supercomputer. Comparing our results to those of the popu- +lar HPO package Optuna, we find that Propulate is consistently drastically faster +without sacrificing solution accuracy. We further show that Propulate scales well +to at least 100 processing elements (PEs) without relevant loss of efficiency, +demonstrating the efficacy of our asynchronous evolutionary approach. +2 +Related Work +Recent progress in ML has triggered heavy use of these techniques with Python +as the de facto standard programming language. Tuning HPs requires solving +high-dimensional optimization problems with ML algorithms as black boxes and +model performance metrics as objective functions (OFs). Most common are +Bayesian optimizers (e.g. Optuna [2], Hyperopt [7], SMAC3 [24,27], Spearmint [32], +GPyOpt [5], and MOE [38]) and bio-inspired methods such as swarm-based (e.g. + +Propulate +3 +FLAPS [39]) and evolutionary (e.g. DEAP [16], MENNDL [40]) algorithms. Below, we +provide an overview of popular HP optimizers in Python, with a focus on state- +dependent parallel algorithms and implementations. A theoretical overview of +parallel GAs can be found in surveys [12,4,3] and books [37,29]. +Optuna adopts various algorithms for HP sampling and pruning of unpromis- +ing trials, including tree-structured Parzen estimators (TPEs), Gaussian pro- +cesses, and covariance matrix adaption evolution strategy. It enables parallel +runs via a relational database server. In the parallel case, an Optuna candidate +obtains information about previous candidates from and stores results to disk. +SMAC3 (Sequential Model-based Algorithm Configuration) combines a ran- +dom-forest based Bayesian approach with an aggressive racing mechanism [24]. +Its parallel variant pSMAC uses multiple collaborating SMAC3 runs which share +their evaluations through the file system. +Spearmint, GPyOpt, and MOE are Gaussian-process based Bayesian optimizers. +Spearmint enables distributed HPO via Sun Grid Engine and MongoDB. GPyOpt +is integrated into the Sherpa package [22], which provides implementations of +recent HP optimizers along with the infrastructure to run them in parallel via +a grid engine and a database server. MOE (Metric Optimization Engine) uses +a one-step Bayes-optimal algorithm to maximize the multi-points expected im- +provement in a parallel setting [38]. Using a REST-based client-server model, +it enables multi-level parallelism by distributing each evaluation and running +multiple evaluations at a time. +Nevergrad [31] and Autotune [25] provide gradient-free and evolutionary +optimizers, including Bayesian, particle swarm, and one-shot optimization. In +Nevergrad, parallel evaluations use several workers via an executor from Python’s +concurrent module. Autotune enables concurrent global and local searches, +cross-method sharing of evaluations, method hybridization, and multi-level par- +allelism. Open Source Vizier [33] is a Python interface for Google’s HPO ser- +vice Vizier. It implements Gaussian process bandits [19] and enables dynamic +optimizer switching. A central database server does the algorithmic proposal +work, clients perform evaluations and communicate with the server via remote +procedure calls. Katib [18] is a cloud-native AutoML project based on the Kuber- +netes container orchestration system. It integrates with Optuna and Hyperopt. +Tune [26] is built on the Ray distributed computing platform. It interfaces with +Optuna, Hyperopt, and Nevergrad and leverages multi-level parallelism. +DEAP (Distributed Evolutionary Algorithms in Python) [16] implements gen- +eral GAs, evolution strategies, multi-objective optimization, and co-evolution of +multi-populations. It enables parallelization via Python’s multiprocessing or +SCOOP module. EvoTorch [36] is built on PyTorch and implements distribution- +and population-based algorithms. Using a Ray cluster, it can scale over mul- +tiple CPUs, GPUs, and computers. MENNDL (Multi-node Evolutionary Neural +Networks for Deep Learning) [40] is a closed-source MPI-parallelized HP op- +timizer for automated network selection. A master node handles the genetic +operations while evaluations are done on the remaining worker nodes. However, +global synchronization hinders optimal resource utilization [40]. + +4 +O. Taubert et al. +Algorithm 1: Basic GA. In each generation, the individuals are eval- +uated in terms of the optimization problem’s OF. Genetic operators +propagate them to the next generation: The selection operator chooses +a portion of the current generation, where better individuals are usually +preferred. To breed new individuals, the genes of two or more parent +individuals from the selected pool are manipulated. While the crossover +operator recombines the parents’ genes, the mutation operator alters +them randomly. This is repeated until a stopping condition is met. +Input: Search-space limits, population size P, termination condition, +selection policy, crossover probability, mutation probability. +1 Initialize population pop of P individuals within search space. +2 while not termination condition do +// OPTIMIZE +3 +Evaluate individuals in pop. +// EVALUATE +4 +Choose parents from pop following selection policy. +// SELECT +5 +foreach individual in pop do +// VARY +6 +if random ≤ crossover probability then +// RECOMBINE +7 +Recombine individuals randomly chosen from parents. +8 +if random ≤ mutation probability then +// MUTATE +9 +Mutate. +10 +Update individual in pop. +Result: Best individual found (i.e., with lowest OF value for minimization). +3 +Propulate Algorithm and Implementation +To alleviate the bottleneck inherent to synchronized parallel genetic algorithms, +our massively parallel genetic optimizer Propulate (propagate and populate) +implements a fully asynchronous island model specifically designed for large- +scale HPC systems. Unlike conventional GAs, Propulate maintains a continu- +ous population of evaluated individuals with a softened notion of the typically +strictly separated generations. This enables asynchronous evaluation, variation, +propagation, and migration of individuals. +Propulate’s basic mechanism is that of Darwinian evolution, i.e., beneficial +traits are selected, recombined, and mutated to breed more fit individuals (see +Algorithm 1). On a higher level, Propulate employs an IM, which combines inde- +pendent evolution of self-contained subpopulations with intermittent exchange of +selected individuals [34]. To coordinate the search globally, each island occasion- +ally delegates migrants to be included in the target islands’ populations. Islands +communicate genetic information competitively, thus increasing diversity among +the subpopulations compared to panmictic models [11]. For synchronous IMs, +this exchange occurs simultaneously after fixed intervals, with no computation +happening in that time. The following hyperparameters characterize IMs: +– Island number and subpopulation sizes +– Migration (pollination) probability +– Number of migrants (pollinators): How many individuals migrate from +the source population at a time. + +Propulate +5 +Fig. 1. Asynchronous propagation. Interaction of two workers on one island. Indi- +viduals bred by worker 1 and 2 are shown in blue and red, respectively. Their origins +are given by a generation sub- and an island superscript. Populations are depicted as +round grey boxes, where most recent individuals have black outlines. Varying evaluation +times are represented by sharp boxes of different widths. We illustrate the asynchronous +propagation and intra-island synchronization of the population using the example of +the blue individual indi1 +g3. This individual is bred by worker 1 in generation 3 by apply- +ing the propagator (yellow) to the worker’s current population. After evaluating indi1 +g3, +worker 1 sends it to all workers on its island and appends it to its population. As no +evaluated individuals dispatched by worker 2 await to be received, worker 1 proceeds +with breeding. Worker 2 receives the blue indi1 +g3 only after finishing the evaluation of +the red indi1 +g2. It then appends both to its population and breeds a new individual for +generation 3. +– Migration (pollination) topology: Directed graph of migration (pollina- +tion) paths between islands. +– Emigration policy: How to select emigrants (e.g., random or best) and +whether to remove them from the source population (actual migration) or +not (pollination). +– Immigration policy: How to insert immigrants into the target population, +i.e., either add them (migration) or replace existing individuals (pollination, +e.g., random or worst). +Propulate’s functional principle is outlined in Algorithm 2. We consider multiple +PEs (or workers) partitioned into islands. Each worker processes one individual +at a time and maintains a population to track evaluated and migrated individuals +on its island. To mitigate the computational overhead of synchronized OF evalu- +ations, Propulate leverages asynchronous propagation of continuous populations +with interwoven, worker-specific generations (see Figure 1). In each iteration, +each worker breeds and evaluates an individual which is added to its population +list. It then sends the individual with its evaluation result to all workers on the + +Breed +Evaluate +Breed +Evaluate +Breed +Evaluate +Append +indg +Append +indg +Append +Propagator +Propagator +Propagator +indl +Worker 1 +Population list +Population list +Population list +ind.2 +indel +indel +indel +indel +indga +indfe +Send +Send +synchronization +Intra-island +indgs +Receive +Receive +Receive +Receive +Breed +Evaluate +Send +Breed +Evaluate +Worker 2 +Append +inde? +Append +Propagator +Propagator +inde +Populationlist +Population list +indei +indel +indfz +indge +. +Optimizationprogress6 +O. Taubert et al. +Algorithm 2: Propulate with pollination. +Input: Search-space limits; hyperparameters n islands, island sizes Pi +(i = 1, . . . , n islands), number of iterations generations, evolutionary +operators (including selection policy, crossover probability, +mutation probability etc.), pollination probability, pollination topology, +emigration policy, immigration policy. +1 Configure n islands islands with Pi workers each. Each worker evaluates one +individual at a time and maintains its own population list pop of evaluated +and migrated individuals on the island. +2 /* START OPTIMIZATION. +*/ +3 for each worker do in parallel +4 +while generation ≤ generations do // Loop over generations. +5 +Breed and evaluate individual. Append it to pop. Send it to other +workers on island to synchronize their populations lists: +evaluate individual() +// BREED AND EVALUATE +6 +Check for and possibly receive individuals bred and evaluated by other +workers on island. Append them to pop: +receive intra isle individuals() +// SYNCHRONIZE +7 +if random ≤ pollination probability then +// EMIGRATE +8 +Choose pollinators from currently active individuals on island +according to emigration policy. Send copies of pollinator(s) to +workers of target islands according to pollination topology: +send emigrants() +9 +Check for and possibly receive pollinators sent by workers from other +islands. Add them to pop. Determine individuals to be replaced by +incoming pollinators according to immigration policy. Send +individuals to be replaced to other workers on island for deactivation: +receive immigrants() +// IMMIGRATE +10 +Check for and possibly receive individuals replaced by pollinators on +other workers on island. Try to deactivate them in pop. If an +individual to be deactivated is not yet in pop, append it to history list +replaced and try again in the next generation: +deactivate replaced individuals() +// SYNCHRONIZE +11 +Go to next generation: generation += 1 +12 +/* OPTIMIZATION DONE: FINAL SYNCHRONIZATION +*/ +13 +Wait for all other workers to finish: MPI.COMM WORLD.barrier() +14 +Final check for incoming messages so all workers hold complete population. +15 +Probe individuals evaluated by other workers on island: +receive intra isle individuals() +16 +Probe for incoming pollinators immigrating from other islands: +receive immigrants() +17 +Probe for individuals replaced by other workers on island to be +deactivated: deactivate replaced individuals() +Result: n individuals with smallest OF values. + +Propulate +7 +same island and, in return, receives evaluated individuals dispatched by them +for a mutual update of their population lists. To avoid explicit synchronization +points, the independently operating workers use asynchronous point-to-point +communication via MPI to share their results. Each one dispatches its result +immediately after finishing an evaluation. Directly afterwards, it non-blockingly +checks for incoming messages from workers of its own island awaiting to be +received. In the next iteration, it breeds a new individual by applying the evo- +lutionary operators to its continuous population list of all evaluated individuals +from any generation on the island. The workers thus proceed asynchronously +without idle times despite the individuals’ varying computational costs. +After the mutual update, asynchronous migration or pollination between is- +lands happens on a per-worker basis with a certain probability. Each worker +selects a number of emigrants from its current population. For actual migra- +tion1, an individual can only exist actively on one island. A worker thus may +only choose eligible emigrants from an exclusive subset of the island’s popu- +lation to avoid overlapping selections by other workers. It then dispatches the +emigrants to the target islands’ workers as specified in the migration topology. +Finally, it sends them to all workers on its island for island-wide deactivation of +emigrated individuals before deactivating them in its own population. +In the next step, the worker probes for and, if applicable, receives immigrants +from other islands. It then checks for individuals emigrated by other workers +of its island and tries to deactivate them in its population. Due to the asyn- +chronicity, individuals might be designated to be deactivated before arriving in +the population. Propulate continuously corrects these synchronization artefacts +during the optimization. +For pollination (Figure 2), identical copies of individuals can exist on multiple +islands. Workers thus can choose emigrating pollinators from any active individ- +uals in their current populations and do not deactivate them upon emigration. +To control the population growth, pollinators replace active individuals in the +target population according to the immigration policy. For proper accounting of +the population, one random worker of the target island selects the individual to +be replaced and informs the other workers accordingly. Individuals to be deac- +tivated that are not yet in the population are cached to be replaced in the next +iteration. This process is repeated until each worker has evaluated a set number +of generations. Finally, the population is synchronized among workers and the +best individuals are returned. +Propulate uses so-called propagators to breed child individuals from an ex- +isting collection of parent individuals. It implements various standard genetic +operators, including uniform, best, and worst selection, random initialization, +stochastic and conditional propagators, point and interval mutation, and several +forms of crossover. In addition, Propulate provides a default propagator: Having +selected two random parents from the breeding pool consisting of a set num- +ber of the currently most fit individuals, uniform crossover and point mutation +1 See +github.com/Helmholtz-AI-Energy/propulate/tree/master/supplementary +for +pseudocode with migration and explanatory figure. + +8 +O. Taubert et al. +Fig. 2. Asynchronous pollination. Consider two islands with N (blue) and M (red) +workers, respectively. We illustrate pollination (dark colors) by tracing worker N on +island 1. After evaluation and mutual intra-island updates (light blue, see Figure 1), +this worker performs pollination: it sends copies of the chosen pollinators to all workers +of each target island, here island 2. The target island’s workers receive the pollinators +asynchronously (dark blue arrows). For proper accounting of the populations, worker +1 on island 2, selects the individual to be replaced and informs all workers on its island +accordingly (middle red arrow). Afterwards, worker N receives incoming pollinators +from island 2 to be included into its population. It then probes for individuals that have +been replaced by other workers on its island, here worker 1, in the meantime and need +to be deactivated. After these pollination-related intra-island population updates, it +breeds the next generation. As pollination does not occur in this generation, it directly +receives pollinators from island 2. This time, worker N chooses the individual to be +replaced. +are performed each with a specified probability. Afterwards, interval mutation +is performed. To prevent premature trapping in a local optimum, a randomly +initialized individual is added with a specified probability instead of one bred +from the current population. +4 +Experimental Evaluation +We evaluate Propulate on various benchmark functions (see Section 4.4) and an +HPO use case in remote sensing classification (see Section 4.5) which provides +a real world application. We compare our results against Optuna, since it is the +most widely used HPO software. +4.1 +Experimental Environment +We ran the experiments on the distributed-memory, parallel hybrid supercom- +puter Hochleistungsrechner Karlsruhe (HoreKa) at the Steinbuch Centre for +Com- puting, Karlsruhe Institute of Technology. Each of its 769 compute nodes + +Worker 1 +Sync? +Send +Receive +Sync? +Propagate +Evaluate +Sync? +Send +Receive +Sync? +Island 1 with +Evaluate +Nworkers +Y +pollinators?N +pollinators?Y +N +Y +pollinators?N +pollinators?Y +Y +. +Worker N +Sync? +Send +Receive +Sync? +Sync? +Send +Receive +Propagate +Evaluate +Sync? +Evaluate +Y +pollinators?Y +pollinators?Y +Y +N +pollinators?N +pollinators?y +N +Pollination +Worker1 +Island 2 with +Send +Receive +Sync? +Propagate +Evaluate +Sync? +Send +Receive +Sync? +M workers +pollinators?Y +pollinators?Y +pollinators?y +N +Propagate +pollinators?N +N +Evaluate +WorkerM +Evaluate +Sync? +Send +Receive +Sync? +Propagate +Evaluate +Sync? +Send +Receive +Sync? +N +pollinators?N +pollinators?N +Y +Y +pollinators?Y +pollinators?Y +OptimizationprogressPropulate +9 +Table 1. Benchmark functions. +Name +Function +Limits Global minimum +Sphere +f1 = x2 +1 + x2 +2 +±5.12 +f (0, 0) = 0 +Rosenbrock f2 = 100 +� +x2 +1 − x2 +�2 + (1 − x1)2 +±2.048 f (1, 1) = 0 +Step +f3 = �5 +i=1 int (xi) +±5.12 +f (xi ≤ −5) = −25 +Quartic +f4 = �30 +i=1 +� +ix4 +i + Ni (0, 1) +� +±1.28 +f (0, ..., 0) = � +i Ni +Rastrigin +f5 = 200 + �20 +i=1 x2 +i − 10 cos (2πxi) +±5.12 +f (0, ..., 0) = 0 +Griewank +f6 = 1 + +1 +4000 +�10 +i=1 x2 +i − �10 +i=1 cos xi +√ +i +±600 +f (0, ..., 0) = 0 +Schwefel +f7 = 10V − �10 +i=1 xi sin +� +|xi| +±500 +f (x∗ +1, ..., x∗ +10) = 0, +with V = 418.982887 +x∗ +i = 420.968746 +Bi-sphere +f8 = min +��30 +i=1 (xi − µ1)2 , +±5.12 +f (µ1, ..., µ1) = 0 +30 + s · �30 +i=1 (xi − µ2)2� +with +µ1 = 2.5, µ2 = − +� +s−1 � +µ2 +1 − 1 +��1/2 , +s = 1 − +� +2 +√ +50 − 8.2 +�−1/2 +Bi-Rastrigin f9 = f8 + 10 �30 +i=1 1 − cos 2π (xi − µ1) +±5.12 +f (µ1, ..., µ1) = 0 +is equipped with two 38-core Intel Xeon Platinum 8368 processors at 2.4 GHz +base and 3.4 GHz maximum turbo frequency, 256 GB (standard) or 512 GB +(high-memory and accelerator) local memory, a local 960 GB NVMe SSD disk, +and two network adapters. 167 of the nodes are accelerator nodes each equipped +with four NVIDIA A100-40 GPUs with 40 GB memory connected via NVLink. +Inter-node communication uses a low-latency, non-blocking NVIDIA Mellanox +InfiniBand 4X HDR interconnect with 200 Gbit/s per port. A Lenovo Xclar- +ity controller measures full node energy consumption, excluding file systems, +networking, and cooling. The operating system is Red Hat Enterprise Linux 8.2. +4.2 +Benchmark Functions +Benchmark functions are used to evaluate optimizers in terms of convergence, +accuracy, and robustness. The informative value of such studies is limited by how +well we understand the characteristics making real-life optimization problems +difficult and our ability to embed these features into benchmark functions [28]. +We use Propulate to optimize a variety of traditional and recent benchmark +functions emulating situations optimizers have to cope with in different kinds of +problems (see Table 1). +– Sphere is smooth, unimodal, strongly convex, symmetric, and thus simple. +– Rosenbrock has a narrow minimum inside a parabola-shaped valley. +– Step represents the problem of flat surfaces. Plateaus pose obstacles to op- +timizers as they lack information about which direction is favorable. +– Quartic is a unimodal function padded with Gaussian noise. As it never +returns the same value on the same point, algorithms that do not perform +well on this test function will do poorly on noisy data. + +10 +O. Taubert et al. +Table 2. Grid search parameters. All experiments use 144 CPUs equally distributed +between two nodes. Random-initialization probability refers to the chance that a new +individual is generated entirely randomly. +Number of islands +2 +4 +8 +16 +32 +Island population size +72 +36 +18 +9 +4 +Migration (pollination) probability +0.1 +0.3 +0.5 +0.7 +0.9 +Pollination +True +False +Crossover probability +0.1 +0.325 +0.55 +0.775 +Point-mutation probability +0.1 +0.325 +0.55 +0.775 +Random-initialization probability +0.1 +0.325 +0.55 +0.775 +– Rastrigin is non-linear and highly multimodal. Its surface is determined +by two external variables, controlling the modulation’s amplitude and fre- +quency. The local minima are located at a rectangular grid with size 1. Their +functional values increase with the distance to the global minimum. +– Griewank’s product creates sub-populations strongly codependent to par- +allel GAs, while the summation produces a parabola. Its local optima lie +above parabola level but decrease with increasing dimensions, i.e., the larger +the search range, the flatter the function. +– Schwefel has a second-best minimum far away from the global optimum. +– Lunacek’s bi-sphere’s [28] landscape structure is the minimum of two +quadratic functions, each creating a single funnel in the search space. The +spheres are placed along the positive search-space diagonal, with the op- +timal and sub-optimal sphere in the middle of the positive and negative +quadrant, respectively. Their distance and the barrier’s height increase with +dimensionality, creating a globally non-separable underlying surface. +– Lunacek’s bi-Rastrigin [28] is a double-funnel version of Rastrigin. This +function isolates global structure as the main difference impacting problem +difficulty on a well understood test case. +4.3 +Meta-Optimizing the Optimizer +Propulate itself has HPs influencing its optimization behavior, accuracy, and +robustness. To explore their effect systematically and give transparent recom- +mendations for default values, we conducted a grid search across the six most +prominent HPs. The search space is shown in Table 2. We ran the grid search +five times for the quartic, Rastrigin, and bi-Rastrigin benchmark functions (see +Table 1 and Section 4.4), each with a different seed consistently used over all +points within a search. All three functions have their global minimum at zero. +They were chosen for their high-dimensional parameter spaces (30, 20, and 30, +respectively) and different levels of difficulty to optimize. For quartic, Propulate +found a minimum below 0.01±0.005 for 80.12 % of all points across the five grid +searches. This increases to 94.94 % for minima found within 0.1 ± 0.05 of the +global minimum. In comparison, the tolerances have to be relaxed considerably + +Propulate +11 +for the more complex Rastrigin and bi-Rastrigin. While only 18.57 % of all grid +points had a function value less than 1.0 ± 0.5 for Rastrigin, only a single point +resulted in an average value of less than 10 for bi-Rastrigin. Although the av- +erage value of bi-Rastrigin was only less than 10 once, we found the minimum +across each of the five searches to be less than 1.0 for 3.31 % of the grid points. +Considering grid points with at least one result smaller than 1.0, 86.61 % used +either 16 or 36 islands, while the remainder used eight. As Propulate initializes +different islands at different positions in the search space, the chance that one of +them is at a very beneficial position increases with the number of islands. This +is further confirmed by a migration probability of 0.7 or 0.9 for 61.41 % of these +points. If one of the islands is well-initialized, it thus will quickly notify others. +With every best grid point using pollination, we clearly find pollination to be +favorable over real migration. To determine the other HPs, we compute the aver- +ages of the results for the top ten grid points across all three functions. The top +ten were determined by grouping over the lowest average and standard deviation +of the function values, sorting by the averages, and sorting by the standard de- +viations. This method reduces the chances of a single run simply benefiting from +an advantageous starting seed. Average crossover, point-mutation, and random- +initialization probabilities are 0.655 ± 0.056, 0.363 ± 0.133, and 0.423 ± 0.135, +respectively. The average number of islands was 28.800 ± 6.009 which equates +to an island population of 5.00 ± 1.043. The average migration probability was +0.527±0.150. These values provide a reasonable starting point towards choosing +default HPs for Propulate (see Table 3). As the grid searches only considered +functions with independent parameters, we assume a relatively high random-ini- +tialization probability to be useful due to the benefits of random search [6]. On +this account, we chose to reduce the default random-initialization probability +to 0.2. As the migration probability might also be lowered artificially by this +phenomenon, we set its default to 0.7. The default probabilities for crossover +and point-mutation were chosen as 0.7 and 0.4, respectively. The island size was +set at four individuals. This is a practical choice as our test system has four +accelerators per node and the number of CPUs per node is a multiple of four. +4.4 +Benchmark Function Optimization +For each function, we ran each ten equivalent Propulate and Optuna optimiza- +tions, using the same compute resources, degree of parallelization, and number of +evaluations. Figure 3 shows the optimization accuracy over walltime comparing +Propulate with default parameters determined from our grid search (see Table 3) +to Optuna’s default optimizer. In terms of accuracy, Propulate and Optuna are +comparable in most experiments. For many functions, e.g. Schwefel, bi-Rastrigin, +and Rastrigin, Propulate even achieves a better OF value. In terms of walltime, +Propulate is consistently at least one order of magnitude faster. This is due to +Propulate’s MPI-based communication over the fast network, whereas Optuna +uses relational databases with SQL and is limited by the slow file system. Since +the functions are cheap to evaluate, optimization and communication dominate +the walltime. In particular for problems where evaluations are cheap compared to + +12 +O. Taubert et al. +Table 3. Propulate HPs for benchmark function minimization. +Number of islands +38 +Island population size +4 +Pollination probability +0.7 +Crossover probability +0.7 +Point-mutation probability +0.4 +Sigma factor +0.05 +Random-initialization probability +0.2 +Generations per worker +256 +Selection policy +Best +Pollination topology +Fully connected +Number of migrants +1 +Emigration policy +Best +Immigration policy +Worst +the search itself, we find that Optuna’s computational efficiency suffers massively +from the frequent file locking inherent to its parallelization strategy, reducing its +usability for large-scale HPC applications. +In addition, we inspected the evolution of the population over walltime for +both Propulate and Optuna. An example for minimizing the Rastrigin function +is shown in Figure 4. Propulate is roughly three orders of magnitude faster and +makes significantly greater progress in terms of both OF values and distance to +the global optimum. Due to this drastic difference in runtime, we measure only +46.27 Wh for Propulate compared to Optuna’s 2646.29 Wh. +4.5 +HP Optimization for Remote Sensing Classification +BigEarthNet [35] is a Sentinel-2 multispectral image dataset in remote sensing. +It comprises 590 326 image patches each of which is assigned one or more of the +19 available CORINE Land Cover map labels [10,35]. Multiple computer vision +networks for BigEarthNet classification have been trained [35], with ResNet- +50 [20] being the most accurate. While a previous Propulate version was used to +optimize a set of HPs and the architecture for this use case [13], a more versatile +and efficient parallelization strategy in the current version makes it worthwhile +to revisit this application. Analogously to [13], we consider different optimizers, +learning rate (LR) schedulers, activation functions, loss functions, number of +filters in each convolutional block, and activation order [21]. The search space +is shown in Table 4. Optimizer parameters, LR functions, and LR warmup are +included as well. We only consider SGD-based optimizers as they share common +parameters and thus exclude Adam-like optimizers from the search. We theorize +that including Adam led to the difficulties seen previously [13]. The training is +exited if the validation loss has not been increasing for ten epochs. We prepared +the data analogously to [13]. The network is implemented in TensorFlow [1]. +For both Propulate and Optuna, we ran each three searches over 24 h on 32 +GPUs. We use 1 − F val +1 +with the validation F1 score as the OF to be minimized. + +Propulate +13 +10 +1 +100 +101 +102 +103 +104 +Walltime / s +10 +7 +10 +5 +10 +3 +10 +1 +101 +103 +Function value +Sphere +Rosenbrock +Quartic +Rastrigin +Griewank +Schwefel +Bi-sphere +Bi-Rastrigin +Fig. 3. Benchmark function minimization accuracy over walltime. Lowest +function values found by Propulate (red) and Optuna (blue) versus walltimes to reach +them, each averaged over ten runs. Step is not shown since both optimizers achieve a +perfect value of −25 within 0.6 s and 278.2 s, respectively. +On average, Optuna achieves its best OF value of (0.39 ± 0.01) h within (7.05 ± +3.14) h. Propulate beats Optuna’s average best after (5.30 ±2.41) h and achieves +its best OF value of (0.36 ± 0.00) within (13.89 ± 5.15) h. +4.6 +Scaling +Finally, we explore Propulate’s scaling behavior for the use case presented in +Section 4.5. Figure 5 shows our results for weak and strong linear scaling. Our +baseline configuration used two nodes. Since each node has four GPUs, we cal- +culate speedup and efficiency with respect to eight workers. For strong scaling, +we fix the total number of evaluations at 512 and increase the number of work- +ers, i.e., GPUs. We average over three runs with different seeds and keep four +workers per island while increasing the number of islands. Speedup increases up +to 128 workers, where we reach approximately half the optimal value. This is an +expected decline since each worker only processes few individuals, so the vari- +ance in evaluation times leads to larger idle times of the faster workers before +the final population synchronization at the end. Additionally, as the number +of workers approaches the total number of evaluations, the randomly initial- +ized evolutionary search in turn approaches a random search. This means that +the search performance is likely to be worse than what the pure compute per- +formance might suggest. It is still possible to apply Propulate on these scales, +but the other search parameters have to be adjusted accordingly as shown in +the weak scaling plot Figure 5 top. Weak efficiency only drops to 95 % at our +largest configuration of 128 workers The early super-scalar behavior is likely due +to the non-sequential baseline. + +14 +O. Taubert et al. +0 +5 +10 +Time / s +101 +102 +Function value +Propulate +0 +5000 +10000 +Time / s +Optuna +100 +101 +Distance to optimum +Maximum value +Maximum distance +Median value +Median distance +Minimum value +Minimum distance +Fig. 4. Evolution of the population over walltime for the Rastrigin func- +tion. Propulate (left) versus Optuna (right). OF values (blue) use the left-hand scale, +distances to the global optimum (purple) use the right-hand scale. Pastel dots show +each individual’s OF value/distance. Solid (dashed) lines show the minimum (median) +value and distance achieved so far. Maximum value (distance) are shown in black. Both +optimizers perform 38 912 evaluations. Note the difference on the time axis. +5 +Conclusion +We presented Propulate, our HPC-adapted, asynchronous genetic optimization +algorithm and software. Our experimental evaluation shows that the fully asyn- +chronous evaluation, propagation, and migration enable a highly efficient and +parallelizable genetic optimization. Harder to quantify than performance but +very important is ease of use. Especially for HPC applications at scale, some +parallelization and distribution models are more suited than others. A purely +MPI-based implementation as in Propulate is not only extremely efficient for +highly parallel and communication-intensive algorithms but also easy to set up +and maintain, since the required infrastructure is commonly available on HPC +systems. This is not the case for any of the other tools investigated, except for +the not publicly available MENNDL. This also facilitates a tighter coupling of in- +dividuals during the optimization, which enables a more efficient evaluation of +candidates and in particular early stopping informed by previously evaluated +individuals in the NAS case. Propulate was already successfully applied to HPO +for various ML models on different HPC machines [13,17]. Another avenue for +future work is including variable-length gene descriptions. Mutually exclusive +genes of different lengths, such as the parameter sets for Adam- and SGD-like +optimizers in our NAS use case, can thus be explored efficiently. While this is +already possible, it requires an inconvenient workaround of including inactive + +Propulate +15 +Table 4. HP search space of ResNet-50 for BigEarthNet classification. +Optimizers +Optimizer parameters +LR warmup parameters +Adagrad +Initial accum. value +� +10−4, 0.5 +� +LR warmup steps +� +100, 104� +SGD +Clipnorm +[−1, −1000] +Initial LR +� +10−5, 10−1� +Adadelta +Clipvalue +[−1, 1000] +Decay steps +� +102, 105� +RMSprop +Use EMA +Boolean +LR warmup power +� +10−1, 101� +EMA momentum +[0.5, 1.0] +EMA overwrite +� +1, 103� +Momentum +[0.0, 1.0] +Nesterov +Boolean +Rho +[0.8, 0.99999] +Epsilon +� +10−9, 10−4� +Loss functions +LR parameters +Binary CE +Categorical CE +Categorical hinge Decay rate +[0.8, 0.9999] +Hinge +KL divergence +Squared hinge +Staircase inverse +Boolean +time decay +Activation functions +Decay rate +[0.1, 0.9] +ELU +ReLU +Softplus +Staircase poly- +Boolean +Exponential +SELU +Softsign +nomial decay +Hard sigmoid Sigmoid +Swish +End LR +� +10−4, 10−2� +Linear +Softmax +Tanh +Power +[0.5, 2.5] +genes and adapting the propagators to manually prevent the evaluation of many +individuals differing only in inactive genes. +Acknowledgments +This work is supported by the Helmholtz AI platform grant and the Helmholtz +Association Initiative and Networking Fund on the HAICORE@KIT partition. +References +1. 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In: Pro- +ceedings of the Workshop on Machine Learning in High-performance Computing +Environments. pp. 1–5 (2015). https://doi.org/10.1145/2834892.2834896 + diff --git a/V9AzT4oBgHgl3EQfJ_vP/content/tmp_files/2301.01091v1.pdf.txt b/V9AzT4oBgHgl3EQfJ_vP/content/tmp_files/2301.01091v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b417f7206b0f51f24bd6f837503b8b2b04adf7fc --- /dev/null +++ b/V9AzT4oBgHgl3EQfJ_vP/content/tmp_files/2301.01091v1.pdf.txt @@ -0,0 +1,1251 @@ +Fitting mixed logit random regret minimization +models using maximum simulated likelihood +Ziyue Zhu +ID +Faculty of Sciences +KU Leuven +Leuven, Belgium +ziyue.zhu16@gmail.com +´Alvaro A. Guti´errez-Vargas +ID +Faculty of Economics and Business +KU Leuven +Leuven, Belgium +alvaro.gutierrezvargas@kuleuven.be +Martina Vandebroek +ID +Faculty of Economics and Business +KU Leuven +Leuven, Belgium +martina.vandebroek@kuleuven.be +Abstract. +This article describes the mixrandregret command, which extends +the randregret command introduced in Guti´errez-Vargas et al. (2021, The Stata +Journal 21: 626–658) incorporating random coefficients for Random Regret Min- +imization models. The newly developed command mixrandregret allows the in- +clusion of random coefficients in the regret function of the classical RRM model +introduced in Chorus (2010, European Journal of Transport and Infrastructure Re- +search 10: 181-196). The command allows the user to specify a combination of +fixed and random coefficients. In addition, the user can specify normal and log- +normal distributions for the random coefficients using the commands’ options. The +models are fitted using simulated maximum likelihood using numerical integration +to approximate the choice probabilities. +Keywords: notag1, mixrandregret, mixrpred, mixrbeta, discrete choice models, +mixed random regret minimization model +1 +Introduction +McFadden (1974) introduced conditional logit models to explain the choice behavior +of individuals and to predict market shares of products and services. The conditional +logit models form the basis for the majority of discrete choice models, which assume +that individuals use a decision rule based on Random Utility Maximization (RUM) +when choosing between various alternatives. In contrast, Chorus et al. (2008) proposed +an alternative decision rule known as Random Regret Minimization (RRM), assuming +that decision-makers aim to minimize regret when making their choices. +McFadden +and Train (2000) extended the random utility model by allowing the parameters to +vary across individuals, leading to the so-called mixed logit model. Similarly, Hensher +et al. (2016) modified the RRM models to include random effects, which account for +preference heterogeneity and allow for correlation among choices made by the same +individual. +arXiv:2301.01091v1 [econ.EM] 3 Jan 2023 + +2 +Fitting mixed logit random regret minimization models +In this article, we extend the command randregret (Guti´errez-Vargas et al. 2021) +into a mixed version called mixrandregret which allows the inclusion of random param- +eters. The new command allows the user to specify normal and log-normally distributed +taste parameters inside the regret function. The parameters of the distribution of the co- +efficients are estimated using Simulated Maximum Likelihood (SML). Specifically, given +that there is no closed-form solution for the choice probabilities, we approximate them +using simulations. +We also developed the mixrpred post-estimation command that +can predict the choice probabilities for each alternative. Additionally, the mixrbeta +post-estimation command allows estimating the individual-level parameters for each in- +dividual. We will illustrate the command’s usage in examples from van Cranenburgh +and Chorus (2018). +2 +Classical Random Regret Models +In contrast to the decision-making process of RUM models, which measure the benefits +of selecting a particular alternative in terms of utility, RRM models focus on the regret +resulting from not-chosen alternatives. Regret occurs when, compared to other avail- +able alternatives, the selected alternative is outperformed by the other alternatives in +some of the attributes (Loomes and Sugden 1982). Accordingly, RRM models assume +that the individuals intend to minimize regret when choosing among alternatives. For- +mally, Chorus et al. (2008) presented an initial model for random regret minimization +models, and Chorus (2010) revised the regret function in order to obtain a smooth like- +lihood function. Accordingly, he proposed (1) to denote the regret of individual n when +choosing alternative i among the J possible alternatives +Rin = +J +� +j̸=i +M +� +m=1 +ln[1 + exp{βm · (xjn,m − xin,m)}] + αi, +(1) +Equation (1) represents the regret that an individual (referred to by n) experiences +when choosing alternative i among J alternatives (referred to by j or i). Additionally, +each alternative is described in terms of the value of M attributes (referred to by m). +Consequently, xin,m represents the values of attribute m of alternative i for individual +n, and βm is the taste parameter of attribute m for individual n. The parameter βm +indicates that for each unit change of attribute m in a non-selected alternative, regret +would either increase (if βm is positive) or decrease (if βm is negative) relative to the level +of the same attribute in the selected alternative. Besides, the inclusion of Alternative +Specific Constants (ASC) in the stated models is possible by simply adding them to +the systematic part of the regret as αi. +The inclusion of the ASC serves the same +purpose as in RUM models, which is to account for omitted attributes for a particular +alternative i. As usual, for identification purposes, we need to exclude one of the ASC +from the model specification, so we define α = (αi, . . . , αJ−1) as the vector of J − 1 +ASC included in the model. A detailed discussion of the ASC in the context of RRM +models see Van Cranenburgh and Prato (2016). Consequently, Rin describes the total +systematic regret for an individual n choosing alternative i. + +Z. Zhu, ´A. A. Guti´errez-Vargas and M. Vandebroek +3 +Similarly to RUM models, we can obtain the random regret function, RRin, by +adding an i.i.d extreme value type I error term to the systematic regret function, Rin, +that will account for the pure random noise and the impact of omitted attributes in the +regret function: RRin = Rin + εin. Mathematically, the minimization of the random +regret function is equivalent to maximizing the negative function, which results in the +conventional closed-form logit formula for the choice probabilities given in equation (2). +Pin = +exp(−Rin) +�J +j=1 exp(−Rjn) +. +(2) +The log-likelihood function of the regret model for N individuals is given by equation +(3), where β = (β1, . . . , βm) is the vector of taste parameters and yin is the dummy +variable that takes the value of 1 when alternative i is chosen by individual n, and 0 +otherwise. +LL(α, β) = +N +� +n=1 +S +� +s=1 +J +� +i=1 +yin × ln (Pin) . +(3) +In the literature, there exist several extensions to the classical RRM models (Chorus +2014; van Cranenburgh et al. 2015). Chorus (2014) proposed the generalized RRM, +which replaces the “1” in the regret function with a new parameter γm denoting the +regret-weight for attribute m. van Cranenburgh et al. (2015) incorporated a scale pa- +rameter into the RRM, which is now referred to as µRRM. The Pure-RRM was proposed +in the same article (van Cranenburgh et al. 2015), as a special case of µRRM when µ +arbitrarily small. For a review that compares the different types of RRM models and +RUM models, see Guti´errez-Vargas et al. (2021). In what follows, we will focus on the +classical regret function of Chorus (2010), but we will allow for the inclusion of random +taste parameters as introduced by Hensher et al. (2016). This model will be referred to +as the Mixed Random Regret Minimization (Mixed RRM) model and takes preference +heterogeneity into consideration by assuming a parametric distribution for the taste +parameters. +3 +Mixed Random Regret Minimization Models +In this section, we describe the Mixed RRM where we (i) allow that the taste pa- +rameters follow a parametric distribution, and (ii) we are able to model data with +panel structure. Consequently, (i) triggers a new sub-index to the taste parameters, +βn = (βn,1, . . . , βn,m), which now follow a parametric distribution f(β|ϕ), where ϕ +are the parameters that describe the distribution1. Hence, βn,m is now an individual- +specific taste parameter that represents the regret sensitivity of individual n to changes +in attribute m. Additionally, (ii) implies that multiple choice situations (referred to by +s) are answered by the same individual, which triggers the inclusion of a new sub-index +for the choice situations in our formulas. Hence, xins,m will now represent the value +of attribute m for alternative i for individual n in choice situation s. Similarly, yins is +1. For instance, if we assume a normal distribution, ϕ would contain its mean and variance. + +4 +Fitting mixed logit random regret minimization models +now a binary variable that takes the value of 1 when individual n choose alternative i in +choice situation s and 0 otherwise. That being said, we will define a new regret function +that considers points (i) and (ii) in equation (4) where Rins describes the systematic +regret for individual n choosing alternative i in choice situation s. +Rins = +J +� +j̸=i +M +� +m=1 +ln[1 + exp{βn,m · (xjns,m − xins,m)}] + αi, +(4) +Similarly, we add the i.i.d extreme value type I error term to the systematic regret +function, and the choice probability is given by equation (5). +Pins = +exp(−Rins) +�J +j=1 exp(−Rjns) +. +(5) +Additionally, the probability of the observed sequence of choices of individual n (condi- +tional on knowing βn) is given by equation (6), which differs from equation (2) in the +sense that equation (6) consider responses from the same individual might be correlated, +but responses from different individuals are treated as independent from one another. +Pn(α, β) = +S +� +s=1 +J +� +j=1 +{Pins}yins. +(6) +The unconditional choice probabilities of the observed sequence of choices are the con- +ditional choice probabilities (see equation 6) integrated over the entire domain of the +distribution. Consequently, the log-likelihood function of the Mixed RRM Model in +equation (7). +LL(α, ϕ) = +N +� +n=1 +ln +�� +β +Pn(α, β)f(β|ϕ)dβ +� +(7) +Given that the integral described in equation (7) does not have a closed-form solution, it +is approximated using simulation (Train 2009). Accordingly, we estimate the model by +SML. Hence, we maximize the simulated log-likelihood function of equation (8) where +R is the number of draws and βr is the rth drawn from f(β|ϕ). Finally, we use Halton +draws to create the draws used to approximate the choice probabilities. +SLL(α, ϕ) = +N +� +n=1 +ln +� +1 +R +R +� +r=1 +Pn(α, βr) +� +(8) +4 +Individual-level Parameters +After maximizing the simulated log-likelihood function to obtain estimates for ˆϕ and +ˆα, we can also obtain estimates for the individual-level parameters. That is to say, we +can estimate the taste parameters for every individual conditional on their sequences + +Z. Zhu, ´A. A. Guti´errez-Vargas and M. Vandebroek +5 +of choices (denoted by yn) and the attribute levels for every alternative and choice set, +denoted by xn, that the individual faced when making the choices. For instance, we can +compute the individual-level parameter ¯βn for every individual n which corresponds to +the mean of the distribution of βn conditional on yn, xn, and our estimated ˆϕ. The +expression for ¯βn is given in equation (9), and its derivation can be found in Train +(2009): +¯βn = +� +β β × Pn(yn|xn, ˆα, β)f(β|ˆϕ)dβ +� +β Pn(yn|xn, ˆα, β)f(β|ˆϕ)dβ +. +(9) +Again, since there is no closed-form solution for the integrals in equation (9), we +approximate them using simulations yielding to equation (10): +� +βn = +R +� +r=1 +� +βr × Pn(yn|xn, ˆα, βr) +�R +r=1 Pn(yn|xn, ˆα, βr) +� +, +(10) +where R is the number of draws and βr is the rth drawn from f(β|ϕ). +5 +Commands +5.1 +mixrandregret +Syntax +mixrandregret depvar +� +indepvars +� � +if +� � +in +� � +weight +� +, id(varname) +group(varname) rand(varlist) alernatives(varname) +� +basealternatives(#) noconstant cluster(varname) robust ln(#) +nrep(#) burn(#) level(#) maximize options +� +depvar equal to 1 identifies the chosen alternative, whereas a 0 indicates that the al- +ternative was not selected. +There is only one chosen alternative for each choice +set. +fweights, iweights, and pweights are allowed (see [U] weight), but they are applied +to decision-makers, not to individual observations. +Description +mixrandregret estimates the mixed random regret minimization model described in +Hensher et al. (2016), which is a mixed version of the classic random regret minimization +model introduced in Chorus (2010). mixrandregret extends the randregret command + +6 +Fitting mixed logit random regret minimization models +(Guti´errez-Vargas et al. 2021) and allows the user to specify normally and log-normally +distributed taste parameters inside the regret function. The command uses simulated +maximum likelihood for estimation (Train 2009). +Options +id(varname) is required and specifies a numeric identifier variable for the decision- +makers. +group(varname) is required and specifies a numeric identifier variable for the choice +occasions. +rand(varlist) is required and specifies the independent variables whose coefficients are +random. The random coefficients can be specified to be normally or log-normally dis- +tributed (see the ln() option). The variables immediately following the dependent +variable in the syntax are specified to have fixed coefficients. +alternatives(varname) is required to identify the alternatives available for each case. +basealternatives(#) sets base Alternative Specific Constants (ASC) if ASC is not +suppressed. +noconstant suppress the ASC. +cluster(varname), robust see [U] estimation. The cluster variable must be numeric. +ln(#) specifies that the last # variables in rand() have log-normally rather than nor- +mally distributed coefficients. The default is ln(0). +nrep(#) specifies the number of Halton draws used for the simulation. The default is +nrep(50). +burn(#) specifies the number of initial elements to be dropped when creating the +Halton sequences. The default is burn(15). Specifying this option helps reduce the +correlation between the sequences in each dimension. +level(#) set the confidence level. The default is level(95). +maximize options difficult, technique(algorithm spec), iterate(#), trace, gradient, +showstep, hessian, tolerance(#), ltolerance(#), gtolerance(#), nrtolerance(#), +from(init specs); see [U] maximize. +5.2 +mixrpred +Syntax +mixrpred newvar +� +if +� � +in +� � +, proba nrep(#) burn(#) +� + +Z. Zhu, ´A. A. Guti´errez-Vargas and M. Vandebroek +7 +Description +Following mixrandregret, mixrpred can be used to obtain the predicted probabilities +by specifying the option proba. +Options +proba calculate the choice probability for each alternative for each choice situation; the +default option. +nrep(#) specifies the number of Halton draws used for the simulation. The default is +nrep(50). +burn(#) specifies the number of initial elements to be dropped when creating the +Halton sequences. The default is burn(15). Specifying this option helps reduce the +correlation between the sequences in each dimension. +5.3 +mixrbeta +Syntax +mixrbeta varlist +� +if +� � +in +� +, saving(filename) +� +, plot nrep(#) burn(#) +replace +� +Description +mixrbeta can be used after mixrandregret to calculate individual-level parameters cor- +responding to the variables in the specified varname using equation (10). The individual- +level parameters are stored in a user-specified data file. +Options +saving(filename) saves individual-level parameters to filename. +plot create the plots of the distribution of individual-level parameters conditional on +the estimates of mixrandregret for individual-level parameters for each individual. +nrep(#) specifies the number of Halton draws used for the simulation. The default is +nrep(50). +burn(#) specifies the number of initial sequence elements to be dropped when creating +the Halton sequences. The default is burn(15). Specifying this option helps reduce +the correlation between the sequences in each dimension. +replace overwrites filename. + +8 +Fitting mixed logit random regret minimization models +6 +Examples +To show how we can fit Mixed RRM Models using mixrandregret, we use data from +van Cranenburgh and Chorus (2018) on a Stated Choice (SC) experiment2. +These +data are collected to analyse the impact of the different decision rules on the statistical +efficiency of the design (Van Cranenburgh and Prato 2016). The participants answered +10 choice situations where they chose from three unlabelled route alternatives with two +generic attributes: travel cost and travel time. The following variables are used in our +illustration: +• altern: identify the alternative faced by the user (sub-index i or j). +• choice: whether the alternative was chosen by the individual (dummy, 1 if cho- +sen). +• id: ID of the individual. +• cs: ID of the choice situation faced by the individual. +• tt: total travel time of the alternative in minutes. +• tc: total travel cost of the alternative in euros. +We follow the data setup in randregret (see [U] randregret), and the setup for +mixrandregret is identical to that required by mixlogit (see [U] mixlogit), which is +the panel representation in terms of individual-alternative. The data set is loaded from +the server to Stata directly as illustrated below. We keep the variables of interest and +list the first 3 observations. The data loaded are in wide format as each row corresponds +to a choice situation. +. scalar server = "https://data.4tu.nl/ndownloader/" +. scalar doi = "files/24015353" +. import delimited "`=server + doi´",clear +(encoding automatically selected: ISO-8859-1) +(29 vars, 1,060 obs) +. keep obs id tt1 tc1 tt2 tc2 tt3 tc3 choice +. list obs id tt1 tc1 tt2 tc2 tt3 tc3 choice in 1/3,sepby(obs) +obs +id +tt1 +tc1 +tt2 +tc2 +tt3 +tc3 +choice +1. +1 +1 +23 +6 +27 +4 +35 +3 +3 +2. +2 +1 +27 +5 +35 +4 +23 +6 +2 +3. +3 +1 +35 +3 +23 +5 +31 +4 +1 +2. You can download the dataset from 4TU ResearchData: https://data.4tu.nl/articles/dataset/Small value- +of-time experiment Netherlands/12681650 + +Z. Zhu, ´A. A. Guti´errez-Vargas and M. Vandebroek +9 +Following the data manipulation in Guti´errez-Vargas et al. (2021), we transform the +data set using the reshape command and present the data in long format below. We +list the first 12 rows, and each row now corresponds to an alternative. The dependent +variable choice is 1 for the chosen alternative in each choice situation, and 0 otherwise. +altern identifies the alternatives in a choice situation; cs identifies the choice situation +faced by the individual; and id identifies the individual. Furthermore, total time and +total cost are obtained from the tt and tc variables. +. rename (choice) +(choice_w) +. qui reshape long tt tc, i(obs) j(altern) +. generate choice = 0 +. replace +choice = 1 if +choice_w==altern +. label define alt_label 1 "First" 2 "Second" 3 "Third" +. label values altern alt_label +. gen cs += obs +. gen total_time += tt +. gen total_cost += tc +. list id cs altern total_time total_cost choice in 1/12, sepby(cs) ab(10) noo +id +cs +altern +total_time +total_cost +choice +1 +1 +First +23 +6 +0 +1 +1 +Second +27 +4 +0 +1 +1 +Third +35 +3 +1 +1 +2 +First +27 +5 +0 +1 +2 +Second +35 +4 +1 +1 +2 +Third +23 +6 +0 +1 +3 +First +35 +3 +1 +1 +3 +Second +23 +5 +0 +1 +3 +Third +31 +4 +0 +1 +4 +First +27 +4 +0 +1 +4 +Second +23 +5 +0 +1 +4 +Third +35 +3 +1 +We begin by fitting a classical RRM Model using the randregret command to obtain +reasonable starting values for mixrandregret. We also declare noncons suppressing the +ASC given that alternatives are non-labeled in the survey. If we have labeled data, we +can specify the base alternative by declaring base() option. As we have repeated choices +from a given individual, the standard errors are corrected by specifying cluster(id). +As expected, both parameter estimates are negative and highly significant, suggesting +that regret decreases as the level of travel time or travel cost increases in a non-chosen +alternative compared with the same attribute level in the chosen one. The coefficients +are saved in init mix rrm for later use as initial values for mixrandregret. +. randregret choice total_time total_cost, group(cs) alternatives(altern) /// +> rrmfn(classic) nocons cluster(id) +Fitting Classic RRM Model +initial: +log likelihood = +-1164.529 + +10 +Fitting mixed logit random regret minimization models +alternative: +log likelihood = -1156.5784 +rescale: +log likelihood = +-1121.29 +Iteration 0: +log likelihood = +-1121.29 +Iteration 1: +log likelihood = -1118.4843 +Iteration 2: +log likelihood = -1118.4784 +Iteration 3: +log likelihood = -1118.4784 +RRM: Classic Random Regret Minimization Model +Case ID variable: cs +Number of cases += +1060 +Alternative variable: altern +Number of obs += +3180 +Wald chi2(2) += +40.41 +Log likelihood = -1118.4784 +Prob > chi2 += +0.0000 +(Std. Err. adjusted for +106 clusters in id) +Robust +choice +Coefficient +std. err. +z +P>|z| +[95% conf. interval] +RRM +total_time +-.102813 +.0182526 +-5.63 +0.000 +-.1385874 +-.0670386 +total_cost +-.417101 +.068059 +-6.13 +0.000 +-.5504943 +-.2837078 +. matrix b_rrm = e(b) +. matrix zero = J(1,1,0.01) +. matrix init_mix_rrm = b_rrm, zero +. matrix li init_mix_rrm +init_mix_rrm[1,3] +RRM: +RRM: +total_time +total_cost +c1 +y1 +-.102813 +-.41710104 +.01 +We then fit a Mixed RRM Model in which the coefficient for total cost is fixed, +but the coefficient for total time is normally distributed. We use the option from() in +mixrandregret to initialize the optimization routine using the values saved in init mix rrm +as the starting point for the mean for the total time parameter. We estimated the +model using 500 Halton draws to approximate the choice probabilities of equation (8). +Additionally, we clustered our standard errors at the individual level using cluster(id). +. mixrandregret choice total_cost, group(cs) alter(altern) rand(total_time) /// +> id(id) nocons cluster(id) nrep(500) from(init_mix_rrm) tech(bhhh) +Iteration 0: +log likelihood = -2850.0956 +Iteration 1: +log likelihood = +-2169.409 +Iteration 2: +log likelihood = -861.11253 +Iteration 3: +log likelihood = -771.96998 +Iteration 4: +log likelihood = -771.20333 +Iteration 5: +log likelihood = -771.09059 +Iteration 6: +log likelihood = +-771.0649 +Iteration 7: +log likelihood = -771.05912 +Iteration 8: +log likelihood = -771.05774 +Iteration 9: +log likelihood = -771.05741 +Iteration 10: +log likelihood = -771.05733 +Iteration 11: +log likelihood = -771.05731 +Case ID variable: cs +Number of cases += +1060 +Alternative variable: altern +Random variable(s): total_time +(Std. Err. adjusted for +106 clusters in id) + +Z. Zhu, ´A. A. Guti´errez-Vargas and M. Vandebroek +11 +Mixed random regret model +Number of obs = +3,180 +Wald chi2(2) += 606.11 +Log likelihood = -771.05731 +Prob > chi2 += 0.0000 +OPG +choice +Coefficient +std. err. +z +P>|z| +[95% conf. interval] +Mean +total_cost +-1.102136 +.0449727 +-24.51 +0.000 +-1.190281 +-1.013991 +total_time +-.3580736 +.0581449 +-6.16 +0.000 +-.4720355 +-.2441117 +SD +total_time +.5068268 +.041366 +12.25 +0.000 +.425751 +.5879027 +The sign of the estimated standard deviations is irrelevant: interpret them as +being positive +. matrix b_mixrrm = e(b) +On average, the regret decreases as the total travel time increases in a non-chosen +alternative, compared to the same level of travel time in the chosen alternative. The +interpretation is similar for the total travel cost attribute. Additionally, we observe that +there is significant regret heterogeneity for total travel time, given that the standard +deviation parameter for total travel time is statistically different from zero. Further- +more, after the estimation of the Mixed RRM Model, we can compute individual-level +parameters using mixrbeta. In the code below, we use equation (10) to approximate +the value for the regret coefficient for each individual using 500 Halton draws. Addi- +tionally, mixrbeta creates a new data set with one observation per individual (id) and +its corresponding parameter estimates. Subsequently, we also display the estimates for +the first five individuals in the sample, where we observe that some of them have a +positive coefficient for the total time attribute. Besides, we plot the individual level +parameters for total time in Figure 1 for all the individuals in the sample and observe +that there are individuals with positive estimates for the total time coefficient, which +is counter-intuitive. +. mixrbeta total_time, nrep(500) replace saving("${graphs_route}\mixRRM_normal_idl") +. use "${graphs_route}\mixRRM_normal_idl", replace +. list id total_time in 1/5 +id +total_time +1. +1 +.37640482 +2. +2 +-.05517462 +3. +3 +.37672848 +4. +4 +.38495822 +5. +5 +.37607978 +One solution to obtain non-positive estimates for the total time coefficient is to use +a bounded distribution. When using mixrandregret, we can specify that a coefficient +is log-normally distributed for this purpose. In our case, since we want a non-positive +distribution for the total time coefficient, we have to multiply the total time attribute + +12 +Fitting mixed logit random regret minimization models +0 +.5 +1 +1.5 +2 +-1.5 +-1 +-.5 +0 +.5 +Density +kdensity total_time +Distribution of Total Time Coefficient +Figure 1: Distribution of Total Time Coefficient (Normal) +for -1 to ensure that it is non-positive. To this end we create the new variable ntt, which +corresponds to the negative of total time. +. gen ntt = -1 * total_time +. mixrandregret choice total_cost, group(cs) alt(altern) rand(ntt) ln(1) id(id) /// +> nocons cluster(id) nrep(500) tech(bhhh) from(b_mixrrm) +Iteration 0: +log likelihood = -994.35461 +Iteration 1: +log likelihood = -858.23241 +Iteration 2: +log likelihood = +-798.4694 +Iteration 3: +log likelihood = -785.66872 +Iteration 4: +log likelihood = -785.30817 +Iteration 5: +log likelihood = -785.27945 +Iteration 6: +log likelihood = -785.27728 +Iteration 7: +log likelihood = -785.27686 +Iteration 8: +log likelihood = -785.27675 +Iteration 9: +log likelihood = -785.27672 +Iteration 10: +log likelihood = -785.27671 +Case ID variable: cs +Number of cases += +1060 +Alternative variable: altern +Random variable(s): ntt +(Std. Err. adjusted for +106 clusters in id) +Mixed random regret model +Number of obs = +3,180 +Wald chi2(2) += 1230.55 +Log likelihood = -785.27671 +Prob > chi2 += +0.0000 +OPG +choice +Coefficient +std. err. +z +P>|z| +[95% conf. interval] +Mean +total_cost +-1.217682 +.0442047 +-27.55 +0.000 +-1.304321 +-1.131042 +ntt +-1.312285 +.1562202 +-8.40 +0.000 +-1.618471 +-1.006099 + +Z. Zhu, ´A. A. Guti´errez-Vargas and M. Vandebroek +13 +SD +ntt +1.363632 +.1185994 +11.50 +0.000 +1.131181 +1.596082 +The sign of the estimated standard deviations is irrelevant: interpret them as +being positive +The estimated ntt parameters are the mean and standard deviation of the natural +logarithm of the coefficient, and we can transform them back to the estimates of the +coefficients themselves. The median of the coefficient is given by exp(bntt), the mean is +given by exp(bntt + s2 +ntt/2), and the standard deviation is given by exp(bntt + s2 +ntt/2) × +� +exp(s2 +ntt) − 1 (Train 2009). The sign change prior to the estimation is reversed by +multiplying the estimates by -1. +. nlcom (mean_time: -1*exp([Mean]_b[ntt]+0.5*[SD]_b[ntt]^2)) +> +(med_time: -1*exp([Mean]_b[ntt])) +> +(sd_time : exp([Mean]_b[ntt]+0.5*[SD]_b[ntt]^2) +> +*sqrt(exp([SD]_b[ntt]^2)-1)) +mean_time: -1*exp([Mean]_b[ntt]+0.5*[SD]_b[ntt]^2) +med_time: -1*exp([Mean]_b[ntt]) +sd_time: exp([Mean]_b[ntt]+0.5*[SD]_b[ntt]^2)*sqrt(exp([SD]_b[ntt]^2)-1) +choice +Coefficient +Std. err. +z +P>|z| +[95% conf. interval] +mean_time +-.682127 +.1587961 +-4.30 +0.000 +-.9933616 +-.3708923 +med_time +-.2692041 +.0420551 +-6.40 +0.000 +-.3516307 +-.1867776 +sd_time +1.588122 +.6295756 +2.52 +0.012 +.3541763 +2.822067 +Again, we calculate individual-level parameters. As we can observe in the listed data +and distribution presented in Figure 2, all individual-level parameters are now negative +as we expected. +. mixrbeta ntt, nrep(500) replace saving("${graphs_route}\mixRRM_ln_idl") +. use "${graphs_route}\mixRRM_ln_idl" , replace +. replace ntt = -1 * ntt /*reverse sign for graph*/ +(106 real changes made) +. list id +ntt in 1/5 +id +ntt +1. +1 +-.04032598 +2. +2 +-.08142616 +3. +3 +-.04047817 +4. +4 +-.04110615 +5. +5 +-.04025335 +We can also generate predictions after running mixrandregret using mixrpred. +To illustrate this command, we rerun the models using mixrandregret with normally +distributed random coefficients, suppressing the output using the quietly command +(see [U] quietly). +Then, using the option proba, we generate the pred p variable +containing the predicted probability for each alternative. +The code and output are +listed below. + +14 +Fitting mixed logit random regret minimization models +0 +.5 +1 +1.5 +-4 +-3 +-2 +-1 +0 +Density +kdensity ntt +Distribution of Total Time Coefficient +Figure 2: Distribution of Total Time Coefficient (Log-normal) +. qui mixrandregret choice total_cost, group(cs) alter(altern) rand(total_time) /// +> id(id) nocons cluster(id) nrep(500) from(init_mix_rrm) tech(bhhh) +. mixrpred pred_p, proba nrep(500) +. list id cs altern total_time total_cost choice pred_p in 151/162, sepby(cs) ab(10) noo +id +cs +altern +total_time +total_cost +choice +pred_p +6 +51 +First +23 +6 +0 +.1516009 +6 +51 +Second +27 +4 +1 +.5547292 +6 +51 +Third +35 +3 +0 +.2936699 +6 +52 +First +27 +5 +0 +.3153724 +6 +52 +Second +35 +4 +1 +.291449 +6 +52 +Third +23 +6 +0 +.3931786 +6 +53 +First +35 +3 +0 +.3134595 +6 +53 +Second +23 +5 +1 +.5523607 +6 +53 +Third +31 +4 +0 +.1341798 +6 +54 +First +27 +4 +0 +.3153724 +6 +54 +Second +23 +5 +1 +.3931786 +6 +54 +Third +35 +3 +0 +.291449 +Additionally, mixrandregret also allows for the inclusion of ASC if users have la- +beled data. Although the data set is unlabeled in this example, we treat it as a labeled +one in that each alternative represents a distinct category. We run the model including +basealternative(1) option, which specify that the first alternative is the reference +group for ASC. +. mixrandregret choice total_cost, group(cs) alt(altern) rand(total_time) id(id) /// + +Z. Zhu, ´A. A. Guti´errez-Vargas and M. Vandebroek +15 +> basealternative(1) cluster(id) nrep(500) tech(bhhh) +Iteration 0: +log likelihood = +-1164.529 +Iteration 1: +log likelihood = -812.87881 +Iteration 2: +log likelihood = -773.05839 +Iteration 3: +log likelihood = +-769.1873 +Iteration 4: +log likelihood = -768.22193 +Iteration 5: +log likelihood = -767.97262 +Iteration 6: +log likelihood = -767.90237 +Iteration 7: +log likelihood = +-767.8867 +Iteration 8: +log likelihood = -767.88268 +Iteration 9: +log likelihood = -767.88165 +Iteration 10: +log likelihood = -767.88138 +Iteration 11: +log likelihood = -767.88131 +Iteration 12: +log likelihood = -767.88129 +Case ID variable: cs +Number of cases += +1060 +Alternative variable: altern +Random variable(s): total_time +(Std. Err. adjusted for +106 clusters in id) +Mixed random regret model +Number of obs = +3,180 +Wald chi2(2) += 465.50 +Log likelihood = -767.88129 +Prob > chi2 += 0.0000 +OPG +choice +Coefficient +std. err. +z +P>|z| +[95% conf. interval] +Mean +total_cost +-1.06784 +.0498243 +-21.43 +0.000 +-1.165494 +-.9701866 +total_time +-.3455217 +.0594409 +-5.81 +0.000 +-.4620237 +-.2290197 +SD +total_time +-.5095087 +.0420965 +-12.10 +0.000 +-.5920163 +-.4270012 +ASC +ASC_2 +.0064798 +.0510223 +0.13 +0.899 +-.0935221 +.1064816 +ASC_3 +.136445 +.0605786 +2.25 +0.024 +.0177131 +.2551768 +The sign of the estimated standard deviations is irrelevant: interpret them as +being positive +7 +Conclusions +This article presents the command mixrandrgret to fit Random Regret Minimization +models with random parameters. +We also developed the post-estimation command +mixrpred for predicting the estimated probabilities. Additionally, the mixrbeta post- +estimation command allows the user to estimate individual-level parameters for the +random coefficients included in the model. +The commands’ usage and options are +illustrated using discrete choice data from van Cranenburgh and Chorus (2018). + +16 +Fitting mixed logit random regret minimization models +8 +Acknowledgments +We thank Michel Meulders, Jan De Spiegeleer, and the participants from the 2022 +London Stata Conference for their helpful comments and constructive suggestions. Ad- +ditionally, substantial portions of our programs were inspired by the book Maximum +Likelihood Estimation with Stata, Fourth Edition by Willian Gould, Jeffrey Pitblado, +and Brian Poi (2010). Finally, many of the previous checks to the data and the construc- +tion of the log-likelihood functions were greatly inspired by the randregret (Guti´errez- +Vargas et al. 2021) and mixlogit (Hole 2007) commands. +9 +Funding +This work was produced while ´Alvaro A. Guti´errez-Vargas was a PhD student at the Re- +search Centre for Operations Research and Statistics (ORSTAT) at KU Leuven funded +by Bijzonder Onderzoeksfonds KU Leuven (Special Research Fund KU Leuven). +10 +Conflict of interest +Ziyue Zhu, ´Alvaro A. Guti´errez-Vargas, and Martina Vandebroek declare no conflicts +of interest. +11 +Contribution +Ziyue Zhu and ´Alvaro A. Guti´errez-Vargas contributed equally to the article by devel- +oping the command and drafting the article. Martina Vandebroek critically commented +on both the article and the command’s functionality. +12 +References +Chorus, C. G. 2010. A new model of random regret minimization. European Journal +of Transport and Infrastructure Research 10(2). +. 2014. A generalized random regret minimization model. Transportation research +part B: Methodological 68: 224–238. +Chorus, C. G., T. A. Arentze, and H. J. Timmermans. 2008. +A random regret- +minimization model of travel choice. Transportation Research Part B: Methodological +42(1): 1–18. +van Cranenburgh, S., and C. Chorus. 2018. Small value-of-time experiment, Netherlands +[Data set]. TU Delft - 4TU.ResearchData . +van Cranenburgh, S., C. A. Guevara, and C. G. Chorus. 2015. New insights on random + +Z. Zhu, ´A. A. Guti´errez-Vargas and M. Vandebroek +17 +regret minimization models. Transportation Research Part A: Policy and Practice +74: 91–109. +Guti´errez-Vargas, ´A. A., M. Meulders, and M. Vandebroek. 2021. randregret: A com- +mand for fitting random regret minimization models using Stata. The Stata Journal +21(3): 626–658. +Hensher, D. A., W. H. Greene, and C. Q. Ho. 2016. Random regret minimization and +random utility maximization in the presence of preference heterogeneity: an empirical +contrast. Journal of Transportation Engineering 142(4): 1–10. +Hole, A. R. 2007. Fitting mixed logit models by using maximum simulated likelihood. +The stata journal 7(3): 388–401. +Loomes, G., and R. Sugden. 1982. Regret theory: An alternative theory of rational +choice under uncertainty. The economic journal 92(368): 805–824. +McFadden, D. 1974. Conditional logit analysis of qualitative choice behavior. In: Zarem- +bka, P., Ed., Frontiers in Econometrics, 105–142. +McFadden, D., and K. Train. 2000. Mixed MNL models for discrete response. Journal +of Applied Econometrics 15(5): 447–470. +Train, K. E. 2009. +Discrete choice methods with simulation. +Cambridge university +press. +Van Cranenburgh, S., and C. G. Prato. 2016. On the robustness of random regret mini- +mization modelling outcomes towards omitted attributes. Journal of choice modelling +18: 51–70. +About the authors +Ziyue Zhu is a master student of statistics and data science at KU Leuven in Belgium. She +earned a Bachelor of Economics from Wuhan University and a Master of Economics from +Barcelona School of Economics. +´Alvaro A. Guti´errez-Vargas is a PhD student at the Research Centre of Operation Research and +Statistics (ORSTAT) at KU Leuven in Belgium. He earned a Bachelor of Science in economics +from the University of Chile. His research interests are mainly methodological and focused on +computational statistics, machine learning, and discrete choice models. He has been published +in The Stata Journal and Journal of Choice Modelling. +Martina Vandebroek is a full professor at the Faculty of Economics and Business at KU Leuven +in Belgium. She earned a PhD in actuarial sciences from KU Leuven. She is interested in the +design of experiments, discrete choice experiments, and multivariate statistics. She has been +published in Transportation Research B, Journal of Choice Modelling, Marketing Science, and +Journal of Statistical Software, among other journals. + diff --git a/V9AzT4oBgHgl3EQfJ_vP/vector_store/index.pkl b/V9AzT4oBgHgl3EQfJ_vP/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..58fbe95712709a5d7fc874c40aa06dc7d68ee9f2 --- /dev/null +++ b/V9AzT4oBgHgl3EQfJ_vP/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9194602d114b017e90720873bfda623d26db1c2786a0cdb28b313fd8868939da +size 93249 diff --git a/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf b/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..20e2c5bf1e4dc4f3d7cfbf57375466d949dce4e2 --- /dev/null +++ b/VtFOT4oBgHgl3EQf6zSP/content/2301.12960v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d036f8787336f417a72164252604526325c7d164b1809afd27ebe3e1193b29ec +size 1105873 diff --git a/VtFOT4oBgHgl3EQf6zSP/vector_store/index.faiss b/VtFOT4oBgHgl3EQf6zSP/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..f629e19509f5e098bed7d28a2f8d291be184c4b9 --- /dev/null +++ b/VtFOT4oBgHgl3EQf6zSP/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a748b03163bd194181e88ff9cf9fd92944e11d1a9a25e1f1cf6b64af5c218117 +size 2752557 diff --git a/W9FQT4oBgHgl3EQfcjYS/vector_store/index.pkl b/W9FQT4oBgHgl3EQfcjYS/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..eac0a769fa9c13ffe01b0b5b7b4916f77dabfcc0 --- /dev/null +++ b/W9FQT4oBgHgl3EQfcjYS/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:568b3393dc32befeb92ab63cd0a39910bf54e1459559200f63c7f7e690e4cba4 +size 165915 diff --git a/WNAyT4oBgHgl3EQfu_m7/content/tmp_files/2301.00624v1.pdf.txt b/WNAyT4oBgHgl3EQfu_m7/content/tmp_files/2301.00624v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..480356fd38fd81949939f74d1ead34d62f88debe --- /dev/null +++ b/WNAyT4oBgHgl3EQfu_m7/content/tmp_files/2301.00624v1.pdf.txt @@ -0,0 +1,2971 @@ +Modular and Incremental Global Model +Management with Extended Generalized +Discrimination Networks +Matthias Barkowsky +matthias.barkowsky@hpi.de +Holger Giese +holger.giese@hpi.de +January 3, 2023 +Abstract +Complex projects developed under the paradigm of model-driven en- +gineering nowadays often involve several interrelated models, which are +automatically processed via a multitude of model operations. Modular +and incremental construction and execution of such networks of models +and model operations are required to accommodate efficient development +with potentially large-scale models. The underlying problem is also called +Global Model Management. +In this report, we propose an approach to modular and incremental +Global Model Management via an extension to the existing technique +of Generalized Discrimination Networks (GDNs). In addition to further +generalizing the notion of query operations employed in GDNs, we adapt +the previously query-only mechanism to operations with side effects to +integrate model transformation and model synchronization. We provide +incremental algorithms for the execution of the resulting extended Gen- +eralized Discrimination Networks (eGDNs), as well as a prototypical im- +plementation for a number of example eGDN operations. +Based on this prototypical implementation, we experiment with an ap- +plication scenario from the software development domain to empirically +evaluate our approach with respect to scalability and conceptually demon- +strate its applicability in a typical scenario. Initial results confirm that +the presented approach can indeed be employed to realize efficient Global +Model Management in the considered scenario. +1 +Introduction +Complex projects developed under the model-driven engineering paradigm nowa- +days often involve several interrelated models, which are inspected, analyzed, +transformed, and synchronized via a multitude of model operations [69]1. An +1Note that references in bold refer to our own publications. +1 +arXiv:2301.00624v1 [cs.SE] 2 Jan 2023 + +effective and efficient management of the resulting sophisticated networks of +model operations is both a crucial prerequisite to successful development projects +and a challenging research problem, known as Global Model Management [10]. +On the one hand, modular and incremental construction of model opera- +tion networks is required in the context of project landscapes that evolve to +accommodate dynamic development processes and changing requirements. On +the other hand, in order to scale to today’s potentially large models and allow +development in teams, modular and incremental execution of these networks is +required, as full re-execution of the entire network in reaction to changes may +result in unacceptable execution times and loss of information [39]. +In this context, model queries, due to being explicitly and implicitly required +by model properties and model consistency checks respectively model transfor- +mations and model synchronizations, play a central role. Solutions thus have +to offer dedicated support for handling potentially complex model queries and +facilitate their modular composition and reuse. +Furthermore, model operations with side-effects, such as model transforma- +tion and synchronization, and their interaction with other model operations pose +a unique challenge regarding the overall goal of guaranteeing the consistency of +a system description that may be distributed over multiple models. +In this report, we propose an approach to Global Model Management that +specifically aims to provide both the required modularity and incrementality. +Our solution is based on an extended notion of Generalized Discrimination Net- +works [45], a mechanism that has previously been implemented in the context of +model driven engineering [7] to allow a modular and incremental specification +and execution of model queries in the form of nested graph conditions [44]. +Therefore, we introduce a more general formalization called extended Gen- +eralized Discrimination Networks (eGDNs), which (i) supports a more flexible +notion of model queries, affording increased expressiveness and (ii) allows the +integration of model operations with side effects into the unifying framework. +In addition, we provide algorithms for the incremental execution of eGDNs. +Furthermore, we integrate a number of typical model operations into a proto- +typical implementation of the approach and use this implementation to perform +an initial evaluation of our technique’s scalability using an application scenario +from the software development domain. This empirical evaluation is comple- +mented by a conceptual evaluation regarding the applicability of eGDNs in a +typical scenario. +The remainder of the report is structured as follows: We briefly reiterate +the basic concepts of models in the form of typed graphs and discrimination +networks in Chapter 2. +After introducing the required concepts, we discuss +requirements of a solution for global model management and related work in +Chapter 3, providing further motivation for the design of a new solution. Our +contribution in the form of extended Generalized Discrimination Networks is +presented in Chapters 4, 5, and 6. Therefore, Chapter 4 provides a definition +of eGDNs along with a graphical notation. Chapter 5 describes the incremen- +tal execution of eGDNs. Chapter 6 then lists a number of examples for eGDN +operations that are part of our prototypical implementation. This prototyp- +2 + +ical implementation is used to perform an initial empirical evaluation of the +presented concepts, which is presented in Chapter 7 along with a conceptual +evaluation of the applicability of eGDNs to an example use case. Finally, Chap- +ter 8 concludes the report and gives an overview of possible directions for future +work. +2 +Preliminaries +In this chapter, we reiterate the basic notions of models in the form of typed +graphs and discrimination networks. +2.1 +Graphs and Models +A graph G = (V G, EG, sG, tG) consists of a set of vertices V G, a set of edges EG, +and two functions sG, tG : EG → V G assigning each edge its source respectively +target vertex [24]. A graph morphism m : G → H between graphs G and H is a +pair of functions mV : V G → V H, mE : EG → EH such that sH ◦mE = mV ◦sG +and tH ◦ mE = mV ◦ tG. +A graph G can be typed over a type graph TG via a morphism typeG : G → +TG that assigns elements from G types defined in TG. This yields a typed +graph GT = (G, typeG). A typed graph morphism mT : GT → HT between two +typed graphs GT = (G, typeG) and HT = (H, typeH) typed over the same type +graph TG is given by a graph morphism m : G → H with typeG = typeH ◦ mT . +In the context of this report, a model is then characterized by a typed graph, +where the type graph effectively acts as a metamodel. Importantly, attributes +for model elements can be realized in the framework of typed graphs by simply +modeling attribute values as dedicated nodes, which leads to the notion of typed +attributed graphs [47]. A modeling language ML is defined by a graph TG and +denotes the set of all possible graphs typed over TG. +Figure 1 shows an example model from the software development domain +in the form of a typed graph G, and the associated metamodel in the form of +the type graph TG, with the typing morphism given by node labels in case of +nodes and implicitly in case of edges. The example model represents the ab- +stract syntax graph (ASG) of a program written in an object-oriented program- +ming language. Nodes in the model represent packages, types, and methods. +Edges represent containment relationships between the different concepts, with +methods contained in types and types contained in packages, and return type +relationships between methods and types. +2.2 +Discrimination Networks +A discrimination network is a graph of nodes representing computation units +and edges representing dependencies between these units. Discrimination net- +works are a popular solution for the incremental execution of model queries such +as the computation of model properties or the checking of model consistency +3 + +Type +Method +Package +types +type +p1:Package +t1:Type +t2:Type +m1:Method +m3:Method +m2:Method +G +TG +methods +Figure 1: Example model and metamodel in the form of typed graph and type +graph from the software development domain +conditions. Therefore, the model query is decomposed into subqueries, which +form the discrimination network’s nodes. +The execution of a subquery can make use of the results computed for an- +other subquery, which is indicated by a dependency relation between the two +subqueries. +The execution of a final discrimination network node yields the +overall query result. +By storing the results of discrimination network nodes +beyond the execution of a query, incremental execution that reuses previously +computed results in subsequent executions is enabled. +Since discrimination networks so far are primarily employed for model query- +ing, current approaches offer only limited or no support for the integration of +model operations with side-effects and thus constitute at best a partial solu- +tion for global model management. However, due to their inherent support for +modularity and incrementality, they offer a promising starting point. +There exist different realizations of the concept of discrimination networks in +the context of model driven engineering, two of which will be briefly presented +in the following subsections. +2.2.1 +RETE nets +RETE nets were initially introduced by Forgy [34] and are characterized by the +fact that nodes are only allowed to have dependencies to at most two other +nodes. Some examples of RETE nodes are: +• input nodes, which correspond to primitive model queries that extract in- +dividual elements, that is, nodes or edges, from a model, and consequently +have no dependencies +• filter nodes, which filter the results of some other subquery by a condition +and consequently have one dependency +• join nodes, which combine the results of two other subqueries into results +for a more complex subquery and consequently have two dependencies +4 + +While the listed node types form the core of incremental model querying so- +lutions such as the well-established VIATRA [78], RETE nets are a flexible +mechanism that allows a multitude of other query-related node types. This is +illustrated by VIATRA’s support for various advanced constructs for specifying +model queries, including negative patterns and certain aggregation operations. +In RETE implementations, results computed by a RETE net’s nodes are +usually stored in memory in so-called indexers, which act as implicit interfaces +between computation nodes. These indexers can also be made explicit by mod- +eling them as part of the RETE net via a different kind of RETE node that +is not associated with any computational functionality, but only serves as a +storage for other nodes’ results. +2.2.2 +Generalized Discrimination Networks +Generalized Discrimination Networks (GDNs) are a less restrictive form of dis- +crimination networks than RETE nets and were developed by Hanson et al. +[45]. Essentially, GDNs drop the limit on the number of a node’s dependencies +of RETE nets and thereby allow for more control over which intermediate query +results are to be stored in memory. +A realization in the context of model querying was presented in [7]. It imple- +ments GDN nodes as model transformation rules that create marking elements +for subquery results directly as part of the queried model. Dependencies between +nodes are realized by considering marking elements created by the required node +in the transformation rule associated with the dependent node. However, while +the approach in [7] is based on a fairly expressive notion of queries in the form +of nested graph conditions, certain query-related operations such as aggregation +are not supported by the underlying formalism. +3 +Requirements for Global Model Management +Nowadays the development of complex systems with models requires Global +Model Management (GMM ) [9, 31] to ensure that the models of different sub- +systems, of different views, and of different domains are properly combined, even +though the models might reside at different levels of abstraction. Indeed, due +to the heterogeneity and complexity of systems such as Cyber-Physical Systems +(CPS), it is no longer feasible to represent the system as a Single Undery- +ing Model (SUM). This is because numerous languages and tools are already +employed independently by domain experts collaborating to build the system. +Redeveloping these tools and thus requiring industry to change its practices is +not conceivable given the required development efforts, but also the strong re- +sistance to change development processes. This is especially relevant in the case +of safety-critical systems that must undergo complex certification processes. +Therefore, many models must be used to represent the system and adequate +GMM is required to ensure that the development activities that operate on the +models are properly coordinated such that the models lead to a proper system +5 + +as a whole, where the different elements and aspects covered by the different +models are correctly integrated and are consistent with each other. +A classification of model integration problems and fundamental integration +techniques has been introduced in [40]. +It highlights the techniques of de- +composition and enrichment, which characterize two orthogonal dimensions of +development where the system is decomposed into subsystems and domains +(horizontal dimension) and into a set of models with increasing level of details +(vertical dimension). This requires coordinating all activities operating on the +models across these dimensions to ensure their consistency. +The development activities for nowadays complex systems are spread across +multiple domains and teams, where each team is using its own set of model- +ing languages thus requiring proper integration of these languages. Indeed, it +has been shown that using a single language to cover all domains would lead +to very large monolithic languages not easily customizable for the development +environments and tools needed by development organizations. These consid- +erations lead to Multi-Paradigm Modeling (MPM ) [77], which advocates the +integration of reusable modular modeling languages instead of large monolithic +languages. Hence, GMM must support integrating with appropriate modularity +not only models but also their modeling languages (hereafter modeling language +integration), in addition to coordinating all activities operating on the models +and specified as model operations / transformations. The execution of these +model operations has to be scalable for being able to handle large models. This +requires incrementality, where only the operations impacted by a model change +are re-executed, thus avoiding the effort to recompute entire models as in the +case of incremental code compilers. +GMM is also known as modeling-in-the-large, which consists of establish- +ing global relationships (e.g. model operations that generated one model from +other models) between macroscopic entities (models and metamodels) while ig- +noring the internal details of these entities [9]. Megamodeling [8, 31] has been +introduced for the purpose of describing these macroscopic entities and their +relations. +Consequently, for modular and incremental global model management solu- +tions for the modular and incremental construction and execution of I) models +and modeling languages integration, II) model operations, and III) megamodels +are required. We will outline in the following that nowadays only preliminary +approaches exist that provide ad hoc solutions for fragments of the sketched +problem and that a solid understanding of the underlying needs and challenges +is currently lacking. In particular, the current approaches do at most offer some +modularity and/or incrementality for a single aspect as modeling languages inte- +gration or model operations. However, support for handling complex modeling +landscapes as a whole in a modular and incremental fashion as required for the +large-scale problems that exist in practice is not offered so far. +In the following, we will discuss the needs in more detail and review how +far existing solutions that address the construction and execution of 1) models +and modeling languages integration, 2) model operations, and 3) megamodels. +The way the existing approaches perform along these dimensions is depicted in +6 + +Table 1, where an empty cell identifies a need that is not addressed, a ˜ denotes +partial fulfilment of the need and a + indicates that the need is addressed +sufficiently2. +This evaluation is discussed in further details in the following +sections. +Approach +Modeling Languages Integration +Model Operations +Megamodels +Const. +Exec. +Const. +Exec. +Const. +Exec. +Links +Int. +MMI +Batch +Inc. +Flow +Ctx. +Batch +Inc. +Mon. +Mod. +Batch +Inc. +Modeling Languages Integration +Blanc et al. [11] ++ ++ +EMF IncQuery [25, 76] ++ ++ +Egyed et al. [43, 23] ++ ++ +Cabot et al. [19] ++ ++ +ACOL [57] +∼ ++ +SmartEMF [58, 50, 73] ++ ++ +Composite +EMF +Mod- +els [52, 21] ++ ++ +EMF Views [27, 17] ++ +∼ ++ +Kompren +[12, +55] +/ +Kompose [33, 54] +∼ +Reuseware +ModelSoc +[51, 60] +∼ +∼ +Ratiu et al. [65] ++ +∼ +∼ +K¨onig et al. [56] ++ ++ +Model Operations +Wires* [66, 80] ++ ++ +ATL Flow [4] ++ ++ +Epsilon [64, 28] ++ ++ ++ +Gaspard2 [29, 37] +∼ ++ +∼ ++ +∼ +Debreceni et al. [22] ++ +∼ ++ ++ +MoTCoF [71] ++ ++ ++ +∼ +MoTE [38][61] ++ ++ ++ +Integration Languages and Others +CyPhy [72, 42] ++ ++ ++ +FUSED [15, 35] ++ +∼ ++ ++ +CONSYSTENT [48, 49] ++ ++ ++ +Megamodels +AM3 [79, 1] ++ ++ ++ ++ +∼ +FTG+PM [59, 5] ++ ++ ++ ++ +MegaL Exp. [32] ++ ++ ++ +GMM* [13] ++ +∼ ++ ++ +∼ ++ ++ +∼ +Seibel et al. [70, 68][6] ++ ++ ++ +∼ +Stevens [75, 74] ++ ++ ++ ++ ++ ++ +Gleitze et al. [41] ++ ++ ++ ++ ++ ++ +Vitruvius [53] ++ ++ ++ ++ ++ ++ ++ ++ ++ +eGDNs ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ +Table 1: Comparison of existing and planned global model management ap- +proaches +3.1 +Models and Modeling Languages Integration: +Con- +struction and Execution +3.1.1 +Construction +The construction of models and modeling languages integration is addressed in +the current approaches in three main ways via (1) linking of models and model +elements, (2) model interfaces and (3) metamodel composition. +(1) Links: +All approaches make use of some kind of trace links between models and their +model elements to integrate models. In this report, we adopt the definitions of +traceability proposed by the Center of Excellence for Software Traceability (Co- +EST) [20]. A trace link is ”...a specified association between a pair of artifacts, +one comprising the source artifact and one comprising the target artifact...”. +Following the CoEST again, trace links are specialized into traces between the +vertical and horizontal dimensions. Hence, a vertical trace ”...links artifacts at +2For convenience, we use the name of the tool or project to identify an approach when it +exists, otherwise the name of the first author of the publication describing the approach is +used. +7 + +different levels of abstraction so as to accommodate lifecycle-wide or end-to-end +traceability, such as from requirements to code...”. An horizontal trace links +”...artifacts at the same level of abstraction, such as: (i) traces between all the +requirements created by ‘Mary’, (ii) traces between requirements that are con- +cerned with the performance of the system, or (iii) traces between versions of a +particular requirement at different moments in time”. +There is a plethora of approaches (e.g., [2, 28, 32, 58, 48, 72, 15] [61]) mak- +ing use of trace links to integrate models. The Atlas Model Weaving (AMW) +language [2] provided one of the first approaches for capturing hierarchical trace- +ability links between models and model elements. The purpose was to support +activities such as automated navigation between elements of the linked models. +In this approach, a generic core traceability language is made available and op- +tionally extended to provide semantics specific to the metamodels of the models +to be linked. Similarly, the Epsilon framework [28] provides a tool (ModeLink) +to establish correspondences between models. MegaL Explorer [32] supports re- +lating heterogeneous software development artifacts which do not necessary have +to be models or model elements using predefined relation types. SmartEMF [58] +is another tool for linking models based on annotations of Ecore metamodels to +specify simple relations between model elements through correspondence rules +for attribute values. Complex relations are specified with ontologies relating +the concepts of the linked languages. +The whole set of combined models is +converted into Prolog facts to support various activities such as navigation, +consistency and user guidance when editing models. The CONSYSTENT tool +and approach [48] make use of a similar idea. However, graph structures and +pattern matching are used to represent the combined models in a common for- +malism and to identify and manage inconsistencies instead of Prolog facts as in +the case of SmartEMF. +There are also a number of approaches such as [72] and [15] that build +on establishing links between models through the use of integration languages +developed for a specific set of integrated modeling languages, where the inte- +gration language embeds constructs specific to the linked languages. This is +also the case for model weaving languages extending the core AMW language. +However, AMW has the advantage of capturing the linking domain with a core +common language. Other means for linking and integrating models are Triple +Graph Grammars (TGG) such as the Model Transformation Engine (MoTE) +tool [61], which similarly requires the specification of some sort of integra- +tion language (correspondence metamodel) specific to the integrated languages. +However, an important asset of this approach is that it automatically establishes +and manages the traceability links and maintains the consistency of the linked +models (model synchronization) in a scalable, incremental manner. +Finally, +in [70, 68][6], an approach is presented to automatically create and maintain +traceability links between models in a scalable manner. While the approach +focuses on traceability management rather than model integration, compared +to integration languages, it relies on link types defined at the model level (and +not at the metamodel / language level), thus avoiding the need to update the +integration language every time a new language must be integrated. More re- +8 + +cently, the concept of reactive links has been presented in [65], which essentially +allows an incremental propagation of attribute value changes between models of +different languages. However, incremental execution is only offered for a limited +notion of consistency. +The comparison of these approaches shows that apart from our own earlier +approach [70, 68][6], all approaches suffer from being dependent on the set +of integrated languages, thus requiring to better support modularity. Further- +more, only our own work [61][70, 68][6] supports automated management of +traceability links. +(2) Interfaces: +In addition to links, a few more sophisticated approaches +(e.g., [57, 50, 52, 51]) introduce a concept of model interface (int. column in +Table 1) for specifying how models can be linked. In [57], the Analysis Con- +straints Optimization Language (ACOL) is proposed, which has been designed +to be pluggable to an Architecture Description Language (ADL). A concept of +interface specific to ACOL is included so that constraints can refer to these in- +terfaces to relate to the model elements expected from the ADL. SmartEMF [50, +73] proposes a more generic concept of model interface to track dependencies +between models and metamodels and provide automated compatibility checks. +Composite EMF Models [52, 21] introduces export and import interfaces to +specify which model elements of a main model (body) should be exposed to +other models (i.e. are part of the public API), and which elements of a body +model are to be required from an export interface. In [51], an approach for the +composition of grammars with explicit variation points (hooks) constituting an +implicit invasive composition interface is presented. +However, while these approaches provide interesting preliminary ideas, they +need to be enriched to cover a larger number of non intrusive model integration +use cases such as for example, specifying modification policies of the linked +model elements required to ensure the models can be kept consistent. They also +lack integration into GMM. +(3) Metamodel Integration: +Some approaches (e.g., [55, 54, 29, 27] [13]) +consider the construction of view metamodels in terms of other metamodels +or language fragments (MMI column in Table 1). In [29], an approach imple- +mented in the Gaspard2 tool [37] is presented where metamodels are artificially +extended for the purpose of combining independent model transformations re- +sulting in an extended transformation for the extended metamodels. In [12], a +language and tool (Kompren) [55] are proposed to specify and generate slices of +metamodels via the selection of classes and properties of an input metamodel. +A reduced metamodel is then produced, which must be completely regenerated +when the input metamodel is changed. Such is the case for the Kompose ap- +proach [54], which on the contrary to Kompren proposes to create compound +metamodels, where a set of visible model elements from each combined meta- +models is selected and optionally related. EMF Views [27, 17] provides similar +approach however without the need to duplicate the metamodel elements as op- +posed to Kompose and Kompren. Indeed, EMF Views allows the specification +of virtual metamodels that only refer to existing metamodel elements instead +of duplicating them. The same principle applies for the given models of the vir- +9 + +tual metamodels, which only refer to elements of the existing integrated models +instead of duplicating them. The defined virtual view metamodels are usable +transparently by tools. Furthermore, the same models can be simultaneously +used by both legacy tools and new tools making use of the virtual metamod- +els, thanks to the non-intrusiveness of the approach. Finally, the Global Model +Management language (GMM*)3 [13] provides means to specify and interpret +reusable language subsets as sets of constraints combined to form subsetted +metamodels. Like for EMF Views, these reduced metamodels can to some ex- +tent be used transparently by tools. Aspect-oriented metamodel composition is +another well-known technique for metamodel composition. However it requires +metamodels to be expressed in a specific aspect-related format, which does not +meet our non-intrusiveness requirement. +While each of these approaches provides interesting support for modular +modeling languages integration, their unification into a common formalism, the +use of an explicit notion of a model interface and their integration into GMM is +lacking, except for subsetted metamodels already integrated within our GMM* +language. Among these approaches, we note that EMF Views provides an ade- +quate starting point for this work, due to its non-intrusiveness property essential +for reusing legacy models and tools. However, in its current implementation, +only changes of attributes of virtual compound models are propagated to the +underlying real models [17]. Other changes propagation as well as metamodel +constraints composition remain to be addressed. The integration of an explicit +metamodel interface construct for governing how metamodels can be composed, +as well as the ability to solve attribute and operation conflicts of merged classes +inspired from the concept of Traits / Mixins developed for object oriented pro- +gramming are required future works for this approach. +Execution of integrated models concerns the evaluation of the well-formedness +constraints of each combined model alone, but also of the combined models as +a whole. To our knowledge, no approach addresses the incremental checking of +well-formedness conditions across the different language fragments of compound +models. However, some approaches on incremental constraints evaluation exist. +In [11], changes on models are expressed as sequences of atomic model opera- +tions to determine which constraint is impacted by the changes, so that only +these constraints need to be re-evaluated. In [25, 76], a graph-based query lan- +guage (EMF-IncQuery) relying on incremental pattern matching for improved +performance is also proposed. In [23], an approach is presented for incremental +evaluation of constraints based on a scope of model elements referenced by the +query and determined during the first query evaluation. This scope is stored +into cache and used to determine which queries need to be re-evaluated accord- +ing for some model changes. In [43], this approach is extended for the case +where the constraints themselves may change besides the constrained models. +Finally in [19], an incremental OCL checker is presented where a simpler OCL +expression and reduced context elements set are computed from an OCL con- +3We use * to distinguish this existing language and tool from the generic Global Model +Management (GMM) acronym. +10 + +straint and a given structural change event. Evaluating this simpler constraint +for the reduced context is sufficient to assert the validity of the initial constraint +and requires significantly less computation resources. +In [56], K¨onig et al. introduce a technique for the checking of consistency +constraints over linked models, which avoids the merging of these models into a +single underlying model to achieve better scalability. However, while formally +defined and proven to be correct, the approach in [56] does not consider incre- +mental consistency checking. +We identified the following requirements as main needs concerning modular- +ity and incrementality of modeling languages integration: +R 1.1 modeling languages integration via integration links and combina- +tion of well-formedness conditions with consistency +R 1.2 interfaces for embedding of modeling languages +Note that concerning Table 1 the requirements cover here Links and Inter- +faces which jointly emulate the less modular direct meta model integration and +that the employed well-formedness conditions and consistency conditions will be +covered when we consider model operations in the next section. Consequently, +as visible in Table 1, there yet does not exists any approach that provides a +combination of all these requirements we target. +3.2 +Model Operations: Construction and Execution +The construction of model operations is addressed in two ways in the liter- +ature. Most approaches combine model operations as model transformations +chains ((1) Flow Composition), where each chained transformation operates at +the granularity of complete models. In order to support reuse and scalability +for complex modeling languages, which are defined by composing them from +simpler modeling languages, a few approaches have considered specifying model +transformations as white boxes. Composed of explicit fine grained operations +processing model elements for a given context, these operations are reusable +across several model transformations ((2) Context Composition). +(1) Flow Composition Approaches: +FUSED (Formal United System Engineering Development) [15] is an integration +language to specify complex relationships between models of different languages. +It supports model transformation chains, but only implicitly via execution of +tools, without explicit representation of the involved transformations and pro- +cessed data. On the contrary, there is a plethora of approaches allowing the ex- +plicit specification and construction of model transformation chains implement- +ing a data flow paradigm. Such is the case of the AtlanMod Megamodel Man- +agement (AM3) tool [1], for which the Atlas Transformation Language (ATL) [3] +is used to specify the model transformations. Besides, a type system has been +developed [79], which enables type checking and inference on artifacts related +11 + +via model transformations. Another similar but less advanced tool is the Ep- +silon Framework [28], which provides model transformation chaining via ANT +tasks. +Wires [66] and ATL Flow [4] are tools providing graphical languages +for the orchestration of ATL model transformations. +The Formalism Trans- +formation Graph + Process Model (FTG+PM) formalism [59] implemented in +the AToMPM (A Tool for Multi-Paradigm Modeling) tool [5] provides similar +functionality. However, it has the advantage of also specifying the complete +modeling process in addition to the involved model transformations. This is +achieved via activity diagrams coupled with model transformation specifications +executed automatically to support the development process. Finally, GMM* +[13] also supports model transformation chaining, but through the specifica- +tion of relations between models of specific metamodels that can be chained. +One advantage of this approach is that automated incremental (re-)execution +of the specified relations between models is provided in response to received +model change events. Incrementality of the execution of the transformations +is also made possible by the integration of the MoTE [61] incremental model +transformation tool into GMM*. +However, while chaining model transformations offers some degree of mod- +ularity of model transformation specifications, apart from GMM*, most ap- +proaches suffer from scalability issues for large models, since the used transfor- +mation tools do not support incremental execution. In addition, the case where a +generated model is modified by hand to add information not expressible with the +language of the original model(s) cannot easily be handled by these approaches, +since regenerating the model modified by hand will destroy the user-specific +information. This need is better supported by context composition approaches. +(2) Context Composition Approaches: +A few approaches allow context composition of model operations (column Ctx. +in Table 1). In [29] as mentioned above, an approach is described to combine +independent model transformations resulting in extended transformations for +corresponding extended metamodels. In [22], an approach is described for spec- +ifying the construction of view models using contextual composition of model +operations (derivation rules) encoded as annotations of queries of the EMF Inc- +Query [25] language. Traceability links between view and source model elements +are automatically established and maintained. The use of EMF IncQuery na- +tively provides incremental execution of the derivation rules to synchronize the +view model with the source model. Some views may be derived from other views +thus allowing flow composition as chains of view models. This approach achieves +results similar to TGGs supporting incrementality, however with the drawback +of being unidirectional. Similarly, but with bi-directionality the MoTCoF lan- +guage [71] allows for both flow and fine grained context composition of model +transformations. An advantage over [29] however is that model transformations +are used as black boxes without the need to adapt the transformations according +to the context. +As can be seen, most approaches only support flow type modularity for model +12 + +operations with batch execution except for our GMM* language thanks to its +integration of MoTE providing incremental execution. This will not scale and +lead to information losses in case of partial model information overlap. Only +a few approaches allow context modularity, which better supports incremental +application where only the impacted operations can be re-applied following a +change in order to avoid the cost of re-computing complete transformations. +Such is the case of MoTCoF, which theoretically permits incremental execu- +tion, but a concrete technical solution is still lacking for it. +To address modularity and incrementality for model operations, we identified +as main needs: +R 2.1 composition of model operations +R 2.2 model operations over integrated models +R 2.3 execution scheme for model operations +Note that concerning Table 1 the requirements cover here Flow and Context +based composition and Batch as well as Incremental Execution at first for all +special cases of model operations and then also for the general case. Conse- +quently, as visible in Table 1, there yet does not exists any approach that fully +cover the envisioned combination of all these requirements we target. +3.3 +Megamodels and other Global Model Management +Approaches +Two strands can be identified for GMM. A first one makes use of (1) model +integration languages, which are defined for a specific set of integrated modeling +languages and tools meaning that the integration language must be updated +every time a new language or tool is used. The second strand attempts to solve +this problem by making use of (2) megamodels providing configurable global +model management. +(1) Integration Languages and other Approaches: +The CyPhy [72] used in the GME modeling tool [42] and FUSED [15, 35] are +examples of model integration languages. But as mentioned above, these lan- +guages must be adapted as soon as a different set of integrated languages and +tools must be used, thus requiring highly skilled developers. Integration lan- +guages are therefore not practical. +Open Services for Lifecycle Collaboration (OSLC) [63] provides standards for +tool integration through the Web. Many specifications are available for change +management, resource previews, linked data, etc. It builds on the W3C linked +data standard, which aims at providing best practices for publishing structured +data on the Web based on the W3C Resource Description Framework (RDF). +RDF is a model for data interchange on the Web where data is represented +as graphs. However, OSLC is more services (and tools) oriented and inherits +13 + +the problems of linked data, which is specific to the Web and therefore does +not separate the concerns of data representation and persistence as opposed to +Model-Driven Engineering (MDE) where an abstract syntax is used indepen- +dently of the way the data is stored. +Another approach making use of these standards is [48] and is implemented +in a tool named CONSYSTENT, used to identify and resolve inconsistencies +across viewpoints due to information overlapping. The information of all mod- +els involved during development is captured in a common RDF graph. The ap- +proach relies on a human4 to specify patterns representing semantic equivalence +links (semantic connections) across the graph models. Inconsistency patterns +based on these semantic connections are continuously checked over the RDF +model for potential matches identifying inconsistencies. +Means to automati- +cally resolve inconsistencies are under development. +However, this approach +necessitating the conversion of all models as a RDF graph is not incremental +and will not scale for large models. +(2) Megamodels: +In this second strand, megamodels serve to capture and manage MDE resources +such as modeling languages, model transformations, model correspondences, +and tools used in modeling environments. +There are several megamodeling +approaches as already mentioned. +AM3 [1] is one of the first ones where a +megamodel is basically a registry for MDE resources. Model transformations +are specified with ATL [3] and model correspondences with the Atlas Model +Weaving (AMW) language [2]. Similarly, FTG+PM [59] as mentioned above +is also a megamodeling language as well as MegaL Explorer +[32] allowing to +model the artifacts used in software development environments and their re- +lations from a linguistic point of view. The involved software languages and +related technologies and technological spaces can be captured with linguistic +relationships between them such as membership, subset, conformance, input, +dependency, definition, etc. Operations between entities can also be captured. +The artifacts do not need to be represented as models, but each entity of the +megamodel can be linked to a Web resource that can be browsed and examined. +However, the language seems to be used mostly for visualization providing a +better understanding of the developments artifacts but cannot be executed to +perform model management. The aforementioned GMM* infrastructure [13] +consists of a megamodeling language inspired from [46]. Metamodels can be de- +clared, as well as relations between models of these metamodels. In particular, +synchronization relations can relate models of two different metamodels making +use of the MoTE TGG engine [61] to transform or synchronize the models. +As mentioned earlier, chains of model transformations can be specified and exe- +cuted incrementally in response to model change events and subsets of modeling +languages can be declared. GMM* is experimented within the Kaolin tool [14] +making use of complex and rich industrial languages such as AADL and VHDL +4An automated method making use of Bayesian Belief Networks is also under study [49]. +14 + +thus challenging GMM for realistic specifications. +A new approach to modeling in the large with bidirectional model trans- +formation has been proposed by Stevens [75]. +The work in [75] presents a +formalized notion of a megamodel in the form of a hypergraph, where models +are represented as nodes that can be connected via hyperedges representing +bidirectional transformations. Incremental execution is generally supported by +the formalism, however, a concrete algorithm is only presented for megamod- +els with a restricted structure for which a certain notion of correctness can be +guaranteed. The author extends her work in [74] by connecting her previous +work to research in the domain of build systems and introducing a so-called +orientation model to steer megamodel execution, relaxing the restrictions on +the megamodel’s structure while maintaining a formal guarantee of correctness. +However, the construction of an orientation model is a manual and potentially +challenging process for complex networks of model operations. Furthermore, +the work in [75, 74] abstracts from the technical realization of model operations +and hence does not explicitly consider how operations such as the computation +of model properties may be composed in a modular manner. +In [41], Gleitze et al. propose an incremental execution strategy for net- +works of model transformations, specifically aiming for a solution that provides +explanations of cases where the strategy failed to produce a consistent result. +While their strategy is applicable to networks with arbitrary structure, only +bidirectional transformations between pairs of models are considered, limiting +the notion of supported model operations. +Recently, significant progress has also been made in the field of model views +[16], which studies how consistent view models can be derived from a system +description consisting of multiple interrelated models and therefore also relates +to Global Model Management. The most comprehensive and advanced model +view technique is probably the Vitruvius approach [53], which relies on a so- +called virtual single underlying model (V-SUM) for the description of the overall +system under development. +The V-SUM is used to integrate the individual +models describing system parts and derive new view models via consistency +relations. Therefore, Vitruvius employs a dedicated incremental algorithm for +executing complex networks of consistency preservation operations. However, +the notion of consistency in [53] is limited to relations between pairs of tuples +of model elements and hence does not support certain model operations such as +computation of model properties using aggregations. Furthermore, intra-model +well-formedness is deliberately not covered and reuse at the mega-model level +is not considered. +However, most of these megamodeling approaches only cover to a certain de- +gree the core ingredients of specifying MDE resources by means of metamodels +and model operations with appropriate modularity and incrementality. Only +fragments of the problem are solved. Furthermore, all these megamodeling lan- +guages are monolithic (column Mon. in Table 1) and as a result, predefined +megamodel fragments cannot be composed and reused to avoid rebuilding com- +plete megamodel specifications from scratch for new projects. We note however +that aspect-oriented metamodel composition may be used as an inspiring point +15 + +and adapted to megamodeling for the specification of distributed megamodels +fragments contributing cross-cutting information in an integrated megamodel. +As for megamodel execution, FTG+PM, [75, 41, 53], GMM*, and [70, 68] +consider automated or semi-automated execution in response to model changes +or modeling events from the tool’s user interfaces. +The related work demonstrates that for global model management, we need +a view that combines all its facets in a mega model. To address modularity and +incrementiality for modamodels we can conclude that the main needs are: +R 3.1 a megamodeling language with +R 3.1.1 support for metamodels, well-formedness, model operations, in- +tegration views, and traceability links +R 3.1.2 a megamodel operation module concept +R 3.2 a robust incremental megamodel execution scheme +R 3.3 megamodel interfaces +R 3.4 an asynchronous incremental megamodel execution scheme +Note that concerning Table 1, the requirements cover here the modular con- +struction as well as incremental execution. As visible in Table 1 there do exist +three approaches that do not support modularity but provide a combination of +all the other requirements we target. However, neither of them provides the +required robust incremental megamodel operation execution scheme. The tech- +nique in [75, 74], while providing formal guarantees regarding correctness and +termination, is limited to networks of model operations in the form of trees of +synchronizations between pairs of models or requires the manual construction +of an orientation model. The Vitruvius approach [53], by virtue of employing +a fixpoint iteration, does not introduce any restrictions regarding the network’s +structure, but consequently does not guarantee termination. +The execution +scheme presented in [41] is applicable to networks of model synchronizations +between pairs of models with arbitrary structure and also guarantees termi- +nation. However, outside of performing the actual execution on the concrete +instance, it provides no means of determining whether a network will eventually +terminate with a correct result. +3.4 +Summary of the state of the art +This survey of the state of the art demonstrates that several approaches address +the needs for modularity and incrementality raised in this report. However, none +of them fulfill these needs at the three levels of model operations, modeling +languages integration and megamodels that we identify as being required all at +once. Moreover, for certain individual aspects of Global Model Management, +solutions with adequate modularity and incrementality do not even exists yet +on their own. This work specifically targets these essential needs that have not +been sufficiently addressed yet. +16 + +4 +Extended Generalized Discrimination +Networks +In this chapter, we introduce a notion of extended Generalized Discrimination +Networks (eGDNs) and explain how the new formalism can be used as a language +for megamodels. +4.1 +Definition of eGDNs +In order to address shortcomings of current solutions and enable the modular +and incremental construction and execution of complex nets of model operations +such as model properties, model consistency operations, model transformations, +and model synchronization, we further generalize the idea of Generalized Dis- +crimination Networks [7] to extended Generalized Discrimination Networks. +Therefore, we introduce a generalized notion of GDN nodes and their inter- +faces. This enables the integration of model operations with side-effects and +allows a more flexible definition of queries in comparison to [7], which also +affords increased expressiveness. +An eGDN G = (O, S, E, s, t) is essentially a bipartite graph with two kinds of +nodes, slot nodes and operation nodes, where O is the set of operation nodes and +S is the set of slot nodes. Operation nodes can be connected to slot nodes and +vice-versa via edges from the set of edges E. The source and target functions +of edges are given by s : E → O ∪ S respectively t : E → O ∪ S. +Operation nodes represent model operations or building blocks thereof, that +is, suboperations. Slot nodes store information used by model operations and +their suboperations in the eGDN. Edges represent dependency relationships be- +tween operation and slot nodes, with the source of an edge representing the +required node and the target of the edge representing the dependent node. +An operation node depending on a slot nodes indicates that the correspond- +ing model operation uses information stored in that slot. A slot node having +a dependency on an operation node means that the operation node’s model +operation modifies the slot’s contents. +We denote the set of dependencies of a slot or operation node n in O ∪ S by +in(n) = {d ∈ O ∪ S|∃e ∈ E : s(e) = d ∧ t(e) = n}. Similarly, we denote the set +of dependent nodes of n by out(n) = {d ∈ O ∪ S|∃e ∈ E : s(e) = n ∧ t(e) = d}. +G is bipartite in the sense that ∀o ∈ O : in(o) ⊆ S ∧ out(o) ⊆ S and ∀s ∈ S : +in(s) ⊆ O ∧ out(s) ⊆ O. For an operation node o ∈ O, we also refer to the set +of slot nodes in(o) as the input slots of o and to the set of slot nodes out(o) as +the output slots of o. +A slot node s is always associated with a modeling language ML or an +ordered set of variables var = {v1, v2, ..., vk} and contains a model (typed +graph) of ML respectively a set of variable assignments for var. A variable +assignment for an ordered set of variables var = {v1, v2, ..., vk} is given by +a tuple in domV (v1) × ... × domV (vk), where dom(vi) denotes the domain of +variable vi, which can either be a set of nodes or edges from one or more +17 + +models or a set of primitives, e.g. +N. +We refer to the set of possible con- +tained assignment sets or models of s as the slot’s domain, which is given +by dom(s) = ML in case s is associated with a modeling language ML or +by dom(s) = P(domV (v1) × ... × domV (vk)) if s is associated with an or- +dered set of variables var = {v1, v2, ..., vk}. +Contents are then assigned to +an eGDN’s slots via a valuation function val : S → � +s∈S dom(s), such that +∀s ∈ S : val(s) ∈ dom(s). +In addition to regular models, we also allow model slots to contain linking +models. The only difference between a regular model and linking model is the +fact that a linking model’s set of vertices may reference vertices from other +regular and linking models as edge targets, thus allowing the establishment of +inter-model connections. Therefore, similarly to linking models, the metamodel +of a linking model, that is, the type graph of a linking model, may refer to +vertices from other type graphs as edge targets. +Regarding operation nodes, we further distinguish between query nodes, +transformation nodes, and mixed nodes. +Query nodes extract information from models and/or other queries’ results. +Therefore, a query node q may have an arbitrary number of input slots and +exactly one output slot. +q’s input slots may contain both models or sets of +variable assignments, whereas q′s output slot may only contain a set of variable +assignments. +Transformation nodes create or modify models based on models and/or query +results. Therefore, a transformation node t may have an arbitrary number of +input and output slots. +t’s input slots may contain both models or sets of +variable assignments, whereas t’s output slots may only contain models. +A mixed node x constitutes a combination of query and transformation nodes +and may have an arbitrary number of input and output slots, which may contain +both models or sets of variable assignments. +Each operation node o with input slots in(o) = {si1, ..., sik} is associated +with a semantics function γS : dom(si1) × ... × dom(sik) → P(F), where F +denotes the set of functions f : out(o) → � +so∈out(o) dom(so) such that ∀so ∈ +out(o) : f(so) ∈ dom(so). Essentially, the semantics function of an operation +node describes a consistency relationship between the operation’s input and +output slots. +To indicate that the contents of the slots adjacent to o are consistent with o’s +semantics function for a valuation function val, we write o.valid(val). Formally, +o.valid(val) ↔ ∃f ∈ γS(val(si1), ..., val(sik)) : ∀so ∈ out(o) : f(so) = val(so). +A valuation function val for an eGDN G = (O, S, E, s, t) is consistent with +G as a whole if it holds that ∀o ∈ O : o.valid(val). +4.2 +eGDNs as Megamodels +Since an eGDN encodes a network of model operations connecting a set of +potentially integrated models, it represents a megamodel. The definition of +eGDNs thus constitutes a language for megamodels. +18 + +Importantly, eGDNs allow the composition of model operations from nodes +that realize suboperations. In addition, they also allow hierarchical composition: +An eGDN (and therefore also a basic GDN or RETE net) can be interpreted as +an eGDN operation node. The input and output slots are given by the input +respectively output slots of its nodes that are connected to another operation +node of the parent eGDN. Any slots of the child eGDN without a connection +to another node of the parent eGDN can act as internal slots of the child and +do not have to be exposed to the parent. However, some such potential internal +slots may also be considered input or output slots if their contents are relevant +to human users. The semantics function of an operation node representing a +sub-eGDN is then implicitly defined by the semantics functions of that eGDN’s +own operation nodes. +In addition to (hierarchical) composability, eGDNs support modularity in +the sense that the semantics of an operation node regarding its output slots +directly depend only on the contents of its immediate input slots. +Thereby, +integrating additional operation nodes (along with additional slot nodes) into +an eGDN only requires appropriate wiring with the node’s input and output +slots, but is completely independent of any other operation nodes. Effectively, +slots thus act as interfaces between model operations. +eGDNs can also enable modularity at the model level by using the results +of query nodes, potentially along with transformation nodes for propagating +changes from the query results back to the base model, as model interfaces or +views. For instance, simple projection queries in combination with access re- +strictions can be employed to implement different visibilities for different roles +in a development process. +Alternatively, dedicated view models in conjunc- +tion with bidirectional model synchronization operations can similarly serve to +implement editable model views in the context of eGDNs. +Figure 2 shows our graphical notation for the visualization of eGDNs. Slot +nodes are depicted as rectangles and labelled “A” in the top right corner in the +case of assignment slots and “M” in the case of model slots. Model slots that +contain linking models are connected to the model slots containing the linked +models via dashed arrows for visual clarity. Operation nodes are visualized as +rectangles with rounded corners, with query nodes such as model properties +or model consistency checks labelled “Q”, transformation nodes such as model +transformations and model synchronizations labelled “T”, and mixed nodes such +as sub-eGDNs labelled “X” in the top right corner. In addition, all nodes are +labeled according to the schema ¡name¿:¡type¿. +Figure 3 shows an example eGDN that consists of three slot nodes and +two operation nodes and realizes a simple chain of model operations. A class +diagram stored in the leftmost slot is transformed into an abstract syntax graph +via a transformation node. Then, a query node extracts some information from +the created abstract syntax graph and makes the query result accessible via an +assignment slot. +19 + +x: +Mixed +X +t: +Trans. +T +Transformation Node +(e.g. model transformation, +model synchronization) +Mixed Node +(e.g. subnet) +q: +Query +Q +Query Node +(e.g. model property, +model consistency) +o: +Operation +Operation Node +(linked to slots) +a: +Assignment +A +m: +Model +M +Assignment Slot +Model Slot +m: +LinkingModel +M +Model Slot +(Linking Model) +Figure 2: Graphical notation for eGDNs +m1: +ClassDiagram +M +t: +Transform +T +q: +Query +Q +m2: +ASG +M +a: +Result +A +Figure 3: Simple example eGDN +5 +Incremental Execution of Extended General- +ized Discrimination Networks +In this chapter, we describe how eGDNs can be executed to restore consistency +in a network of models and model operations in reaction to external changes. +5.1 +Definitions regarding Incremental Execution +As a result of edit operations by a user, a model M in the model slot of an eGDN +can undergo changes. In this context, a change corresponds to the creation or +deletion of a vertex or an edge and is characterized by an atomic model delta +of one of four types: +• δV ++ is a single-element tuple (v), with v a vertex; applying δV ++ to M modifies +M into M ′ = (V M ∪ {v}, EM, sM, tM) +• δV +− is a single-element tuple (v), with v ∈ V M, applying δV +− to M modifies +M into M ′ = (V M \ {v}, EM, sM, tM) +• δE ++ is a tuple (e, s, t), with e an edge and s, t ∈ V M; applying δE ++ to M +modifies M into M ′ = (V M, EM ∪ {e}, sM ∪ {(e, s)}, tM ∪ {(e, t)}) +• δE +− is a single-element tuple (e), with e ∈ EM; applying δE +− to M modifies +M into M ′ = (V M, EM \ {e}, sM \ {(e, sM(e))}, tM \ {(e, tM(e))}) +20 + +Importantly, this notion of atomic deltas can also cover the case of changes +to attribute values in models in the form of typed attributed graphs [47]. In this +context, attributes can be modeled via dedicated vertices representing attribute +values and edges representing the assignment of these values to attributes of +regular vertices. +Note that we do not allow implicit deletion of edges. If a vertex is deleted, it +must not have any adjacent edges, that is, all adjacent edges have to be deleted +previously. Similarly, if an edge is created, adjacent vertices have to be present +in the model already. +Changes to the assignment set A in a slot s can similarly be described by +atomic slot deltas: +• δA ++ is a single-element tuple (a), where a ∈ dom(s); applying δA ++ to the +assignment set A modifies A into A′ = A ∪ {a} +• δA +− is a single-element tuple (a), where a ∈ dom(s); applying δA +− to the +assignment set A modifies A into A′ = A \ {a} +For a slot node s, we denote the set of all possible atomic deltas over dom(s) +by dom∆(s) and the set of all possible sequences of elements in dom∆(s) by +S(dom∆(s)). +Atomic deltas can be applied to a model or assignment set via an apply +procedure. We overload this procedure to also work with a sequence of atomic +deltas, in which case the procedure applies the individual deltas in the order +specified by the sequence. +We say that a sequence of atomic deltas ∆ is minimal for the contents of +a slot node s, iff for all possible contents v ∈ dom(s), it holds that ∄∆′ ∈ +S(dom∆(s)) : apply(v, ∆′) = apply(v, ∆), where we only consider equality of +graphs up to isomorphism. +To enable reacting to model changes with an eGDN G = (O, S, E, s, t), an +operation node o ∈ O with input slots in(o) = {si1, ..., sik} and output slots +out(o) = {so1, ..., sol} can be equipped with an update procedure. This pro- +cedure is parametrized with a valuation function for G and realizes a func- +tion γδ : dom(si1) × ... × dom(sik) × S(dom∆(si1)) × ... × S(dom∆(sik)) × +dom(so1) × ... × dom(sol) → F∆, with F∆ the set of functions f∆ : out(o) → +� +soi∈out(o) S(dom∆(soi)) such that ∀soi ∈ out(o) : f∆(soi) ∈ S(dom∆(soi)). +To store deltas to react to later, o is also extended by an array o.∆ that +caches sequences of atomic deltas for its input and output slots, which can +in practice be collected via a notification mechanism and the observer design +pattern [36]. +Calling o.update(val) with val a valuation function for G’s slots then yields +the value of γδ parametrized according to val and the cached sequences of deltas: +o.update(val) = γδ(val(si1), ..., val(sik), o.∆[si1], ..., o.∆[sik], val(so1), ..., val(sol)). +Intuitively, the update procedure of an operation node should produce a +sequence of deltas for the node’s output slots that update the contents of these +output slots to be consistent with the current contents of the operation node’s +21 + +input slots. Therefore, in addition to the contents of slots adajcent to o, an +update procedure may also consider additional information in the form of deltas +to input slots to enable a more efficient realization. +Formally, an update procedure update of operation node o with input slots +in(o) = {si1, ..., sik}, output slots out(o) = {so1, ..., sol}, and associated function +γδ is correct iff for parameters ∆1 ∈ S(dom∆(si1)), ..., ∆k ∈ S(dom∆(sik)), +vi1 ∈ dom(si1), ..., vik ∈ dom(sik), and vo1 ∈ dom(so1), ..., vl ∈ dom(sol), +∃f ∈ γS(vi1, ..., vik) : ∀soi ∈ out(o) : apply(voi, f∆(so)) = f(soi), +with f∆ = γδ(∆1, ..., ∆k, vi1, ..., vik, vo1, ..., vol) and ∀i ∈ [1, k], j ∈ [1, l] : +sii = soj → vii = voj. +In many cases, the efficient realization of an update procedure requires a +relaxed notion of correctness, which requires the contents of the output slot to +be consistent with the contents of the input slots before the application of the +deltas according to o’s semantics function. In the following, we will refer to this +relaxed notion of correctness as conditional correctness, which is formally given +by +∃v′ +i1 ∈ dom(si1), ..., v′ +ik ∈ dom(sik) : +(apply(v′ +i1, ∆1) = vi1 ∧ ... ∧ apply(v′ +ik, ∆k) = vik∧ +∃f ′ ∈ γS(v′ +i1, ..., v′ +ik) : +(∀sii ∈ in(o) ∩ out(o) : v′ +ii = f ′(sii)∧ +∀soi ∈ out(o) \ in(o) : voi = f ′(soi))) +→ +∃f ∈ γS(vi1, ..., vik) : ∀soi ∈ out(o) : apply(voi, f∆(so)) = f(soi), +with f∆ = γδ(∆1, ..., ∆k, vi1, ..., vik, vo1, ..., vol) and ∀i ∈ [1, k], j ∈ [1, l] : +sii = soj → vii = voj. +We say that the realization of an update procedure of an operation node o +is fully incremental iff for a valuation function val and cached deltas ∆1, ..., ∆k +with � +i∈{1,...,k} |∆i| = 1, that is, a single atomic delta as an input, (i) the +runtime complexity is in O(|∆o|), with ∆o = � +so∈out(o) o.update(val)(so), and +(ii) the produced sets of deltas for each output slot are minimal. +In some cases, as with bidirectional or in-place model transformations, op- +eration nodes may be connected to a slot via both an incoming and an outgoing +edge, making such a slot simultaneously an input and output slot to the same +operation node. Such an operation node may as a result exhibit recursive be- +havior, since an application of its update procedure can also change the contents +of the operation node’s input slots and thus necessitate further calls to update +to restore consistency. In this context, we call an update procedure of an oper- +ation node o is non-recursive, if, after one execution of o’s update function and +22 + +subsequent application of the resulting deltas to o’s output slot values, a second +execution with updated slot values never yields any new deltas. +Formally, an update procedure of an operation node o with input slots +in(o) = {si1, ..., sik} and output slots out(o) = {so1, ..., sol}, is non-recursive, +if for any possible parametrization vi1 ∈ dom(si1), ..., vik ∈ dom(sik), ∆1 ∈ +S(dom∆(si1)), ..., ∆k ∈ S(dom∆(sik)), and vo1 ∈ dom(so1), ..., vol ∈ dom(sol), it +holds that +∀so ∈ out(o) : γδ(∆′ +1, ..., ∆′ +k, v′ +i1, ..., v′ +ik, v′ +o1, ..., v′ +ol)(so) = ∅, +where +∆′ +i = +� +f∆(si) +if sii ∈ out(o) +∆i +otherwise +and +v′ +ii = +� +apply(vii, f∆(si)) +if sii ∈ out(o) +vii +otherwise +and +v′ +oi = apply(voi, f∆(so1)), +with f∆ = γδ(∆1, ..., ∆k, vi1, ..., vik, vo1, ..., vol). +The potential update directions of an update procedure of operation node o +for a set of input slots Si ⊆ in(o) are given by o.dir∆(o, Si), where for a slot +so ∈ out(o), +so ∈ o.dir∆(o, Si) ↔∃∆1 ∈ S(dom∆(si1)), ..., ∆k ∈ S(dom∆(sik)), +vi1 ∈ dom(si1), ..., vik ∈ dom(sik), +vo1 ∈ dom(so1), ..., vol ∈ dom(sol) : +∀sii ∈ in(o) \ Si : ∆i = ∅∧ +γδ(∆1, ..., ∆k, vi1, ..., vik, vo1, ..., vol)(so) ̸= ∅ +Intuitively, o.dir∆(o, Si) thus denotes the subset of output slots for which +o’s update procedure may generate deltas if the contents of at most the input +slots in Si have changed. +A function dir∆ for potential update directions is monotonic by definition +in the sense that ∀Si1, Si2 ⊆ in(o) : Si1 ⊆ Si2 → o.dir∆(o, Si1) ⊆ o.dir∆(o, Si2). +We say that dir∆ is union monotonic if it furthermore holds that ∀Si1, Si2 ⊆ +in(o) : o.dir∆(Si1) ∪ o.dir∆(Si2) = o.dir∆(Si1 ∪ Si2). +In the following, we present algorithms for the incremental execution of +an eGDN based on the update procedures of its operation nodes. For these +algorithms, we assume that deltas cached in the input eGDN are consistent in +the sense that they correspond to a modification from slot contents that were +23 + +consistent with the semantics functions of all operations in the eGDN to the +current contents. Intuitively, this assumption simply implies that the presented +algorithms can only produce consistent slot contents if the slot contents were +previously consistent at some point and all changes since then have been tracked +and cached in the eGDN. +5.2 +Incremental Execution with Guaranteed Termination +Given a correct update function for each operation node, an input eGDN G = +(O, S, E, s, t) can be executed incrementally in the context of a valuation func- +tion val via Algorithm 1. Therefore, Algorithm 1 first derives an ordering of +G’s operation nodes and then updates the val function by executing the nodes’ +update functions, applying the resulting deltas to the appropriate slots, and +updating the cached deltas. +Importantly, the employed ordering has to guarantee correct results in the +sense that the contents of G’s slots after the execution must be consistent with +the semantics functions of all of its operation nodes, that is, it must hold that +∀o ∈ O : o.valid(val). +If G takes the form of a directed acyclic graph and operation nodes do not +share output slots, such an ordering can be obtained by simply sorting G’s +operation nodes topologically. However, requiring DAG structure represents a +substantial restriction, as it effectively prohibits bidirectional transformations +where some input slots are also output slots. Moreover, the assumption regard- +ing the complete absence of shared output slots, while required to prevent over- +writing of operation’s results, is another obstacle to realizing several desirable +use cases, for instance those involving chains of bidirectional transformations. +Based on the properties of an eGDN’s operation nodes with respect to non- +recursiveness and potential update directions, an appropriate order can also be +found for certain cyclical eGDNs, with a relaxed assumption regarding shared +output slots. Algorithm 2 represents an analysis for an eGDN G that contains +only nodes with non-recursive update procedures and a set of slots Si with ini- +tially modified contents. If successful, the algorithm returns an execution order +that can be used instead of the topological ordering in Algorithm 1. Impor- +tantly, the computed ordering still yields a valuation function that is consistent +with all operations’ semantics. +The algorithm first creates an array C with one cell per operation node in O +and initializes it with empty sets. It also initializes a queue Q with all operation +nodes that are connected to a slot in Si and, for each such operation node, +stores the set of its input slots that are also in Si in the corresponding cell in +C. Then, a slightly modified breadth-first search is performed over the eGDN +structure using the initialized queue Q to essentially simulate an execution of +G without concrete inputs. +Therefore, the procedure loops until Q is empty. In each loop execution, the +first operation node o in Q is dequeued. Then, all output slot nodes for which +deltas could be produced due to the execution of o′s update procedure So are +obtained based on o’s potential update directions and the set of slots that might +24 + +Procedure ExecuteIncrementalDAG(G = (O, S, E, s, t), val) +Input +: G: The eGDN +val: A valuation function for G’s slots +1 +D ← FindValidUpdateOrder(O, {s ∈ S|∃o ∈ O : o.∆[s] ̸= ∅}); +2 +if D ̸= null then +3 +foreach o ∈ D do +4 +∆o ← o.update(val); +5 +foreach so ∈ out(o) do +6 +val(so) ← apply(val(so), ∆o(so)); +7 +foreach o′ ∈ out(so) do +8 +o′.∆[so] ∪ ∆o; +9 +end +10 +end +11 +foreach s ∈ in(s) ∪ out(s) do +12 +o.∆[s] ← ∅; +13 +end +14 +end +15 +end +Algorithm 1: Incremental algorithm for executing an eGDN based on an +ordering of its operation nodes +currently contain unhandled deltas, which is retrieved from C. Afterwards, all +operation nodes o′ connected to a slot in So are added to Q if they are not +yet contained. Also, the set of o’s input slots with potentially unhandled deltas +stored in C is updated based on So. An exception is made for the currently +considered node o, which is never added to the queue again and whose set of +input slots with potentially unhandled deltas is reset to the empty set, exploiting +the assumption that all update procedures in the eGDN are non-recursive. +During execution, the algorithm keeps track of the dependencies between G’s +operations in a trigger graph GT . Execution aborts by returning null as soon as +a cyclical dependency is detected, which may indicate a potential infinite loop +in G’s execution for the initially populated slots Si. This also guarantees that +after a full execution of the loop in line 10, GT is a DAG. +Finally, a topological ordering of GT , is returned as a possible canonic execu- +tion order that, under the mentioned assumptions, produces a valuation function +for the input eGDN’s slots that is consistent with the semantics functions of all +of the eGDN’s operation nodes. +While the presented algorithm is formulated to handle incremental changes +to a network of models and model operations, the batch case that requires an +initial execution of model operations to derive corresponding query results and +transformed models for an initial set of existing models can be handled in a +straightforward manner. Therefore, an incremental construction of the initially +existing models can be emulated by deriving trivial sequences of corresponding +creation operations, which can act as the starting point for the algorithm. This +25 + +Procedure FindValidUpdateOrder(G = (O, S, E, s, t), Si) +Input +: G: The eGDN +Si: The set of initially changed slots +1 +C ← new Array(|O|); +2 +C.init(∅); +3 +Q ← new Queue; +4 +GT = new Graph; +5 +foreach o ∈ in(Si) ∪ out(Si) do +6 +Q.enqueue(o); +7 +C[o] ← Si ∩ in(o); +8 +end +9 +GT .addV ertices(Q); +10 +while ¬Q.isEmpty() do +11 +o ← Q.dequeue(); +12 +So ← o.dir∆(o, C[o]); +13 +Oo ← out(So) ∪ in(So) \ {o}; +14 +foreach o′ ∈ Oo do +15 +if ¬o′ ∈ Q then +16 +Q.enqueue(o′); +17 +end +18 +C[o′] ← C[o′] ∪ (So ∩ in(o′)); +19 +GT .addV ertexIfNotExists(o′); +20 +GT .createEdgeIfNotExists(o, o′); +21 +if GT .hasCycle() then +22 +return null; +23 +end +24 +end +25 +C[o] ← ∅; +26 +end +27 +return SortTopologically(GT ); +Algorithm 2: Static analysis algorithm for finding an eGDN update order +26 + +only requires the assumption that the case where all slots of an eGDN are empty +constitutes a consistent valuation regarding the semantics of all of the eGDN’s +operations, which seems reasonable. The additional assumption is essentially +required to satisfy the rerquirement regarding consistency of initially cached +deltas with the current state. +Termination +By including the additional termination criterion in the loop in line 10 of Algo- +rithm 2 that requires the constructed dependency graph to be acyclic, Algorithm +2 is guaranteed to terminate. +Theorem 1. Algorithm 2 always terminates. +Proof. Except for the loop in line 10, all loops only iterate over finite sets, and all +individual operations always terminate. The loop in line 10 also always termi- +nates due to the termination criterion regarding cyclical dependencies between +the eGDN’s operation nodes: Since one operation node is removed from Q in +each loop iteration, termination is only threatened if operation nodes keep get- +ting added to Q. Since there is only a finite number of operation nodes, infinite +behavior can only occur as a result of cycles in the modified breadth-first search. +However, such cycles are detected via GT and immediately lead to abortion of +the execution. +Consequently, Algorithm 1 is also guaranteed to terminate if the execution +of the input eGDN’s update procedures always terminates. +Theorem 2. For an input eGDN G, Algorithm 1 always terminates if the +update procedures of G’s operation nodes always terminate. +Proof. According to Theorem 1, Algorithm 2 always terminates by either abort- +ing or returning a sequence of operation nodes. Such a sequence being returned +implies that the sequence is finite. The loop in line 3 is thus only executed for +finitely many iterations. Since all other loops only iterate over finite sets and +all individual operations always terminate due to the assumption regarding G’s +update procedures, Algorithm 1 always terminates. +Correctness +The following theorem states the correctness of a canonic execution order re- +sulting from an execution of Algorithm 2 for the case that all dir∆ functions +are union monotonic. +Theorem 3. For inputs G = (O, S, E, s, t) and val, if all update procedures in +G are correct and non-recursive, all dir∆ functions in G are union-monotonic, +and if the valuation function before the application of the deltas cached in G +was consistent with the semantics of G’s operation nodes, Algorithm 1 aborts or +produces a final valuation function val such that ∀o ∈ O : o.valid(val). +27 + +Proof. If Algorithm 1 does not abort, a canonic execution order R for G’s op- +eration nodes has been generated by topologically sorting the resulting directed +acyclic dependency graph GT of a terminating execution of Algorithm 2. +Due to the non-recursiveness of G’s operation nodes, we know that after +executing an operation node o via Algorithm 1, it holds that o.valid(val). Thus, +for an operation node o, ¬o.valid(val) can only hold after executing the entire +sequence R if there exists some operation node o’ that comes after o in R and +that changes the contents of a slot adjacent to o or if o /∈ R. Considering that +all operation nodes that have an adjacent slot with initially modified contents +are initially added to GT , the algorithm has terminated, and that prior to the +cached modifications of G’s slots, slot contents were consistent with all operation +nodes’ semantics, for a node o /∈ R, ¬o.valid(val) can also only hold if there is +some operation node o′ ∈ R that changes the contents of a slot s adjacent to o. +In either case, we know that there cannot exist an edge from o′ to o in +GT , because o′ either comes after o in the topological ordering or because the +addition of such an edge would have caused o to be added to GT and conse- +quently R. This means that, due to the definition of o′.dir∆ and because of +the assumed union monotonicity, there must be a slot s′ with o′.∆[s] ̸= ∅ and +s ∈ o′.dir∆({s′}) before executing o′ that was never in the set of slots C[o′] +when o′ was dequeued in line 11 of Algorithm 2. Since slots are only removed +from C[o′] when o′ is dequeued and corresponding edges are added, we know +that s′ /∈ Si and thus o′.∆[s′] = ∅ at the start of Algorithm 1, as otherwise, o′ +would have been added to Q and the edge between o′ and o would eventually +have been created. +There hence must be a node o′′ that comes before o′ in R that modified +the contents of s′. Also for o′′, there must be a slot s′′ with o′′.∆[s′′] ̸= ∅ and +s′ ∈ o′′.dir∆({s′′}) before executing o′′ that was never in C[o′′] whenever o′′ +was dequeued (because otherwise, the edge between o′ and o would have been +created eventually). Therefore, again, there must be an operation node before +o′′ in R that modified the contents of s′′ and for which the same constraints +apply as for o′′. Ultimately, this implies that for the first operation node in the +sequence, there must be a predecessor that changes the contents of some slot +node, which is obviously a contradiction. +Hence, there cannot be an operation node in R whose execution changes the +contents of a slot adjacent to a previous operation node in R or an operation +node not contained in R. Consequently, we know that after executing R, ∀o ∈ +O : o.valid(val). +If the eGDN’s update functions are only conditionally correct, an additional +constraint has to be introduced regarding eGDN structure to guarantee correct- +ness. Namely, operation nodes may not share output slots if the output slots +are not also input slots of all sharing operation nodes, and output slots of a +node that are not simultaneously input slots of the same node may not have +their contents modified by users. +Intuitively, these conditions impose the restriction on the eGDN structure +that the contents of an operation node’s output slot may not be modified by +28 + +another operation node or a user without that operation node being able to pick +up on and handle the changes. +Corollary 1. Assuming that ∀o1, o2 ∈ O : o1 ̸= o2 → ∀s ∈ out(o1) ∩ out(o2) : +s ∈ in(o1) ∧ s ∈ in(o2) and ∀o ∈ O : ∀s ∈ out(o) : o.∆[s] = ∅, for inputs G = +(O, S, E, s, t) and val, if all update procedures in G are conditionally correct and +non-recursive, all dir∆ functions in G are union-monotonic, and if the valuation +function before the deltas cached in G was consistent with the semantics of G’s +operation nodes, Algorithm 1 aborts or produces a final valuation function val +such that ∀o ∈ O : o.valid(val). +Proof. From the additional assumptions regarding output slots of G’s operation +nodes it follows directly that the condition in the definition of conditional cor- +rectness is never violated. Thus, the statement from Theorem 3 also applies for +the case of conditionally correct update functions. +Notably, the order in which operation nodes are added to the queue Q in +lines 5 and 14 of Algorithm 2 is undefined. Since the order of operation nodes +in Q affects the behavior of the algorithm, this might mean that Algorithm 2 is +ultimately not deterministic. +We can however show that, if Algorithm 2 does not abort due to cycles +in GT , the final dependency graph GT is uniquely defined, independently of +the order in which operation nodes are added to Q. Thus, the only remaining +nondeterminism in Algorithm 2 affecting the result stems from the topological +sorting at the end of the algorithm, which is an inherently nondeterministic +operation. +Theorem 4. For inputs G = (O, S, E, s, t) and Si, the dependency graph GT +after a full execution of the loop in line 10 of Algorithm 2 is uniquely defined +up to isomorphism if all dir∆ functions in G are union monotonic. +Proof. The set of vertices initially added to GT is uniquely determined by +in(Si) ∪ out(Si). Since additional vertices are only ever added in conjunction +with the creation of an edge, the set of vertices added during the execution of +the loop in line 10 is determined by the set of added edges. +To show the unique determination of added edges by the algorithm’s inputs, +we show that in a terminating execution of the loop, the initial set Si in con- +junction with the eGDN G uniquely determines a set of pairs of operation nodes +(o1, o2), between which directed edges are created in GT . +Si uniquely determines the set of operation nodes OQ = in(Si) ∪ out(Si) +that is initially added to Q. +For each of these operation nodes oQ ∈ OQ, +due to the monotonicity of oQ.dir∆ and because slots are only removed from +C[oQ] after oQ has been dequeued and processed, at least the edges for pairs +edgesS(oQ, Si) = {(oQ, oT )|oT ∈ out(So) ∪ in(So) \ {oQ}} are added to GT , +where So = oQ.dir∆(Si ∩ in(oQ)) when oQ is dequeued. +According to the +assumption regarding union monotonicity, we can also write edgesS(oQ, Si) = +edges∅(oQ) ∪ � +si∈Si∩in(oQ) edgesN(oQ, si), with edges∅(oQ) = {(oQ, oT )|oT ∈ +29 + +out(oQ.dir∆(∅)) ∪ in(oQ.dir∆(∅)) \ {oQ}} and edgesN(oQ, si) = {(oQ, oT )|oT ∈ +out(oQ.dir∆(in(oQ) ∩ {si})) ∪ in(oQ.dir∆(in(oQ) ∩ {si})) \ {oQ}}. +In addition, the modification of C and Q that takes place for each dequeued +oQ ∈ OQ may cause the addition of further edges down the line. Specifically, for +each so ∈ oQ.dir∆({si}) and each oT ∈ out(so)∪in(so)\{oQ}, oT , if not already +contained, is added to Q and subsequently handled in the same way as oQ, with +si guaranteed to be in C[oT ] at that moment. This will cause the addition of all +edges corresponding to the pairs edgesN(oT , so) and again trigger the addition +of further edges. Due to the monotonicity of oT .dir∆ and because slots are only +removed from C[oT ] when oT is dequeued, the addition of these edges happens +independently from any other modifications to C[oT ] that might be made in the +meantime. Furthermore, due to the assumption regarding union monotonicity +of oT .dir∆, a combination of modifications of C[oT ] cannot yield any additional +edges compared to what is yielded for the individual members of C[oT ]. +Because neither can C[oT ] be modified in any other way, nor can edges be +added to GT in any other way, the set of pairs of operation nodes (o1, o2) between +which directed edges are created in GT is given by the function edges(Si) = +� +oQ∈in(Si)∪out(Si)(edges∅(oQ)∪� +si∈Si∩in(oQ) edgesR(oQ, si)), with edgesR(oQ, si) = +edgesN(oQ, si)∪� +oT ∈OT ))\{oQ} +� +so∈oQ.dir∆(in(oQ)∩{si} edgesR(oT , so), where OT +is given by OT = out(oQ.dir∆(in(oQ) ∩ {si})) ∪ in(oQ.dir∆(in(oQ) ∩ {si}. +The loop terminating due to Q becoming empty implies that all nodes ever +added to Q have been processed and hence all corresponding edges have been +added to GT . Since it is ensured that for each pair of operation nodes (o1, o2), +only one corresponding edge is added, we know that regardless of the concrete +processing order, GT always contains exactly one directed edge for each pair +(o1, o2) ∈ edges(Si). +Since the set of added vertices is uniquely determined by the set of added +edges and each vertex can only be added once, the set of GT ’s vertices is uniquely +defined for inputs G and Si. +The graph GT at the end of a full execution of the loop in line 10 of Algorithm +2 is hence uniquely defined for inputs G and Si, regardless of the order in which +operation nodes are added to Q in lines 5 and 14. +The fact that GT is uniquely defined by the inputs G and Si also implies +that if an execution of Algorithm 1 terminates without aborting, so does any +possible execution for the same inputs. +Theorem 5. An execution of the loop in line 10 of Algorithm 1 terminates +without aborting for inputs G = (O, S, E, s, t) and Si if and only if any other +execution for the same inputs also terminates without aborting. +Proof. According to Theorem 1, the loop in line 10 of Algorithm 1 always ter- +minates, either because of a violation of the looping condition or because the +loop aborts. Since the loop aborts if and only if a cycle is detected in GT at any +point and edges are never removed from GT , it follows that the loop terminates +without aborting if and only if the set of edges added to GT during the loop +execution does not form cycles. Since the set of edges added to GT during the +30 + +loop execution is functionally determined by only the inputs G and Si, it hence +follows that, if an execution of the loop terminates without aborting for G and +Si, any execution with the same inputs will also terminate without aborting. +Furthermore, we can show that if there exists an execution sequence for +G that guarantees correct results in the worst case and that executes every +operation node at most once, Algorithm 2 finds such a sequence. +Theorem 6. For an input eGDN G = (O, S, E, s, t) with correct and non- +recursive update procedures with union monotonic dir∆ functions and a set +of slots Si ⊆ S with initially modified contents for a valuation function val, +assuming that +1. for any operation node o1 ∈ O, for any execution of o1.update(val′) with +a valuation function val′ and deltas for input slots S∆, it holds that ∀so ∈ +o1.dir∆(S∆) : o1.update(val′)(so) ̸= ∅, +2. for a second node o2 ∈ O with o1 ̸= o2, it holds that ∃so ∈ out(o1)∩in(o2) : +o1.update(val′)(so) ̸= ∅ → ¬o2.valid(val′′), where for s ∈ S +val′′(s) = +� +apply(val′(s), o1.update(val′)(s)) +if s ∈ out(o1) ∩ in(o2) +val′(s) +otherwise +(1) +and +3. it holds that ∀o ∈ in(Si) ∪ out(Si) : ¬o.valid(val), +if there exists a sequence that guarantees a correct resulting valuation function if +executed via Algorithm 1 and that only contains each node o ∈ O once, Algorithm +2 returns such a sequence. +Proof. Under the given assumptions, the set of edges in GT created by Algo- +rithm before termination or abortion represents a subset of all relations between +pairs of operation nodes (o1, o2), where o1’s update procedure has to be executed +at least once to produce a correct final valuation function and that execution +modifies the contents of a slot adjacent to o2, necessitating the subsequent exe- +cution of o2 according to assumption (2). +This is due to the fact that, to restore consistency, all operation nodes in +o ∈ in(Si) ∪ out(Si) have to be executed at least once according to assumptions +(2) and (3). All these operation nodes o1 are initially added to the queue Q in +Algorithm 2. Each execution of an operation node o1, according to assumption +(1), modifies all slots in o1.dir∆(Si ∩ in(o1)), which necessitates a subsequent +execution of all operation nodes o2 ∈ in(o1.dir∆(Si ∩in(o1)))∪out(o1.dir∆(Si ∩ +in(o1))) according to assumption (2). Algorithm 2 creates edges for all these +pairs (o1, o2) when o1 is dequeued. +The subsequent execution of any operation node o2 similarly necessitates the +execution of all nodes o3 ∈ in(o2.dir∆(Si ∩ in(o2))) ∪ out(o2.dir∆(Si ∩ in(o2))), +which is also reflected by the edges created in Algorithm 2 when o2 is dequeued, +31 + +and so on. Since the algorithm creates no additional edges due to the assumption +regarding union monotonicity of the dir∆ functions, all edges in GT represent +such necessary relationships on the ordering of operation nodes5. +Since Algorithm 2 always produces a correct sequence of operation nodes +if GT is acyclic, we can assume that in the case where the algorithm does not +produce an ordering, there is at least one cycle in GT . There hence cannot exist +a sequence of the operation nodes involved in this cycle where each node is only +contained once and each node is executed at least once after its predecessor in +the cycle. Thus, by contraposition it follows that, if there exists a sequence of +operation nodes that guarantees correct results and where each operation node +is only contained once, Algorithm 2 finds such a sequence. +Note that there may be finite orders of operation node executions that guar- +antee correct results based on the assumptions in Theorem 6 that are not found +by Algorithm 2. However, these orders require that at least one operation node +is executed at least twice. +5.3 +Incremental Execution of Arbitrary eGDNs +If the eGDN is not a DAG and no suitable ordering of its operation nodes can +be found via Algorithm 2, incremental execution can instead be achieved via a +simple fixpoint iteration as in Algorithm 3. +Algorithm 3 first initializes the set of operation nodes that require execution +D with the set of all operation nodes in the input eGDN for which there are +changes in one of the node’s input or output slots. Then, the algorithm iterates +until a fixpoint is reached. +Therefore, a set of operation nodes that will require execution in the next +iteration Dn is initialized with the empty set. Afterwards, for each operation +node o that is due for execution in the current iteration, that node is removed +from the set D. Then, o’s update procedure is called to compute a set of changes +to the contents of o’s output slots to make them consistent with the semantics +of o. +For each output slot so of o that update has computed changes for, these +changes are subsequently applied and appropriately registered at each operation +node o′ for which so is an input slot. If any such o′ is not still due for execution in +the current iteration, it is marked for execution in the next iteration by adding +it to Dn. Operation nodes for which so is an output slot are similarly marked +for execution. Finally, after all operation nodes in D have been considered, Dn +replaces D and a new iteration starts if Dn is not empty. +Analogously to Algorithm 1, Algorithm 3 can handle the batch case of an +initial eGDN execution for existing models by encoding such existing models as +sequences of element creations. +5As a side note, since operation nodes can be dequeued/executed with different sets of +potentially modified input slots, an edge between nodes (o1, o2) in GT does not necessarily +mean that o2 has to be executed after any execution of o1, but only that such a subsequent +execution is necessary at least once. +32 + +Procedure ExecuteIncremental(G = (O, S, E, s, t), val) +Input +: G: The eGDN +val: A valuation function for G’s slots +1 +D ← {o ∈ O|∃s ∈ in(o) ∪ out(o) : o.∆[s] ̸= ∅}; +2 +while D ̸= ∅ do +3 +Dn ← ∅; +4 +foreach o ∈ D do +5 +D ← D \ {o}; +6 +∆o ← o.update(val); +7 +foreach s ∈ in(s) ∪ out(s) do +8 +o.∆[s] ← ∅; +9 +end +10 +foreach so ∈ out(o) do +11 +if ∆o(so) ̸= ∅ then +12 +val(so) ← apply(val(so), ∆o(so)); +13 +foreach o′ ∈ out(so) do +14 +o′.∆[so] ∪ ∆o; +15 +if o′ /∈ D then +16 +Dn ← Dn ∪ {o′}; +17 +end +18 +end +19 +foreach o′ ∈ in(so) do +20 +if o′ ̸= o ∧ o′ /∈ D then +21 +Dn ← Dn ∪ {o′}; +22 +end +23 +end +24 +end +25 +end +26 +end +27 +D ← Dn; +28 +end +Algorithm 3: Incremental algorithm for eGDN execution +33 + +Termination +In contrast to Algorithm 1, Algorithm 3 is not guaranteed to terminate, since +cyclical transitive dependencies of operation nodes may cause infinite cycles of +changes to the contents of some slot node. Without restricting developers in +what kinds of eGDNs they are allowed to specify, this problem is inevitable. +In practice however, termination of networks of model operations like eGDNs +can be achieved despite the presence of cyclical structures. In some cases for +instance, cycles at the network level do not necessarily correspond to actual +cyclical dependencies of model operations if the involved model operations only +affect distinct parts of slot contents, such as elements of certain, distinct types. +In some cases, a restructuring of the eGDN may remove cycles at the struc- +tural level while preserving semantics, for instance by converting in-place model +transformations without an effective reflexive dependency into a model trans- +formation with distinct input and output models. +Moreover, cycles of model operations may exhibit monotonic behavior, for +instance by deleting certain elements in each iteration that are never recreated, +thus guaranteeing convergence. Ultimately however, it remains the responsibil- +ity of the developers to create networks of model operations that do not lead to +infinite loops in execution. +Correctness +If Algorithm 3 terminates, the resulting valuation function is guaranteed to be +consistent with the semantics of all operation nodes in the input eGDN. +Theorem 7. For inputs G = (O, S, E, s, t) and val, if Algorithm 3 terminates, +all employed update procedures are correct and non-recursive, and if the valua- +tion function before the application of the deltas cached in G was consistent with +the semantics of G’s operation nodes, the algorithm produces a final valuation +function val such that ∀o ∈ O : o.valid(val). +Proof. We show that the invariant (1) ∀o ∈ O : ¬o.valid(val) → o ∈ D holds +for the loop in line 2 via induction over the number of loop iterations. +The base case for invariant (1) holds due to the initialization of D and the +assumption regarding the initial cached deltas and previous valuation function. +To show the induction step for invariant (1), we first show that under the +induction assumption, the invariant (2) ∀o ∈ O : ¬o.valid(val) → o ∈ D ∪ Dn +holds for the loop in line 10. This can also be done via induction. +The base case for invariant (2) holds due to the induction assumption of (1). +The induction step holds for invariant (2) since in each iteration of the +inner loop, only one operation node o is executed via its update procedure +and removed from D, updating val and the cached deltas in the process. If the +execution of o does not change the contents of one of its own input slots, we know +that afterwards, o.valid(val) due to the assumption regarding correctness of +update procedures and because the cached deltas are always updated correctly. +Otherwise, o is added to Dn in the loop in line 13. The loops in line 13 and 19 +34 + +also add all operation nodes o′ to Dn for which the result of o′.valid(val) may +have been impacted by the update to val. Thus, given the induction assumption, +at the end of the loop in line 10, we again have ∀o ∈ O : ¬o.valid(val) → o ∈ +D ∪ Dn and hence the induction step holds. +Since at the end of the loop in line 10, D = ∅, we know that ∀o ∈ O : +¬o.valid(val) → o ∈ Dn. Because at the end of the iteration of the loop in line +2, the set D is replaced by Dn, the induction step for (1) holds. +Since the loop in line 2 is only left when D = ∅ after the replacement with +Dn, we know that, if the algorithm terminates, ∀o ∈ O : o.valid(val). +Similar to Algorithm 1, the algorithm also yields correct results if all em- +ployed update procedures are at least conditionally correct, the eGDN’s nodes +do not share output slots that are not also input slots to all sharing nodes, and +there are no deltas for an output slot of a node that is not simultaneously an +input slot. +Corollary 2. Assuming that ∀o1, o2 ∈ O : o1 ̸= o2 → ∀s ∈ out(o1) ∩ out(o2) : +s ∈ in(o1) ∧ s ∈ in(o2) and ∀o ∈ O : ∀s ∈ out(o) : o.∆[s] = ∅, for inputs G = +(O, S, E, s, t) and val, if Algorithm 3 terminates, all employed update procedures +are conditionally correct and non-recursive, and if the valuation function before +the deltas cached in G was consistent with the semantics of G’s operation nodes, +it produces a final valuation function val such that ∀o ∈ O : o.valid(val). +Proof. From the additional assumptions regarding output slots of G’s operation +nodes, it follows directly that the condition in the definition of conditional cor- +rectness is never violated. Thus, the statement from Theorem 7 also applies for +the case of conditionally correct update functions. +5.4 +Development with eGDNs +Since Algorithm 2 considers only the eGDN structure and no concrete slot con- +tents, it can be employed as a tool for statically analyzing eGDNs. In particular, +via the algorithm, configurations of slots with modified contents can be analyzed +regarding termination of a corresponding eGDN execution. For instance, the +algorithm can be used to check whether termination is guaranteed if a specific +individual model is modified. +If this is the case for all user-editable models, a conservative approach that +always guarantees terminating eGDN executions and correct results while avoid- +ing the exponential effort of executing the analysis for every combination of user- +editable models would be enforcing a direct propagation policy. Under this +policy, after modifying a single model, the corresponding changes would imme- +diately be propagated to restore consistency. Only after that, the modification +of a different model would be permitted. +Furthermore, Algorithm 2 can be adapted to return the set of slots closure∆(Si) +that may be automatically modified by eGDN operations if the eGDN were to +be executed via Algorithm 1 with initially modified slots Si. This enables col- +laborative development of a network of models managed via an eGDN with +35 + +guaranteed termination and conflict-free consistency restoration via a propa- +gation closure locking policy. For a set of already modified slots S∆, this +policy would only allow modification of the contents of another slot s if, for +the set S∆ ∪ {s}, Algorithm 2 produces an execution order. Furthermore, to +guarantee that no user edits are overwritten, the policy would check whether +S∆ ∪ {s} ∩ closure∆(S∆ ∪ {s}) = ∅. Note that the restrictions of this policy +would also apply in the case where the same user wants to edit the contents of +multiple slots. +Since Algorithm 3 does not guarantee termination, careful consideration is +required if an eGDN cannot be executed via Algorithm 1. However, if developers +are confident that their eGDN is guaranteed to terminate despite cyclical de- +pendencies at the structural level, Algorithm 3 can be used as a fallback option +for eGDN execution. +The presented algorithms also enable the treatment of sub-eGDNs as oper- +ation nodes of a parent eGDN, as they essentially provide a realization of the +required update procedure. +6 +Implementation +We have prototypically implemented a number of concrete example operation +node types for the construction of eGDNs for usage in the context of the Eclipse +Modeling Framework (EMF) [26]. In addition to listing the implemented oper- +ations’ names, Table 2 also provides brief descriptions of their behavior. Table +3 characterizes our implementations in terms of the properties defined in this +report. +Non-recursiveness: The update procedure of TGG Snychronisation oper- +ations is non-recursive if the slots containing source, target, and correspondence +model are distinct. The non-recursiveness of composite nodes depends on the +exact composition of the sub-eGDN. All other nodes’ update procedures are +only guaranteed to be non-recursive if their input and output slots are distinct. +The checkmark symbol ✓indicates non-recursive update procedures under this +assumption. +Potential Population/Update Directions: +The potential update di- +rections of the TGG Synchronization (↔) can be characterized as follows (under +the assumption of distinct slots for source, target, and correspondence model): +If the set of considered input slots is empty, no modifications will be made to +the contents of any output slot. If the set of considered input slots contains only +the source model, the operation will only modify the target model and corre- +spondence model and vice-versa. In all other cases, all models may be modified. +The potential update directions of composite nodes are determined by the exact +structure of the sub-eGDN. All other nodes may modify the contents of all of +their output slots for any set of considered input slots. The potential update +direction function dir∆ of all example nodes is union monotonic. +Correctness: The update procedures of all operation implementations are +only conditionally correct. Effectively, this means that operations may not share +36 + +output slots and no user edits are allowed to output slots of operation nodes, +unless the shared or edited output slot is also simultaneously an input slot of +the concerned operation nodes. +Incrementality: +The checkmark symbol ✓indicates a fully incremental +update procedure under the assumption of ideal data structures. Also opera- +tions which are listed as not fully incremental support incremental execution +to some extent. The degree of incrementality depends on the operation and its +concrete inputs. Naturally, our implementation of the Expression node is only +fully incremental if the evaluation of the considered expression has a runtime +complexity in O(1). In the case of the Pattern Matching and TGG Synchroniza- +tion node, a fully incremental execution can be achieved for certain input models +and patterns respectively TGGs. The degree of incrementality of the execution +of an eGDN or sub-eGDN depends on which slots are designated the eGDN’s +interface slots, as well as the contained operation nodes and their composition. +While the Group Expression node also has a partially incremental update pro- +cedure, due to the handling of collections via the employed OCL-interpreter, a +fully incremental execution is usually not possible. +As interfaces between these operations, that is, slot nodes, our implemen- +tation employs regular EMF models for model slots and hash-based indices for +assignment slots. While the choice of hash-based indices over array-based in- +dices means that the theoretically fully incremental operation implementations +may not be fully incremental in conjunction with our slot implementations, +hash-based data structures are usually preferable in practice due to their lower +memory footprint and exhibit acceptable performance in most scenarios. +Figure 3 shows a more complex version of the example eGDN from Figure +3 that can be realized using the introduced example eGDN nodes from Table +2. +The transformation from class diagram to abstract syntax graph is now +concretely realized via a unidirectional TGG Synchronization. The query op- +eration that was previously represented by a single query node is decomposed +into a complex network of subqueries. This sub-eGDN consists of two Pattern +Matching nodes labeled “x → y” that look for primitive patterns consisting of +a single edge, one Group Count and one Group Sum node visualized as nodes +labeled “COUNT (X)” respectively “SUM (X)”, and a Join node labeled “▷◁”. +Alternatively, the Pattern Matching nodes could also be realized as Edge Inputs. +7 +Evaluation +In this chapter, we report on an initial empirical evaluation based on our proto- +typical implementation. Moreover, we describe how eGDNs can be employed in +a typical application scenario, evaluating the developed approach with respect +to the requirements from Chapter 3. +37 + +Name +Description +RETE Nodes +Node Input +extracts individual nodes of a given type from a model +Edge Input +extracts individual edges of a given type from a model +Join +performs a natural join of assignments stored in two +input assignment slots +Anti-Join +performs an anti-join of a left input assignment slot +against a right input assignment slot +GDN Nodes +Pattern Matching +finds matches for a given pattern into a model; supports +additional constraints formulated in OCL [62]; supports +constraints regarding the existence/absence of matches +for other patterns via dependencies to related assign- +ment slots +Property Computation Nodes +Expression +computes the value of an OCL [62] expression for indi- +vidual assignments +Group Expression +computes the value of an OCL [62] expression for col- +lections of assignments grouped by certain variables +Group Count +counts the number of assignments in collections of as- +signments grouped by certain variables +Group Sum +computes the sum of numerical values of a specific vari- +able in collections of assignments grouped by certain +variables +Transformation Nodes +TGG Sync. (→) +performs +unidirectional +model +synchronization +of +changes from a source to a target and associated cor- +respondence model via a triple graph grammar [67] +TGG Sync. (↔) +performs bidirectional model synchronization of changes +between a source, target, and associated correspondence +model via a triple graph grammar [67] +Composite Nodes +eGDN +executes a sub-eGDN to update the contents of exposed +slots via Algorithm 1 or Algorithm 3 +Table 2: Example eGDN node types +38 + +Name +Non-recursive +Directions +Correct +Incremental +RETE Nodes +Node Input +✓ +all +cond. +✓ +Edge Input +✓ +all +cond. +✓ +Join +✓ +all +cond. +✓ +Anti-Join +✓ +all +cond. +✓ +GDN Nodes +Pattern Matching +✓ +all +cond. +(✓) +Property Computation Nodes +Expression +✓ +all +cond. +(✓) +Group Expression +✓ +all +cond. +∼ +Group Count +✓ +all +cond. +✓ +Group Sum +✓ +all +cond. +✓ +Transformation Nodes +TGG Sync. (→) +✓ +all +cond. +(✓) +TGG Sync. (↔) +✓ +* +cond. +(✓) +Composite Nodes +eGDN +? +? +cond. +(✓) +Table 3: Properties of example eGDN node types +m1: +ClassDiagram +M +t: +TGG Sync. +(→) +T +q3: +COUNT +(per Type) +Q +m2: +ASG +M +a5: +Assignment +A +q1: +Type → +Method +Q +a1: +Assignment +A +q2: +Package → +Type +Q +a2: +Assignment +A +a3: +Assignment +A +q4: +Q +a4: +Assignment +A +q5: +SUM +(per Package) +Q +q: +eGDN +Q +m3: +Corr. +Model +M +Figure 4: Complex example eGDN +39 + +7.1 +Evaluation of Performance +For an initial empirical evaluation of the proposed approach, we perform an +experiment inspired by an application scenario from the software development +domain, where an evolving class diagram serves as the basis for generating +object-oriented code, which is subsequently analyzed to compute code metrics. +Therefore, we have implemented a simple model transformation from Ecore +models [26] to Java abstract syntax graphs [18] via a triple graph grammar. +For each class in the class diagram, the transformation creates an interface in +the Java abstract syntax graph in a first package, along with an implementation +class in a second package. Also, for each attribute of a class in the class diagram, +the transformation creates a corresponding field and associated getter and setter +methods in the corresponding interface and class in the abstract syntax graph. +In addition, we have realized a model query composed of several subqueries, +which counts the number of methods in all types of a Java package. The trans- +formation and query are integrated into an eGDN, which yields the structure +displayed in Figure 4. +Using our prototypical implementation, which is available under [30], we +assign a real-world Ecore model [18] to the class diagram model slot and per- +form an initial population of the remaining slots via Algorithm 1. To evaluate +the scalability of the eGDN, we then apply a number of synthetic updates to +the model in the class diagram slot, each of which adds an attribute to each +class in the model, and measure the time required for the eGDN to process +each such update via Algorithm 1 (“INCREMENTAL”). We compare this to +a baseline, where instead, we perform a full recomputation of both the model +transformation’s and the query’s results via non-incremental implementations +of the corresponding operations (“BATCH”).6 +Figure 5 displays the execution times for the first 30 updates. After an initial +phase comprising the first 5 updates, where execution time decreases from up- +date to update, the execution time for processing an update to the class diagram +via the strategy INCREMENTAL does not change much. In particular, there +does not seem to be any trend of increasing execution time related to the growth +of the class diagram as additional updates are being performed. In contrast, the +execution time of BATCH increases from update to update as the class diagram +grows. While it starts out similar to the execution time of INCREMENTAL +(larger by factor 1.6), by update 30 the execution time of BATCH has increased +to factor 80 compared to the execution time of INCREMENTAL. +The measurements thus indicate that incremental eGDN execution via IN- +CREMENTAL efficiently handles updates to the class diagram, in the sense that +execution time only seems to depend on the actual changes rather than the size +of the model, indeed affording incrementality. Therefore, eGDNs seem to consti- +tute a suitable formalism for a scalable, modular and incremental realization of +6All experiments were performed on a Linux SMP Debian 4.19.67-2 machine with Intel +Xeon E5-2630 CPU (2.3 GHz clock rate) and 386 GB system memory running OpenJDK +version 11.0.6. +Reported execution time measurements correspond to the mean execution +time of 10 runs of the respective experiment. +40 + +0 +5000 +10000 +15000 +20000 +5 +10 +15 +20 +25 +30 +execution time (ms) +update number +INCREMENTAL +BATCH +Figure 5: Execution time measurements for class diagram updates +networks of model operations for this scenario. The decreasing execution times +per update during the initial phase of the experiment can likely be attributed +to warming-up effects of the Java virtual machine. +The internal validity of our results is mostly threatened by unexpected be- +havior of the Java virtual machine, most notably garbage collection. To miti- +gate such effects, the reported execution time measurements were obtained as +the arithmetic mean of multiple runs of the experiment, with the standard de- +viation of the overall execution time always below 5% of the overall execution +time. +The synthetic updates used in the experiment pose a threat to external +validity. However, the experiment is inspired by a real-world application scenario +and uses a real-world model as its basis and demonstrates the applicability of the +eGDN approach in this scenario. The synthetic updates only serve the purpose +of allowing a systematic evaluation of our technique’s scalability. We hence do +not make any quantitative claims regarding our approach in practical application +scenarios, but merely consider our experimental results as an indicator for the +presented approach’s potential. We furthermore do not make claims regarding +the generalizability of the approach to other application domains, which would +require further evaluation and is left for future work. +7.2 +Evaluation of Applicability +In order to investigate the applicability of the developed technique, we consider +the following extended example scenario that requires global model manage- +ment: A class diagram, adhering to a metamodel similar to the one displayed +in Figure 1, is used to model the structure of a software system under develop- +ment by means of classes contained in packages. Classes may contain methods, +which may in turn reference classes as the method’s return type. OCL expres- +sions in a separate model are used to describe the behavior of some of the class +diagram’s methods. Therefore, the OCL model has its own representation of +types corresponding to the classes in the class diagram. This correspondence +is captured by means of a linking model, which simply contains dedicated link +vertices. A link vertex can either have edges to a class from the class diagram +41 + +and the corresponding type from the OCL model or edges to a method in the +class diagram and the corresponding expression, that is, implementation, in the +OCL model. +We consider the following use cases for this setup: +• Consistency Checking: The developers want to run automatic and in- +cremental consistency checks that verify that the return type of a method +in the class diagram matches the corresponding type of the method’s OCL- +implementation. The developed consistency check should also work for +similar setups that use a different expression language than OCL for the +method implementations. An implementation for this use case thus re- +quires a solution satisfying the requirements R 1.1, R 1.2, R 2.2, and +R 2.3. +• Code Generation: The developers want to automatically and incremen- +tally generate Java code in the form of an ASG from the class diagram. +In addition, Java implementations for the class diagram’s methods should +be generated from the methods’ OCL implementations. In the end, the +resulting Java code fragments for the two models should be integrated and +analyzed for some code metrics. An implementation for this use case thus +requires a solution satisfying the requirements R 2.1, R 2.2, and R 2.3. +• Megamodel Reuse: After developing the automatic consistency check- +ing and code generation, the developers want to reuse the same two op- +erations in another project with a similar set of models. An implementa- +tion for this use case thus requires a solution satisfying the requirements +R 3.1.2, R 3.2, R 3.3, and R 3.4. +In order to allow global model management for all three use cases, a solu- +tion also has to enable the modeling of a network of different kinds of model +operations over a set of potentially integrated models, that is, a solution has to +satisfy requirement R 3.1.1. +Figures 7, 6, and 8 visualize example eGDN implementations for the Con- +sistency Checking, Code Generation, and Megamodel Reuse use cases, +respectively. +As displayed in Figure 7, the Consistency Checking use case is realized +via four Pattern Matching query nodes that extract certain simple patterns +from the base models and make them accessible in a generalized format via +assignment slots. Then, a complex query operation, which is composed of three +Join query operations and an Anti-Join query operation (labelled ▷), realizes +the actual consistency check by finding all the combinations of a method from +the class diagram, its implementation from the OCL model, and the associated +return class respectively expression type, where the return class and expression +type do not correspond. Thus, the eGDN-based approach in this case fulfills +the requirements R 1.1 and R 2.2, as it implements a consistency check over +a set of models integrated via integration links. +Furthermore, the resulting +implementation is reusable for different modeling languages that offer similar +42 + +functionality via the generic interface provided by the assignment slots a1, a2, +a3, and a7, satisfying requirement R 1.2. The example eGDN also demonstrates +how more complex model operations can be composed from simpler operations, +satisfying requirement R 2.1, and provides an incremental execution scheme for +these operations, satisfying requirement R 2.3. In particular, via Algorithm 2, +it can be verified that Algorithm 1 provides a means of executing the eGDN that +guarantees both correct results and termination for changes to any combination +of the three base models. +The eGDN shown in Figure 6 realizes the Code Generation use case via a +combination of two TGG Synchronizations that translate the class diagram and +OCL model into Java ASGs. The two Java models are integrated via a dedicated +linking model, which is produced by a unidirectional model transformation from +the original linking model and the correspondence models created by the TGG +Synchronizations. +Finally, query operations can be executed over the ASGs +to compute code metrics. Using Algorithm 2 to analyze the eGDN, it can be +determined that terminating execution via Algorithm 1 can be guaranteed for +changes to any combination of the three base models. This example shows how +the eGDN provides a unified, modular notion of model operations along with +an incremental execution scheme and demonstrates the composition of model +operations, satisfying requirements R 2.1, R 2.2, and R 2.3. +Together, the eGDNs in Figure 7 and 6 also illustrate how eGDNs can be +used as a megamodeling language, supporting different kinds of model opera- +tions, including model properties (like the metrics computed in the Code Gener- +ation use case), model consistency (like the consistency condition in the Consis- +tency Checking use case), and model transformation and synchronization (like +the transformation and synchronizations in the Code Generation use case). It +also shows how integration views (like the cross-model consistency query results +in slot a8 in Figure 7) and traceability links (like the correspondence models +produced by the TGG Synchronizations) can be represented in the language. +While not present in the example eGDNs, the class diagram and OCL meta- +model are models themselves and could simply be made explicit by including +them in dedicated model slots. Well-formedness conditions for metamodels or +regular models can be realized and treated as regular query operations. eGDNs +thus satisfy the requirement R 3.1.1. +Finally, the eGDN realization of the Megamodel Reuse use case in Figure +8 considers the eGDNs from Figure 7 and 6 as operation nodes in an overarching +eGDN. This exemplifies how eGDNs offer modularity and incrementality at the +megamodel level by considering sub-eGDNs as regular operations that can be +executed via the general execution scheme, which also permits the accumulation +of several changes before execution. The example thus illustrates the satisfaction +of requirements R 3.1.2, R 3.2, and R 3.4. The eGDN also demonstrates how +slots act as interfaces for these megamodel operations, satisfying requirement +R 3.3. +Thus, eGDNs can be employed to realize the functionality required by the +three example use cases, satisfying the requirements regarding model operations, +modeling languages integration, and megamodels introduced in Section 3 in this +43 + +m1: +ClassDiagram +M +m2: +OCL +M +m3: +LinkingModel +M +t1: +TGG Sync. +(↔) +T +t2: +TGG Sync. +(↔) +T +m5: +Corr. +OCL +M +m6: +ASG +M +m7: +ASG +M +t3: +Transform +T +m8: +LinkingModel +M +m4: +Corr. +ClassDiagram +M +q1: +Query +Q +q2: +Query +Q +metric1: +Assignment +A +metric2: +Assignment +A +codegen & analysis: +eGDN +X +Figure 6: Sample eGDN realizing the Code Generation use case +scenario. +Table 4 summarizes the coverage of the requirements by the example use +cases and the eGDN approach, with “◦” denoting that the realization of a use +case relates to a requirement and “✓” indicating that a requirement is satisfied +by eGDNs in this scenario. +44 + +m1: +ClassDiagram +M +m2: +OCL +M +m3: +LinkingModel +M +q1: +Method → +Class +Q +q2: +Expression → +Type +Q +q3: +Method ← Link → Expression +Q +q4: +Q +q5: +Q +a1: +Assignment +A +a3: +Assignment +A +a2: +Assignment +A +a4: +Assignment +A +a5: +Assignment +A +q6: +Q +a6: +Assignment +A +q8: +▷ +Q +q7: +Class ← Link → Type +Q +a7: +Assignment +A +a8: +Assignment +A +consistency: +eGDN +Q +Figure 7: Sample eGDN realizing the Consistency Checking use case +45 + +m1: +ClassDiagram +M +m2: +OCL +M +m3: +LinkingModel +M +q1: +Method → +Class +Q +q7: +Class ← Link → Type +Q +q3: +Method ← Link → Expression +Q +a1: +Assignment +A +a3: +Assignment +A +a7: +Assignment +A +q2: +Expression → +Type +Q +a2: +Assignment +A +consistency: +eGDN +Q +m6: +ASG +M +m7: +ASG +M +m8: +LinkingModel +M +metric1: +Assignment +A +metric2: +Assignment +A +a8: +Assignment +A +codegen +& analysis: +eGDN +X +Figure 8: Sample eGDN realizing the Megamodel Reuse use case +46 + +Consistency Checking +Code Generation +Megamodel Reuse +eGDNs +R 1.1: modeling languages integration +◦ +✓ +R 1.2: interfaces for embedding of modeling languages +◦ +✓ +R 2.1: composition of model operations +◦ +✓ +R 2.2: model operations over integrated models +◦ +◦ +✓ +R 2.3: execution scheme for model operations +◦ +◦ +✓ +R 3.1.1: megamodeling language +◦ +◦ +◦ +✓ +R 3.1.2: megamodel operation module concept +◦ +✓ +R 3.2: robust megamodel execution scheme +◦ +✓ +R 3.3: megamodel interfaces +◦ +✓ +R 3.4: asynchronous megamodel execution scheme +◦ +✓ +Table 4: Coverage of requirements from Chapter 3 +8 +Conclusion +In this report, we have developed a further generalization of the GDN mech- +anism called eGDNs, which enables the modular and incremental construction +and execution of complex networks of model operations, including model prop- +erties, model consistency, model transformation and model synchronization. In +addition to a formal definition of eGDNs, we have provided incremental algo- +rithms for their execution. Moreover, we have presented a number of example +eGDN nodes that we have prototypically implemented in order to perform an +initial empirical evaluation of the approach regarding scalability. Our experi- +ments, which are based on an application scenario from the software develop- +ment domain, indicate that the introduced technique can be employed to realize +efficient Global Model Management. Moreover, we have conceptually evaluated +our approach against identified requirements of global model management solu- +tions. +In future work, we plan to perform a more extensive evaluation with respect +to both expressiveness and performance of eGDNs in real application scenar- +ios. +This may also involve the implementation of additional types of eGDN +nodes and may ultimately result in the implementation of true tool support +for the specification and execution of eGDNs. Furthermore, we will investigate +how the presented concepts can be extended to the case of evolving modeling +landscapes that consist of multiple distinct versions. 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In: Software +& Systems Modeling 12 (2013), pp. 105–119. url: http://dx.doi.org/ +10.1007/s10270-011-0191-2. +[80] +Wires Project Homepage. http://atenea.lcc.uma.es/index.php/ +Main_Page/Resources/Wires*. +54 + diff --git a/WNAyT4oBgHgl3EQfu_m7/content/tmp_files/load_file.txt b/WNAyT4oBgHgl3EQfu_m7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..98f4be7a46b1f92107381581d51f3c266bc37cbe --- /dev/null +++ b/WNAyT4oBgHgl3EQfu_m7/content/tmp_files/load_file.txt @@ -0,0 +1,1738 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf,len=1737 +page_content='Modular and Incremental Global Model Management with Extended Generalized Discrimination Networks Matthias Barkowsky matthias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='barkowsky@hpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='de Holger Giese holger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='giese@hpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='de January 3, 2023 Abstract Complex projects developed under the paradigm of model-driven en- gineering nowadays often involve several interrelated models, which are automatically processed via a multitude of model operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Modular and incremental construction and execution of such networks of models and model operations are required to accommodate efficient development with potentially large-scale models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The underlying problem is also called Global Model Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In this report, we propose an approach to modular and incremental Global Model Management via an extension to the existing technique of Generalized Discrimination Networks (GDNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In addition to further generalizing the notion of query operations employed in GDNs, we adapt the previously query-only mechanism to operations with side effects to integrate model transformation and model synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We provide incremental algorithms for the execution of the resulting extended Gen- eralized Discrimination Networks (eGDNs), as well as a prototypical im- plementation for a number of example eGDN operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Based on this prototypical implementation, we experiment with an ap- plication scenario from the software development domain to empirically evaluate our approach with respect to scalability and conceptually demon- strate its applicability in a typical scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Initial results confirm that the presented approach can indeed be employed to realize efficient Global Model Management in the considered scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 1 Introduction Complex projects developed under the model-driven engineering paradigm nowa- days often involve several interrelated models, which are inspected, analyzed, transformed, and synchronized via a multitude of model operations [69]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' An 1Note that references in bold refer to our own publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='00624v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='SE] 2 Jan 2023 effective and efficient management of the resulting sophisticated networks of model operations is both a crucial prerequisite to successful development projects and a challenging research problem, known as Global Model Management [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' On the one hand, modular and incremental construction of model opera- tion networks is required in the context of project landscapes that evolve to accommodate dynamic development processes and changing requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' On the other hand, in order to scale to today’s potentially large models and allow development in teams, modular and incremental execution of these networks is required, as full re-execution of the entire network in reaction to changes may result in unacceptable execution times and loss of information [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In this context, model queries, due to being explicitly and implicitly required by model properties and model consistency checks respectively model transfor- mations and model synchronizations, play a central role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Solutions thus have to offer dedicated support for handling potentially complex model queries and facilitate their modular composition and reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Furthermore, model operations with side-effects, such as model transforma- tion and synchronization, and their interaction with other model operations pose a unique challenge regarding the overall goal of guaranteeing the consistency of a system description that may be distributed over multiple models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In this report, we propose an approach to Global Model Management that specifically aims to provide both the required modularity and incrementality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Our solution is based on an extended notion of Generalized Discrimination Net- works [45], a mechanism that has previously been implemented in the context of model driven engineering [7] to allow a modular and incremental specification and execution of model queries in the form of nested graph conditions [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, we introduce a more general formalization called extended Gen- eralized Discrimination Networks (eGDNs), which (i) supports a more flexible notion of model queries, affording increased expressiveness and (ii) allows the integration of model operations with side effects into the unifying framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In addition, we provide algorithms for the incremental execution of eGDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Furthermore, we integrate a number of typical model operations into a proto- typical implementation of the approach and use this implementation to perform an initial evaluation of our technique’s scalability using an application scenario from the software development domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This empirical evaluation is comple- mented by a conceptual evaluation regarding the applicability of eGDNs in a typical scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The remainder of the report is structured as follows: We briefly reiterate the basic concepts of models in the form of typed graphs and discrimination networks in Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' After introducing the required concepts, we discuss requirements of a solution for global model management and related work in Chapter 3, providing further motivation for the design of a new solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Our contribution in the form of extended Generalized Discrimination Networks is presented in Chapters 4, 5, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, Chapter 4 provides a definition of eGDNs along with a graphical notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Chapter 5 describes the incremen- tal execution of eGDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Chapter 6 then lists a number of examples for eGDN operations that are part of our prototypical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This prototyp- 2 ical implementation is used to perform an initial empirical evaluation of the presented concepts, which is presented in Chapter 7 along with a conceptual evaluation of the applicability of eGDNs to an example use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Finally, Chap- ter 8 concludes the report and gives an overview of possible directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 2 Preliminaries In this chapter, we reiterate the basic notions of models in the form of typed graphs and discrimination networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1 Graphs and Models A graph G = (V G, EG, sG, tG) consists of a set of vertices V G, a set of edges EG, and two functions sG, tG : EG → V G assigning each edge its source respectively target vertex [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A graph morphism m : G → H between graphs G and H is a pair of functions mV : V G → V H, mE : EG → EH such that sH ◦mE = mV ◦sG and tH ◦ mE = mV ◦ tG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A graph G can be typed over a type graph TG via a morphism typeG : G → TG that assigns elements from G types defined in TG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This yields a typed graph GT = (G, typeG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A typed graph morphism mT : GT → HT between two typed graphs GT = (G, typeG) and HT = (H, typeH) typed over the same type graph TG is given by a graph morphism m : G → H with typeG = typeH ◦ mT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In the context of this report, a model is then characterized by a typed graph, where the type graph effectively acts as a metamodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Importantly, attributes for model elements can be realized in the framework of typed graphs by simply modeling attribute values as dedicated nodes, which leads to the notion of typed attributed graphs [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A modeling language ML is defined by a graph TG and denotes the set of all possible graphs typed over TG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Figure 1 shows an example model from the software development domain in the form of a typed graph G, and the associated metamodel in the form of the type graph TG, with the typing morphism given by node labels in case of nodes and implicitly in case of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The example model represents the ab- stract syntax graph (ASG) of a program written in an object-oriented program- ming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Nodes in the model represent packages, types, and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Edges represent containment relationships between the different concepts, with methods contained in types and types contained in packages, and return type relationships between methods and types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2 Discrimination Networks A discrimination network is a graph of nodes representing computation units and edges representing dependencies between these units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Discrimination net- works are a popular solution for the incremental execution of model queries such as the computation of model properties or the checking of model consistency 3 Type Method Package types type p1:Package t1:Type t2:Type m1:Method m3:Method m2:Method G TG methods Figure 1: Example model and metamodel in the form of typed graph and type graph from the software development domain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, the model query is decomposed into subqueries, which form the discrimination network’s nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The execution of a subquery can make use of the results computed for an- other subquery, which is indicated by a dependency relation between the two subqueries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The execution of a final discrimination network node yields the overall query result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' By storing the results of discrimination network nodes beyond the execution of a query, incremental execution that reuses previously computed results in subsequent executions is enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since discrimination networks so far are primarily employed for model query- ing, current approaches offer only limited or no support for the integration of model operations with side-effects and thus constitute at best a partial solu- tion for global model management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, due to their inherent support for modularity and incrementality, they offer a promising starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' There exist different realizations of the concept of discrimination networks in the context of model driven engineering, two of which will be briefly presented in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1 RETE nets RETE nets were initially introduced by Forgy [34] and are characterized by the fact that nodes are only allowed to have dependencies to at most two other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Some examples of RETE nodes are: input nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' which correspond to primitive model queries that extract in- dividual elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' nodes or edges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' from a model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' and consequently have no dependencies filter nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' which filter the results of some other subquery by a condition and consequently have one dependency join nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' which combine the results of two other subqueries into results for a more complex subquery and consequently have two dependencies 4 While the listed node types form the core of incremental model querying so- lutions such as the well-established VIATRA [78],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' RETE nets are a flexible mechanism that allows a multitude of other query-related node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This is illustrated by VIATRA’s support for various advanced constructs for specifying model queries, including negative patterns and certain aggregation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In RETE implementations, results computed by a RETE net’s nodes are usually stored in memory in so-called indexers, which act as implicit interfaces between computation nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' These indexers can also be made explicit by mod- eling them as part of the RETE net via a different kind of RETE node that is not associated with any computational functionality, but only serves as a storage for other nodes’ results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2 Generalized Discrimination Networks Generalized Discrimination Networks (GDNs) are a less restrictive form of dis- crimination networks than RETE nets and were developed by Hanson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Essentially, GDNs drop the limit on the number of a node’s dependencies of RETE nets and thereby allow for more control over which intermediate query results are to be stored in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A realization in the context of model querying was presented in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' It imple- ments GDN nodes as model transformation rules that create marking elements for subquery results directly as part of the queried model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Dependencies between nodes are realized by considering marking elements created by the required node in the transformation rule associated with the dependent node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, while the approach in [7] is based on a fairly expressive notion of queries in the form of nested graph conditions, certain query-related operations such as aggregation are not supported by the underlying formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 3 Requirements for Global Model Management Nowadays the development of complex systems with models requires Global Model Management (GMM ) [9, 31] to ensure that the models of different sub- systems, of different views, and of different domains are properly combined, even though the models might reside at different levels of abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Indeed, due to the heterogeneity and complexity of systems such as Cyber-Physical Systems (CPS), it is no longer feasible to represent the system as a Single Undery- ing Model (SUM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This is because numerous languages and tools are already employed independently by domain experts collaborating to build the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Redeveloping these tools and thus requiring industry to change its practices is not conceivable given the required development efforts, but also the strong re- sistance to change development processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This is especially relevant in the case of safety-critical systems that must undergo complex certification processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, many models must be used to represent the system and adequate GMM is required to ensure that the development activities that operate on the models are properly coordinated such that the models lead to a proper system 5 as a whole, where the different elements and aspects covered by the different models are correctly integrated and are consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A classification of model integration problems and fundamental integration techniques has been introduced in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' It highlights the techniques of de- composition and enrichment, which characterize two orthogonal dimensions of development where the system is decomposed into subsystems and domains (horizontal dimension) and into a set of models with increasing level of details (vertical dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This requires coordinating all activities operating on the models across these dimensions to ensure their consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The development activities for nowadays complex systems are spread across multiple domains and teams, where each team is using its own set of model- ing languages thus requiring proper integration of these languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Indeed, it has been shown that using a single language to cover all domains would lead to very large monolithic languages not easily customizable for the development environments and tools needed by development organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' These consid- erations lead to Multi-Paradigm Modeling (MPM ) [77], which advocates the integration of reusable modular modeling languages instead of large monolithic languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Hence, GMM must support integrating with appropriate modularity not only models but also their modeling languages (hereafter modeling language integration), in addition to coordinating all activities operating on the models and specified as model operations / transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The execution of these model operations has to be scalable for being able to handle large models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This requires incrementality, where only the operations impacted by a model change are re-executed, thus avoiding the effort to recompute entire models as in the case of incremental code compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' GMM is also known as modeling-in-the-large, which consists of establish- ing global relationships (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' model operations that generated one model from other models) between macroscopic entities (models and metamodels) while ig- noring the internal details of these entities [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Megamodeling [8, 31] has been introduced for the purpose of describing these macroscopic entities and their relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Consequently, for modular and incremental global model management solu- tions for the modular and incremental construction and execution of I) models and modeling languages integration, II) model operations, and III) megamodels are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We will outline in the following that nowadays only preliminary approaches exist that provide ad hoc solutions for fragments of the sketched problem and that a solid understanding of the underlying needs and challenges is currently lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In particular, the current approaches do at most offer some modularity and/or incrementality for a single aspect as modeling languages inte- gration or model operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, support for handling complex modeling landscapes as a whole in a modular and incremental fashion as required for the large-scale problems that exist in practice is not offered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In the following, we will discuss the needs in more detail and review how far existing solutions that address the construction and execution of 1) models and modeling languages integration, 2) model operations, and 3) megamodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The way the existing approaches perform along these dimensions is depicted in 6 Table 1, where an empty cell identifies a need that is not addressed, a ˜ denotes partial fulfilment of the need and a + indicates that the need is addressed sufficiently2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This evaluation is discussed in further details in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Approach Modeling Languages Integration Model Operations Megamodels Const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Exec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Exec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Exec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Links Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' MMI Batch Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Flow Ctx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Batch Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Batch Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Modeling Languages Integration Blanc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' [11] + + EMF IncQuery [25, 76] + + Egyed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' [43, 23] + + Cabot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' [19] + + ACOL [57] ∼ + SmartEMF [58, 50, 73] + + Composite EMF Mod- els [52, 21] + + EMF Views [27, 17] + ∼ + Kompren [12, 55] / Kompose [33, 54] ∼ Reuseware ModelSoc [51, 60] ∼ ∼ Ratiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' [65] + ∼ ∼ K¨onig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' [56] + + Model Operations Wires* [66, 80] + + ATL Flow [4] + + Epsilon [64, 28] + + + Gaspard2 [29, 37] ∼ + ∼ + ∼ Debreceni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' [22] + ∼ + + MoTCoF [71] + + + ∼ MoTE [38][61] + + + Integration Languages and Others CyPhy [72, 42] + + + FUSED [15, 35] + ∼ + + CONSYSTENT [48, 49] + + + Megamodels AM3 [79, 1] + + + + ∼ FTG+PM [59, 5] + + + + MegaL Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' [32] + + + GMM* [13] + ∼ + + ∼ + + ∼ Seibel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' [70, 68][6] + + + ∼ Stevens [75, 74] + + + + + + Gleitze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' [41] + + + + + + Vitruvius [53] + + + + + + + + + eGDNs + + + + + + + + + + + + + Table 1: Comparison of existing and planned global model management ap- proaches 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1 Models and Modeling Languages Integration: Con- struction and Execution 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1 Construction The construction of models and modeling languages integration is addressed in the current approaches in three main ways via (1) linking of models and model elements, (2) model interfaces and (3) metamodel composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (1) Links: All approaches make use of some kind of trace links between models and their model elements to integrate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In this report, we adopt the definitions of traceability proposed by the Center of Excellence for Software Traceability (Co- EST) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A trace link is ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='a specified association between a pair of artifacts, one comprising the source artifact and one comprising the target artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Following the CoEST again, trace links are specialized into traces between the vertical and horizontal dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Hence, a vertical trace ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='links artifacts at 2For convenience, we use the name of the tool or project to identify an approach when it exists, otherwise the name of the first author of the publication describing the approach is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 7 different levels of abstraction so as to accommodate lifecycle-wide or end-to-end traceability, such as from requirements to code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' An horizontal trace links ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='artifacts at the same level of abstraction, such as: (i) traces between all the requirements created by ‘Mary’, (ii) traces between requirements that are con- cerned with the performance of the system, or (iii) traces between versions of a particular requirement at different moments in time”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' There is a plethora of approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', [2, 28, 32, 58, 48, 72, 15] [61]) mak- ing use of trace links to integrate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The Atlas Model Weaving (AMW) language [2] provided one of the first approaches for capturing hierarchical trace- ability links between models and model elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The purpose was to support activities such as automated navigation between elements of the linked models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In this approach, a generic core traceability language is made available and op- tionally extended to provide semantics specific to the metamodels of the models to be linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Similarly, the Epsilon framework [28] provides a tool (ModeLink) to establish correspondences between models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' MegaL Explorer [32] supports re- lating heterogeneous software development artifacts which do not necessary have to be models or model elements using predefined relation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' SmartEMF [58] is another tool for linking models based on annotations of Ecore metamodels to specify simple relations between model elements through correspondence rules for attribute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Complex relations are specified with ontologies relating the concepts of the linked languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The whole set of combined models is converted into Prolog facts to support various activities such as navigation, consistency and user guidance when editing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The CONSYSTENT tool and approach [48] make use of a similar idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, graph structures and pattern matching are used to represent the combined models in a common for- malism and to identify and manage inconsistencies instead of Prolog facts as in the case of SmartEMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' There are also a number of approaches such as [72] and [15] that build on establishing links between models through the use of integration languages developed for a specific set of integrated modeling languages, where the inte- gration language embeds constructs specific to the linked languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This is also the case for model weaving languages extending the core AMW language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, AMW has the advantage of capturing the linking domain with a core common language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Other means for linking and integrating models are Triple Graph Grammars (TGG) such as the Model Transformation Engine (MoTE) tool [61], which similarly requires the specification of some sort of integra- tion language (correspondence metamodel) specific to the integrated languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, an important asset of this approach is that it automatically establishes and manages the traceability links and maintains the consistency of the linked models (model synchronization) in a scalable, incremental manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Finally, in [70, 68][6], an approach is presented to automatically create and maintain traceability links between models in a scalable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' While the approach focuses on traceability management rather than model integration, compared to integration languages, it relies on link types defined at the model level (and not at the metamodel / language level), thus avoiding the need to update the integration language every time a new language must be integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' More re- 8 cently, the concept of reactive links has been presented in [65], which essentially allows an incremental propagation of attribute value changes between models of different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, incremental execution is only offered for a limited notion of consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The comparison of these approaches shows that apart from our own earlier approach [70, 68][6], all approaches suffer from being dependent on the set of integrated languages, thus requiring to better support modularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Further- more, only our own work [61][70, 68][6] supports automated management of traceability links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (2) Interfaces: In addition to links, a few more sophisticated approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', [57, 50, 52, 51]) introduce a concept of model interface (int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' column in Table 1) for specifying how models can be linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In [57], the Analysis Con- straints Optimization Language (ACOL) is proposed, which has been designed to be pluggable to an Architecture Description Language (ADL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A concept of interface specific to ACOL is included so that constraints can refer to these in- terfaces to relate to the model elements expected from the ADL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' SmartEMF [50, 73] proposes a more generic concept of model interface to track dependencies between models and metamodels and provide automated compatibility checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Composite EMF Models [52, 21] introduces export and import interfaces to specify which model elements of a main model (body) should be exposed to other models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' are part of the public API), and which elements of a body model are to be required from an export interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In [51], an approach for the composition of grammars with explicit variation points (hooks) constituting an implicit invasive composition interface is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, while these approaches provide interesting preliminary ideas, they need to be enriched to cover a larger number of non intrusive model integration use cases such as for example, specifying modification policies of the linked model elements required to ensure the models can be kept consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' They also lack integration into GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (3) Metamodel Integration: Some approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', [55, 54, 29, 27] [13]) consider the construction of view metamodels in terms of other metamodels or language fragments (MMI column in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In [29], an approach imple- mented in the Gaspard2 tool [37] is presented where metamodels are artificially extended for the purpose of combining independent model transformations re- sulting in an extended transformation for the extended metamodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In [12], a language and tool (Kompren) [55] are proposed to specify and generate slices of metamodels via the selection of classes and properties of an input metamodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A reduced metamodel is then produced, which must be completely regenerated when the input metamodel is changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Such is the case for the Kompose ap- proach [54], which on the contrary to Kompren proposes to create compound metamodels, where a set of visible model elements from each combined meta- models is selected and optionally related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' EMF Views [27, 17] provides similar approach however without the need to duplicate the metamodel elements as op- posed to Kompose and Kompren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Indeed, EMF Views allows the specification of virtual metamodels that only refer to existing metamodel elements instead of duplicating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The same principle applies for the given models of the vir- 9 tual metamodels, which only refer to elements of the existing integrated models instead of duplicating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The defined virtual view metamodels are usable transparently by tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Furthermore, the same models can be simultaneously used by both legacy tools and new tools making use of the virtual metamod- els, thanks to the non-intrusiveness of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Finally, the Global Model Management language (GMM*)3 [13] provides means to specify and interpret reusable language subsets as sets of constraints combined to form subsetted metamodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Like for EMF Views, these reduced metamodels can to some ex- tent be used transparently by tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Aspect-oriented metamodel composition is another well-known technique for metamodel composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However it requires metamodels to be expressed in a specific aspect-related format, which does not meet our non-intrusiveness requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' While each of these approaches provides interesting support for modular modeling languages integration, their unification into a common formalism, the use of an explicit notion of a model interface and their integration into GMM is lacking, except for subsetted metamodels already integrated within our GMM* language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Among these approaches, we note that EMF Views provides an ade- quate starting point for this work, due to its non-intrusiveness property essential for reusing legacy models and tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, in its current implementation, only changes of attributes of virtual compound models are propagated to the underlying real models [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Other changes propagation as well as metamodel constraints composition remain to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The integration of an explicit metamodel interface construct for governing how metamodels can be composed, as well as the ability to solve attribute and operation conflicts of merged classes inspired from the concept of Traits / Mixins developed for object oriented pro- gramming are required future works for this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Execution of integrated models concerns the evaluation of the well-formedness constraints of each combined model alone, but also of the combined models as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' To our knowledge, no approach addresses the incremental checking of well-formedness conditions across the different language fragments of compound models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, some approaches on incremental constraints evaluation exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In [11], changes on models are expressed as sequences of atomic model opera- tions to determine which constraint is impacted by the changes, so that only these constraints need to be re-evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In [25, 76], a graph-based query lan- guage (EMF-IncQuery) relying on incremental pattern matching for improved performance is also proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In [23], an approach is presented for incremental evaluation of constraints based on a scope of model elements referenced by the query and determined during the first query evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This scope is stored into cache and used to determine which queries need to be re-evaluated accord- ing for some model changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In [43], this approach is extended for the case where the constraints themselves may change besides the constrained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Finally in [19], an incremental OCL checker is presented where a simpler OCL expression and reduced context elements set are computed from an OCL con- 3We use * to distinguish this existing language and tool from the generic Global Model Management (GMM) acronym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 10 straint and a given structural change event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Evaluating this simpler constraint for the reduced context is sufficient to assert the validity of the initial constraint and requires significantly less computation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In [56], K¨onig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' introduce a technique for the checking of consistency constraints over linked models, which avoids the merging of these models into a single underlying model to achieve better scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, while formally defined and proven to be correct, the approach in [56] does not consider incre- mental consistency checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We identified the following requirements as main needs concerning modular- ity and incrementality of modeling languages integration: R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1 modeling languages integration via integration links and combina- tion of well-formedness conditions with consistency R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2 interfaces for embedding of modeling languages Note that concerning Table 1 the requirements cover here Links and Inter- faces which jointly emulate the less modular direct meta model integration and that the employed well-formedness conditions and consistency conditions will be covered when we consider model operations in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Consequently, as visible in Table 1, there yet does not exists any approach that provides a combination of all these requirements we target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2 Model Operations: Construction and Execution The construction of model operations is addressed in two ways in the liter- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Most approaches combine model operations as model transformations chains ((1) Flow Composition), where each chained transformation operates at the granularity of complete models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In order to support reuse and scalability for complex modeling languages, which are defined by composing them from simpler modeling languages, a few approaches have considered specifying model transformations as white boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Composed of explicit fine grained operations processing model elements for a given context, these operations are reusable across several model transformations ((2) Context Composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (1) Flow Composition Approaches: FUSED (Formal United System Engineering Development) [15] is an integration language to specify complex relationships between models of different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' It supports model transformation chains, but only implicitly via execution of tools, without explicit representation of the involved transformations and pro- cessed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' On the contrary, there is a plethora of approaches allowing the ex- plicit specification and construction of model transformation chains implement- ing a data flow paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Such is the case of the AtlanMod Megamodel Man- agement (AM3) tool [1], for which the Atlas Transformation Language (ATL) [3] is used to specify the model transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Besides, a type system has been developed [79], which enables type checking and inference on artifacts related 11 via model transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Another similar but less advanced tool is the Ep- silon Framework [28], which provides model transformation chaining via ANT tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Wires [66] and ATL Flow [4] are tools providing graphical languages for the orchestration of ATL model transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The Formalism Trans- formation Graph + Process Model (FTG+PM) formalism [59] implemented in the AToMPM (A Tool for Multi-Paradigm Modeling) tool [5] provides similar functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, it has the advantage of also specifying the complete modeling process in addition to the involved model transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This is achieved via activity diagrams coupled with model transformation specifications executed automatically to support the development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Finally, GMM* [13] also supports model transformation chaining, but through the specifica- tion of relations between models of specific metamodels that can be chained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' One advantage of this approach is that automated incremental (re-)execution of the specified relations between models is provided in response to received model change events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Incrementality of the execution of the transformations is also made possible by the integration of the MoTE [61] incremental model transformation tool into GMM*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, while chaining model transformations offers some degree of mod- ularity of model transformation specifications, apart from GMM*, most ap- proaches suffer from scalability issues for large models, since the used transfor- mation tools do not support incremental execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In addition, the case where a generated model is modified by hand to add information not expressible with the language of the original model(s) cannot easily be handled by these approaches, since regenerating the model modified by hand will destroy the user-specific information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This need is better supported by context composition approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (2) Context Composition Approaches: A few approaches allow context composition of model operations (column Ctx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In [29] as mentioned above, an approach is described to combine independent model transformations resulting in extended transformations for corresponding extended metamodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In [22], an approach is described for spec- ifying the construction of view models using contextual composition of model operations (derivation rules) encoded as annotations of queries of the EMF Inc- Query [25] language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Traceability links between view and source model elements are automatically established and maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The use of EMF IncQuery na- tively provides incremental execution of the derivation rules to synchronize the view model with the source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Some views may be derived from other views thus allowing flow composition as chains of view models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This approach achieves results similar to TGGs supporting incrementality, however with the drawback of being unidirectional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Similarly, but with bi-directionality the MoTCoF lan- guage [71] allows for both flow and fine grained context composition of model transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' An advantage over [29] however is that model transformations are used as black boxes without the need to adapt the transformations according to the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' As can be seen, most approaches only support flow type modularity for model 12 operations with batch execution except for our GMM* language thanks to its integration of MoTE providing incremental execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This will not scale and lead to information losses in case of partial model information overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Only a few approaches allow context modularity, which better supports incremental application where only the impacted operations can be re-applied following a change in order to avoid the cost of re-computing complete transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Such is the case of MoTCoF, which theoretically permits incremental execu- tion, but a concrete technical solution is still lacking for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' To address modularity and incrementality for model operations, we identified as main needs: R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1 composition of model operations R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2 model operations over integrated models R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3 execution scheme for model operations Note that concerning Table 1 the requirements cover here Flow and Context based composition and Batch as well as Incremental Execution at first for all special cases of model operations and then also for the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Conse- quently, as visible in Table 1, there yet does not exists any approach that fully cover the envisioned combination of all these requirements we target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3 Megamodels and other Global Model Management Approaches Two strands can be identified for GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A first one makes use of (1) model integration languages, which are defined for a specific set of integrated modeling languages and tools meaning that the integration language must be updated every time a new language or tool is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The second strand attempts to solve this problem by making use of (2) megamodels providing configurable global model management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (1) Integration Languages and other Approaches: The CyPhy [72] used in the GME modeling tool [42] and FUSED [15, 35] are examples of model integration languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' But as mentioned above, these lan- guages must be adapted as soon as a different set of integrated languages and tools must be used, thus requiring highly skilled developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Integration lan- guages are therefore not practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Open Services for Lifecycle Collaboration (OSLC) [63] provides standards for tool integration through the Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Many specifications are available for change management, resource previews, linked data, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' It builds on the W3C linked data standard, which aims at providing best practices for publishing structured data on the Web based on the W3C Resource Description Framework (RDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' RDF is a model for data interchange on the Web where data is represented as graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, OSLC is more services (and tools) oriented and inherits 13 the problems of linked data, which is specific to the Web and therefore does not separate the concerns of data representation and persistence as opposed to Model-Driven Engineering (MDE) where an abstract syntax is used indepen- dently of the way the data is stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Another approach making use of these standards is [48] and is implemented in a tool named CONSYSTENT, used to identify and resolve inconsistencies across viewpoints due to information overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The information of all mod- els involved during development is captured in a common RDF graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The ap- proach relies on a human4 to specify patterns representing semantic equivalence links (semantic connections) across the graph models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Inconsistency patterns based on these semantic connections are continuously checked over the RDF model for potential matches identifying inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Means to automati- cally resolve inconsistencies are under development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, this approach necessitating the conversion of all models as a RDF graph is not incremental and will not scale for large models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (2) Megamodels: In this second strand, megamodels serve to capture and manage MDE resources such as modeling languages, model transformations, model correspondences, and tools used in modeling environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' There are several megamodeling approaches as already mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' AM3 [1] is one of the first ones where a megamodel is basically a registry for MDE resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Model transformations are specified with ATL [3] and model correspondences with the Atlas Model Weaving (AMW) language [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Similarly, FTG+PM [59] as mentioned above is also a megamodeling language as well as MegaL Explorer [32] allowing to model the artifacts used in software development environments and their re- lations from a linguistic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The involved software languages and related technologies and technological spaces can be captured with linguistic relationships between them such as membership, subset, conformance, input, dependency, definition, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Operations between entities can also be captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The artifacts do not need to be represented as models, but each entity of the megamodel can be linked to a Web resource that can be browsed and examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, the language seems to be used mostly for visualization providing a better understanding of the developments artifacts but cannot be executed to perform model management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The aforementioned GMM* infrastructure [13] consists of a megamodeling language inspired from [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Metamodels can be de- clared, as well as relations between models of these metamodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In particular, synchronization relations can relate models of two different metamodels making use of the MoTE TGG engine [61] to transform or synchronize the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' As mentioned earlier, chains of model transformations can be specified and exe- cuted incrementally in response to model change events and subsets of modeling languages can be declared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' GMM* is experimented within the Kaolin tool [14] making use of complex and rich industrial languages such as AADL and VHDL 4An automated method making use of Bayesian Belief Networks is also under study [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 14 thus challenging GMM for realistic specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A new approach to modeling in the large with bidirectional model trans- formation has been proposed by Stevens [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The work in [75] presents a formalized notion of a megamodel in the form of a hypergraph, where models are represented as nodes that can be connected via hyperedges representing bidirectional transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Incremental execution is generally supported by the formalism, however, a concrete algorithm is only presented for megamod- els with a restricted structure for which a certain notion of correctness can be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The author extends her work in [74] by connecting her previous work to research in the domain of build systems and introducing a so-called orientation model to steer megamodel execution, relaxing the restrictions on the megamodel’s structure while maintaining a formal guarantee of correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, the construction of an orientation model is a manual and potentially challenging process for complex networks of model operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Furthermore, the work in [75, 74] abstracts from the technical realization of model operations and hence does not explicitly consider how operations such as the computation of model properties may be composed in a modular manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In [41], Gleitze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' propose an incremental execution strategy for net- works of model transformations, specifically aiming for a solution that provides explanations of cases where the strategy failed to produce a consistent result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' While their strategy is applicable to networks with arbitrary structure, only bidirectional transformations between pairs of models are considered, limiting the notion of supported model operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Recently, significant progress has also been made in the field of model views [16], which studies how consistent view models can be derived from a system description consisting of multiple interrelated models and therefore also relates to Global Model Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The most comprehensive and advanced model view technique is probably the Vitruvius approach [53], which relies on a so- called virtual single underlying model (V-SUM) for the description of the overall system under development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The V-SUM is used to integrate the individual models describing system parts and derive new view models via consistency relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, Vitruvius employs a dedicated incremental algorithm for executing complex networks of consistency preservation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, the notion of consistency in [53] is limited to relations between pairs of tuples of model elements and hence does not support certain model operations such as computation of model properties using aggregations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Furthermore, intra-model well-formedness is deliberately not covered and reuse at the mega-model level is not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, most of these megamodeling approaches only cover to a certain de- gree the core ingredients of specifying MDE resources by means of metamodels and model operations with appropriate modularity and incrementality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Only fragments of the problem are solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Furthermore, all these megamodeling lan- guages are monolithic (column Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' in Table 1) and as a result, predefined megamodel fragments cannot be composed and reused to avoid rebuilding com- plete megamodel specifications from scratch for new projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We note however that aspect-oriented metamodel composition may be used as an inspiring point 15 and adapted to megamodeling for the specification of distributed megamodels fragments contributing cross-cutting information in an integrated megamodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' As for megamodel execution, FTG+PM, [75, 41, 53], GMM*, and [70, 68] consider automated or semi-automated execution in response to model changes or modeling events from the tool’s user interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The related work demonstrates that for global model management, we need a view that combines all its facets in a mega model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' To address modularity and incrementiality for modamodels we can conclude that the main needs are: R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1 a megamodeling language with R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1 support for metamodels, well-formedness, model operations, in- tegration views, and traceability links R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2 a megamodel operation module concept R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2 a robust incremental megamodel execution scheme R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3 megamodel interfaces R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='4 an asynchronous incremental megamodel execution scheme Note that concerning Table 1, the requirements cover here the modular con- struction as well as incremental execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' As visible in Table 1 there do exist three approaches that do not support modularity but provide a combination of all the other requirements we target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, neither of them provides the required robust incremental megamodel operation execution scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The tech- nique in [75, 74], while providing formal guarantees regarding correctness and termination, is limited to networks of model operations in the form of trees of synchronizations between pairs of models or requires the manual construction of an orientation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The Vitruvius approach [53], by virtue of employing a fixpoint iteration, does not introduce any restrictions regarding the network’s structure, but consequently does not guarantee termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The execution scheme presented in [41] is applicable to networks of model synchronizations between pairs of models with arbitrary structure and also guarantees termi- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, outside of performing the actual execution on the concrete instance, it provides no means of determining whether a network will eventually terminate with a correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='4 Summary of the state of the art This survey of the state of the art demonstrates that several approaches address the needs for modularity and incrementality raised in this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, none of them fulfill these needs at the three levels of model operations, modeling languages integration and megamodels that we identify as being required all at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Moreover, for certain individual aspects of Global Model Management, solutions with adequate modularity and incrementality do not even exists yet on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This work specifically targets these essential needs that have not been sufficiently addressed yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 16 4 Extended Generalized Discrimination Networks In this chapter, we introduce a notion of extended Generalized Discrimination Networks (eGDNs) and explain how the new formalism can be used as a language for megamodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1 Definition of eGDNs In order to address shortcomings of current solutions and enable the modular and incremental construction and execution of complex nets of model operations such as model properties, model consistency operations, model transformations, and model synchronization, we further generalize the idea of Generalized Dis- crimination Networks [7] to extended Generalized Discrimination Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, we introduce a generalized notion of GDN nodes and their inter- faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This enables the integration of model operations with side-effects and allows a more flexible definition of queries in comparison to [7], which also affords increased expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' An eGDN G = (O, S, E, s, t) is essentially a bipartite graph with two kinds of nodes, slot nodes and operation nodes, where O is the set of operation nodes and S is the set of slot nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Operation nodes can be connected to slot nodes and vice-versa via edges from the set of edges E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The source and target functions of edges are given by s : E → O ∪ S respectively t : E → O ∪ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Operation nodes represent model operations or building blocks thereof, that is, suboperations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Slot nodes store information used by model operations and their suboperations in the eGDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Edges represent dependency relationships be- tween operation and slot nodes, with the source of an edge representing the required node and the target of the edge representing the dependent node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' An operation node depending on a slot nodes indicates that the correspond- ing model operation uses information stored in that slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A slot node having a dependency on an operation node means that the operation node’s model operation modifies the slot’s contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We denote the set of dependencies of a slot or operation node n in O ∪ S by in(n) = {d ∈ O ∪ S|∃e ∈ E : s(e) = d ∧ t(e) = n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Similarly, we denote the set of dependent nodes of n by out(n) = {d ∈ O ∪ S|∃e ∈ E : s(e) = n ∧ t(e) = d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' G is bipartite in the sense that ∀o ∈ O : in(o) ⊆ S ∧ out(o) ⊆ S and ∀s ∈ S : in(s) ⊆ O ∧ out(s) ⊆ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For an operation node o ∈ O, we also refer to the set of slot nodes in(o) as the input slots of o and to the set of slot nodes out(o) as the output slots of o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A slot node s is always associated with a modeling language ML or an ordered set of variables var = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vk} and contains a model (typed graph) of ML respectively a set of variable assignments for var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A variable assignment for an ordered set of variables var = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vk} is given by a tuple in domV (v1) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' × domV (vk), where dom(vi) denotes the domain of variable vi, which can either be a set of nodes or edges from one or more 17 models or a set of primitives, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We refer to the set of possible con- tained assignment sets or models of s as the slot’s domain, which is given by dom(s) = ML in case s is associated with a modeling language ML or by dom(s) = P(domV (v1) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' × domV (vk)) if s is associated with an or- dered set of variables var = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Contents are then assigned to an eGDN’s slots via a valuation function val : S → � s∈S dom(s), such that ∀s ∈ S : val(s) ∈ dom(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In addition to regular models, we also allow model slots to contain linking models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The only difference between a regular model and linking model is the fact that a linking model’s set of vertices may reference vertices from other regular and linking models as edge targets, thus allowing the establishment of inter-model connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, similarly to linking models, the metamodel of a linking model, that is, the type graph of a linking model, may refer to vertices from other type graphs as edge targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Regarding operation nodes, we further distinguish between query nodes, transformation nodes, and mixed nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Query nodes extract information from models and/or other queries’ results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, a query node q may have an arbitrary number of input slots and exactly one output slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' q’s input slots may contain both models or sets of variable assignments, whereas q′s output slot may only contain a set of variable assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Transformation nodes create or modify models based on models and/or query results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, a transformation node t may have an arbitrary number of input and output slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' t’s input slots may contain both models or sets of variable assignments, whereas t’s output slots may only contain models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A mixed node x constitutes a combination of query and transformation nodes and may have an arbitrary number of input and output slots, which may contain both models or sets of variable assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Each operation node o with input slots in(o) = {si1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', sik} is associated with a semantics function γS : dom(si1) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' × dom(sik) → P(F), where F denotes the set of functions f : out(o) → � so∈out(o) dom(so) such that ∀so ∈ out(o) : f(so) ∈ dom(so).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Essentially, the semantics function of an operation node describes a consistency relationship between the operation’s input and output slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' To indicate that the contents of the slots adjacent to o are consistent with o’s semantics function for a valuation function val, we write o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Formally, o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val) ↔ ∃f ∈ γS(val(si1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', val(sik)) : ∀so ∈ out(o) : f(so) = val(so).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A valuation function val for an eGDN G = (O, S, E, s, t) is consistent with G as a whole if it holds that ∀o ∈ O : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2 eGDNs as Megamodels Since an eGDN encodes a network of model operations connecting a set of potentially integrated models, it represents a megamodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The definition of eGDNs thus constitutes a language for megamodels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 18 Importantly, eGDNs allow the composition of model operations from nodes that realize suboperations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In addition, they also allow hierarchical composition: An eGDN (and therefore also a basic GDN or RETE net) can be interpreted as an eGDN operation node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The input and output slots are given by the input respectively output slots of its nodes that are connected to another operation node of the parent eGDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Any slots of the child eGDN without a connection to another node of the parent eGDN can act as internal slots of the child and do not have to be exposed to the parent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, some such potential internal slots may also be considered input or output slots if their contents are relevant to human users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The semantics function of an operation node representing a sub-eGDN is then implicitly defined by the semantics functions of that eGDN’s own operation nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In addition to (hierarchical) composability, eGDNs support modularity in the sense that the semantics of an operation node regarding its output slots directly depend only on the contents of its immediate input slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Thereby, integrating additional operation nodes (along with additional slot nodes) into an eGDN only requires appropriate wiring with the node’s input and output slots, but is completely independent of any other operation nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Effectively, slots thus act as interfaces between model operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' eGDNs can also enable modularity at the model level by using the results of query nodes, potentially along with transformation nodes for propagating changes from the query results back to the base model, as model interfaces or views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For instance, simple projection queries in combination with access re- strictions can be employed to implement different visibilities for different roles in a development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Alternatively, dedicated view models in conjunc- tion with bidirectional model synchronization operations can similarly serve to implement editable model views in the context of eGDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Figure 2 shows our graphical notation for the visualization of eGDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Slot nodes are depicted as rectangles and labelled “A” in the top right corner in the case of assignment slots and “M” in the case of model slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Model slots that contain linking models are connected to the model slots containing the linked models via dashed arrows for visual clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Operation nodes are visualized as rectangles with rounded corners, with query nodes such as model properties or model consistency checks labelled “Q”, transformation nodes such as model transformations and model synchronizations labelled “T”, and mixed nodes such as sub-eGDNs labelled “X” in the top right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In addition, all nodes are labeled according to the schema ¡name¿:¡type¿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Figure 3 shows an example eGDN that consists of three slot nodes and two operation nodes and realizes a simple chain of model operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A class diagram stored in the leftmost slot is transformed into an abstract syntax graph via a transformation node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Then, a query node extracts some information from the created abstract syntax graph and makes the query result accessible via an assignment slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 19 x: Mixed X t: Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' T Transformation Node (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' model transformation, model synchronization) Mixed Node (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' subnet) q: Query Q Query Node (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' model property,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' model consistency) o: Operation Operation Node (linked to slots) a: Assignment A m: Model M Assignment Slot Model Slot m: LinkingModel M Model Slot (Linking Model) Figure 2: Graphical notation for eGDNs m1: ClassDiagram M t: Transform T q: Query Q m2: ASG M a: Result A Figure 3: Simple example eGDN 5 Incremental Execution of Extended General- ized Discrimination Networks In this chapter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' we describe how eGDNs can be executed to restore consistency in a network of models and model operations in reaction to external changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1 Definitions regarding Incremental Execution As a result of edit operations by a user, a model M in the model slot of an eGDN can undergo changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In this context, a change corresponds to the creation or deletion of a vertex or an edge and is characterized by an atomic model delta of one of four types: δV + is a single-element tuple (v), with v a vertex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' applying δV + to M modifies M into M ′ = (V M ∪ {v}, EM, sM, tM) δV − is a single-element tuple (v), with v ∈ V M, applying δV − to M modifies M into M ′ = (V M \\ {v}, EM, sM, tM) δE + is a tuple (e, s, t), with e an edge and s, t ∈ V M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' applying δE + to M modifies M into M ′ = (V M, EM ∪ {e}, sM ∪ {(e, s)}, tM ∪ {(e, t)}) δE − is a single-element tuple (e), with e ∈ EM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' applying δE − to M modifies M into M ′ = (V M, EM \\ {e}, sM \\ {(e, sM(e))}, tM \\ {(e, tM(e))}) 20 Importantly, this notion of atomic deltas can also cover the case of changes to attribute values in models in the form of typed attributed graphs [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In this context, attributes can be modeled via dedicated vertices representing attribute values and edges representing the assignment of these values to attributes of regular vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Note that we do not allow implicit deletion of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' If a vertex is deleted, it must not have any adjacent edges, that is, all adjacent edges have to be deleted previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Similarly, if an edge is created, adjacent vertices have to be present in the model already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Changes to the assignment set A in a slot s can similarly be described by atomic slot deltas: δA + is a single-element tuple (a), where a ∈ dom(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' applying δA + to the assignment set A modifies A into A′ = A ∪ {a} δA − is a single-element tuple (a), where a ∈ dom(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' applying δA − to the assignment set A modifies A into A′ = A \\ {a} For a slot node s, we denote the set of all possible atomic deltas over dom(s) by dom∆(s) and the set of all possible sequences of elements in dom∆(s) by S(dom∆(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Atomic deltas can be applied to a model or assignment set via an apply procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We overload this procedure to also work with a sequence of atomic deltas, in which case the procedure applies the individual deltas in the order specified by the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We say that a sequence of atomic deltas ∆ is minimal for the contents of a slot node s, iff for all possible contents v ∈ dom(s), it holds that ∄∆′ ∈ S(dom∆(s)) : apply(v, ∆′) = apply(v, ∆), where we only consider equality of graphs up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' To enable reacting to model changes with an eGDN G = (O, S, E, s, t), an operation node o ∈ O with input slots in(o) = {si1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', sik} and output slots out(o) = {so1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', sol} can be equipped with an update procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This pro- cedure is parametrized with a valuation function for G and realizes a func- tion γδ : dom(si1) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' × dom(sik) × S(dom∆(si1)) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' × S(dom∆(sik)) × dom(so1) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' × dom(sol) → F∆, with F∆ the set of functions f∆ : out(o) → � soi∈out(o) S(dom∆(soi)) such that ∀soi ∈ out(o) : f∆(soi) ∈ S(dom∆(soi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' To store deltas to react to later, o is also extended by an array o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆ that caches sequences of atomic deltas for its input and output slots, which can in practice be collected via a notification mechanism and the observer design pattern [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Calling o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='update(val) with val a valuation function for G’s slots then yields the value of γδ parametrized according to val and the cached sequences of deltas: o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='update(val) = γδ(val(si1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', val(sik), o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[si1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[sik], val(so1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', val(sol)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Intuitively, the update procedure of an operation node should produce a sequence of deltas for the node’s output slots that update the contents of these output slots to be consistent with the current contents of the operation node’s 21 input slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, in addition to the contents of slots adajcent to o, an update procedure may also consider additional information in the form of deltas to input slots to enable a more efficient realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Formally, an update procedure update of operation node o with input slots in(o) = {si1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', sik}, output slots out(o) = {so1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', sol}, and associated function γδ is correct iff for parameters ∆1 ∈ S(dom∆(si1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', ∆k ∈ S(dom∆(sik)), vi1 ∈ dom(si1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vik ∈ dom(sik), and vo1 ∈ dom(so1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vl ∈ dom(sol), ∃f ∈ γS(vi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vik) : ∀soi ∈ out(o) : apply(voi, f∆(so)) = f(soi), with f∆ = γδ(∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', ∆k, vi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vik, vo1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vol) and ∀i ∈ [1, k], j ∈ [1, l] : sii = soj → vii = voj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In many cases, the efficient realization of an update procedure requires a relaxed notion of correctness, which requires the contents of the output slot to be consistent with the contents of the input slots before the application of the deltas according to o’s semantics function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In the following, we will refer to this relaxed notion of correctness as conditional correctness, which is formally given by ∃v′ i1 ∈ dom(si1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', v′ ik ∈ dom(sik) : (apply(v′ i1, ∆1) = vi1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' ∧ apply(v′ ik, ∆k) = vik∧ ∃f ′ ∈ γS(v′ i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', v′ ik) : (∀sii ∈ in(o) ∩ out(o) : v′ ii = f ′(sii)∧ ∀soi ∈ out(o) \\ in(o) : voi = f ′(soi))) → ∃f ∈ γS(vi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vik) : ∀soi ∈ out(o) : apply(voi, f∆(so)) = f(soi), with f∆ = γδ(∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', ∆k, vi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vik, vo1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vol) and ∀i ∈ [1, k], j ∈ [1, l] : sii = soj → vii = voj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We say that the realization of an update procedure of an operation node o is fully incremental iff for a valuation function val and cached deltas ∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', ∆k with � i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=',k} |∆i| = 1, that is, a single atomic delta as an input, (i) the runtime complexity is in O(|∆o|), with ∆o = � so∈out(o) o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='update(val)(so), and (ii) the produced sets of deltas for each output slot are minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In some cases, as with bidirectional or in-place model transformations, op- eration nodes may be connected to a slot via both an incoming and an outgoing edge, making such a slot simultaneously an input and output slot to the same operation node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Such an operation node may as a result exhibit recursive be- havior, since an application of its update procedure can also change the contents of the operation node’s input slots and thus necessitate further calls to update to restore consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In this context, we call an update procedure of an oper- ation node o is non-recursive, if, after one execution of o’s update function and 22 subsequent application of the resulting deltas to o’s output slot values, a second execution with updated slot values never yields any new deltas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Formally, an update procedure of an operation node o with input slots in(o) = {si1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', sik} and output slots out(o) = {so1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', sol}, is non-recursive, if for any possible parametrization vi1 ∈ dom(si1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vik ∈ dom(sik), ∆1 ∈ S(dom∆(si1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', ∆k ∈ S(dom∆(sik)), and vo1 ∈ dom(so1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vol ∈ dom(sol), it holds that ∀so ∈ out(o) : γδ(∆′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', ∆′ k, v′ i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', v′ ik, v′ o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', v′ ol)(so) = ∅, where ∆′ i = � f∆(si) if sii ∈ out(o) ∆i otherwise and v′ ii = � apply(vii, f∆(si)) if sii ∈ out(o) vii otherwise and v′ oi = apply(voi, f∆(so1)), with f∆ = γδ(∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', ∆k, vi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vik, vo1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The potential update directions of an update procedure of operation node o for a set of input slots Si ⊆ in(o) are given by o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(o, Si), where for a slot so ∈ out(o), so ∈ o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(o, Si) ↔∃∆1 ∈ S(dom∆(si1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', ∆k ∈ S(dom∆(sik)), vi1 ∈ dom(si1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vik ∈ dom(sik), vo1 ∈ dom(so1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vol ∈ dom(sol) : ∀sii ∈ in(o) \\ Si : ∆i = ∅∧ γδ(∆1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', ∆k, vi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vik, vo1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=', vol)(so) ̸= ∅ Intuitively, o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(o, Si) thus denotes the subset of output slots for which o’s update procedure may generate deltas if the contents of at most the input slots in Si have changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A function dir∆ for potential update directions is monotonic by definition in the sense that ∀Si1, Si2 ⊆ in(o) : Si1 ⊆ Si2 → o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(o, Si1) ⊆ o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(o, Si2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We say that dir∆ is union monotonic if it furthermore holds that ∀Si1, Si2 ⊆ in(o) : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(Si1) ∪ o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(Si2) = o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(Si1 ∪ Si2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In the following, we present algorithms for the incremental execution of an eGDN based on the update procedures of its operation nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For these algorithms, we assume that deltas cached in the input eGDN are consistent in the sense that they correspond to a modification from slot contents that were 23 consistent with the semantics functions of all operations in the eGDN to the current contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Intuitively, this assumption simply implies that the presented algorithms can only produce consistent slot contents if the slot contents were previously consistent at some point and all changes since then have been tracked and cached in the eGDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2 Incremental Execution with Guaranteed Termination Given a correct update function for each operation node, an input eGDN G = (O, S, E, s, t) can be executed incrementally in the context of a valuation func- tion val via Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, Algorithm 1 first derives an ordering of G’s operation nodes and then updates the val function by executing the nodes’ update functions, applying the resulting deltas to the appropriate slots, and updating the cached deltas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Importantly, the employed ordering has to guarantee correct results in the sense that the contents of G’s slots after the execution must be consistent with the semantics functions of all of its operation nodes, that is, it must hold that ∀o ∈ O : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' If G takes the form of a directed acyclic graph and operation nodes do not share output slots, such an ordering can be obtained by simply sorting G’s operation nodes topologically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, requiring DAG structure represents a substantial restriction, as it effectively prohibits bidirectional transformations where some input slots are also output slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Moreover, the assumption regard- ing the complete absence of shared output slots, while required to prevent over- writing of operation’s results, is another obstacle to realizing several desirable use cases, for instance those involving chains of bidirectional transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Based on the properties of an eGDN’s operation nodes with respect to non- recursiveness and potential update directions, an appropriate order can also be found for certain cyclical eGDNs, with a relaxed assumption regarding shared output slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Algorithm 2 represents an analysis for an eGDN G that contains only nodes with non-recursive update procedures and a set of slots Si with ini- tially modified contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' If successful, the algorithm returns an execution order that can be used instead of the topological ordering in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Impor- tantly, the computed ordering still yields a valuation function that is consistent with all operations’ semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The algorithm first creates an array C with one cell per operation node in O and initializes it with empty sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' It also initializes a queue Q with all operation nodes that are connected to a slot in Si and, for each such operation node, stores the set of its input slots that are also in Si in the corresponding cell in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Then, a slightly modified breadth-first search is performed over the eGDN structure using the initialized queue Q to essentially simulate an execution of G without concrete inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, the procedure loops until Q is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In each loop execution, the first operation node o in Q is dequeued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Then, all output slot nodes for which deltas could be produced due to the execution of o′s update procedure So are obtained based on o’s potential update directions and the set of slots that might 24 Procedure ExecuteIncrementalDAG(G = (O, S, E, s, t), val) Input : G: The eGDN val: A valuation function for G’s slots 1 D ← FindValidUpdateOrder(O, {s ∈ S|∃o ∈ O : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[s] ̸= ∅});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 2 if D ̸= null then 3 foreach o ∈ D do 4 ∆o ← o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='update(val);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 5 foreach so ∈ out(o) do 6 val(so) ← apply(val(so), ∆o(so));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 7 foreach o′ ∈ out(so) do 8 o′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[so] ∪ ∆o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 9 end 10 end 11 foreach s ∈ in(s) ∪ out(s) do 12 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[s] ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 13 end 14 end 15 end Algorithm 1: Incremental algorithm for executing an eGDN based on an ordering of its operation nodes currently contain unhandled deltas, which is retrieved from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Afterwards, all operation nodes o′ connected to a slot in So are added to Q if they are not yet contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Also, the set of o’s input slots with potentially unhandled deltas stored in C is updated based on So.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' An exception is made for the currently considered node o, which is never added to the queue again and whose set of input slots with potentially unhandled deltas is reset to the empty set, exploiting the assumption that all update procedures in the eGDN are non-recursive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' During execution, the algorithm keeps track of the dependencies between G’s operations in a trigger graph GT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Execution aborts by returning null as soon as a cyclical dependency is detected, which may indicate a potential infinite loop in G’s execution for the initially populated slots Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This also guarantees that after a full execution of the loop in line 10, GT is a DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Finally, a topological ordering of GT , is returned as a possible canonic execu- tion order that, under the mentioned assumptions, produces a valuation function for the input eGDN’s slots that is consistent with the semantics functions of all of the eGDN’s operation nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' While the presented algorithm is formulated to handle incremental changes to a network of models and model operations, the batch case that requires an initial execution of model operations to derive corresponding query results and transformed models for an initial set of existing models can be handled in a straightforward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, an incremental construction of the initially existing models can be emulated by deriving trivial sequences of corresponding creation operations, which can act as the starting point for the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This 25 Procedure FindValidUpdateOrder(G = (O, S, E, s, t), Si) Input : G: The eGDN Si: The set of initially changed slots 1 C ← new Array(|O|);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='init(∅);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 3 Q ← new Queue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 4 GT = new Graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 5 foreach o ∈ in(Si) ∪ out(Si) do 6 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='enqueue(o);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 7 C[o] ← Si ∩ in(o);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 8 end 9 GT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='addV ertices(Q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 10 while ¬Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='isEmpty() do 11 o ← Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dequeue();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 12 So ← o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(o, C[o]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 13 Oo ← out(So) ∪ in(So) \\ {o};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 14 foreach o′ ∈ Oo do 15 if ¬o′ ∈ Q then 16 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='enqueue(o′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 17 end 18 C[o′] ← C[o′] ∪ (So ∩ in(o′));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 19 GT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='addV ertexIfNotExists(o′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 20 GT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='createEdgeIfNotExists(o, o′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 21 if GT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='hasCycle() then 22 return null;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 23 end 24 end 25 C[o] ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 26 end 27 return SortTopologically(GT );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Algorithm 2: Static analysis algorithm for finding an eGDN update order 26 only requires the assumption that the case where all slots of an eGDN are empty constitutes a consistent valuation regarding the semantics of all of the eGDN’s operations, which seems reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The additional assumption is essentially required to satisfy the rerquirement regarding consistency of initially cached deltas with the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Termination By including the additional termination criterion in the loop in line 10 of Algo- rithm 2 that requires the constructed dependency graph to be acyclic, Algorithm 2 is guaranteed to terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Algorithm 2 always terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Except for the loop in line 10, all loops only iterate over finite sets, and all individual operations always terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The loop in line 10 also always termi- nates due to the termination criterion regarding cyclical dependencies between the eGDN’s operation nodes: Since one operation node is removed from Q in each loop iteration, termination is only threatened if operation nodes keep get- ting added to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since there is only a finite number of operation nodes, infinite behavior can only occur as a result of cycles in the modified breadth-first search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, such cycles are detected via GT and immediately lead to abortion of the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Consequently, Algorithm 1 is also guaranteed to terminate if the execution of the input eGDN’s update procedures always terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For an input eGDN G, Algorithm 1 always terminates if the update procedures of G’s operation nodes always terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' According to Theorem 1, Algorithm 2 always terminates by either abort- ing or returning a sequence of operation nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Such a sequence being returned implies that the sequence is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The loop in line 3 is thus only executed for finitely many iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since all other loops only iterate over finite sets and all individual operations always terminate due to the assumption regarding G’s update procedures, Algorithm 1 always terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Correctness The following theorem states the correctness of a canonic execution order re- sulting from an execution of Algorithm 2 for the case that all dir∆ functions are union monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For inputs G = (O, S, E, s, t) and val, if all update procedures in G are correct and non-recursive, all dir∆ functions in G are union-monotonic, and if the valuation function before the application of the deltas cached in G was consistent with the semantics of G’s operation nodes, Algorithm 1 aborts or produces a final valuation function val such that ∀o ∈ O : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 27 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' If Algorithm 1 does not abort, a canonic execution order R for G’s op- eration nodes has been generated by topologically sorting the resulting directed acyclic dependency graph GT of a terminating execution of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Due to the non-recursiveness of G’s operation nodes, we know that after executing an operation node o via Algorithm 1, it holds that o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Thus, for an operation node o, ¬o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val) can only hold after executing the entire sequence R if there exists some operation node o’ that comes after o in R and that changes the contents of a slot adjacent to o or if o /∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Considering that all operation nodes that have an adjacent slot with initially modified contents are initially added to GT , the algorithm has terminated, and that prior to the cached modifications of G’s slots, slot contents were consistent with all operation nodes’ semantics, for a node o /∈ R, ¬o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val) can also only hold if there is some operation node o′ ∈ R that changes the contents of a slot s adjacent to o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In either case, we know that there cannot exist an edge from o′ to o in GT , because o′ either comes after o in the topological ordering or because the addition of such an edge would have caused o to be added to GT and conse- quently R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This means that, due to the definition of o′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆ and because of the assumed union monotonicity, there must be a slot s′ with o′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[s] ̸= ∅ and s ∈ o′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆({s′}) before executing o′ that was never in the set of slots C[o′] when o′ was dequeued in line 11 of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since slots are only removed from C[o′] when o′ is dequeued and corresponding edges are added, we know that s′ /∈ Si and thus o′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[s′] = ∅ at the start of Algorithm 1, as otherwise, o′ would have been added to Q and the edge between o′ and o would eventually have been created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' There hence must be a node o′′ that comes before o′ in R that modified the contents of s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Also for o′′, there must be a slot s′′ with o′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[s′′] ̸= ∅ and s′ ∈ o′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆({s′′}) before executing o′′ that was never in C[o′′] whenever o′′ was dequeued (because otherwise, the edge between o′ and o would have been created eventually).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, again, there must be an operation node before o′′ in R that modified the contents of s′′ and for which the same constraints apply as for o′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Ultimately, this implies that for the first operation node in the sequence, there must be a predecessor that changes the contents of some slot node, which is obviously a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Hence, there cannot be an operation node in R whose execution changes the contents of a slot adjacent to a previous operation node in R or an operation node not contained in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Consequently, we know that after executing R, ∀o ∈ O : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' If the eGDN’s update functions are only conditionally correct, an additional constraint has to be introduced regarding eGDN structure to guarantee correct- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Namely, operation nodes may not share output slots if the output slots are not also input slots of all sharing operation nodes, and output slots of a node that are not simultaneously input slots of the same node may not have their contents modified by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Intuitively, these conditions impose the restriction on the eGDN structure that the contents of an operation node’s output slot may not be modified by 28 another operation node or a user without that operation node being able to pick up on and handle the changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Assuming that ∀o1, o2 ∈ O : o1 ̸= o2 → ∀s ∈ out(o1) ∩ out(o2) : s ∈ in(o1) ∧ s ∈ in(o2) and ∀o ∈ O : ∀s ∈ out(o) : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[s] = ∅, for inputs G = (O, S, E, s, t) and val, if all update procedures in G are conditionally correct and non-recursive, all dir∆ functions in G are union-monotonic, and if the valuation function before the deltas cached in G was consistent with the semantics of G’s operation nodes, Algorithm 1 aborts or produces a final valuation function val such that ∀o ∈ O : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' From the additional assumptions regarding output slots of G’s operation nodes it follows directly that the condition in the definition of conditional cor- rectness is never violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Thus, the statement from Theorem 3 also applies for the case of conditionally correct update functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Notably, the order in which operation nodes are added to the queue Q in lines 5 and 14 of Algorithm 2 is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since the order of operation nodes in Q affects the behavior of the algorithm, this might mean that Algorithm 2 is ultimately not deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We can however show that, if Algorithm 2 does not abort due to cycles in GT , the final dependency graph GT is uniquely defined, independently of the order in which operation nodes are added to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Thus, the only remaining nondeterminism in Algorithm 2 affecting the result stems from the topological sorting at the end of the algorithm, which is an inherently nondeterministic operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For inputs G = (O, S, E, s, t) and Si, the dependency graph GT after a full execution of the loop in line 10 of Algorithm 2 is uniquely defined up to isomorphism if all dir∆ functions in G are union monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The set of vertices initially added to GT is uniquely determined by in(Si) ∪ out(Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since additional vertices are only ever added in conjunction with the creation of an edge, the set of vertices added during the execution of the loop in line 10 is determined by the set of added edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' To show the unique determination of added edges by the algorithm’s inputs, we show that in a terminating execution of the loop, the initial set Si in con- junction with the eGDN G uniquely determines a set of pairs of operation nodes (o1, o2), between which directed edges are created in GT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Si uniquely determines the set of operation nodes OQ = in(Si) ∪ out(Si) that is initially added to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For each of these operation nodes oQ ∈ OQ, due to the monotonicity of oQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆ and because slots are only removed from C[oQ] after oQ has been dequeued and processed, at least the edges for pairs edgesS(oQ, Si) = {(oQ, oT )|oT ∈ out(So) ∪ in(So) \\ {oQ}} are added to GT , where So = oQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(Si ∩ in(oQ)) when oQ is dequeued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' According to the assumption regarding union monotonicity, we can also write edgesS(oQ, Si) = edges∅(oQ) ∪ � si∈Si∩in(oQ) edgesN(oQ, si), with edges∅(oQ) = {(oQ, oT )|oT ∈ 29 out(oQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(∅)) ∪ in(oQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(∅)) \\ {oQ}} and edgesN(oQ, si) = {(oQ, oT )|oT ∈ out(oQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(in(oQ) ∩ {si})) ∪ in(oQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(in(oQ) ∩ {si})) \\ {oQ}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In addition, the modification of C and Q that takes place for each dequeued oQ ∈ OQ may cause the addition of further edges down the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Specifically, for each so ∈ oQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆({si}) and each oT ∈ out(so)∪in(so)\\{oQ}, oT , if not already contained, is added to Q and subsequently handled in the same way as oQ, with si guaranteed to be in C[oT ] at that moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This will cause the addition of all edges corresponding to the pairs edgesN(oT , so) and again trigger the addition of further edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Due to the monotonicity of oT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆ and because slots are only removed from C[oT ] when oT is dequeued, the addition of these edges happens independently from any other modifications to C[oT ] that might be made in the meantime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Furthermore, due to the assumption regarding union monotonicity of oT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆, a combination of modifications of C[oT ] cannot yield any additional edges compared to what is yielded for the individual members of C[oT ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Because neither can C[oT ] be modified in any other way, nor can edges be added to GT in any other way, the set of pairs of operation nodes (o1, o2) between which directed edges are created in GT is given by the function edges(Si) = � oQ∈in(Si)∪out(Si)(edges∅(oQ)∪� si∈Si∩in(oQ) edgesR(oQ, si)), with edgesR(oQ, si) = edgesN(oQ, si)∪� oT ∈OT ))\\{oQ} � so∈oQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(in(oQ)∩{si} edgesR(oT , so), where OT is given by OT = out(oQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(in(oQ) ∩ {si})) ∪ in(oQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(in(oQ) ∩ {si}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The loop terminating due to Q becoming empty implies that all nodes ever added to Q have been processed and hence all corresponding edges have been added to GT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since it is ensured that for each pair of operation nodes (o1, o2), only one corresponding edge is added, we know that regardless of the concrete processing order, GT always contains exactly one directed edge for each pair (o1, o2) ∈ edges(Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since the set of added vertices is uniquely determined by the set of added edges and each vertex can only be added once, the set of GT ’s vertices is uniquely defined for inputs G and Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The graph GT at the end of a full execution of the loop in line 10 of Algorithm 2 is hence uniquely defined for inputs G and Si, regardless of the order in which operation nodes are added to Q in lines 5 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The fact that GT is uniquely defined by the inputs G and Si also implies that if an execution of Algorithm 1 terminates without aborting, so does any possible execution for the same inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' An execution of the loop in line 10 of Algorithm 1 terminates without aborting for inputs G = (O, S, E, s, t) and Si if and only if any other execution for the same inputs also terminates without aborting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' According to Theorem 1, the loop in line 10 of Algorithm 1 always ter- minates, either because of a violation of the looping condition or because the loop aborts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since the loop aborts if and only if a cycle is detected in GT at any point and edges are never removed from GT , it follows that the loop terminates without aborting if and only if the set of edges added to GT during the loop execution does not form cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since the set of edges added to GT during the 30 loop execution is functionally determined by only the inputs G and Si, it hence follows that, if an execution of the loop terminates without aborting for G and Si, any execution with the same inputs will also terminate without aborting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Furthermore, we can show that if there exists an execution sequence for G that guarantees correct results in the worst case and that executes every operation node at most once, Algorithm 2 finds such a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For an input eGDN G = (O, S, E, s, t) with correct and non- recursive update procedures with union monotonic dir∆ functions and a set of slots Si ⊆ S with initially modified contents for a valuation function val, assuming that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' for any operation node o1 ∈ O, for any execution of o1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='update(val′) with a valuation function val′ and deltas for input slots S∆, it holds that ∀so ∈ o1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(S∆) : o1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='update(val′)(so) ̸= ∅, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' for a second node o2 ∈ O with o1 ̸= o2, it holds that ∃so ∈ out(o1)∩in(o2) : o1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='update(val′)(so) ̸= ∅ → ¬o2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val′′), where for s ∈ S val′′(s) = � apply(val′(s), o1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='update(val′)(s)) if s ∈ out(o1) ∩ in(o2) val′(s) otherwise (1) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' it holds that ∀o ∈ in(Si) ∪ out(Si) : ¬o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val), if there exists a sequence that guarantees a correct resulting valuation function if executed via Algorithm 1 and that only contains each node o ∈ O once, Algorithm 2 returns such a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Under the given assumptions, the set of edges in GT created by Algo- rithm before termination or abortion represents a subset of all relations between pairs of operation nodes (o1, o2), where o1’s update procedure has to be executed at least once to produce a correct final valuation function and that execution modifies the contents of a slot adjacent to o2, necessitating the subsequent exe- cution of o2 according to assumption (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This is due to the fact that, to restore consistency, all operation nodes in o ∈ in(Si) ∪ out(Si) have to be executed at least once according to assumptions (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' All these operation nodes o1 are initially added to the queue Q in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Each execution of an operation node o1, according to assumption (1), modifies all slots in o1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(Si ∩ in(o1)), which necessitates a subsequent execution of all operation nodes o2 ∈ in(o1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(Si ∩in(o1)))∪out(o1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(Si ∩ in(o1))) according to assumption (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Algorithm 2 creates edges for all these pairs (o1, o2) when o1 is dequeued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The subsequent execution of any operation node o2 similarly necessitates the execution of all nodes o3 ∈ in(o2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(Si ∩ in(o2))) ∪ out(o2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='dir∆(Si ∩ in(o2))), which is also reflected by the edges created in Algorithm 2 when o2 is dequeued, 31 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since the algorithm creates no additional edges due to the assumption regarding union monotonicity of the dir∆ functions, all edges in GT represent such necessary relationships on the ordering of operation nodes5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since Algorithm 2 always produces a correct sequence of operation nodes if GT is acyclic, we can assume that in the case where the algorithm does not produce an ordering, there is at least one cycle in GT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' There hence cannot exist a sequence of the operation nodes involved in this cycle where each node is only contained once and each node is executed at least once after its predecessor in the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Thus, by contraposition it follows that, if there exists a sequence of operation nodes that guarantees correct results and where each operation node is only contained once, Algorithm 2 finds such a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Note that there may be finite orders of operation node executions that guar- antee correct results based on the assumptions in Theorem 6 that are not found by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, these orders require that at least one operation node is executed at least twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3 Incremental Execution of Arbitrary eGDNs If the eGDN is not a DAG and no suitable ordering of its operation nodes can be found via Algorithm 2, incremental execution can instead be achieved via a simple fixpoint iteration as in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Algorithm 3 first initializes the set of operation nodes that require execution D with the set of all operation nodes in the input eGDN for which there are changes in one of the node’s input or output slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Then, the algorithm iterates until a fixpoint is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, a set of operation nodes that will require execution in the next iteration Dn is initialized with the empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Afterwards, for each operation node o that is due for execution in the current iteration, that node is removed from the set D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Then, o’s update procedure is called to compute a set of changes to the contents of o’s output slots to make them consistent with the semantics of o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For each output slot so of o that update has computed changes for, these changes are subsequently applied and appropriately registered at each operation node o′ for which so is an input slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' If any such o′ is not still due for execution in the current iteration, it is marked for execution in the next iteration by adding it to Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Operation nodes for which so is an output slot are similarly marked for execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Finally, after all operation nodes in D have been considered, Dn replaces D and a new iteration starts if Dn is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Analogously to Algorithm 1, Algorithm 3 can handle the batch case of an initial eGDN execution for existing models by encoding such existing models as sequences of element creations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 5As a side note, since operation nodes can be dequeued/executed with different sets of potentially modified input slots, an edge between nodes (o1, o2) in GT does not necessarily mean that o2 has to be executed after any execution of o1, but only that such a subsequent execution is necessary at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 32 Procedure ExecuteIncremental(G = (O, S, E, s, t), val) Input : G: The eGDN val: A valuation function for G’s slots 1 D ← {o ∈ O|∃s ∈ in(o) ∪ out(o) : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[s] ̸= ∅};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 2 while D ̸= ∅ do 3 Dn ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 4 foreach o ∈ D do 5 D ← D \\ {o};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 6 ∆o ← o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='update(val);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 7 foreach s ∈ in(s) ∪ out(s) do 8 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[s] ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 9 end 10 foreach so ∈ out(o) do 11 if ∆o(so) ̸= ∅ then 12 val(so) ← apply(val(so), ∆o(so));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 13 foreach o′ ∈ out(so) do 14 o′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[so] ∪ ∆o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 15 if o′ /∈ D then 16 Dn ← Dn ∪ {o′};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 17 end 18 end 19 foreach o′ ∈ in(so) do 20 if o′ ̸= o ∧ o′ /∈ D then 21 Dn ← Dn ∪ {o′};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 22 end 23 end 24 end 25 end 26 end 27 D ← Dn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 28 end Algorithm 3: Incremental algorithm for eGDN execution 33 Termination In contrast to Algorithm 1, Algorithm 3 is not guaranteed to terminate, since cyclical transitive dependencies of operation nodes may cause infinite cycles of changes to the contents of some slot node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Without restricting developers in what kinds of eGDNs they are allowed to specify, this problem is inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In practice however, termination of networks of model operations like eGDNs can be achieved despite the presence of cyclical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In some cases for instance, cycles at the network level do not necessarily correspond to actual cyclical dependencies of model operations if the involved model operations only affect distinct parts of slot contents, such as elements of certain, distinct types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In some cases, a restructuring of the eGDN may remove cycles at the struc- tural level while preserving semantics, for instance by converting in-place model transformations without an effective reflexive dependency into a model trans- formation with distinct input and output models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Moreover, cycles of model operations may exhibit monotonic behavior, for instance by deleting certain elements in each iteration that are never recreated, thus guaranteeing convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Ultimately however, it remains the responsibil- ity of the developers to create networks of model operations that do not lead to infinite loops in execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Correctness If Algorithm 3 terminates, the resulting valuation function is guaranteed to be consistent with the semantics of all operation nodes in the input eGDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For inputs G = (O, S, E, s, t) and val, if Algorithm 3 terminates, all employed update procedures are correct and non-recursive, and if the valua- tion function before the application of the deltas cached in G was consistent with the semantics of G’s operation nodes, the algorithm produces a final valuation function val such that ∀o ∈ O : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We show that the invariant (1) ∀o ∈ O : ¬o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val) → o ∈ D holds for the loop in line 2 via induction over the number of loop iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The base case for invariant (1) holds due to the initialization of D and the assumption regarding the initial cached deltas and previous valuation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' To show the induction step for invariant (1), we first show that under the induction assumption, the invariant (2) ∀o ∈ O : ¬o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val) → o ∈ D ∪ Dn holds for the loop in line 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This can also be done via induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The base case for invariant (2) holds due to the induction assumption of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The induction step holds for invariant (2) since in each iteration of the inner loop, only one operation node o is executed via its update procedure and removed from D, updating val and the cached deltas in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' If the execution of o does not change the contents of one of its own input slots, we know that afterwards, o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val) due to the assumption regarding correctness of update procedures and because the cached deltas are always updated correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Otherwise, o is added to Dn in the loop in line 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The loops in line 13 and 19 34 also add all operation nodes o′ to Dn for which the result of o′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val) may have been impacted by the update to val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Thus, given the induction assumption, at the end of the loop in line 10, we again have ∀o ∈ O : ¬o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val) → o ∈ D ∪ Dn and hence the induction step holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since at the end of the loop in line 10, D = ∅, we know that ∀o ∈ O : ¬o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val) → o ∈ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Because at the end of the iteration of the loop in line 2, the set D is replaced by Dn, the induction step for (1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since the loop in line 2 is only left when D = ∅ after the replacement with Dn, we know that, if the algorithm terminates, ∀o ∈ O : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Similar to Algorithm 1, the algorithm also yields correct results if all em- ployed update procedures are at least conditionally correct, the eGDN’s nodes do not share output slots that are not also input slots to all sharing nodes, and there are no deltas for an output slot of a node that is not simultaneously an input slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Assuming that ∀o1, o2 ∈ O : o1 ̸= o2 → ∀s ∈ out(o1) ∩ out(o2) : s ∈ in(o1) ∧ s ∈ in(o2) and ∀o ∈ O : ∀s ∈ out(o) : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='∆[s] = ∅, for inputs G = (O, S, E, s, t) and val, if Algorithm 3 terminates, all employed update procedures are conditionally correct and non-recursive, and if the valuation function before the deltas cached in G was consistent with the semantics of G’s operation nodes, it produces a final valuation function val such that ∀o ∈ O : o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='valid(val).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' From the additional assumptions regarding output slots of G’s operation nodes, it follows directly that the condition in the definition of conditional cor- rectness is never violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Thus, the statement from Theorem 7 also applies for the case of conditionally correct update functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='4 Development with eGDNs Since Algorithm 2 considers only the eGDN structure and no concrete slot con- tents, it can be employed as a tool for statically analyzing eGDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In particular, via the algorithm, configurations of slots with modified contents can be analyzed regarding termination of a corresponding eGDN execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For instance, the algorithm can be used to check whether termination is guaranteed if a specific individual model is modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' If this is the case for all user-editable models, a conservative approach that always guarantees terminating eGDN executions and correct results while avoid- ing the exponential effort of executing the analysis for every combination of user- editable models would be enforcing a direct propagation policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Under this policy, after modifying a single model, the corresponding changes would imme- diately be propagated to restore consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Only after that, the modification of a different model would be permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Furthermore, Algorithm 2 can be adapted to return the set of slots closure∆(Si) that may be automatically modified by eGDN operations if the eGDN were to be executed via Algorithm 1 with initially modified slots Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This enables col- laborative development of a network of models managed via an eGDN with 35 guaranteed termination and conflict-free consistency restoration via a propa- gation closure locking policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For a set of already modified slots S∆, this policy would only allow modification of the contents of another slot s if, for the set S∆ ∪ {s}, Algorithm 2 produces an execution order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Furthermore, to guarantee that no user edits are overwritten, the policy would check whether S∆ ∪ {s} ∩ closure∆(S∆ ∪ {s}) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Note that the restrictions of this policy would also apply in the case where the same user wants to edit the contents of multiple slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Since Algorithm 3 does not guarantee termination, careful consideration is required if an eGDN cannot be executed via Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, if developers are confident that their eGDN is guaranteed to terminate despite cyclical de- pendencies at the structural level, Algorithm 3 can be used as a fallback option for eGDN execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The presented algorithms also enable the treatment of sub-eGDNs as oper- ation nodes of a parent eGDN, as they essentially provide a realization of the required update procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 6 Implementation We have prototypically implemented a number of concrete example operation node types for the construction of eGDNs for usage in the context of the Eclipse Modeling Framework (EMF) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In addition to listing the implemented oper- ations’ names, Table 2 also provides brief descriptions of their behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Table 3 characterizes our implementations in terms of the properties defined in this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Non-recursiveness: The update procedure of TGG Snychronisation oper- ations is non-recursive if the slots containing source, target, and correspondence model are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The non-recursiveness of composite nodes depends on the exact composition of the sub-eGDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' All other nodes’ update procedures are only guaranteed to be non-recursive if their input and output slots are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The checkmark symbol ✓indicates non-recursive update procedures under this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Potential Population/Update Directions: The potential update di- rections of the TGG Synchronization (↔) can be characterized as follows (under the assumption of distinct slots for source, target, and correspondence model): If the set of considered input slots is empty, no modifications will be made to the contents of any output slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' If the set of considered input slots contains only the source model, the operation will only modify the target model and corre- spondence model and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In all other cases, all models may be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The potential update directions of composite nodes are determined by the exact structure of the sub-eGDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' All other nodes may modify the contents of all of their output slots for any set of considered input slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The potential update direction function dir∆ of all example nodes is union monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Correctness: The update procedures of all operation implementations are only conditionally correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Effectively, this means that operations may not share 36 output slots and no user edits are allowed to output slots of operation nodes, unless the shared or edited output slot is also simultaneously an input slot of the concerned operation nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Incrementality: The checkmark symbol ✓indicates a fully incremental update procedure under the assumption of ideal data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Also opera- tions which are listed as not fully incremental support incremental execution to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The degree of incrementality depends on the operation and its concrete inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Naturally, our implementation of the Expression node is only fully incremental if the evaluation of the considered expression has a runtime complexity in O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In the case of the Pattern Matching and TGG Synchroniza- tion node, a fully incremental execution can be achieved for certain input models and patterns respectively TGGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The degree of incrementality of the execution of an eGDN or sub-eGDN depends on which slots are designated the eGDN’s interface slots, as well as the contained operation nodes and their composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' While the Group Expression node also has a partially incremental update pro- cedure, due to the handling of collections via the employed OCL-interpreter, a fully incremental execution is usually not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' As interfaces between these operations, that is, slot nodes, our implemen- tation employs regular EMF models for model slots and hash-based indices for assignment slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' While the choice of hash-based indices over array-based in- dices means that the theoretically fully incremental operation implementations may not be fully incremental in conjunction with our slot implementations, hash-based data structures are usually preferable in practice due to their lower memory footprint and exhibit acceptable performance in most scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Figure 3 shows a more complex version of the example eGDN from Figure 3 that can be realized using the introduced example eGDN nodes from Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The transformation from class diagram to abstract syntax graph is now concretely realized via a unidirectional TGG Synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The query op- eration that was previously represented by a single query node is decomposed into a complex network of subqueries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This sub-eGDN consists of two Pattern Matching nodes labeled “x → y” that look for primitive patterns consisting of a single edge, one Group Count and one Group Sum node visualized as nodes labeled “COUNT (X)” respectively “SUM (X)”, and a Join node labeled “▷◁”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Alternatively, the Pattern Matching nodes could also be realized as Edge Inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 7 Evaluation In this chapter, we report on an initial empirical evaluation based on our proto- typical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Moreover, we describe how eGDNs can be employed in a typical application scenario, evaluating the developed approach with respect to the requirements from Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 37 Name Description RETE Nodes Node Input extracts individual nodes of a given type from a model Edge Input extracts individual edges of a given type from a model Join performs a natural join of assignments stored in two input assignment slots Anti-Join performs an anti-join of a left input assignment slot against a right input assignment slot GDN Nodes Pattern Matching finds matches for a given pattern into a model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' supports additional constraints formulated in OCL [62];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' supports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='constraints regarding the existence/absence of matches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='for other patterns via dependencies to related assign- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='ment slots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Property Computation Nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Expression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='computes the value of an OCL [62] expression for indi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='vidual assignments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Group Expression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='computes the value of an OCL [62] expression for col- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='lections of assignments grouped by certain variables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Group Count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='counts the number of assignments in collections of as- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='signments grouped by certain variables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Group Sum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='computes the sum of numerical values of a specific vari- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='able in collections of assignments grouped by certain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='variables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Transformation Nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='TGG Sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (→) performs unidirectional model synchronization of changes from a source to a target and associated cor- respondence model via a triple graph grammar [67] TGG Sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (↔) performs bidirectional model synchronization of changes between a source, target, and associated correspondence model via a triple graph grammar [67] Composite Nodes eGDN executes a sub-eGDN to update the contents of exposed slots via Algorithm 1 or Algorithm 3 Table 2: Example eGDN node types 38 Name Non-recursive Directions Correct Incremental RETE Nodes Node Input ✓ all cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' ✓ Edge Input ✓ all cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' ✓ Join ✓ all cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' ✓ Anti-Join ✓ all cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' ✓ GDN Nodes Pattern Matching ✓ all cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (✓) Property Computation Nodes Expression ✓ all cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (✓) Group Expression ✓ all cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' ∼ Group Count ✓ all cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' ✓ Group Sum ✓ all cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' ✓ Transformation Nodes TGG Sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (→) ✓ all cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (✓) TGG Sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (↔) ✓ cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (✓) Composite Nodes eGDN ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (✓) Table 3: Properties of example eGDN node types m1: ClassDiagram M t: TGG Sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (→) T q3: COUNT (per Type) Q m2: ASG M a5: Assignment A q1: Type → Method Q a1: Assignment A q2: Package → Type Q a2: Assignment A a3: Assignment A q4: Q a4: Assignment A q5: SUM (per Package) Q q: eGDN Q m3: Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Model M Figure 4: Complex example eGDN 39 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1 Evaluation of Performance For an initial empirical evaluation of the proposed approach, we perform an experiment inspired by an application scenario from the software development domain, where an evolving class diagram serves as the basis for generating object-oriented code, which is subsequently analyzed to compute code metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, we have implemented a simple model transformation from Ecore models [26] to Java abstract syntax graphs [18] via a triple graph grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' For each class in the class diagram, the transformation creates an interface in the Java abstract syntax graph in a first package, along with an implementation class in a second package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Also, for each attribute of a class in the class diagram, the transformation creates a corresponding field and associated getter and setter methods in the corresponding interface and class in the abstract syntax graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In addition, we have realized a model query composed of several subqueries, which counts the number of methods in all types of a Java package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The trans- formation and query are integrated into an eGDN, which yields the structure displayed in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Using our prototypical implementation, which is available under [30], we assign a real-world Ecore model [18] to the class diagram model slot and per- form an initial population of the remaining slots via Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' To evaluate the scalability of the eGDN, we then apply a number of synthetic updates to the model in the class diagram slot, each of which adds an attribute to each class in the model, and measure the time required for the eGDN to process each such update via Algorithm 1 (“INCREMENTAL”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We compare this to a baseline, where instead, we perform a full recomputation of both the model transformation’s and the query’s results via non-incremental implementations of the corresponding operations (“BATCH”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='6 Figure 5 displays the execution times for the first 30 updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' After an initial phase comprising the first 5 updates, where execution time decreases from up- date to update, the execution time for processing an update to the class diagram via the strategy INCREMENTAL does not change much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In particular, there does not seem to be any trend of increasing execution time related to the growth of the class diagram as additional updates are being performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In contrast, the execution time of BATCH increases from update to update as the class diagram grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' While it starts out similar to the execution time of INCREMENTAL (larger by factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='6), by update 30 the execution time of BATCH has increased to factor 80 compared to the execution time of INCREMENTAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The measurements thus indicate that incremental eGDN execution via IN- CREMENTAL efficiently handles updates to the class diagram, in the sense that execution time only seems to depend on the actual changes rather than the size of the model, indeed affording incrementality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, eGDNs seem to consti- tute a suitable formalism for a scalable, modular and incremental realization of 6All experiments were performed on a Linux SMP Debian 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='67-2 machine with Intel Xeon E5-2630 CPU (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3 GHz clock rate) and 386 GB system memory running OpenJDK version 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Reported execution time measurements correspond to the mean execution time of 10 runs of the respective experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 40 0 5000 10000 15000 20000 5 10 15 20 25 30 execution time (ms) update number INCREMENTAL BATCH Figure 5: Execution time measurements for class diagram updates networks of model operations for this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The decreasing execution times per update during the initial phase of the experiment can likely be attributed to warming-up effects of the Java virtual machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The internal validity of our results is mostly threatened by unexpected be- havior of the Java virtual machine, most notably garbage collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' To miti- gate such effects, the reported execution time measurements were obtained as the arithmetic mean of multiple runs of the experiment, with the standard de- viation of the overall execution time always below 5% of the overall execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The synthetic updates used in the experiment pose a threat to external validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' However, the experiment is inspired by a real-world application scenario and uses a real-world model as its basis and demonstrates the applicability of the eGDN approach in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The synthetic updates only serve the purpose of allowing a systematic evaluation of our technique’s scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We hence do not make any quantitative claims regarding our approach in practical application scenarios, but merely consider our experimental results as an indicator for the presented approach’s potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We furthermore do not make claims regarding the generalizability of the approach to other application domains, which would require further evaluation and is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2 Evaluation of Applicability In order to investigate the applicability of the developed technique, we consider the following extended example scenario that requires global model manage- ment: A class diagram, adhering to a metamodel similar to the one displayed in Figure 1, is used to model the structure of a software system under develop- ment by means of classes contained in packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Classes may contain methods, which may in turn reference classes as the method’s return type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' OCL expres- sions in a separate model are used to describe the behavior of some of the class diagram’s methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Therefore, the OCL model has its own representation of types corresponding to the classes in the class diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This correspondence is captured by means of a linking model, which simply contains dedicated link vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' A link vertex can either have edges to a class from the class diagram 41 and the corresponding type from the OCL model or edges to a method in the class diagram and the corresponding expression, that is, implementation, in the OCL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' We consider the following use cases for this setup: Consistency Checking: The developers want to run automatic and in- cremental consistency checks that verify that the return type of a method in the class diagram matches the corresponding type of the method’s OCL- implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The developed consistency check should also work for similar setups that use a different expression language than OCL for the method implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' An implementation for this use case thus re- quires a solution satisfying the requirements R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1, R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2, R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2, and R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Code Generation: The developers want to automatically and incremen- tally generate Java code in the form of an ASG from the class diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In addition, Java implementations for the class diagram’s methods should be generated from the methods’ OCL implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In the end, the resulting Java code fragments for the two models should be integrated and analyzed for some code metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' An implementation for this use case thus requires a solution satisfying the requirements R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1, R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2, and R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Megamodel Reuse: After developing the automatic consistency check- ing and code generation, the developers want to reuse the same two op- erations in another project with a similar set of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' An implementa- tion for this use case thus requires a solution satisfying the requirements R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2, R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2, R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3, and R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In order to allow global model management for all three use cases, a solu- tion also has to enable the modeling of a network of different kinds of model operations over a set of potentially integrated models, that is, a solution has to satisfy requirement R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Figures 7, 6, and 8 visualize example eGDN implementations for the Con- sistency Checking, Code Generation, and Megamodel Reuse use cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' As displayed in Figure 7, the Consistency Checking use case is realized via four Pattern Matching query nodes that extract certain simple patterns from the base models and make them accessible in a generalized format via assignment slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Then, a complex query operation, which is composed of three Join query operations and an Anti-Join query operation (labelled ▷), realizes the actual consistency check by finding all the combinations of a method from the class diagram, its implementation from the OCL model, and the associated return class respectively expression type, where the return class and expression type do not correspond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Thus, the eGDN-based approach in this case fulfills the requirements R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1 and R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2, as it implements a consistency check over a set of models integrated via integration links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Furthermore, the resulting implementation is reusable for different modeling languages that offer similar 42 functionality via the generic interface provided by the assignment slots a1, a2, a3, and a7, satisfying requirement R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The example eGDN also demonstrates how more complex model operations can be composed from simpler operations, satisfying requirement R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1, and provides an incremental execution scheme for these operations, satisfying requirement R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In particular, via Algorithm 2, it can be verified that Algorithm 1 provides a means of executing the eGDN that guarantees both correct results and termination for changes to any combination of the three base models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The eGDN shown in Figure 6 realizes the Code Generation use case via a combination of two TGG Synchronizations that translate the class diagram and OCL model into Java ASGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The two Java models are integrated via a dedicated linking model, which is produced by a unidirectional model transformation from the original linking model and the correspondence models created by the TGG Synchronizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Finally, query operations can be executed over the ASGs to compute code metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Using Algorithm 2 to analyze the eGDN, it can be determined that terminating execution via Algorithm 1 can be guaranteed for changes to any combination of the three base models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This example shows how the eGDN provides a unified, modular notion of model operations along with an incremental execution scheme and demonstrates the composition of model operations, satisfying requirements R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1, R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2, and R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Together, the eGDNs in Figure 7 and 6 also illustrate how eGDNs can be used as a megamodeling language, supporting different kinds of model opera- tions, including model properties (like the metrics computed in the Code Gener- ation use case), model consistency (like the consistency condition in the Consis- tency Checking use case), and model transformation and synchronization (like the transformation and synchronizations in the Code Generation use case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' It also shows how integration views (like the cross-model consistency query results in slot a8 in Figure 7) and traceability links (like the correspondence models produced by the TGG Synchronizations) can be represented in the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' While not present in the example eGDNs, the class diagram and OCL meta- model are models themselves and could simply be made explicit by including them in dedicated model slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Well-formedness conditions for metamodels or regular models can be realized and treated as regular query operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' eGDNs thus satisfy the requirement R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Finally, the eGDN realization of the Megamodel Reuse use case in Figure 8 considers the eGDNs from Figure 7 and 6 as operation nodes in an overarching eGDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This exemplifies how eGDNs offer modularity and incrementality at the megamodel level by considering sub-eGDNs as regular operations that can be executed via the general execution scheme, which also permits the accumulation of several changes before execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The example thus illustrates the satisfaction of requirements R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2, R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2, and R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' The eGDN also demonstrates how slots act as interfaces for these megamodel operations, satisfying requirement R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Thus, eGDNs can be employed to realize the functionality required by the three example use cases, satisfying the requirements regarding model operations, modeling languages integration, and megamodels introduced in Section 3 in this 43 m1: ClassDiagram M m2: OCL M m3: LinkingModel M t1: TGG Sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (↔) T t2: TGG Sync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' (↔) T m5: Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' OCL M m6: ASG M m7: ASG M t3: Transform T m8: LinkingModel M m4: Corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' ClassDiagram M q1: Query Q q2: Query Q metric1: Assignment A metric2: Assignment A codegen & analysis: eGDN X Figure 6: Sample eGDN realizing the Code Generation use case scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Table 4 summarizes the coverage of the requirements by the example use cases and the eGDN approach, with “◦” denoting that the realization of a use case relates to a requirement and “✓” indicating that a requirement is satisfied by eGDNs in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='m1: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='ClassDiagram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='m2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='OCL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='m3: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='LinkingModel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='q1: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Method → ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='q2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Expression → ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='q3: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Method ← Link → Expression ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='codegen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='& analysis: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='eGDN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Figure 8: Sample eGDN realizing the Megamodel Reuse use case ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Consistency Checking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Code Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='Megamodel Reuse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='eGDNs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1: modeling languages integration ✓ R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2: interfaces for embedding of modeling languages ✓ R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1: composition of model operations ✓ R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2: model operations over integrated models ✓ R 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3: execution scheme for model operations ✓ R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1: megamodeling language ✓ R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2: megamodel operation module concept ✓ R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='2: robust megamodel execution scheme ✓ R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='3: megamodel interfaces ✓ R 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content='4: asynchronous megamodel execution scheme ✓ Table 4: Coverage of requirements from Chapter 3 8 Conclusion In this report, we have developed a further generalization of the GDN mech- anism called eGDNs, which enables the modular and incremental construction and execution of complex networks of model operations, including model prop- erties, model consistency, model transformation and model synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In addition to a formal definition of eGDNs, we have provided incremental algo- rithms for their execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Moreover, we have presented a number of example eGDN nodes that we have prototypically implemented in order to perform an initial empirical evaluation of the approach regarding scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Our experi- ments, which are based on an application scenario from the software develop- ment domain, indicate that the introduced technique can be employed to realize efficient Global Model Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Moreover, we have conceptually evaluated our approach against identified requirements of global model management solu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In future work, we plan to perform a more extensive evaluation with respect to both expressiveness and performance of eGDNs in real application scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' This may also involve the implementation of additional types of eGDN nodes and may ultimately result in the implementation of true tool support for the specification and execution of eGDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Furthermore, 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Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' Springer Berlin / Heidelberg, 2010, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' 555–579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' [39] Holger Giese and Robert Wagner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' “From model transformation to in- cremental bidirectional model synchronization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} +page_content=' In: Software & Systems Modeling 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQfu_m7/content/2301.00624v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Alejandro Ibarra1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' and Satoshi Shirai3 1Physik-Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Technische Universität München,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' James-Franck-Straße,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Germany 2Max-Planck-Institut für Physik (Werner-Heisenberg-Institut),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Föhringer Ring 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='80805 München,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Germany 3Kavli Institute for the Physics and Mathematics of the Universe (WPI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='The University of Tokyo Institutes for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Kashiwa 277-8583,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Japan Abstract In some scenarios,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' the dark matter particle predominantly scatters inelastically with the target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' producing a heavier neutral particle in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In this class of scenarios, the reach in parameter space of direct detection experiments is limited by the velocity of the dark matter particle, usually taken as the escape velocity from the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' On the other hand, it has been argued that a fraction of the dark matter particles in the Solar System could be bound to the envelope of the Local Group or to the Virgo Supercluster, and not to our Galaxy, and therefore could carry velocities larger than the escape velocity from the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In this paper we estimate the enhancement in sensitivity of current direct detection experiments to inelastic dark matter scatterings with nucleons or electrons due to the non-galactic diffuse components, and we discuss the implications for some well motivated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 1 Introduction The existence of dark matter in galaxies, clusters of galaxies and the Universe at large scale is by now established by their gravitational effects on ordinary matter (for reviews, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' [1– 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' If the dark matter is constituted by new particles, it is plausible that they could interact with the ordinary matter through other interactions aside from gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' A promising avenue to probe these putative interactions consists in the search for nuclear or electron recoils induced by dark matter particles entering a dedicated detector at the Earth [5, 6] (for reviews, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' [7–9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' This search strategy, denominated direct detection, has seen an impressive increase in sensitivity since it was first proposed more than three decades ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Yet, no conclusive dark matter signal has been found to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Assuming that the dark matter scatters elastically with the nucleus, current direct detection experiments restrict the spin-independent interaction cross-section to be smaller than ∼ 1 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='00870v1 [hep-ph] 2 Jan 2023 zeptobarn in the mass range ∼ 10 GeV - 1 TeV [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' These stringent constraints put pressure on several well motivated dark matter scenarios, especially those for which the dark matter particle couples at tree level with the valence quarks in models addressing the electroweak hierarchy problem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' On the other hand, there are many other dark matter scenarios, arguably also well motivated theoretically, which are largely unconstrained by current searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In this paper we will focus on scenarios where the dark matter cannot scatter elastically with a nucleus (or an electron), so that the stringent limits on the elastic scattering cross-section do not necessarily hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' This seemingly strong assumption naturally arises in some models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For instance, the elastic scattering mediated by vector current is forbidden for Majorana dark matter χ, due to the Majorana nature of fermion: ¯χγµχ = 0 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' However, Majorana dark matter particles may leave an imprint in direct search experiments if they could scatter inelastically producing a heavier Majorana fermion χ′ in the final state, since there is an off-diagonal fermion current ¯χ′γµχ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' This scenario is approximately realized in the Minimal Supersymmetric Standard Model, when the lightest supersymmetric particle is almost a pure Higgsino state, and the other supersymmetric particles are very heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In this case, the elastic scattering of the Higgsino dark matter is suppressed by the large sfermion and gaugino masses, while it has a large inelastic scattering cross section by the electroweak gauge interactions [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Scenarios of inelastic dark matter have also been motivated phenomenologically, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' in [12–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The kinematics of the inelastic scattering differs from the one in the elastic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In order to allow the production of a heavier neutral particle in the final state, the velocity of the incoming dark matter particle must be larger than a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Therefore, as the mass difference between the initial and final neutral particles increases, faster and faster dark matter particles are necessary in order to open kinematically the inelastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For dark matter particles bound to our galaxy, and which have speeds smaller than the escape velocity from the Milky Way, vesc = 544 km/s [23, 24], the inelastic scattering off a nucleus is kinematically allowed when the mass difference between the two states is δm < 1/2µv2 esc, with µ the reduced mass of the DM-nucleus system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' for the scattering off an electron, the inelastic channel is open when δm < 1/2µev2 esc − |Enl|, where µe is the reduced mass of the DM-electron system, and |Enl| is the binding energy of an electron in the (n, l) shell of the target nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In practice, experiments can only detect recoiling nuclei/ionized electrons within a given energy range, therefore the mass difference that can be probed in direct searches is smaller than this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In this letter we argue that the parameter space of inelastic dark matter scenarios that can be probed in direct search experiments is larger than the one previously considered in the literature, that implicitly assumes that the Milky Way is an isolated galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Instead, the Milky Way is one among the various members of the Local Group, which include M31, M33 and several dwarf galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' It has been argued that the Local Group contains a diffuse dark matter component, which is not bound to any individual galaxy, and which is distributed roughly homogeneously over the Local Group [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Notably, a non-negligible fraction of the dark matter particles in the Solar System is expected to be associated to this non-galactic diffuse component, rather than to the Milky Way halo, and could have velocities larger than the escape velocity from the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Consequently, the mass splitting that could be probed in experiments correspondingly increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Likewise, the Local Group is one among the many groups of galaxies embedded in the Virgo Supercluster, which could also contain a diffuse component [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Although the fraction of dark matter particles in the Solar System associated to the Virgo Supercluster is fairly small, they have very large velocities, and could be pivotal in generating a signal in direct search experiments when the inelastic scattering is kinematically 2 inaccessible for the dark matter bound to the Milky Way and to the Local Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In section 2, we present the non-galactic dark matter flux at Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In section 3, we derive constraints on inelastic dark matter from nuclear recoil searches, and in section 4, we derive constraints from electron recoil searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Finally, in section 5, we present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 2 Dark matter flux at Earth A correct description of the dark matter flux at Earth is crucial for assessing the prospects for detection of a given dark matter model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The largest contribution to the flux is expected to arise from dark matter particles in the Milky Way halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The local density of dark matter particles and their velocity distribution is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' However, it is common in the literature to adopt the Standard Halo Model (SHM), characterized by a local density ρloc SHM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='3 GeV/cm3 and an isotropic velocity distribution described by a Maxwell-Boltzmann distribution truncated at the escape velocity of the Milky Way [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In the galactic frame, the velocity distribution reads: fSHM(⃗v) = 1 (2πσ2 v)3/2Nesc exp � − v2 2σ2 v � for v ≤ vesc , (1) where v = |⃗v|, σv ≈ 156 km/s is the velocity dispersion [30, 31], and vesc = 544 km/s is the escape velocity from our Galaxy [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Further, Nesc is a normalization constant, given by: Nesc = erf � vesc √ 2σv � − � 2 π vesc σv exp � −v2 esc 2σ2 v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (2) For our chosen parameters, Nesc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The contribution to the local dark matter flux from the Milky Way halo then reads: FSHM(⃗v) = ρloc SHM mDM vfSHM(⃗v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (3) It is plausible that the dark matter flux at Earth also contains a contribution from dark mat- ter particles not bound to the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Astronomical observations indicate the presence of diffuse dark matter components homogeneously distributed between clusters and Superclusters of galaxies [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Since these dark matter particles are not gravitationally bound to the Milky Way, they carry larger velocities than the escape velocity of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In this work, we consider the contribution to the dark matter flux from the Local Group and from the Virgo Supercluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The dark matter particles from the Local Group contribute at the Solar System with a local density of ρLG ∼ 10−2 GeV/cm3, and are expected to move isotropically with a narrow velocity distribution, σv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='LG ∼ 20 km/s, and with mean velocity vLG ∼ 600 km/s [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The contribution from the Local Group to the dark matter flux at the location of the Solar System then reads: FLG(⃗v) = ρloc LG mDM δ(v − vLG) 4πv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (4) Dark matter particles bound to the Virgo Supercluster give a small contribution to the local dark matter density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Observations indicate that the average density in the diffuse component 3 of the Virgo Supercluster is close to the cosmological value ∼ 10−6 GeV/cm3 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' However, the gravitational focusing due to the Local Group leads to an increase in the density at the location of the Sun by a factor ∼ 1 + v2 esc/v2 σVS, where vσVS is the velocity dispersion of the dark matter particles from the Virgo Supercluster [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' This value is highly uncertain, but it is expected to be comparable to that of the observable members of the Supercluster, which ranges from vσVS ∼ 50 km/s to vσVS ∼ 500 km/s [28, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' We consider for concreteness an enhancement on the local density of dark matter particles from the Virgo Supercluster of ∼ 10, consistent with the value of the velocity dispersion of the observable members of the Supercluster, which leads to ρloc VG ∼ 10−5 GeV/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Current knowledge on the dark matter velocity distribution in the Virgo Supercluster is much poorer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Following [33], we assume that the dark matter particles have the typical velocities of the members of the Virgo Supercluster, corresponding to (at least) vVS ∼ 1000 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The contribution to the dark matter flux at the location of the Solar System from the Virgo Supercluster can then be written as: FVS(⃗v) = ρloc VS mDM δ(v − vVS) 4πv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (5) The total (galactic plus non-galactic) dark matter flux at the Solar System is therefore approximately given by: F(⃗v) = FSHM(⃗v) + FLG(⃗v) + FVS(⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (6) Following [33], we adopt values for the local density of each component such that the total sum yields the canonical value of the local density used by direct detection experiments ρloc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='3 GeV/cm3, namely ρloc SHM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='26 GeV/cm3 (∼ 88%), ρloc LG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='037 GeV/cm3 (∼ 12%), and ρloc VS = 10−5 GeV/cm3 (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='00003%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 3 Impact on nuclear recoils The differential rate of nuclear recoils induced by inelastic up-scatterings of dark matter parti- cles traversing a detector at the Earth is given by: dR dER = � i ξi mAi � v≥vi min(ER) d3vF(⃗v + ⃗v⊙) dσi dER (v, ER) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (7) Here, ⃗v is the dark matter velocity in the rest frame of the detector, F(⃗v + ⃗v⊙) is the dark matter flux in the detector frame, and ⃗v⊙ is the velocity of the Sun with respect to the Galactic frame with |⃗v⊙| ≈ 232 km/s [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For the inelastic scattering with mass splitting between two dark matter states, δDM, the minimum velocity necessary to induce a recoil with energy ER of the nucleus i with mass mAi and mass fraction ξi in the detector reads vi min(ER) = 1 � 2ERmAi �ERmAi µAi + δDM � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (8) Further, for spin-independent interactions, the differential dark matter-nucleus cross section reads, dσSI i dER (v, ER) = mAi 2µ2 Aiv2σSI 0,iF 2 i (ER) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (9) 4 Here mAi is mass of the nucleus i, µAi is the reduced mass of the dark matter-nucleus i system and F 2 i (ER) is the nuclear form-factor, for which we adopt the Helm prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Besides, σSI 0,i is the spin-independent dark matter-nucleus scattering cross section at zero momentum transfer, which depends on the details of the dark matter model and the target nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' From the differential rate, one can calculate the total recoil rate using: R = � ∞ 0 dER ϵi(ER) dR dER , (10) where ϵi(ER) is the efficiency of that experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Finally, the total number of expected recoil events is N = R · E, with E the exposure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' mass multiplied by live-time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In our analysis, we will consider two scenarios for the coupling of dark matter to nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' First, we will consider a Majorana dark matter candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In this case σSI 0,i = 4µ2 Ai π � Zif p S + (Ai − Zi)f n S �2 , (11) where f p S and f n S parametrize the strength of the scalar interactions to the proton and the neutron (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' [7, 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' It is common to write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (11) as σSI 0,i = µ2 Ai µ2 p � Zi + (Ai − Zi)f n S f p S �2 σDM,p , (12) with µp the reduced mass of the DM-proton system and σDM,p an effective DM-proton inter- action cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Within the Majorana dark matter scenario, we will consider in particular the widely adopted benchmark case where the interaction is “isoscalar”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' when the dark matter couples with equal strength to protons and neutrons, for which σSI 0,i = µ2 Ai µ2 p A2 i σDM,p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (13) We will also consider a scenario where the dark matter has hypercharge Y , and interacts with the quarks via the exchange of a Z boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In this case, σSI 0,i has the same form as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (11), replacing the scalar couplings by the corresponding vector couplings, f p,n S → f p,n V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For interactions with the Z boson, f p V and f n V are explicitly given by: f p V = GFζY 2 √ 2 (1 − 4 sin2 θW) , f n V = −GFζY 2 √ 2 , (14) with ζ = 1 (ζ = 2) for fermionic (bosonic) dark matter [5, 21, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In this scenario, the dark matter-nucleus cross section can be related to the dark matter-proton cross-section through: σSI 0,i = µ2 Ai µ2 p � Zi − (Ai − Zi) (1 − 4 sin2 θW) �2 σDM,p , (15) which is independent of the dark matter hypercharge and spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' To assess the impact of the non-galactic diffuse components for direct detection experiments, we plot in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 1 the differential rate of inelastic scatterings in the LUX-ZEPLIN experiment 5 10 20 30 40 50 60 70 80 ER [keV] 10−6 10−4 10−2 100 102 104 106 dR dER [keV−1] mDM = 1 TeV σDM−p = 10−38cm2 LUX-ZEPLIN (SHM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' δDM = 100 keV) LUX-ZEPLIN (SHM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' δDM = 200 keV) LUX-ZEPLIN (SHM+Non-galactic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' δDM = 100 keV) LUX-ZEPLIN (SHM+Non-galactic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' δDM = 200 keV) CEνNS (Solar neutrinos) Figure 1: Differential rate for the inelastic scattering of a Majorana dark matter candidate in the “isoscalar” scenario with mass mDM = 1 TeV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' for δDM = 100 keV (light blue) and 200 keV (dark blue),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' for a dark matter flux at Earth as modelled by the Standard Halo Model (dotted line) or including also the contribution from the non-galactic diffuse dark matter component (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For the plots it was assumed σDM,p = 10−38 cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' for the “isoscalar” scenario, assuming mDM = 1 TeV and σDM,p = 10−38 cm2, for δDM = 100 keV (light blue) and 200 keV (dark blue), including in the flux only the contribution from dark matter bound to the Milky Way (dotted lines), as commonly assumed in the literature, and including the contribution from the non-galactic diffuse component (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The impact of the non-galactic component in the differential rate is apparent from the figure, and increases the number of events at all recoil energies, especially in the region with low ER which is not kinematically accessible to the galactic dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The non-galactic dark matter, therefore, has implications not only for enhancing the sensitivity of the experiment, but also for the interpretation of a putative dark matter signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Current direct search experiments have not observed a significant excess of nuclear recoils, which allows to derive upper limits on the dark matter nucleon cross section for given com- binations of the dark matter mass and mass splitting between the dark matter particle and the neutral particle in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In Figure 2, we show upper limits on the dark matter- proton spin-independent scattering cross section versus mass splitting for mDM = 1 TeV from LUX-ZEPLIN (blue) [10], PICO60 (green) [38], CRESST-II (red) [39], and from a radiopurity measurement in a CaWO4 crystal (orange) [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The dotted lines represent the limits ob- tained considering the galactic dark matter (described by the SHM) as the only contribution to the dark matter flux, while the solid lines were obtained including also the contributions to the flux from the non-galactic diffuse component in the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In the upper left plot, we show the limits for a Majorana dark matter candidate in the “isoscalar” scenario, and in the upper right plot, the most conservative limit for the Majorana dark matter, without making assumptions on the coupling strengths, derived following the approach of [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Lastly, in the lower plot we show the limits for a scenario where the dark matter interacts with the nucleus via the exchange of a Z-boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In the latter plot we also show the dark matter-proton scattering cross-section for scenarios of a fermionic dark matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' and Y = 1/2 (corresponding to the well motivated scenario of the Higgsino dark matter in the limit of high scale supersymmetry [12]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 6 0 200 400 600 800 1000 1200 δDM [keV] 10−48 10−46 10−44 10−42 10−40 10−38 10−36 10−34 10−32 10−30 σSI DM−p[cm2] mDM = 1 TeV Majorana DM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' f n = f p LUX-ZEPLIN (SHM) LUX-ZEPLIN (SHM + Non-galactic) PICO60 (SHM) PICO60 (SHM + Non-galactic) CRESST II (SHM) CRESST II (SHM + Non-galactic) CaWO4 (SHM) CaWO4 (SHM + Non-galactic) 0 200 400 600 800 1000 1200 δDM [keV] 10−48 10−46 10−44 10−42 10−40 10−38 10−36 10−34 10−32 10−30 σSI DM−p[cm2] mDM = 1 TeV Majorana DM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' f n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' f p free ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='LUX-ZEPLIN (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='LUX-ZEPLIN (SHM + Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='PICO60 (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='PICO60 (SHM + Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='CRESST II (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='CRESST II (SHM + Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='CaWO4 (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='CaWO4 (SHM + Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='δDM [keV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='σSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='DM−p[cm2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='mDM = 1 TeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='Y=1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='Y=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='Y=3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='Z-boson mediation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='LUX-ZEPLIN (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='LUX-ZEPLIN (SHM + Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='PICO60 (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='PICO60 (SHM + Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='CRESST II (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='CRESST II (SHM + Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='CaWO4 (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='CaWO4 (SHM + Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='Figure 2: 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L upper limits on the spin-independent dark matter-proton inelastic cross section for a dark matter mass of 1 TeV as a function of the mass splitting, from LUX-ZEPLIN (blue), PICO60 (green), CRESST-II (red and orange) and from a CaWO4 detector radiopurity measurement (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' We show the limits for three different scenarios: Majorana dark matter with scalar interactions f p = f n (upper left plot), arbitrary f p and f n (upper right plot), and dark matter interacting via the Z-boson (lower plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In the lower plot, we also show for reference the predicted value of the cross-section with a xenon target for scenarios of fermionic dark matter with hypercharge Y = 1/2, 1, 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Y = 1 and Y = 3/2 (which correspond to different scenarios of minimal dark matter [37]), for a xenon target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For other targets, the expected cross section for mDM = 1 TeV scales as ∼ Ai/Zi, being indistinguishable in the Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' As seen in the plots, for all the scenarios the non-galactic diffuse component enhances the sensitivity of experiments to inelastic dark matter, allowing to probe larger mass splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For instance, for our representative dark matter mass of 1 TeV, the LUX-ZEPLIN experiment is insensitive to dark matter particles of the Milky Way scattering inelastically if the mass difference with the neutral particle in the final state is δDM ≳ 300 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' However, the presence of dark matter in the Solar System from the envelope of the Local Group extends the reach up to δDM ≃ 330 keV and allows to probe uncharted parameter space for large mass splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 7 Concretely, the LUX-ZEPLIN experiment sets for the isoscalar scenario the limit σSI DM−p ≲ 10−44 cm2 for δDM = 250 keV, which is about three orders of magnitude stronger than the limit obtained assuming that all dark matter is bound to the Milky Way, and only a factor of 100 weaker than the limit on the elastic scattering cross-section i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' for δDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For the interaction mediated by the Z-boson the upper limit is σSI DM−p ≲ 10−44 cm2, and the most conservative limit without making assumptions on the form of the interaction is σSI DM−p ≲ 10−40 cm2, obviously much weaker than for concrete scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The dark matter particles from the Virgo Supercluster extend the reach to even larger mass differences, up to δDM ≃ 450 keV and sets for the isoscalar scenario the limit σSI DM−p ≲ 5 × 10−40 cm2 for δDM = 450 keV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' for the interaction mediated by the Z-boson the upper limit is σSI DM−p ≲ 10−41 cm2, while the model independent limit is σSI DM−p ≲ 5 × 10−36 cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Similar conclusions apply for the PICO and CRESST experiments, and from the radiopurity measurements on a CaWO4 target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' It is interesting to note the complementarity of the different experiments in probing the parameter space of inelastic dark matter scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Both in the scenario of a Majorana dark matter with f n = f p and for the scenario with Z-boson mediation, LUX-ZEPLIN is the most sensitive probe for small δDM, whereas the radiopurity measurements on a CaWO4 is the most sensitive probe for large δDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' PICO-60 is relevant for intermediate values of δDM, and is in fact the most sensitive current probe of some well motivated dark matter scenarios, as suggested by the gray lines in the Figure, which correspond to the expected cross-section for different scenarios of electroweakly interacting fermionic dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The complementarity of experiments in probing these scenarios is investigated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The dotted lines show the upper limit on the mass splitting as a function of the dark matter mass assuming the Standard Halo Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Under this common assumption, LUX-ZEPLIN is the most constraining experiment over the whole parameter space considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' However, when including the non- galactic components, different experiments contribute to set the upper limit, as reflected by the breaks in the solid lines in the Figure: LUX-ZEPLIN remains as the most sensitive experiment for small dark matter masses, while PICO-60 is the best experiment for larger masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Further, the dark matter mass at which PICO-60 becomes the leading experiment becomes larger and larger as the dark matter hypercharge increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' As seen in the Figure, for this class of scenarios the non-galactic components in the dark matter flux enhance the sensitivity of experiments to the mass splitting by a factor ∼ 2 for mDM = 100 GeV - 1 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' It is noteworthy the pivotal role of the radiopurity measurements on a CaWO4 target to probe large mass splittings in inelastic dark matter scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' This can be understood from the expression for the minimum DM velocity required to induced a recoil with energy ER, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Let us consider a velocity distribution where the maximum speed is v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Then, for an experiment capable of detecting a recoil of a nucleus Ai with energy ER, the maximum mass splitting that can be probed is: δDM ≤ � 2ERmAiv∗ − ERmAi µAi ≤ 1 2µAiv2 ∗ , (16) where the absolute maximum is reached when ER = µ2 Aiv2 ∗/(2mAi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' This is shown in Figure 4, for v∗ = 764 km/s, v∗ = 820 km/s, v∗ = 1220 km/s (solid lines), corresponding respectively to the maximal velocity at the Earth of dark matter particles bound to the Milky Way (described by the Standard Halo Model), from the Local Group envelope and from the Virgo Supercluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The plot also shows the range of recoil energies that can be detected by the CRESST-II ex- periment and by the radiopurity measurements in CaWO4 crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' As seen in the plot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' while 8 102 103 104 mDM [GeV] 100 200 300 400 500 δDM [keV] Upper limits at 90% CL from LZ+PICO60+CaWO4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Dirac dark matter Y = 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' SHM Y = 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' SHM + Non-galactic Y = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' SHM Y = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' SHM + Non-galactic Y = 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' SHM Y = 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' SHM + Non-galactic Figure 3: Upper limits on the mass splitting for electroweakly charged (pseudo-)dirac dark matter as a function of the dark matter mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' for different choices of the hypercharge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' and including in the flux only the Standard Halo Model component (dotted lines) or also the non- galactic diffuse components (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' CRESST-II can only probe up to δDM ∼ 700 keV, the radiopurity measurements allow to probe up to δDM ∼ 1200 keV, when including the flux component from the dark matter bound to the Virgo Supercluster (however with a lower sensitivity due to the smaller exposure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' From this plot it follows that the CRESST experiment would have an enhanced sensitivity to inelastic dark matter scenarios if the window of recoil energies used in the analysis were extended to larger values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Let us note that for low dark matter masses, extending the search window to higher recoil energies would not help in probing larger values of the mass splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' This is illustrated in the Figure for mDM = 100 GeV, from where it is apparent that to increase the reach in mass splittings it is necessary to extend the search to lower recoil energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Finally, we show in Figure 5 the isocontours with the 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' upper limits on the cross- section for different dark matter masses and mass splittings, from LUX-ZEPLIN (top panels), PICO60 (middle panels) and from radiopurity measurements on a CaWO4 target (bottom panels), considering that all dark matter in the Solar System is bound to the Milky Way, as commonly assumed (left panels), and including the non-galactic components (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The enhancement in sensitivity is clear from the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 4 Impact on electron recoils The differential ionization rate induced by dark matter-electron inelastic scattering in liquid xenon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' with mass splitting between the two dark matter states given by δDM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' reads: dRion dlnEer = NT � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='l � v≥vnl min(Eer) d3vF(⃗v + ⃗v⊙) dσnl ion dlnEer (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Eer) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='(17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='where NT is the number of target nuclei and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='vnl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='min(Eer) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='mDM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='(Eer + |Enl| + δDM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='(18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='ER [keV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='1400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='δDM [keV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='mDM = 100 GeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='CaWO4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='CRESST-II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='SHM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='SHM+LG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='SHM+LG+VS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='ER [keV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='1400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='δDM [keV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='mDM = 1 TeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='CaWO4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='CRESST-II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='SHM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='SHM+LG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='SHM+LG+VS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='Figure 4: Values of the mass splitting δDM that can produce a recoil energy in a 184W target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='for mDM = 100 GeV (left plot) and mDM = 1 TeV (right plot) when the maximal velocity of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='the dark matter particles at Earth is v∗ = 764 km/s (dotted lines),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' v∗ = 820 km/s (dashed lines) and v∗ = 1220 km/s (solid lines),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' corresponding respectively to dark matter bound to the Milky Way (described by the Standard Halo Model),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' bound to the Local Group and bound to the Virgo Supercluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For comparison, we also show the range of recoil energies that can be detected by the CRESST-II experiment (red band) and by the CaWO4 radiopurity measurement (yellow band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' is the minimum dark matter velocity necessary to ionize a bound electron in the (n, l) shell of a xenon atom (with energy Enl), giving a free electron with energy Eer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Further, dσnl ion/dlnEer is the differential ionization cross section, given by: dσnl ion dlnEer (v, Eer) = ¯σDM−e 8µ2 DM,ev2 � qnl max qnl min dqq ��f nl ion(k′, q) ��2 |FDM(q)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (19) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' µDM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='e is the reduced mass of the dark matter-electron system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' ¯σDM−e is the dark matter- free electron scattering cross section at fixed momentum transfer q = αme,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' ��f nl ion(k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' q) ��2 is the ionization form factor of an electron in the (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' l) shell with final momentum k′ = √2meEer and momentum transfer q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' and FDM(q) is a form factor that encodes the q-dependence of the squared matrix element for dark matter-electron scattering and depends on the mediator under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The maximum and minimum values of the momentum transfer needed to ionize a bound electron in the (n, l) shell recoil with energy Eer from the interaction of a dark matter particle with speed v are: qnl max min(Eer) = mDMv � �1 ± � 1 − �vnl min(Eer) v �2� � , (20) with vnl min(Eer) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Finally, the total number of expected ionization events reads N = Rion · E, with Rion the total ionization rate, calculated from integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' (17) over the experimentally measured recoil energies, and E the exposure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' mass multiplied by live-time) of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 10 102 103 104 mDM [GeV] 100 200 300 400 500 600 δDM [keV] Upper limits at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L from LUX-ZEPLIN, SHM, Isoscalar 10−47 10−45 10−43 10−41 10−39 10−37 σDM−p 102 103 104 mDM [GeV] 100 200 300 400 500 600 δDM [keV] Upper limits at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L from LUX-ZEPLIN, Non-galactic, Isoscalar 10−47 10−45 10−43 10−41 10−39 10−37 σDM−p 102 103 104 mDM [GeV] 100 200 300 400 500 600 δDM [keV] Upper limits at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L from PICO60, SHM, Isoscalar 10−45 10−43 10−41 10−39 10−37 10−35 σDM−p 102 103 104 mDM [GeV] 100 200 300 400 500 600 δDM [keV] Upper limits at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L from PICO60, Non-galactic, Isoscalar 10−45 10−43 10−41 10−39 10−37 10−35 σDM−p 102 103 104 mDM [GeV] 200 400 600 800 1000 1200 1400 δDM [keV] Upper limits at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L from CaWO4, SHM, Isoscalar 10−41 10−39 10−37 10−35 10−33 10−31 10−29 10−27 σDM−p 102 103 104 mDM [GeV] 200 400 600 800 1000 1200 1400 δDM [keV] Upper limits at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L from CaWO4, Non-galactic, Isoscalar 10−41 10−39 10−37 10−35 10−33 10−31 10−29 10−27 σDM−p Figure 5: Isocontours of the 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' upper limits on the spin-independent dark matter-proton inelastic cross-section for the isoscalar scenario (f p = f n) in the parameter space spanned by the dark matter mass and mass splitting, from LUX-ZEPLIN (top panels), PICO60 (middle panels) and radiopurity measurements in a CaWO4 target (lower panels), assuming that all dark matter in the Solar System is bound to the Milky Way (left panels) or including the non-galactic diffuse component (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 11 In semiconductor detectors, the electron excitation rate induced by dark matter-electron inelastic scatterings, with a mass splitting δDM, reads [43, 44] R = 1 ρT ¯σDM−e µ2 DM,e π α � d3vF(⃗v + ⃗v⊙) v � d3q (2π)3q2 |FDM(q)|2 � dω 2π 1 1 − e−βω Im � −1 ϵ(ω, ⃗q) � δ � ω + δDM + q2 2mχ − ⃗q · ⃗v � , (21) where w is the energy deposited in the material, ⃗q is the momentum transfer of the process, and ρT is the target density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The rate involves an integration of the Electronic Loss Function (ELF) of the target material, which we calculate with DarkELF [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For the dielectric function ϵ(ω, q), we use the Lindhard method, which treats the target as a non-interacting Fermi liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Finally, the total number of events reads N = R · E, with E the exposure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' mass multiplied by live-time) of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The non-observation of a significant excess of electron recoils in a given experiment allows to set upper limits on the dark matter-electron scattering cross section, for a given dark matter mass and a given mass splitting between the dark matter particle and the heavier neutral state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' We show in Figure 6, upper limits on the inelastic dark matter-electron cross section versus mass splitting for a fixed dark matter mass of mDM = 1 GeV from XENON1T [45](blue lines), and from the semiconductor experiment SENSEI [46](purple lines), both when considering the SHM flux only (solid lines), and when including the non-galactic components to the dark matter flux (dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In the upper plots, we take the form factor FDM = α2m2 e/q2, corresponding to an ultralight or massless mediator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' In the middle plots, we take the form factor FDM = αme/q, corresponding to an electric dipole interaction, and in the lower plots we take the form factor FDM = 1, corresponding to a heavy mediator [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' As can be seen in the Figure, the non-galactic components enhance the sensitivity to the mass splitting of both XENON1T and SENSEI by a factor of ∼ 2, compared to the sensitivity estimated from considering just the galactic component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' This conclusion holds independently of the choice of the dark matter form factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Further, the reach in cross-section is enhanced due to the non-galactic components, especially at low mass splittings, being the effect stronger for XENON1T than for SENSEI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For comparison, we also show as a grey band the cross section for which the observed dark matter abundance is reproduced via freeze-in in the case of an ultralight mediator [49], or via freeze-out in the case of a heavy mediator [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Clearly, the non-galactic dark matter components allow to probe larger values of the mass splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 5 Conclusions We have investigated the impact of a non-galactic diffuse dark matter component inside the Solar System for the detection of the inelastic scattering of a dark matter particle in direct search experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Concretely, we have considered the contribution to the dark matter flux from dark matter particles in the envelope of the Local Group and from the Virgo Supercluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Their speeds in the galactic frame are ∼ 600 km/s and ∼ 1000 km/s, respectively, which are larger than the maximal speed of dark matter particles bound to the Milky Way, ∼ 540 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' As a result, the region of parameter space that can be probed with current experiments is larger than reported in previous works, that implicitly assumed that the Milky Way is an isolated galaxy in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='δDM [eV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='¯σe[cm2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='FDM = α2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='e/q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='mDM = 1 GeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='Freeze-in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='Ultralight mediator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='SENSEI (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='SENSEI (SHM+Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='XENON1T (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='XENON1T (SHM+Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='δDM [eV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='¯σe[cm2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='FDM = αme/q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='mDM = 1 GeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='Dipole interaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='SENSEI (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='SENSEI (SHM+Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='XENON1T (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='XENON1T (SHM+Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='δDM [eV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10−26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='¯σe[cm2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='FDM = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='mDM = 1 GeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='Freeze-out (Pseudo-Dirac fermion) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='Massive mediator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='SENSEI (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='SENSEI (SHM+Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='XENON1T (SHM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='XENON1T (SHM+Non-galactic) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='Figure 6: 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L upper limits on the spin-independent dark matter-electron inelastic cross section for a dark matter mass of 1 GeV, as a function of the mass splitting, from XENON1T (blue) and SENSEI (purple), when the dark matter-electron interaction is mediated by an ultralight dark photon (upper left plot), by a dipole operator (upper right plot), or by a heavy mediator (lower plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For nuclear recoils, the non-galactic component expands the reach in mass splitting at the LUX-ZEPLIN, PICO60, and CRESST-II experiments by a factor ∼ 2 in the mass range mDM = 10 GeV- 10 TeV, and enhances significantly the reach in cross-section, especially close to the kinematic threshold for the galactic dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For instance, for mDM = 1 TeV and δDM = 250 keV, the sensitivity to the cross-section improves by about three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' We have also stressed the relevance of experiments capable of detecting high recoil energies for probing the parameter space of inelastic dark matter scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' We have illustrated this capability with the radiopurity measurements in CaWO4 crystals performed by the CRESST collaboration, and which allows to probe up to δDM ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='2 MeV (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='4 MeV) for mDM = 1 TeV (10 TeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For electron recoils, the conclusions are analogous, allowing to increase reach in mass splitting of the XENON1T and SENSEI experiments also by a factor ∼ 2 for dark matter 13 10−2 10−1 100 101 mDM [GeV] 5 10 15 20 25 30 δDM [eV] Upper limits at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L from SENSEI, SHM, Massive mediator 10−37 10−35 10−33 10−31 10−29 10−27 10−25 ¯σDM−e 10−2 10−1 100 101 mDM [GeV] 5 10 15 20 25 30 δDM [eV] Upper limits at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L from SENSEI, Non-galactic, Massive mediator 10−37 10−35 10−33 10−31 10−29 10−27 10−25 ¯σDM−e 10−2 10−1 100 101 mDM [GeV] 100 200 300 400 500 600 δDM [eV] Upper limits at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L from XENON1T, SHM, Massive mediator 10−41 10−40 10−39 10−38 10−37 10−36 10−35 10−34 ¯σDM−e 10−2 10−1 100 101 mDM [GeV] 100 200 300 400 500 600 δDM [eV] Upper limits at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L from XENON1T, Non-galactic, Massive mediator 10−41 10−40 10−39 10−38 10−37 10−36 10−35 10−34 ¯σDM−e Figure 7: Isocontours of the 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' upper limits on the dark matter-electron inelastic scat- tering cross-section for the heavy mediator scenario (FDM = 1) in the parameter space spanned by the dark matter mass and mass splitting, from SENSEI (top panels), and XENON1T (lower panels), assuming that all dark matter in the Solar System is bound to the Milky Way (left panels) or including the non-galactic component diffuse (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' masses in the range mDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='01 GeV-10 GeV, Acknowledgments The work of GH and AI was supported by the Collaborative Research Center SFB1258 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2094 - 390783311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' The work of SS is supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technol- ogy (MEXT), Japan, 18K13535, 20H01895, 20H05860 and 21H00067, and by World Premier International Research Center Initiative (WPI), MEXT, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 14 A Derivation of upper limits from direct detection exper- iments To derive upper limits on the inelastic dark matter-nucleon scattering cross section, as a function of the dark matter mass and/or the dark matter mass splitting, we follow a poissonian-likelihood approach, and we calculate the rates for the different experiments/detectors independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For the LUX-ZEPLIN experiment, we use the data from [10], with an exposure of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='904 tonne×year, a region of interest extending from 2 keV to 70 keV, and the efficiency function reported by the collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Given the agreement of the number of signal events with the background prediction reported by the collaboration, we take a 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' upper limit on the number of signal events of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For the PICO-60 experiment, we use the results from [38], corresponding to an exposure of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='356 kg×year, a region of interest extending from 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='5 keV to 100 keV, and the efficiency function reported by the collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Since PICO-60 observed no signal events, we take a 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' upper limit on the number of signal events of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For CRESST-II, we use the published data [39], corresponding to an exposure of 52 kg×days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' We do not consider as signal events those belonging to the acceptance region of the experiment at low recoil energies, but instead, we consider the recoil energy region extending from 30 keV to 120 keV, which gives an upper limit of 4 signal events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Finally, for the CaWO4 radiopurity measurement from [40], we take an exposure of 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='10 kg×days, with a recoil energy region extending from 300 keV to 2000 keV, and a number of 3 signal events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For the inelastic dark matter-electron scattering cross-section, we derive upper limits at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='L at fixed momentum transfer q = αme using data from XENON1T [45] and SENSEI [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' We consider the observed event rate XENON1T between 150-3000 photoelectrons (PE), which corresponds to the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='18 keVee to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='5 keVee (kiloelectronvolt electron equivalent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' We take the efficiency function from [45], an exposure of 22 ± 3 tonne-days and an upper limit on the number of events of 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' For SENSEI, we sum-up the observed events in the energy bins ranging from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='91 eV to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='31 eV, resulting in an upper limit of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='957 events per gram day of exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' Further, we use the efficiency reported by the collaboration in every energy bin [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' References [1] Gerard Jungman, Marc Kamionkowski, and Kim Griest.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='1007/JHEP04(2022)060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' arXiv: 2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content='13422 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf'} diff --git a/XtAzT4oBgHgl3EQfKfuP/vector_store/index.faiss b/XtAzT4oBgHgl3EQfKfuP/vector_store/index.faiss new file mode 100644 index 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+Abstract. This article is a generalization of a result in Quillen’s note [Qui96] “Module theory over non- +unital rings” giving a one-to-one correspondence between bilocalization of abelian categories of modules +and idempotent ideals of the base ring. Faltings [Fal88]; Gabber and Ramero [GR03] established almost +mathematics, the same as Quillen’s bilocalization of a category of modules by nil modules. +In this paper, by using the theory of Smith ideals mentioned in Hovey [Hov14], we consider almost +mathematics of symmetric monoidal pointed model categories and prove a weak analogue of the one-to- +one correspondence in Quillen [Qui96]. +1. Introduction +Almost mathematics is firstly introduced by Faltings [Fal88], proving almost purity in his article. Gab- +ber and Ramero [GR03] provided a detailed foundation of almost mathematics; they called it almost ring +theory: almost modules, almost algebras, and almost homotopy algebras. Let V be a unital commutative +ring with an idempotent ideal I of V. A V-module M is said to be almost I-zero (or simply, almost zero) +if M is killed by I, and almost mathematics is a theory of algebra working by localizing the category of +V-modules by the Serre subcategory of almost zero modules. +Independently, Quillen considered linear algebra over non-unital rings, the same as almost mathe- +matics in his unpublished note [Qui96]: Almost mathematics in Quillen [Qui96] is characterized as +bilocalization (both localization and colocalization) of an abelian category of modules. +Theorem 1.1. ([Qui96] Proposition 6.5) Let V be a commutative ring with a multiplicative unit. There +is a one-to-one correspondence between Serre subcategories S of ModV which the localization F : +ModV → ModV/S is also a colocalization, and idempotent ideals of V. +This article aims to prove a symmetric monoidal model categorical analogue of Theorem 1.1. To intro- +duce a theory of idempotent ideals for model categories, we use the theory of Smith ideals Hovey [Hov14]. +In the theory of algebra, a two-sided ideal of a ring can be defined to be a kernel of some ring homo- +morphism. We can interpret that Smith ideals theory is based on the correspondence between two-sided +ideals of rings and kernels of ring homomorphisms (See Hovey [Hov14, Section 1]). +A category is pointed if it has an initial object ∅ and a final object ∗ such that the unique morphism +∅ → ∗ is an isomorphism. We call the initial object the zero object of a pointed category, and 0 denotes it. +Throughout this paper, we consider pointed (model) categories. Let C be a pointed symmetric monoidal +Date: January 12, 2023. +Key words and phrases. Symmetric monoidal model categories, Smith ideals, almost mathematics, Bousfield localization +and colocalization. +1 + +category with a monoidal unit object V. Let ∆1 denote the category with two objects 0 and 1, and only +one non-trivial morphism 0 → 1. The functor category Fun(∆1, C) from ∆1 to C is called the arrow +category of C, and Ar(C) denotes it. By Hovey [Hov14, Theorem 1.2], the category Ar(C) has two +distinct symmetric monoidal structures derived from C’s: For any two morphisms in C, f : X0 → X1 and +g : Y0 → Y1, one has a commutative square: +X0 ⊗ Y0 +f⊗id � +id⊗g +� +X1 ⊗ Y0 +id⊗g +� +X0 ⊗ Y1 +f⊗id +�X1 ⊗ Y1. +One of them is the diagonal (or tensor) product monoidal structure is defined by f ⊗g as the composition +f ⊗ g = ( f ⊗ id) ◦ (id ⊗ g) = (id ⊗ g) ◦ ( f ⊗ id) : X0 ⊗ Y0 → X1 ⊗ Y1. +The other is the push-out product monoidal structure is defined by the induced morphism: +f□g : (X0 ⊗ Y1) ∐X0⊗Y0 (X1 ⊗ Y0) → X1 ⊗ Y1. +We can define Smith ideals by the push-out product monoidal structure: A Smith ideal j : I → R is +defined to be a commutative monoid object of Ar(C) with respect to the push-out product monoidal +structure. In the case of a pointed symmetric monoidal category C, the cokernel functor +cok : Ar(C) ∋ ( f : X → Y) �→ (Y → Coker( f)) ∈ Ar(C) +is symmetric monoidal from the pushout product monoidal structure to the diagonal product monoidal +structure. Further, the right adjoint of cok is the kernel functor +ker : Ar(C) ∋ ( f : X → Y) �→ (Ker( f) → X) ∈ Ar(C). +(See Hovey [Hov14, Theorem 1.4]). In the special case, a Smith ideal j : I → R with an isomorphism +j → (ker ◦ cok)( j) is isomorphic to the kernel of the monoid morphism cok( j) : R → Coker( j). +For any class T of morphisms in a category, �T (resp. T�) denotes the class of morphisms which have +right (resp. left) lifting property with respect to all morphisms in T. Then �(T�) is said to be the weakly +saturated class of morphisms generated by T. In this paper, we always assume that all model categories +are cofibrantly generated: there exist small subsets I and J of the set of morphisms of the model category +such that the collection of all cofibrations (resp. trivial cofibrations) is the weakly saturated class of +morphisms generated by I (resp. J). +In this paper, we define a homotopical analogue of almost mathematics; idempotent ideals, almost +zero-modules, and almost isomorphisms by using the Smith ideal theory of model categories. In Sec- +tion 2, we recall the definition of Smith ideals of model categories and some properties of the homotopy +cokernel and kernel functors to work in this paper. In Section 3, we introduce the theory of almost +mathematics of pointed symmetric monoidal model categories; homotopically idempotent Smith ideals, +homotopically almost zero objects, and homotopically almost weak equivalences. +As the main results, we prove a weak version of a model categorical analogue of Theorem 1.1: Given +a pointed symmetric monoidal model category with a homotopically idempotent Smith ideal of the +monoidal unit object, we obtain that the Bousfield localization on the model category by homotopically +2 + +almost weak equivalences is also a Bousfield colocalization (Theorem 3.13). Conversely, in the case that +the monoidal unit generates the stable symmetric monoidal model category in the sense of MacLane [?, +Chapter V, Section 7], we can construct a homotopically idempotent ideal (Theorem 4.5). +Acknowledgements. I would like to thank a colleague for giving me an innovative idea to use Hovey’s +Smith ideal theory. +2. Smith ideal theory of symmetric monoidal model categories +Following Hovey [Hov14], we explain concise of the theory of Smith ideals to generalize almost +mathematics theory. +2.1. The arrow categories of pointed model categories. A symmetric monoidal model category M is +a model category with a symmetric monoidal structure − ⊗ − : M × M → M such that, for any object M +of M, those functors (−) ⊗ M and M ⊗ (−) are left Quillen functors on themselves. This paper assumes +that symmetric monoidal model categories are always closed. That is, Madmits internal hom-objects +MapM(M, N) for any couple (M, N) of objects of M. +The arrow category Ar(M) has the evaluation functors Evi( f : X0 → X1) = Xi (i = 0, 1), which has a +left adjoint and a right adjoint: +Evi : Ar(M) +�M : Li, Ui (i = 0, 1), +�� +where Li denotes the left adjoint of Evi and Ui the right adjoint. +The category Ar(M) has two canonical model structures the injective model structure and the projec- +tive model structure induced by M’s: +Definition 2.1. Let M be a model category. The arrow category Ar(M) has the following two model +structures. +• (Injective model structure) A morphism α : ( f : X0 → X1) → (g : Y0 → Y1) is a cofibrations +(resp. weak equivalence) in Ar(M) if and only if so is each Evi(α) for i = 0, 1. Fibrations are +morphisms with the right lifting property for all trivial cofibrations. +• (Projective model structure) A morphism α : ( f : X0 → X1) → (g : Y0 → Y1) is a fibrations +(resp. weak equivalence) in Ar(M) if and only if so is each Evi(α) for i = 0, 1. Cofibrations are +morphisms with the right lifting property for all trivial fibrations. +By the definition of the injective and the projective model structures, for each i = 0, 1, Evi is both a +left Quillen functor with respect to the injective model structure and a right Quillen functor with respect +to the projective model structure. Therefore one has the following: +Proposition 2.2. Let M be a model category. +(1) The injective model structure of Ar(M) admits a Quillen adjunction Li : M ⇄ Ar(M) : Evi for each +i = 0, 1. +(2) The projective model structure of Ar(M) admits a Quillen adjunction Evi : Ar(M) ⇄ M : Ui for +each i = 0, 1. +□ +3 + +In the allow category Ar(M), fibrations of the injective model structure and cofibrations of the projec- +tive model structure are characterized as follows: +Proposition 2.3 ([Hov14] Thereom 2.1(2) and Therem 3.1 (2)). Let M be a model category. +(1) On the injective model structure of Ar(M), α : f → g is a (trivial) fibration if and only if so are +Ev1(α) and the induced morphism ( f, Ev1(α)) : Ev1( f) → Ev0(g) ×Ev1(g) Ev1( f). In particular, if α +is a (trivial) fibration, so is Evi(α) for each i = 0, 1. +(2) On the projective model structure of Ar(M), α : f → g is a (trivial) cofibration if and only if so are +Ev0(α) and the induced morphism (Ev0(α), g) : Ev1( f) ∐Ev0(f) Ev0(g) → Ev1(g). In particular, if α +is a (trivial) cofibration, so is Evi(α) for each i = 0, 1. +□ +In a pointed model category M, we consider a homotopically commutative diagram: +X +f +� +� +Y +g +� +0 +�Z. +If the diagram is a homotopy Cartesian square, then X is said to be a homotopy kernel of g, and if it +is a homotopy coCartesian sequence, then Z is a homotopy cokernel of f. We can consider homotopy +image objects and homotopy coimage objects as additive categories: Let f : X → Y be a morphism. The +homotopy image object of f is the homotopy kernel of Y → Coker f, and Im( f) denotes it. Dually, the +homotopy coimage object is the homotopy cokernel of Ker f → X, and Coim( f) denotes it. +2.2. The two monoidal structure of the arrow category of symmetric monoidal pointed model +categories. A symmetric monoidal category can be regarded as a symmetric multicategory (See Le- +inster [Lei04, Example 2.1.3 and Section 3.3]). Let 1 denote the terminal multicategory. A commutative +monoid object R of M is a covariant functor from 1 to M of symmetric multi-categories [Lei04, Example +2.1.1]. Let CAlg(M) denote the category of commutative monoid object of M. Directly explaining, the +commutative monoid R is equipped with a unit morphism η : V → R and a multiplication µ : R ⊗ R → R +satisfying commutativity, associativity, and unity. +On the arrow category Ar(M), those functors cok : Ar(M) → Ar(M) and ker : Ar(M) → Ar(M) +are defined as follows: For a morphism f : X → Y in M, the arrow cok( f) is Y → Coker f and ker( f) +Ker( f) → X. Then the pair +cok : Ar(M) ⇄ Ar(M) : ker +is a Quillen adjunction. The diagonal monoidal structure is compatible with the injective model structure, +and the push-out monoidal structure is compatible with the projective model structure: +Theorem 2.4 ([Hov14] Theorem 2.1 (4), Theorem 3.1 (5), and c.f. Theorem 1.4). Let M be a cofi- +brantly generated symmetric monoidal model category. Then the category Ar(M) has the following two +symmetric monoidal model structures: +(1) The injective model structure of Ar(M) is symmetric monoidal with respect to the diagonal +monoidal model structure. +4 + +(2) The projective model structure of Ar(M) is symmetric monoidal with respect to the push-out +monoidal model structure. +(3) Further, if M is pointed, then the Quillen adjunction +cok : Ar(M) ⇄ Ar(M) : ker +is compatible with those monoidal model structures the push-out and the diagonal. +proof. The assertions (1) and (2) are Hovey [Hov14, Theorem 2.1 (4)] and [Hov14, Theorem 3.1 (5)], +respectively. The third is obtained by the homotopical analogues of the proof of Hovey [Hov14, Theorem +1.4]: Let V be a unit object of M. Then U0(V) = (0 → V) and L0(V) = idV : V → V are unit +objects of the push-out monoidal structure and the diagonal monoidal structure, respectively. The functor +cok : Ar(M) → Ar(M) sends U0(V) to (V → Coker(0 → V)) ≃ (idV : V → V) = L0(V). Therefore, +the functor cok is homotopically unital. For two morphisms f : X0 → X1 and g : Y0 → Y1, we show +that those objects cok( f)□cok(g) and cok( f ⊗ g) are isomorphic in the homotopy category of Ar(M). +Consider the following commutative diagram: +X1 ⊗ Y1 +X0 ⊗ Y1 +� +�0 +Coker( f) ⊗ Y1 +X1 ⊗ Y0 +� +� +X0 ⊗ Y0 +� +� +� +� +0 +� +� +Coker( f) ⊗ Y0 +id⊗g +� +� +X1 ⊗ Y0 +X1 ⊗ Y0 +� +�0 +0 +X1 ⊗ Y1 +(X1 ⊗ Y1) ∐X0⊗Y0 (X1 ⊗ Y0) +� +f□g +� +0, +where the bottom horizontal diagram of the out of squares is induced by the homotopy push-outs of +the vertical arrows in the squares, and the right vertical diagram is induced by the homotopy-push outs +of horizontal arrows in the squares. Since the homotopy colimit of the whole squares are uniquely +determined up to homotopy equivalence, those push-outs of the diagrams out of the squares are weakly +equivalent: one has a zig-zag of weak equivalences +Coker( f□g) ← Z → Coker( f) ⊗ Coker(g), +where Z denotes the homotopy colimit of whole the squares. +□ +Definition 2.5. Let M be a pointed symmetric monoidal model category. A Smith ideal in M is a monoid +object j : I → R in the symmetric monoidal model category Ar(M) with respect to the push-out product +monoidal model structure. +A stable model category is a model category whose homotopy category is triangulated. In the case +that M is further stable, the cokernel functor is a left Quillen equivalence: +Theorem 2.6 ([Hov14] Theorem 4.3). Let M be a cofibrantly generated stable symmetric monoidal +model category. +• The cokernel functor cok : Ar(M) → Ar(M) is a left Quillen equivalence form the projective +model structure to the injective model structure. +5 + +• If M is a cofibrantly generated symmetric monoidal stable model category, the cokernel functor +is a Quillen equivalence from the model category of Smith ideals to the model category of monoid +morphisms. +□ +3. Almost mathematics of pointed symmetric monoidal model categories +On the model category setting, we will consider almost mathematics theory. Fix a pointed closed +symmetric monoidal model category M with a zero object 0, a unital commutative monoid object V, and +a Smith ideal j : I → V. +Definition 3.1. A Smith ideal j : I → V is homotopically idempotent if it satisfies the following proper- +ties: +• The homotopically coCartesian square +I +j +� +� +V +cok(j) +� +0 +�Coker( j) +is also homotopy Cartesian. +• The push-out product µ j : j□ j → j is a weak equivalence in the model category Ar(M). Equiv- +alently, the homotopy cokernel I ∐I⊗I 0 of the product µ : I ⊗ I → I is contractible. +• The tensor product ˜I := I ⊗ I is homotopically flat. That is, the functor ˜I ⊗ (−) : M → M +preserves all finite homotopy limits. +Remark 3.2. Let V be a unital commutative ring and m an idempotent ideal of V. In almost mathematics +theory, we usually consider the case ˜m = m ⊗ m is a flat V-module. Therefore, in this paper, we include +the homotopically flatness of ˜I in the definition of homotopically idempotentness. +Definition 3.3. Let j : I → V be a homotopically idempotent Smith ideal of V. For any M ∈ M, We say +that M is homotopically almost zero if the homotopy image of the composition µM : I⊗M → V⊗M → M +is contractible. +Let V/I denote the homotopy cokernel of j : I → V. By the homotopically idempotentness of I, one +has I ⊗ V/I ≃ I ⊗ (V ∐I 0) ≃ I ∐I⊗I 0 ≃ 0, implying that V/I is homotopically almost zero. Furthermore, +since I ⊗ I is homotopically flat and j : I → V is the homotopy kernel of cok( j) : V → Coker( j), the +composition ε′ +˜I ◦ (µ ⊗ j) : ˜I ⊗ I → ˜I ⊗ V → ˜I is a weak equivalence, where ε′ +˜I is the inverse of the unit +isomorphism ε˜I : ˜I → ˜I ⊗ V. +Definition 3.4. A morphism f : M → N in M is a homotopically almost weak equivalence, if the +homotopy kernel and cokernel are both homotopically almost zero. +Definition 3.5. An object M of M is homotopically almost local if, for any homotopically almost weak +equivalence f : N1 → N2, the induced map f∗ : HomHoM(M, N1) → HomHoM(M, N2) is an isomor- +phism. Dually, M is homotopically almost colocal if the induced morphism f ∗ : HomHoM(M, N2) → +HomHoM(M, N1) is an isomorphism. +6 + +Proposition 3.6. Let M be a pointed symmetric monoidal model category with a monoidal unit V and a +homotopically idempotent Smith ideal j : I → V. Then the following conditions are equivalent: +(1) An object M is homotopically almost local. +(2) For any homotopically almost zero object N, the set HomM(M, N) has only one point in the +homotopy category. +Dually, the following conditions are equivalent: +(1)’ An object M is homotopically almost colocal. +(2)’ For any homotopically almost zero object N, the set HomM(N, M) has only one point in the +homotopy category. +proof. Let N be a homotopically almost zero object. Then the trivial morphisms 0 → N and N → 0 are +almost weak equivalences by definition. Therefore, the implication (1) to (2) (resp. (1)’ to (2)’) is trivial. +We assume that the homotopy kernel and cokernel of f : X → Y are homotopically almost zero, and we +have zig-zags of weak equivalences Y ≃ X∐X Y ≃ (Y ∐X 0)∐X ← X and X ≃ X×Y Y ≃ (X×Y 0)×Y → Y. +Therefore, under the condition (2) (resp. (2)’), the induced map by f +f∗ : HomM(M, X) → HomM(M, Y) +(resp. f ∗ : HomM(Y, M) → HomM(X, M)) +is an isomorphism in the homotopy category of M. +□ +Definition 3.7. An object M of M is homotopically firm if the product morphism µM : I ⊗ M → M is a +weak equivalence. Dually, M is homotopically closed if the induced morphism µ∗ +M : M → MapM(I, M) +is a weak equivalence. +Lemma 3.8. Let j : I → V be a homotopically idempotent Smith ideal. Then an object M is homotopi- +cally almost zero if and only if ˜I ⊗ M is contractible. +proof. Since j : I → V is homotopically idempotent, the morphism µM : I ⊗ M → M induces an weak +equivalence id˜I ⊗ µM : ˜I ⊗ I ⊗ M → ˜I ⊗ M. If µM is null-homotopic, the ˜I ⊗ M is contractible. +Conversely, if the bar construction ˜I ⊗ M is contractible, the morphism µM ◦ (µ ⊗ idM) : I ⊗ I ⊗ M → +I ⊗ M → M is null-homotopic. Since the homotopy cokernel of µ ⊗ idM is contractible, the induced +morphism Coker(µM◦(µ⊗idM)) → Coker(µM) is a weak equivalence. Therefore the canonical morphism +cok(µM) : M → Coker(µM) is also a weak equivalence, entailing µM : I ⊗ M → M is null-homotopic. □ +We recall the definition of lax Quillen adjunctions: +Definition 3.9 ([SS03a] Definition 3.6). Let L : M ⇄ N : R be a Quillen adjunction of monoidal model +categories. Then we say that (L, R) is lax Quillen monoidal if it satisfies the following properties: +(1) The right Quillen functor R is lax monoidal. +(2) For any cofibrant objects M and N in M, the induced morphism +∇M, N : L(M ⊗ N) → L(M) ⊗ L(N) +by the canonical morphism ∆M, N : M ⊗ N → R(L(M)) ⊗ R(L(N)) → R(L(M) ⊗ L(N)) is a weak +equivalence. +7 + +(3) For any cofibrant replacement c : C(1M) → 1M of the monoidal unit 1M of M, the composition +L(C(1M)) → L(1M) → 1M +is a weak equivalence. +Proposition 3.10. Assume that ˜I is flat. Let M be an object of M. Then the following conditions are +equivalent: +(1) The object M is almost colocal. +(2) The object M is homotopically firm. +(3) The canonical morphism ˜I ⊗ M → M is a weak equivalence. +proof. Since ˜I ⊗ I → ˜I is a weak equivalence, I ⊗ ˜I ⊗ M → ˜I ⊗ M is also a weak equivalence. Therefore +the conditions (2) and (3) are equivalent. +For any homotopically almost zero object N, the identity morphism on N factors through V/I ⊗ N. A +homotopical section s : V/I ⊗ N → N corresponds to the morphism s∗ : N → MapM(V/I, N). If M is +firm, since V/I ⊗ M is contractible, any morphism M → MapM(V/I, N) is null-homotopic. Hence, any +morphism from M to N is also null-homotopic. This means that M is almost colocal. +Finally, we assume that M is almost colocal. Then µM : I ⊗ M → M has a homotopical section. +Equivalently, any morphism from coker(µM) is null-homotopic. Therefore coker(µM) is contractible and +µM : I ⊗ M → M is a weak equivalence. +□ +Dually, we have the following: +Proposition 3.11. Let M be an object of M. Then the following conditions are equivalent: +(1) The object M is almost local. +(2) The object M is homotopically closed. +(3) The canonical morphism M → MapM(˜I, M) is a weak equivalence. +□ +Corollary 3.12. Let j : I → V be a homotopically idempotent Smith ideal. Then the functor ˜I ⊗ (−) : +M → M is a left lax monoidal Quillen functor. +proof. Indeed, the product µ˜I : (˜I ⊗ M) ⊗ (˜I ⊗ N) → ˜I ⊗ M ⊗ N is a weak equivalence, letting the functor +˜I ⊗ (−) : M → M be homotopically monoidal. +□ +Theorem 3.13. Let M be a pointed symmetric monoidal model category and V a unit object. Assume that +V has an idempotent Smith ideal j : I → V and ˜I is flat. Then the Bousfield localization La : M → Ma +by almost weak equivalence admits homotopically fully faithful left Quillen adjoint and right Quillen +adjoint. Furthermore, let Mfirm and Mclosed denote the full subcategory spanned by all firm objects and +closed objects, respectively. Then the left Quillen functor ˜I ⊗ (−) : M → Mfirm and the right Quillen +functor MapM(˜I, −) : M → Mclosed are a left adjoint and a right adjoint of La: +Mclosed La +�Ma +MapM(˜I, −) +� +˜I⊗(−)�Mfirm +La +� +8 + +proof. We only prove that the right Quillen adjunction ˜I ⊗ (−) : Ma ⇄: Mfirm is a Quillen equivalence. +By Lemma 3.8 and the homotopically idempotentness of the functor ˜I ⊗ (−), the natural transformation +˜I ⊗ (−) → IdM is a homotopically almost equivalence. Therefore the unit IdMa → La ◦ ˜I ⊗ (−) is a weak +equivalence. By the equivalence of conditions (2) and (3) of Proposition 3.10, the counit ˜I ⊗ (−) ◦ La → +IdMfirm is a weak equivalence. +□ +Corollary 3.14. Let V be a commutative unital monoid object of M and j : I → V a homotopically +idempotent Smith ideal. Then ˜I ⊗ M is contractible if and only if so is MapM(˜I, M). +□ +4. Bousfield bilocalization of stable symmetric monoidal model categories generated by the +monoidal unit +Following the previous section, we consider a stable symmetric monoidal model category M with a +unit object V. Given a Bousfield bilocalization functor F : M → M, we show that F determines an +idempotent Smith ideal j : I → V. Let F∗ : M → M denote the homotopically fully faithful left adjoint +of F and F! the homotopically fully faithful right adjoint: +M +F +�M +F! +� +F∗ �M. +F +� +Further, we assume that, in this section, the left adjoint F∗ : M → M preserves all finite homotopy limits +and +F∗ : M ⇄ M : F +is a lax Quillen monoidal adjunction and the counit c : F∗ ◦ F → IdM induces a Smith ideal cV : +F∗(F(V)) → V. Let j : I → V denote the homotopy image of cV. Since M is stable, by Theorem 2.6, the +unit morphism cV → j is a projective weak equivalence in the arrow category of Ar(M). We will prove +that I is homotopically idempotent. By the assumption of F∗, one has a weak equivalence: +∇ : F∗(F(V) ⊗ F(V)) → F∗(F(V)) ⊗ F∗(F(V)). +The product µ : F∗(F(V)) ⊗ F∗(F(V)) → F∗(F(V)) is homotopic to the push-out of ∇ along the isomor- +phism F∗(F(µ)) : F∗(F(V) ⊗ F(V)) → F∗(F(V)), letting µ : I ⊗ I → I be a weak equivalence. +Following MacLane [Mac88], we recall the definition of generators of categories: Let C be a category +and G a class of objects of C. The set G is a generator of C if for any parallel of morphisms h, h′ : X → Y +in C, h � h′ implies that there exists S ∈ G and a morphism f : S → X such that h ◦ f � h′ ◦ f. +Definition 4.1. Let M be a model category with a class G of objects of M. We say that G generates M if +the equivalence class of G generates the homotopy category of M. +Lemma 4.2. Let M be a pointed closed symmetric monoidal model category with a class G of objects +of M. Assume that G generates M. Then an object M is contractible if and only if so is the internal +hom-object MapM(S, M) for any S ∈ G. +proof. The if-direction is clear. Let M be an object of M such that the internal-hom object MapM(S, M) +is contractible for any S ∈ G. Since G generates M, for any object N and morphism f : N → M, +9 + +the condition that f is not null-homotopic implies that there exists an object S ∈ G and a morphism +i : S → N and f ◦ i : S → M is not null-homotopic, contradicting the generating condition. +□ +Lemma 4.3. Let M be a pointed closed symmetric monoidal model category generated by the monoidal +unit V and F : M → M a symmetric monoidal Bousfield bilocalization, and F∗ : M → M denote the +homotopically fully faithful left adjoint and F! : M → M the homotopically fully faithful right adjoint. +Then, for any object M, F(M) is contractible if and only if so is F∗(F(V)) ⊗ M. +proof. Since the induced morphism F(cV) is to homotopic the identity on F(V), the if-direction is clear. +Assume that F(M) is contractible. Since F is monoidal, the counit cV : F∗(F(V)) → V induces a +weak equivalence F(M) → F(F∗(F(V)) ⊗ M). Therefore, by Lemma 4.2, the internal-hom object +MapM(V, F!(F(F∗(F(V) ⊗ M)))) is contractible. Note that, for any object N, one has a zig-zag of iso- +morphisms: +HomHo(M)(V, MapM(F∗(F(V)), N)) ≃ HomHo(M)(F∗(F(V)), N) +≃ HomHo(M)(V, F!F(N)) ≃ HomHo(M)(V, MapM(V, F!(F(N))))). +Applying the case N = F∗(F(V)) ⊗ M to the isomorphisms, we obtain that the internal-hom object +MapM(F∗(F(V)), F∗(F(V) ⊗ M))) is contractible, implying that, by the canonical isomorphism +HomHo(M)(F∗(F(V)) ⊗ M, F∗(F(V)) ⊗ M) ≃ HomHo(M)(M, MapM(F∗(F(V)), F∗(F(V)) ⊗ M)))), +F∗(F(V)) ⊗ M is also contractible. +□ +Proposition 4.4. Let M be a stable closed symmetric monoidal model category generated by the monoidal +unit V and F : M → M a symmetric monoidal Bousfield bilocalization, and F∗ : M → M denote the +homotopically fully faithful left adjoint. Then, for any object M of M, the counit cM : F∗(F(M)) → M of +M induces a zig-zag of weak equivalences: +F∗(F(M)) ≃ F∗(F(V) ⊗ F(M)) ≃ F∗(F(V)) ⊗ F∗(F(M)) → F∗(F(V)) ⊗ M. +proof. Let C denote the homotopy cokernel of the counit cM : F∗(F(M)) → M. Then F(C) is con- +tractible. By Lemma 4.3, the homotopy cokernel C ⊗ F(F∗(V)) of the induced morphism idF∗(F(V)) ⊗cM : +F∗(F(V)) ⊗ F∗(F(M)) → F∗(F(V)) ⊗ M is also contractible, implying that idF∗(F(V)) ⊗ cM : F∗(F(V)) ⊗ +F∗(F(M)) → F∗(F(V)) ⊗ M is a weak equivalence. +□ +In the case that M is stable and F∗ : M → M preserves all finite homotopy limits, the composition +F∗ ◦ F also preserves all finite homotopy limits. Immediately, by Proposition 4.4, the counit F∗(F(V)) is +a flat object of M if the monoidal unit V generates M. +Thus, we obtain the following theorem: +Theorem 4.5. Let M be a stable closed symmetric monoidal model category with a monoidal unit object +V and F : M → M a symmetric monoidal Bousfield bilocalization, and F∗ : M → M denote the +homotopically fully faithful left adjoint of F and c : F∗ ◦ F → Id the counit. Assume that the left adjoint +F∗ : M → M preserves all finite homotopy limits and the monoidal unit V generates M. Then the counit +morphism cV : F∗(F(V)) → V of the Quillen adjunction (F∗, F) determines a homotopically idempotent +Smith ideal j : I → V. +□ +10 + +References +[Fal88] +Faltings, Gerd: p-adic Hodge theory. In: J. Amer. Math. Soc. 1 (1988), Nr. 1, S. 255–299. – ISSN 0894–0347 +[GR03] +Gabber, Ofer ; Ramero, Lorenzo: Lecture Notes in Mathematics. Bd. 1800: Almost ring theory. Springer-Verlag, +Berlin, 2003. – vi+307 S. – ISBN 3–540–40594–1 +[Hir03] +Hirschhorn, Philip S.: Math. Surv. Monogr.. Bd. 99: Model categories and their localizations. Providence, RI: +American Mathematical Society (AMS), 2003. – ISBN 0–8218–3279–4 +[Hov99] Hovey, Mark: Mathematical Surveys and Monographs. Bd. 63: Model categories. American Mathematical Society, +Providence, RI, 1999. – xii+209 S. – ISBN 0–8218–1359–5 +[Hov14] Hovey, Mark: Smith ideals of structured ring spectra. Available at:https://arxiv.org/abs/1401.2850, 2014 +[Lei04] +Leinster, Tom: London Mathematical Society Lecture Note Series. Bd. 298: Higher operads, higher categories. +Cambridge University Press, Cambridge, 2004. – xiv+433 S. http://dx.doi.org/10.1017/CBO9780511525896. +http://dx.doi.org/10.1017/CBO9780511525896. – ISBN 0–521–53215–9 +[Lur09] +Lurie, Jacob: Annals of Mathematics Studies. Bd. 170: Higher topos theory. Princeton, NJ : Princeton University +Press, 2009. – xviii+925 S. – ISBN 978–0–691–14049–0; 0–691–14049–9 +[Lur17] +Lurie, Jacob: Higher Algebra. available at:https://www.math.ias.edu/˜lurie/papers/HA.pdf, 2017 +[Mac88] MacLane, Saunders: Grad. Texts Math.. Bd. 5: Categories for the working mathematician. 4th corrected printing. +New York etc.: Springer-Verlag, 1988. – ISBN 3–540–90035–7 +[Qui96] Quillen, Daniel: Module theory over nonunital rings. Available at: +https://ncatlab.org/nlab/files/QuillenModulesOverRngs.pdf, 1996 +[SS03a] Schwede, Stefan ; Shipley, Brooke: Equivalences of monoidal model categories. In: Algebr. Geom. Topol. 3 (2003), S. +287–334. http://dx.doi.org/10.2140/agt.2003.3.287. – DOI 10.2140/agt.2003.3.287. – ISSN 1472–2747 +[SS03b] Schwede, Stefan ; Shipley, Brooke: Stable model categories are categories of modules. In: Topology 42 (2003), Nr. 1, +S. 103–153. http://dx.doi.org/10.1016/S0040-9383(02)00006-X. – DOI 10.1016/S0040–9383(02)00006– +X. – ISSN 0040–9383 +National institute of technology, Ube college, 2-14-1, Tokiwadai, Ube, Yamaguchi, JAPAN 755-8555. +Email address: ykato@ube-k.ac.jp +11 + diff --git a/cNE4T4oBgHgl3EQfow2y/content/tmp_files/2301.05187v1.pdf.txt b/cNE4T4oBgHgl3EQfow2y/content/tmp_files/2301.05187v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..97cc09beaeefba94d0c878eec62cbd317a201580 --- /dev/null +++ b/cNE4T4oBgHgl3EQfow2y/content/tmp_files/2301.05187v1.pdf.txt @@ -0,0 +1,1657 @@ +WIRE: Wavelet Implicit Neural Representations +Vishwanath Saragadam, Daniel LeJeune, Jasper Tan, Guha Balakrishnan, +Ashok Veeraraghavan, Richard G. Baraniuk +Rice University +https://vishwa91.github.io/wire +Abstract +Implicit neural representations (INRs) have recently ad- +vanced numerous vision-related areas. INR performance +depends strongly on the choice of the nonlinear activation +function employed in its multilayer perceptron (MLP) net- +work. A wide range of nonlinearities have been explored, +but, unfortunately, current INRs designed to have high ac- +curacy also suffer from poor robustness (to signal noise, pa- +rameter variation, etc.). Inspired by harmonic analysis, we +develop a new, highly accurate and robust INR that does not +exhibit this tradeoff. Wavelet Implicit neural REpresen- +tation (WIRE) uses a continuous complex Gabor wavelet +activation function that is well-known to be optimally con- +centrated in space-frequency and to have excellent biases +for representing images. A wide range of experiments (im- +age denoising, image inpainting, super-resolution, com- +puted tomography reconstruction, image overfitting, and +novel view synthesis with neural radiance fields) demon- +strate that WIRE defines the new state of the art in INR +accuracy, training time, and robustness. +1. Introduction +Implicit neural representations (INRs), which learn a +continuous function over a set of data points, have emerged +as a promising general-purpose signal processing frame- +work. An INR consists of a multilayer perceptron (MLP) +with alternating linear layers and element-wise nonlinear +activation functions. Thanks to the MLP, INRs do not share +the locality biases that can limit the performance of convo- +lutional neural networks (CNNs). Consequently, INRs have +advanced the state of the art in numerous vision-related ar- +eas, including computer graphics [24, 29, 30], image pro- +cessing [11], inverse problems [43], and signal representa- +tions [42]. +Currently, INRs still face a number of obstacles that limit +their use. First, for some applications, especially those with +high-dimensional data such as 3D volumes, fitting an INR +to high accuracy can still take too long (tens of seconds) for +WIRE +Gauss +SIREN +Ground truth +1.0 +0.0 +Spatially +spread- +out +error +Spatially +compact +but large +error +Spatially +compact +and small +error +Error map +SIREN sin(𝜔0𝑥) +Gauss 𝑒− 𝑠0𝑥 2 +WIRE 𝑒j𝜔0𝑥𝑒−|𝑠0𝑥|2 +𝜎 𝑥 +𝜎 𝑥 +Re 𝜎 𝑥 +𝑥 +𝑥 +𝑥 +Nonlinearity for Implicit Neural Representations +Approximation accuracy with various nonlinearities +Figure 1. Wavelet implicit neural representation (WIRE). We +propose a new nonlinearity for implicit neural representations +(INRs) based on the continuous complex Gabor wavelet that has +high representation capacity for visual signals. The top row vi- +sualizes two commonly used nonlinearities: SIREN with sinu- +soidal nonlinearity and Gaussian nonlinearity, and WIRE that uses +a continuous complex Gabor wavelet. WIRE benefits from the fre- +quency compactness of sine, and spatial compactness of a Gaus- +sian nonlinearity. The bottom row shows error maps for approx- +imating an image with strong edges. +SIREN results in global +ringing artifacts while Gaussian nonlinearity leads to compact but +large error at edges. WIRE produces results with the smallest and +most spatially compact error. This enables WIRE to learn repre- +sentations rapidly and accurately, while being robust to noise and +undersampling of data. +real time applications. Second, INRs are not robust to sig- +nal noise or insufficient measurements. Indeed, most works +on INRs in the literature assume virtually no signal noise +and large amounts of data. We find in our own experiments +that current INR methods are ineffective for tasks such as +denoising or super-resolution. Finally, INRs still have room +for improvement in representational accuracy, especially for +fine details. +1 +arXiv:2301.05187v1 [cs.CV] 5 Jan 2023 + +In this paper, we develop a new, faster, more accurate, +and robust INR that addresses these issues and takes INR +performance to the next level. To achieve this, we take in- +spiration from harmonic analysis and reconsider the nonlin- +ear activation function used in the MLP. Recent work has +shown that an INR can be interpreted as a structured signal +representation dictionary [53], where the activation nonlin- +earity dictates the atoms of the dictionary. For example, +the sine activation creates a pseudo-Fourier transform rep- +resentation of the signal that is maximally concentrated in +the frequency domain [53]. +An important conclusion one can draw from the past four +decades of harmonic analysis research is that Fourier meth- +ods are suboptimal for representing the kinds of signals that +feature in typical vision tasks [26]. These kinds of signals, +e.g., natural images from photographs, are much more con- +cisely and robustly represented using wavelet atoms that are +optimally concentrated in space–frequency. Sparse compo- +sitions of wavelet atoms are known to have excellent bi- +ases for representing images; cf. the seminal work in com- +puter vision (e.g., Laplacian pyramid), computational neu- +roscience [32], and the JPEG2000 compression standard. +In this paper, we introduce Wavelet Implicit neural +REpresentation (WIRE), a new INR based on a complex +Gabor wavelet activation function (see Figure 1). Through +a wide range of experiments, we demonstrate that WIRE +defines the new state of the art in INR accuracy, train- +ing time, and robustness. We showcase that WIRE’s in- +creased robustness is particularly useful for solving difficult +vision inverse problems, including image denoising (robust- +ness), image inpainting and super-resolution (superior inter- +polation), and 2D computed tomography (CT) reconstruc- +tion (solving higher-dimensional inverse problems). WIRE +also outperforms other INRs for signal representation tasks +such as overfitting images and learning point cloud occu- +pancy volumes. Finally, we show that WIRE enables faster, +more robust novel view synthesis with neural radiance fields +(NeRF) [29] from critically few training views. +2. Prior Work +Regularization for inverse problems. +Inverse problems +involve estimating a signal from a linear or nonlinear set +of measurements. +Inevitably, the measurements are de- +graded by noise (such as camera readout or photon noise), +or the problem is ill-conditioned, necessitating regulariza- +tion. +There are many forms of regularization, including +ridge regression, Lasso [46], total variation (TV) [10], and +sparsity-based [7] techniques that seek to penalize the ℓ1 +norm the signal or some transform thereof. +In the past +decade, data-driven regularization, including overcomplete +dictionary-based [4] and generative network-based [31, 36, +37] ones, have been developed. The classical model-based +approaches are inadequate for severely ill-conditioned prob- +lems, while the data-driven ones critically depend on data. +Convolutional neural networks (CNNs). +CNNs, the +most popular neural network architectures in computer vi- +sion for the past decade, have been shown to exhibit strong +implicit biases that favor image-like signals. This has been +demonstrated with works like deep image prior (DIP) [48] +and its variations [15, 21] that produce remarkable results +on image-related linear inverse problems without any prior +training data. However, such CNN-based priors are tied to a +discrete grid-like signal representation which is not applica- +ble to problems such as novel view synthesis, or for solving +ordinary and partial differential equations, and not scalable +for very high dimensional signals such as 3D tomographic +volumes, gigapixel images, or large point clouds. +Deep image prior. +Neural networks, and particularly +CNNs, exhibit implicit biases due to their specific archi- +tectures (such as a UNet [38]), implying that even untrained +neural networks can be used for regularization. This was +leveraged to build a deep image prior (DIP) [48] that pro- +duces outputs that tend to look like images. The key idea +is to recast regularization as optimizing for the weights of +the network for each instance of the problem. The perfor- +mance of DIPs is considerably superior to classical regu- +larization approaches. +However, DIPs exhibit good per- +formance only when over-parameterized and are tied to a +grid-like discretized representation of the signal, implying +DIPs do not scale to high dimensional signals such as point +clouds with a large number of points. The issue of compu- +tational cost has been addressed to a certain extent by the +deep decoder [21] and the DeepTensor [39], but they still +need the signal to be defined as a regular data grid such as a +2D matrix or 3D tensor. +Implicit representations. +INRs are continuous learned +function approximators based on multilayer perceptrons +(MLPs). The continuous nature of INRs is particularly ap- +pealing when dealing with irregularly sampled signals such +as a point clouds. Since its first widespread usage in novel +view synthesis in graphics [29], INRs have pervaded nearly +all fields of vision and signal processing including render- +ing [24], computational imaging [6, 12], medical imag- +ing [51], and virtual reality [16]. +The popular choice of the ReLU nonlinearity in stan- +dard neural networks has been empirically shown to result +in poor approximation accuracy in INRs. This has been +remedied by several modifications to the MLP including +the so-called positional encoding [30, 44], as well as var- +ious choices of nonlinearity such as the sinusoidal func- +tion [42] and the Gaussian function [34]. A closely related +work is the Gabor wavelet-based multiplicative filter net- +works (MFN), where the output after each layer is multi- +2 + +plied by a Gabor filter. The output then results in a combina- +tion of exponentially many Gabor wavelets, thereby result- +ing in large capacity. Numerous architectural changes have +also been proposed that leverage multiscale properties of vi- +sual signals to accelerate the INR training procedure includ- +ing adaptive block decomposition [27], kilo-NeRF [35], and +predicting the Laplacian pyramid of the signal [40]. +INRs can now train on signals nearly instantly [30] +thanks to these numerous advances. However, the high ca- +pacity of such INRs precludes robustness — implying that +the signal representation is brittle, resulting in overfitting to +both noise and signal equally. In this paper, we propose the +complex Gabor wavelet as a nonlinearity, which is uniquely +well-suited to induce robustness in INRs. +Wavelet transform and the Gabor wavelet. +The Fourier +transform decomposes the signal as a sum of sinusoids +with infinite space support, implying that there is no no- +tion of spatial compactness. The wavelet transform reme- +dies this by decomposing the signal as a linear combina- +tion of translated and scaled versions of a short oscillating +pulse called a wavelet. Wavelets typically result in faster ap- +proximation rates for signals and images than Fourier trans- +form [17], and hence they are often used for image com- +pression [18, 41] and as a robust prior for inverse problems +of images [20] and videos [49]. In this paper, we show that +wavelets are a universally superior choice for the nonlin- +earity in INRs due to their compact support in space and +frequency and therefore faster approximation rates. +3. Wavelet Implicit Representations +3.1. INR details +Consider an INR function Fθ : RDi �→ RDo mapping Di +input dimensions to Do output dimensions, where θ repre- +sents the MLP’s tunable parameters. The goal is to con- +struct Fθ such that it approximates a function g(x) of in- +terest, i.e., g(x) ≈ Fθ(x). For example, g(x) may simply +be a ground truth image, represented as a function mapping +coordinates to pixel values. Modeling Fθ(·) as an M-layer +MLP, the output at each layer is given by +ym = σ(Wmym−1 + bm), +(1) +where σ is the nonlinearity (or nonlinear activation func- +tion); Wm, bm are weights and biases for the mth layer; +y0 = x ∈ RDi is the input coordinate and yM+1 = +WM+1yM + bM+1 is the final output. +The nonlinear activation σ plays a key role in the rep- +resentation capacity of the INR (see Fig. 1). +Two lead- +ing choices include the periodic σ(x) = sin(ω0x) used +in SIREN [42], and the Gaussian nonlinearity σ(x) = +e−(s0x)2 used by Ramasinghe et. al. [34]—both result in +significantly higher representation accuracy than ReLU. +However, their high representation capacity is also a draw- +back, since they can represent noise with nearly equal ac- +curacy as an image. Our goal is to propose a nonlinearity σ +that is well-suited for visual signals such as images, videos, +and 3D volumes but poorly fits noise-like signals. +3.2. WIRE +Armed with the insight that a Gabor wavelet achieves op- +timal time-frequency compactness, we propose the wavelet +implicit representation (WIRE) that uses the continuous +complex Gabor wavelet ψ for its activation nonlinearity: +σ(x) = ψ(x; ω0, s0) = ejω0xe−|s0x|2, +(2) +where ω0 controls the frequency of the wavelet and s0 con- +trols the spread (or width). The first layer activations have +the form +y1 = ψ(W1x + b1; ω0, s0), +(3) +which are copies of the mother Gabor wavelet ψ at scales +and shifts determined by W1 and b1. Hence the building +blocks of WIRE are drawn from a dictionary of wavelet +atoms. We let the weights of the INR as well as the outputs +be complex-valued to preserve phase relationships through- +out, and we represent real signals by simply taking the real +part of the output and discarding the imaginary part. Just as +wavelets combine space and frequency compactness, WIRE +enjoys the advantages of periodic nonlinearities such as +SIREN due to the complex exponential term and the spa- +tial compactness from the Gaussian window term; recall +Figure 1. Additionally, unlike SIREN, WIRE does not re- +quire a carefully chosen set of initial weights (see Fig. 3) +due to the Gaussian window, which creates a spatially com- +pact output at each layer and produces high quality results +with the default neural network initialization of uniformly +random weights independent of the parameters ω0, s0. +3.3. Implicit bias of WIRE +Neural tangent kernel perspective. +As stated, we seek +an INR that fits visual signals well but fits noise poorly +in comparison. Inspired by [53], who proposed to com- +pare eigenfunctions of the empirical neural tangent ker- +nel (NTK) [22] of INRs to understand their approximation +properties, we compare the fitting of noisy natural images +using NTK gradient flow. The NTK gradient flow of INRs +accurately captures the behavior of early training of neural +networks, and so in tasks such as denoising where we reg- +ularize via early stopping, the early training behavior deter- +mines the implicit bias. In the lazy training regime of wide +neural networks [25], the fit image at time t ≥ 0 has value +Fθt(x) = [(I − e−tK)g](x), +(4) +3 + + + + + + + + + + + + + + + + +(a) Implicit bias with NTK +gradient flow + + + + + + + + + + + + + + + + + +(b) Implicit bias in standard INRs. +Figure 2. +Implicit bias in denoising (a) The empirical NTK +of finite-width INRs provides an insight into the implicit bias +of INRs. Early trajectories of NTK gradient flow show WIRE +converging to the image faster than the noise, outperforming all +other nonlinearities. Bars indicate one standard deviation over the +dataset. (b) Early iterations of standard training are reflected well +by the relative performances of NTK gradient flow from part (a). +Furthermore, WIRE maintains its advantage against other nonlin- +earities throughout the remainder of training. +where I is the identity operator, K is the NTK operator on +the image’s spatial domain, and g is the image being fit. +In Fig. 2a, we apply NTK gradient flow using the em- +pirical finite-width NTK to a denoising task, fitting the +original image with N(0, 0.052) i.i.d. pixel-wise additive +noise. Due to the computational intensity of evaluating the +NTK, we evaluate on 64 × 64 × 3 images from Tiny Ima- +geNet [2]. Comparing WIRE to other INRs, we see that, as +desired, WIRE prefers to learn the signal in the image early +in training rather than the noise, converging orders of mag- +nitude faster to essentially any given peak signal-to-noise- +ratio (PSNR). +Empirical evaluation. +We perform a denoising task anal- +ogous to the NTK-based analysis for real INRs on the 24 +768 × 512 × 3 images from the Kodak Lossless True Color +Image Suite [1], again with N(0, 0.052) additive noise, in +Fig. 2b. We apply the same INRs as in the NTK example, +but train with ordinary neural network gradient optimiza- +tion instead of NTK gradient flow. Again, WIRE drastically +outperforms other INRs, converging an order of magnitude +faster to the same PSNR. +3.4. Choosing the parameters ω0, s0 +WIRE’s performance is primarily decided by the con- +stants ω0, s0 that control frequency of the sinusoid and +width of the Gaussian, respectively. +WIRE outperforms +both the SIREN and Gaussian nonlinearities across a broad +range of parameters. Figure 3 shows the approximation ac- +curacy achieved by WIRE for various parameters. We set +the number of hidden layers to three, and number of hidden +SIREN (𝑠0 = 0) +Image Representation +Image denoising +Gauss (𝜔0 = 0) +PSNR +(dB) +Figure 3. WIRE is robust to the choice of parameters. The plot +above shows accuracy for image representation and denoising with +various settings of ω0 and s0. The boxes show special cases with +ω = 0 corresponding to Gaussian nonlinearity, and s0 = 0 cor- +responding to SIREN. WIRE achieves higher accuracy than both +SIREN and Gauss on image representation as well as image de- +noising tasks (marked by white cross). Further, WIRE achieves +super performance for a large choice of parameters ω0, s0 imply- +ing that WIRE is not overly sensitive to the hyperparameters. +features to 256. When ω0 = 0, we used a Gaussian non- +linearity. When s0 = 0, we used a sinusoidal nonlinearity. +When both parameters were zero, we used a ReLU nonlin- +earity. For the denoising task, we added photon noise equiv- +alent to a maximum of 50 photons per pixel. We observe +from Fig. 3 that WIRE outperforms SIREN, Gauss, and +ReLU. Moreover, the performance is superior for a large +swath of values of ω0, s0 for both image representation and +denoising. The reduced sensitivity to the exact values of +ω0, s0 implies that WIRE can be used without precise in- +formation about image or noise statistics. +Alternate forms of WIRE. +For problems where com- +plex weights are infeasible, WIRE can be instantiated as +the imaginary (or real) part of the complex Gabor wavelet, +ψ(x; ω0, s0) = sin(ω0x)e−(s0x)2. Note that setting s0 = 0 +results in the sine nonlinearity used in SIREN [42] and +ω0 = 0 results in Gaussian nonlinearity [34] implying +WIRE inherits the favorable properties of previously pro- +posed nonlinearities. Another embodiment of WIRE is a +Constant-Q Gabor wavelet where ω0s0 = Q, which re- +sults in constant fractional bandwidth (ω/δω). Constant- +Q Gabor wavelets are often used in music analysis [8, 47], +wavelet transforms [26], and the Laplace transform. Hav- +ing only a single parameter makes hyperparameter tuning +simpler with a fixed Q. +4 + +3.5. Multidimensional localization +As defined above, WIRE applies the Gabor mother +wavelet ψ element-wise to output of the linear transfor- +mation Wmym−1. Hence, the output of each unit is spa- +tially localized only in the single direction determined by +the corresponding row of Wm (see Fig. 11 top). +While +such highly anisotropic spatial localization is well-suited +for certain kinds of data, many natural data (e.g., photo- +graphic images) are best represented using a combination +of atoms with isotropic and anisotropic spatial localization +(e.g., wavelets and curvelets [9] or wavelets and wedgelets +[50]). To achieve spatial localization along multiple direc- +tions, we augment the Gabor mother wavelet with Dm − 1 +additional Gaussian windows: +ym = ψ(W (1) +m ym−1 + b(1) +m ; w0, s0) +· e− �Dm +k=2 |s0(W (k) +m ym−1+b(k) +m )|2. +(5) +In two-dimensional settings, such as with natural image +data, the resulting first-layer activations will resemble a +mixture of Gabor wavelets and curvelets (see Fig. 11 bot- +tom). As we will see below in Section 4.4, the more diverse +spatial localization of the resulting 2D WIRE representa- +tion significantly benefits its performance (see Fig. 12 and +Fig.13). +4. Experiments +WIRE learns representations for all signal classes faster +than state-of-the-art techniques. In addition, WIRE is well- +suited to solve a large class of inverse problems where the +number of measurements is far fewer than the dimensional- +ity of the signal, or when the measurements are corrupted +by noise. For all the experiments below, we implemented +the optimization procedure in PyTorch [33] and used the +Adam optimizer [23]. Code was executed on a system unit +equipped with 64GB RAM, and an Nvidia RTX 2080 Ti +graphical processing unit (GPU) with 8GB memory. Unless +specified, we used an ℓ2 loss function between the measure- +ments and the outputs of INR. No other regularization was +used. We used a learning rate scheduler which decayed the +initial learning rate by 0.1 at the end of training epochs. +4.1. Signal representation +A common feature enabled by INRs is representation of +signals. We evaluate two tasks for this experiment: repre- +senting images and representing occupancy volumes [28]. +In both cases, we used an MLP with three hidden layers +with a width of 300 features for all nonlinearities. +For +WIRE, we reduced the number of parameters by half to ac- +count for the doubling due to real and imaginary parts. We +did so by reducing the number of hidden features by +√ +2. +The parameters for each nonlinearity and the learning rate + + + + + + + + + + + + + + +(a) Image representation + + + + + + + + + + + + + + + + +(b) Volume representation +Figure 4. WIRE learns faster. The two plots above show rep- +resentation accuracy for an image (top row in Fig. 5) and an oc- +cupancy volume (bottom row in Fig. 5) over time. Owing to the +high approximation capacity of Gabor wavelets for visual signals, +WIRE achieves high accuracy at a faster rate, making it an appro- +priate choice for representing visual signals. +were chosen to obtain fastest approximation rate. Specif- +ically, we chose ω0 = 20, s0 = 10 for WIRE, ω0 = 40 +for SIREN, and s0 = 30 for Gaussian. We also compare +against multiplicative frequency networks (MFN) [19]. For +the occupancy volume, we sampled over a 512 × 512 × 512 +grid with each voxel within the volume assigned a 1, and +voxels outside the volume assigned a 0. We evaluated the +PSNR and structural similarity (SSIM) [52] for images and +intersection over union (IOU) for the occupancy volumes. +Figure 4 shows the approximation accuracy as a func- +tion of time for an image (Kodak dataset) and an occupancy +volume (Thai statue). WIRE not only achieves the highest +accuracy, but it does so at a much faster rate than other ap- +proaches. Figure 5 visualizes the final representation of the +example image after 1.6 minutes, and the 3D mesh of the +Thai Statue constructed with marching cubes after 30 min- +utes. WIRE achieves the highest accuracy both for images +(43.2dB) and for the occupancy volume (0.99), underlying +our hypothesis that INRs equipped with a Gabor nonlinear- +ity have higher approximation accuracy. +4.2. Solving inverse problems of 2D images +WIRE’s inductive bias favors images, and hence can be +used for solving linear inverse problems. To demonstrate +the advantages of WIRE as a strong prior for images, we +showcase its performance on image denoising, single image +super resolution, and multiimage super resolution. +Image denoising. +To evaluate the robustness of INRs for +representing noisy signals, we learned a representation on a +high resolution color image from the DIV2K dataset [3]. +We simulated photon noise with an independently dis- +tributed Poisson random variable at each pixel with a max- +imum mean photon count of 30, and a readout count of 2, +resulting in an input PSNR of 17.6 dB. We then learned a +5 + +Ground truth +WIRE (0.99) +ReLU + Pos. Enc (0.98) SIREN (0.97) +Gauss (0.97) +MFN (0.94) +Ground +truth +WIRE +(43.2dB) +ReLU + Pos. Enc +(32.1dB) +SIREN +(42.4dB) +Gauss +(40.0dB) +MFN +(32.1dB) +Figure 5. WIRE has high representation capacity. The results above show image representation in the first row and meshes generated +with occupancy volumes in the second row with various nonlinearities. WIRE achieves highest representation accuracy for both data, +underlining its advantages as a signal model. +representation on this noisy image with various nonlineari- +ties. In all cases, we chose an MLP with two hidden layers +and 256 features per layer. We also compared the denoising +result with deep image prior (DIP) [48]. Figure 6 visualizes +the final result for each nonlinearity along with metrices for +each result. WIRE produces the sharpest image with least +amount of residual noise. Qualitatively, WIRE’s result is +similar to DIP’s, implying that WIRE enjoys inductive bi- +ases that make it a good choice for inverse problems. +Image super resolution. +INRs function as interpolatants, +and hence super resolution is expected to benefit from INRs +with good implicit biases. We evaluate this hypothesis by +implementing 4× super resolution on a DIV2K image. The +forward operator can be cast as y = A4x where A4 imple- +ments a 4× downsampling operator (without aliasing). We +then solved for the sharp image by modeling x as output of +an INR. Figure 7 visualizes the result on super resolution of +image of a butterfly with various approaches. WIRE pro- +duces the sharpest result with crisp details on the butterfly’s +antenna and on the wings. As with denoising, WIRE results +are similar to DIP, establishing the ubiquity of WIRE. +INRs are particularly advantageous when data interpola- +tion needs to be performed on an irregular grid. An example +of such settings is multi-image super resolution where the +images are shifted and rotated with respect to each other. +Figure 21 shows an example of 4× super resolution with +four images (and hence 25% compression) from the Kodak +dataset [1] simulated with a small sub-pixel motion between +them. The forward operator is then yk = Ak +4x where Ak +4 +encodes the downsampling, and translation and rotation for +the kth image. The visualizations in the figure demonstrate +that WIRE achieves the highest accuracy and is qualitatively +better at reconstructing high frequency components. In con- +trast, the Gaussian nonlinearity leads to a blurry reconstruc- +tion, while SIREN results in ringing artifacts. +Computed tomography (CT) reconstruction. +Strong +signal priors are critical for solving underconstrained prob- +lems, and CT reconstruction is one such example. We em- +6 + +Ground truth +Noisy image +MFN +DIP +WIRE +Gauss +ReLU + Pos. Enc. +SIREN +17.6dB +0.34 +28.1dB +0.85 +30.1dB +0.93 +29.7dB +0.93 +29.2dB +0.89 +26.6dB +0.90 +30.2dB +0.93 +Figure 6. WIRE is robust to noise. A powerful feature uniquely enabled by WIRE is the robustness to noisy data. Here, we show an +image representation with added shot noise, resulting in an input PSNR of 17.6dB. Among the various approaches, WIRE results in the +highest PSNR and SSIM of any representation, thereby naturally resulting in denoising. +Ground truth +Bilinear interpolation +MFN +DIP +WIRE +Gauss +ReLU + Pos. Enc. +SIREN +26.4dB +0.92 +21.9dB +0.77 +26.9dB +0.93 +26.1dB +0.91 +26.3dB +0.92 +25.6dB +0.90 +27.3dB +0.93 +Figure 7. WIRE for single image super resolution. The figure above shows results for a 4× single image super resolution with various +approaches. Thanks to its strong implicit bias, WIRE results in the sharpest reconstruction with quantitatively higher reconstruction metrics. +ulated 100 CT measurements of a 256 × 256 x-ray col- +orectal image [13]. Figure 9 shows the final reconstruction +with various approaches. WIRE results in the sharpest re- +construction with clearly pronounced features. SIREN per- +forms the second best but has striation artifacts that are ex- +pected from an unregularized reconstruction. The Gaussian +nonlinearity results in overly smooth results. WIRE can +hence be used as a robust prior for inverse problems with +noisy and undersampled measurements. +4.3. Learning neural radiance fields +INRs have been leveraged successfully for novel-view +synthesis with neural radiance fields (NeRF) [29]. Given +images from a sparse set of view points, the goal is to render +an image from a different view point that is not in the train- +ing set. NeRF achieves this by training a common INR that +takes 3D location and viewing directions as inputs (hence +a 5D input), and produces transmission and color at that +location. Images are then produced by integrating along +lines that pass through each view’s lens (pinhole). The sim- +plest NeRF architecture consists of positional encoding, and +two MLPs equipped with ReLU for transmission and color +values. +We show that WIRE without any positional en- +coding produces higher quality results within fewer epochs. +We trained NeRFs for reconstruction on the synthetic drum +dataset [29]. Each image was downsampled to a resolution +7 + +Q00Ground truth +Bicubic (4x) +WIRE +ReLU + Pos. Enc +SIREN +Gauss +18.8dB +0.58 +23.2dB +0.80 +21.7dB +0.71 +22.2dB +0.74 +22.4dB +0.75 +Figure 8. Multi-image super resolution. INRS are particularly appealing for handling data on an irregular grid, such as images captured +with multiple sub-pixel shifts. The figure above shows 4× super resolution with 4 images captured with varying sub-pixel shifts and +rotations. We then solved a joint inverse problem where the high resolution image is modeled as the output of an INR. WIRE produces the +best reconstruction both quantitatively and qualitatively, implying that WIRE has favorable interpolation properties for visual signals. +Ground truth +TV-regularized +MFN +DIP +WIRE +Gauss +ReLU + Pos. Enc. +SIREN +30.8dB +0.75 +18.1dB +0.23 +31.9dB +0.82 +29.2dB +0.73 +28.5dB +0.71 +30.3dB +0.76 +32.3dB +0.81 +Figure 9. Computed tomography reconstruction. Inverse problems with noisy undersampled data require a strong signal prior for +robust reconstruction. Here, we show CT-based reconstruction with 100 angles for a 256 × 256 image (2.5× compression) with various +approaches. WIRE results in sharp reconstruction, exposing features that are blurry, or with ringing artifacts in reconstructions with other +approaches. WIRE is hence a strong signal prior for images, and can solve a large class of inverse problems. +WIRE (24.4dB) +ReLU+Pos.Enc (21.1dB) +SIREN (24.2dB) +Gauss (22.9dB) +Figure 10. Novel-view synthesis with neural radiance fields. INRs have shown most promise in novel-view synthesis where the transmit- +tance and color at each 3D voxel is modeled as output of INRs. Here, we show that WIRE is well-suited for novel-view synthesis with no +additional positional encoding. WIRE not only achieves higher accuracy (+0.2dB) with fewer epochs, but captures details that are missed +out by other nonlinearities, such as the rod connecting the ride cymbal to its stand and the anisotropic reflections on the cymbals. +8 + +24 +23 +(dB) +22 +PSNR ( +21 +20 +-WIRE + e -ReLU + Pos. Enc. +-A--SIREN +19 +........ Gauss +0 +500 +1000 +1500 +EpochReal +WIRE +Imaginary +Real +Imaginary +2D WIRE +Figure 11. First layer outputs with multi-dimensional WIRE. +The figure above shows outputs after first hidden layer with WIRE +and 2D WIRE. We observe that 2D WIRE has spatially compact +outputs due to the second Gaussian window, while WIRE has elon- +gated structures orthogonal to the Gaussian window. +of 200 × 200. To show the advantages of WIRE, we trained +the radiance field with only 25 images instead of the default +100 images. We used the “torch-NGP” codebase [45] for +training the NeRF model. For all experiments, we chose a +4-layered MLP with a width of 128 features for each layer. +Parameters and learning rate were chosen to achieve fastest +rate of increase of approximation accuracy on the validation +dataset. Figure 10 shows results with various nonlinearities. +WIRE produces highest accuracy (+0.2dB) with fastest rate +of increase. WIRE learns features absent in outputs of other +nonlinearities such as the rod connecting the ride cymbal to +its stand and the anisotropic reflections on the cymbal. +4.4. Multi-dimensional WIRE comparisons +As we discussed in Section 3.5, WIRE can be instanti- +ated as a multi-dimensional non-linearity. One advantage +of a two-dimensional WIRE is that its activations tend to be +compact along both spatial axes (see Fig. 11). This enables +a more accurate fit for signals that are composed of spatially +compact structures, such as many natural images. Figure 12 +demonstrates the advantage of 2D WIRE’s more general +spatial localization for representing an image of point sin- +gularities. 2D WIRE results in a sharper representation than +1D WIRE, which blurs out some of the points. +(a) Ground truth +(b) 2D WIRE +(c) WIRE + + + + + + + + + + + + + + + + + + + +(d) Accuracy vs. time +Figure 12. Multi-dimensional localization. The spatially com- +pact nature of 2D WIRE enables representing sparse images. In +this example, we show representation of a 256 × 256 image with +256 non-zero values. 2D WIRE represents each dot sharply while +WIRE tends to blur the features along one of the two axes. Both +WIRE and 2D WIRE converge rapidly compared to other ap- +proaches, as visualized in the plot in the bottom right corner. +The spatial localization of multi-dimensional WIRE is +also of great advantage for solving inverse problems more +robustly. Figure 13 compares WIRE and 2D WIRE on de- +noising, CT reconstruction, and 4× super-resolution tasks. +In all tasks, ω0 and s0 of the mother wavelet were chosen +for best performance, and the total number of parameters +were the same for WIRE and 2D WIRE. Across the board, +2D WIRE achieves higher accuracy than WIRE. 2D WIRE +also learns sharper features for the CT reconstruction of +lungs while WIRE’s tend to be more blurry. Similarly, in +the super-resolution task, 2D WIRE learns improved high +frequency features to represent the center of the flower. +5. Conclusions +We have proposed and validated the advantages of WIRE +that equips INRs with a complex Gabor wavelet activation +nonlinearity. We have shown with an extensive set of ex- +periments that WIRE (a) has higher representation capac- +ity, (b) achieves higher accuracy at a faster rate, and (c) has +strong inductive biases that make it compelling for solv- +ing challenging inverse problems. +Other activation non- +linearities have largely complementary strengths: SIREN +9 + +国国国WIRE +24.3dB +0.78 +CT with 50 projections +26.9dB +0.84 +2D WIRE +31.9dB +0.972 +4X Super-resolution +32.1dB +0.974 +28.1dB +0.95 +Denoising +28.5dB +0.96 +Ground truth +Figure 13. Performance with multi-dimensional WIRE. The +figure above shows various linear inverse problems solved with +INRs equipped with WIRE and 2D WIRE. Across the board, 2D +WIRE achieves higher performance in terms of PSNR and SSIM. +Visually, we observe that CT reconstruction and super resolution is +significantly sharper with 2D WIRE owing to the compact nature +of activations. +has high representation capacity and trains fast, but is poor +at regularizing inverse problems. Positional encoding has +lower capacity but is a good choice for novel-view synthe- +sis. Gaussian nonlinearity is more favorable for denoising +tasks. However, WIRE inherits the best properties of all of +the above activation nonlinearities, and hence is the current +go-to INR solution for signal representation and solving in- +verse problems. +6. Acknowledgements +This work was supported by NSF grants CCF-1911094, +IIS-1838177, and IIS-1730574; ONR grants N00014-18- +12571, N00014-20-1-2534, and MURI N00014-20-1-2787; +AFOSR grant FA9550-22-1-0060; and a Vannevar Bush +Faculty Fellowship, ONR grant N00014-18-1-2047. +A. Appendix 1: Experimental Details +A.1. WIRE initialization +INRs like SIREN [42] strongly depend on initialization +to obtain accurate representation. WIRE does not require +any initialization except for the default uniform weights. +However, since WIRE consists of a complex sinusoidal +term, it marginally benefits from SIREN-like initialization. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Image representation +Image denoising +Figure 14. Effect of initialization. The plots show approximation +accuracy for image representation and image denoising (20dB in- +put PSNR) across training epochs with SIREN-like weight initial- +ization and standard initialization. WIRE is robust to the initial +weights, but marginally benefits from a SIREN-like initialization. +To understand the dependence, we evaluated approximation +accuracy for image representation (no noise), and image de- +noising (20dB image noise). Here, a SIREN-like weight +initialization implies the first layer weights are drawn from +U(−1/N, 1/N) and the weights of the rest of the layers +are drawn from U(− +� +6/(ω0N), +� +6/(ω0N)), where N +is the number of input features and U(a, b) is a uniform +distribution over [a, b]. A normal weight initialization in- +volves drawing weights from U(−1/ +√ +N, 1/ +√ +N) for all +layers. Fig. 14 compares the representation accuracy for +SIREN-like and standard initialization. In both cases, we +see that the trends are nearly similar; SIREN-like initializa- +tion results in up to 1dB higher accuracy. Hence WIRE is +largely robust to initial parameters which enables easy tun- +ing across a large range of hyperparameters ω0, s0. +A.2. WIRE layer visualizations +Gabor wavelets uniquely enable space–frequency local- +ization, a property we observe is inherited by WIRE. To +evaluate this hypothesis, we visualized the output WIRE +composed of an MLP with two hidden layers and 181 hid- +den features each. We then learned a representation for a +Siemens star test image that consists of all spatial frequen- +cies and orientations. Fig. 15 visualizes the input image real +and imaginary outputs of 64 hidden features with least vari- +ance. The output of each first layer feature consists of one- +dimensional Gabor wavelets at various orientations, while +the outputs of second layer consist of sparsely populated +images. +Fig. 16 visualizes outputs at each layer for various non- +linearities and the final approximated image for the Siemens +star test image. The sparse outputs of second layer are ev- +idently unique to WIRE. Gauss has outputs that look less +spares, while SIREN and ReLU with positional encoding +result in dense outputs. This has a direct consequence on +approximation capacity for high frequency parts of the sig- +10 + +Real part +Layer 1 +Imaginary part +Layer 2 +Input image +(256x256) +Each patch 256x256 +Figure 15. +Layer outputs for WIRE. The input image is a +Siemens star test image that contains all spatial frequencies and +all angles. The patches show outputs (same size as image) of each +hidden feature in layer one and two for a two hidden-layer MLP +equipped with WIRE nonlinearity. WIRE results in sparse images +which enables high representational capacity for images, as shown +in Fig. 16. +nal. The final result in the bottom row shows that the sparse +nature of outputs of WIRE enables high approximation ac- +curacy with qualitatively better features at the center of the +image which consists of highest spatial frequenices. Gauss +follows next as it results in the second most sparse outputs +at each layer. SIREN and ReLU with positional encoding +alike produce blurry outputs at the center, primarily due to +the non-compact nature of outputs. WIRE’s ability to de- +compose images as a linear combination of sparse images +results in high representational capacity for the same num- +ber of parameters, as we verify empirically in the next sec- +tion. +A.3. Sensitivity to training parameters +WIRE is a promising INR model that achieves high rep- +resentation accuracy and is robust to a wide range of train- +ing parameters. We demonstrate the efficacy of WIRE in +this section with several sensitivity analyses. +Effect of learning rate. +WIRE performs well for a large +range of learning rates. +To understand the performance +trends, we learned an image representation with added noise +(20dB input PNSR) with various nonlinearities. We used a +2-layer MLP with 256 hidden features per layer. Fig. 17 +shows the maximum representation PSNR with varying +learning rates. WIRE has a stable and significantly higher +accuracy compared to other approaches. Interestingly, the +highest accuracy is achieved at a high learning rate of +2 × 10−2. This behavior is also observed with deep im- +age prior [48] where a larger learning rate enabled stronger +regularization. Such a similar behavior implies WIRE en- +joys strong inductive biases and hence is amenable to solve +inverse problems. +Effect of number of layers. +Fig. 18a shows a plot of rep- +resentation accuracy of an image for varying number of +hidden layers with various nonlinearities. +In each case, +the number of hidden features were set to 256. +We re- +duced the learning rate with increasing layers to avoid diver- +gence. WIRE uniformly outperforms other approaches (ex- +cept with 0 hidden features), as is to be expected as Gabor +wavelets enable high approximation accuracies for images. +Interestingly, for a large number of hidden layers (≥ 3), +WIRE performance is similar to SIREN and Gaussian non- +linearity. This is to be expected as the network has a large +capacity with so many layers. However, a large number +of layers is computationally expensive and often results in +an unstable learning regime. WIRE therefore is a reliable +choice for small to medium number of hidden layers for +most cases. +Effect of number of features. +Fig. 18b shows approx- +imation accuracy for image representation with varying +number of hidden features. In all cases, the number of hid- +den layers were fixed to be two. The performance of WIRE +is similar to other nonlinearities at very low number of hid- +den features, where all models similarly lack sufficient rich- +ness. For higher than 128 features, WIRE outperforms other +approaches with MFN [19] coming a close second. +A.4. Inverse problems +Computed tomographic reconstruction. +We showed in +Section 4 that computed tomography (CT) benefits from in- +ductive biases of INRs. Here, we study the effect of num- +ber of measurements. Fig. 19a shows the ground truth im- +age we used in our experiments. We denoised the original +512 × 512 image [5] with BM3D [14] (σ = 0.1) to remove +streak artifacts. We then simulated CT reconstruction with +varying numbers of projections. In each case, we used an +MLP with three hidden layers and 256 hidden features per +layer. We sampled the INR on a regular grid to first gen- +erate the image, and then use Radon transform to obtain +the sinogram. From the accuracy plot in Fig. 19b, we see +that WIRE achieves higher PSNR than any other nonlinear- +ity. Fig. 20 visualizes the reconstruction with varying num- +ber of projections for each nonlinearity. The reconstruction +is visually superior even with small number of projections, +which is particularly beneficial for reducing exposure to x- +rays during capture. +11 + +WIRE (real part) +Gauss +SIREN +ReLU + Pos. Enc. +Layer 1 +Layer 2 +Layer 3/output +40.9dB +0.99 +33.9dB +0.98 +31.9dB +0.96 +30.5dB +0.92 +Figure 16. Visualization of hidden layer outputs. The figure above visualizes outputs of hidden features in the two layers for the Siemens +sector test image shown in Fig. 15. WIRE uniquely results in sparse images, which enables high accurate representation of high frequency +parts of the image (center of the sector). +Multi-image super-resolution. +We showed a result on +multi-image super-resolution in Section 4. Here, we pro- +vide more details about the experiment. Figure 21 shows +the 512×768 dimensional ground truth image from the Ko- +dak dataset [1]. We simulated a total of four low-resolution +images by modifying each 4× downsampled image by a +small translation and rotation, thereby resulting in sub-pixel +motion between the frames. We assumed the transforma- +tion Ak between the high-resolution frame x and each low- +resolution frame yk was known. We represented the high +resolution x as output of an INR. In each case, the INR +had three hidden layers with 256 hidden features. We then +solved a linear inverse problem to estimate the high reso- +lution image. Figure 21 shows the reconstructed output for +each nonlinearity and their metrics. The inset shows re- +construction of spokes in the motorcycle. Visually, WIRE +generates the sharpest features without any ringing artifacts. +Moreover, WIRE results in 1dB or better reconstruction ac- +curacy, and 0.04 higher SSIM. +A.5. Neural radiance fields +Implementation details. +For all experiments, we used +the torch-ngp package [45] that implements a wide va- +riety of approaches for training neural radiance fields. The +architecture consists of two networks that predict transmit- +tance (sigma) and the color at each voxel respectively. Each +of the two networks consisted of an MLP with four hid- +den layers and 182 hidden features each. The color MLP +took position (x,y,z) and direction (θ, φ) as inputs, while +the transmittance MLP took only the position as input. As +with all other experiments, we used 182/ +√ +2 = 128 hidden +features for WIRE to account for parameter doubling due +12 + + + + + + + + + + + + + + + + + + +Figure 17. Effect of learning rate. The plot above shows approx- +imation accuracy for representing a noisy image (input PSNR of +20dB) with various nonlinearities. WIRE is robust to learning rate, +and produces best results with high learning rate of 2 × 10−2. + + + + + + + + + + + + + + + +(a) PSNR vs. number of features + + + + + + + + + + + + + + + + + + + +(b) PSNR vs. number of layers +Figure 18. Effect of number of parameters. The plot above +shows approximation accuracy for representing an image with +varying number of (a) hidden features and (b) hidden layers. +WIRE outperforms other nonlinearities with 128 or more hidden +features, and one or more layers and is nearly the same as SIREN +and Gaussian nonlinearities for more than 3 layers. +to complex weights. We downsampled the images by 4× +to ensure that the model and training data fit in the graphi- +cal processing unit’s (GPU) memory. In the results shown +in Fig. 10, we used a total of 25 randomly chosen images +to train the NeRF, and then validated it on 100 images. We +used a learning rate of 4×10−4 for WIRE and 2.5×10−4 for +all other nonlinearities and reduced it to 0.1× initial value +over a total of 2500 training epochs. Except for ReLU, we +did not use any form of positional encoding with other non- +linearities as we wished to demonstrate the capacity of each +nonlinearity by itself. +(a) Ground truth image + + + + + + + + + + + + + + + + + + + + + +(b) PSNR vs. number of prjections +Figure 19. CT with varying number of projections. (a) shows +the 512×512 ground truth x-ray image of lungs [5] we used for our +CT experiments. We denoised the original image to remove streak +artifacts. (b) shows accuracy as a function of number of measure- +ments with various nonlinearities. Across the board, WIRE out- +performs all other approaches by a considerable margin. +Effect of number of images. +Fig. 22a shows accuracy vs. +number of epochs for the drums dataset when trained with +all 100 images. WIRE results in highest accuracy within +2500 epochs and converges more rapidly than other ap- +proaches. Fig. 22b shows accuracy as a function of num- +ber of training images. WIRE achieves 0.1dB higher than +the next competitor SIREN for 25, 50, and 75 images, and +0.4dB higher when trained with 100 images. +Fig. 23 visualizes one of the reconstructed views for the +drums. We varied the number of images from 25 to 100 and +then rendered the image from a novel view. Visually, WIRE +generated the most pleasing results including sharp features +of the cymbals and their stands, and the smooth membrane +on the drum. In contrast, Gaussian nonlinearity results in +cloudy artifacts, while SIREN has high frequency artifacts, +especially at lower numbers of images. ReLU+positional +encoding requires all 100 images and considerably more +than 2500 epochs to reconstruct the components. 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Vision and Pat- +tern Recognition (CVPR), 2022. 2, 3 +17 + diff --git a/d9E4T4oBgHgl3EQfpw0m/vector_store/index.pkl b/d9E4T4oBgHgl3EQfpw0m/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..de3b74cab2c91a9a35c6d2c7737e1f1ca849e8be --- /dev/null +++ b/d9E4T4oBgHgl3EQfpw0m/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:38abb55f8b27c57da7ae212332203c54242c0840c1823f22b498e0f52af8fab0 +size 69840 diff --git a/dNE2T4oBgHgl3EQfGAZK/content/tmp_files/2301.03652v1.pdf.txt b/dNE2T4oBgHgl3EQfGAZK/content/tmp_files/2301.03652v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d8655ab364f85498b36835e4f011033b3a42b331 --- /dev/null +++ b/dNE2T4oBgHgl3EQfGAZK/content/tmp_files/2301.03652v1.pdf.txt @@ -0,0 +1,817 @@ +On The Fragility of Learned Reward Functions +Lev McKinney∗ +University of Toronto +lev.mckinney@mail.utoronto.ca +Yawen Duan∗ +University of Cambridge +yd338@cam.ac.uk +David Krueger +University of Cambridge +david.scott.krueger@gmail.com +Adam Gleave +University of California, Berkeley +gleave@berkeley.edu +Abstract +Reward functions are notoriously difficult to specify, especially for tasks with +complex goals. Reward learning approaches attempt to infer reward functions from +human feedback and preferences. Prior works on reward learning have mainly +focused on the performance of policies trained alongside the reward function. +This practice, however, may fail to detect learned rewards that are not capable of +training new policies from scratch and thus do not capture the intended behavior. +Our work focuses on demonstrating and studying the causes of these relearning +failures in the domain of preference-based reward learning. We demonstrate with +experiments in tabular and continuous control environments that the severity of +relearning failures can be sensitive to changes in reward model design and the +trajectory dataset composition. Based on our findings, we emphasize the need for +more retraining-based evaluations in the literature. +1 +Introduction +Reward functions for most real-world tasks are difficult or impossible to specify procedurally. +Specifically, hand-designed reward functions frequently misspecify the task [18]. The field of reward +learning attempts to overcome this challenge by designing algorithms to infer reward functions +from data. These learned reward functions aim to succinctly represent the desired behaviors [22], +drastically reduce the amount of human feedback required to learn a task [8] and allow practitioners +to generalize these behaviors to new environments [10]. +One of the most promising approaches is to learn reward functions from binary human preferences +over trajectory segments where these segments are collected online using a sampler agent trained to +optimize the learned reward [8]. This form of preference-based reward learning is already being used +to train large language models to summarize [41] and become more helpful and harmless [3]. +Prior work has typically focused on the performance of the sampler agent [19, 8]. Unfortunately, +the sampler agent performing the correct behavior does not guarantee that a robust reward function +has been uncovered. In particular, when using reinforcement learning to train a randomly initialized +relearner agent on the learned reward, the reward may fail to induce the correct behavior despite the +sampler agent behaving well [15]. If we only require a policy that works passably well in the exact +training environment, this may not be an issue because we can use the sampler agent and throw away +the learned reward. We argue, however, that such a method cannot be accurately described as learning +a reward function. At most, it is a preference-based policy learning technique, using reward functions +to give a helpful inductive bias during training. Moreover, it is desirable for many applications to +1Equal contribution. Work done during internship at Center for Human-Compatible AI, UC Berkeley. +Preprint. Under review. +arXiv:2301.03652v1 [cs.LG] 9 Jan 2023 + +truly uncover a reward function. For example, we might wish to train a new policy using a learned +reward function with a more powerful R.L. algorithm or different agent architecture than was used +during the initial reward learning process. +Past work has preformed preliminary investigations into the robustness of learned rewards in toy +environments [28], in Atari [15] and for fine-tuning language models [3, 32]. However, these +investigations are typically only reported short sections of their respective papers. +Inspired by this work, our paper empirically examines the relearner performance for learned reward +functions. Since we have access to a ground truth reward in our synthetic experiments, we define +poor relearner performance as achieving relatively low ground truth returns. Our results show that +relearing can produce very different policies than the sampler, frequently achieving low ground-truth +returns. Thus, we argue that current preference-based reward learning methods may produce reward +functions that are not reliable as signals for policy relearning. +Our paper makes three key contributions: +• We demonstrate that state-of-the-art reward learning algorithms can produce reward models +that fail to train new agents from scratch in tabular and continuous control settings; +• We show that the severity can increase as the trajectory dataset concentrates on high reward +regions; +• Finally, as an example of how these relearning failures can be sensitive to changes in reward +model design, we demonstrate that reward ensembles can effect relearning failures. +2 +Related Work +Preference-based reward learning +Our primary focus is on methods that learn from preference +comparisons between two trajectories [2, 36, 29, 8]. Preference comparison is one of the most +scalable reward learning methods, successfully applied to fine-tune large transformer language +models [24, 32, 21, 3] to enhance their performance at certain tasks. Note that trajectory comparison +methods contain more information about the reward than demonstrations, so they tend to produce +better results when available [30]. However, note that these methods may still fare poorly when the +human preference feedback does not match their model of human rationality [20]. +Other reward learning approaches +Many other methods have been developed to learn reward +functions from human data [16]. One of the most popular is Inverse reinforcement learning (IRL) [22] +methods that infer a reward function from demonstrations [1, 27, 39, 38, 40, 9, 10]. T-REX [6] is a +hybrid approach, learning from a ranked set of demonstrations. An alternative approach learns from +“sketches” of cumulative reward over an episodeCabi et al. [7]. +Reward hacking +Pan et al. provides the first systematic empirical study of reward hacking: RL +agents exploiting misspecified reward functions [25]. Notably, they find that increasing agent capabil- +ities, such as by increasing the RL policy’s model size, can sometimes lead to worse performance on +the ground truth reward, while performance on the misspecified proxy reward increases. In contrast +to our work, Pan et al. only study reward hacking in hand-designed rewards designed to illustrate the +phenomenon, whereas we investigate this phenomenon in learned rewards. +Reward hacking has also been studied from a theoretical perspective. Under the framework of general +principle agent problems Zhuang and Hadfield-Menell examines the case where the agent’s utility +function can only account for a limited subspace of the set of attributes that make the true utility [37]. +The authors proceed to show that, within their model, an optimal state under this proxy utility can +have arbitrarily low ground truth utility, assuming the attributes that make up the reward exhibit +a condition analogous to decreasing marginal utility and increasing opportunity cost. Skalse et al. +instead propose a formal definition of reward hacking [31]. In our paper, however, we focus on +more concrete cases of relearning failures and practically attainable measures, such as relearner +performance being close to sampler performance. +The most closely related work is by Ibarz et al., which evaluates their learned reward functions by +freezing them and training a new policy, analogous to our relearner evaluations [15, Section 3.2]. +However, this study was only a small, half-page section of their paper, and they did not examine +factors that may increase or decrease the chances/severity of relearning failures. +2 + +Another important related work is by Reddy et al., who observes that rewards can fail to generalize +due to a lack of informative trajectories in their training data [28]. They attempt to ameliorate this +by querying humans on diverse hypothetical trajectories generated from a model. However, their +method requires a world model and primarily focuses on taking advantage of this model to improve +reward quality. In addition, they assume the user provides feedback through quantitative reward +labels, whereas we focus on the more realistic and widely used preference comparison setting. +Finally, past work has found that language models trained on a preference-based reward model can +learn to exploit their reward model [32, Section 4.3]. In a similar vein, Bai et al. found that the ability +of their reward model to correctly predict human preferences over a pair of inputs degraded as those +inputs where perceived as more rewarding by the model [3, Section 4.2]. However, none of these +works have offered much analysis of what leads to reward hacking or in general relearning failures, +beyond training against the learned reward for to long [32, Section 4.3] or a lack of data from off +distribution [28]. +Retraining and transfer in IRL domain +There has also been multiple works exploring relearning +and transfer when learning rewards learned from expert demonstrations i.e. inverse reinforcement +learning (IRL). Fu et al. [10] propose an IRL method to learn state-only reward functions disentangled +from transition dynamics and preform experiments on transferring their learned rewards to new agents +and environments. Ni et al. [23] derive an analytic gradient estimator for an arbitrary f-divergence +between expert and on policy distributions with respect to the reward functions parameters. In their +relearning evaluations, they find that there method produces relearners that match expert performance. +Finally, Wang et al. [34] borrows methods from random network distillation to directly estimate the +expert distribution with only expert data. This process, removes the need for a sampler, obviating the +issue of relearning failures. In contrast to these IRL methods, our work focuses on the more scalable +preference-based reward learning setting. +3 +Background +Deep RL from Human Preferences +We follow the framework of learning a preference model ˆrφ +from trajectory segment comparisons. Our method is the closest to deep reinforcement learning +from human preferences [8]. It consists of four phases iterated: trajectory collection, preference +elicitation, reward inference and policy optimization. During trajectory collection, the current +policy, initially a random policy, samples rollouts from the environment collecting trajectory segments +σi = (s0, a0, s1, a1, · · · , sn) without reward labels and stores them in B. In phase two, the algorithm, +elicits preferences y ∈ {≻, ≺, ≡} for randomly selected pairs of segments (σ1, σ2) ∈ B from a +labeler — human or synthetic 1. The preferences are then stored in a preference dataset D. The +algorithm assumes these preferences have been sampled from the Bradly-Terry model [4], +P(σ1 ≻ σ2) = +exp +�� +s,a,s′∈σ1 r(s, a, s′) +� +exp +�� +s,a,s′∈σ1 r(s, a, s′) +� ++ exp +�� +s,a,s′∈σ2 r(s, a, s′) +�. +(1) +a widely used approximate model for human data in the preference based reward learning literature +[8, 15, 19]. In the third phase reward inference, the reward ˆrφ is fit by using Adam [17] to minimize +the negative log likelihood of ˆrφ under D. The fourth and final phase of each iteration consists of +policy optimization. In this stage, we can apply existing deep reinforcement learning algorithms to +improve our policies expected return under the learned reward, and the process repeats. +4 +Training and Evaluation Procedure +Reward Learning +We train reward models with synthetic data that is sampled from the Bradley- +Terry model of Eq. 1 with r set to the ground truth reward. In the tabular setting, we train the sampler +policy using soft Q-Learning [13] and the learned reward networks simply take the current state as a +one-hot vector for input. In the continuous control setting, we use soft actor-critic (SAC) [14] from +Stable-Baselines3 [26] and the learned reward networks receive the observation, action and next +observation as input. See Appendix A for further details. +1We follow Ibarz et al. in selecting preference pairs to query uniformly at random [15] +3 + +0 +10 +20 +30 +40 +50 +Number of Iterations +0 +2000 +4000 +6000 +8000 +10000 +12000 +Ground Truth Return +RL Budget +0.5M +1M +2M +4M +8M +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Number of Timesteps +1e6 +0 +1000 +2000 +3000 +4000 +5000 +6000 +Ground Truth Return +RL Budget +0.5M +1M +2M +4M +8M +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +Number of Preference Pairs +0 +2 +4 +6 +8 +10 +12 +14 +Segment Pair Average Return +RL Budget +1M +8M +(a) Reward learning curves +(b) Relearning curves +(c) Preference datasets +Figure 1: Anti-correlated sampeler and relearner ground truth returns in HalfCheetah. (a) x-axis +represents the number of iterations of each run. See section 3 RL budget is the total number of RL +timesteps available to the sampler. (b) x-axis represents the number of timesteps during relearning. +In plots (a-b), for each RL budget setting, we performed ten runs of reward learning, and for each of +these, we ran five relearning evaluations for a total of 50 relearning runs. Solid lines and shaded lines +represent the mean and 90% confidence respectively. (c) Scatterplot of average ground truth reward +of each segment pair in the example preference datasets with 1M and 8M RL budgets. +Reward Ensembles +Christiano et al. and Lee et al. use reward ensembles to estimate the uncertainty +of the learned reward [8, 19]. We explore how these ensembles may have another benefit, reducing +the variance of off-distribution transitions. As in prior works, we train each ensemble member on +bootstrapped datasets, normalize their outputs separately and use their mean as the reward. +Relearning +We freeze the learned reward and train a new, randomly initialized, relearner policy to +evaluate our reward functions. We evaluate this policy under the ground truth reward. This is similar +to the method employed by Ibarz et al. [15, Section 3.2] to study reward hacking. In the continuous +control settings this consist of training a new agent from the learned reward function using the same +R.L. algorithm as the sampler, then evaluating it under the ground truth reward. In the tabular setting, +we simply solve for the soft-optimal policy (α = 0.1) [13] under the learned reward function. +5 +Experiments +First, we investigate the occurrence of relearning failures in the continuous control domain. We use +HalfCheetah environment as our test bed since it has been used in past works on preference-based +reward learning [8]. Here we find that increasing the number of training timesteps the sampler takes +between sampling trajectories for labeling increases the severity of relearning failures. Next, we focus +on the effects of reward model design and observe that reward ensembles may reduce reward hacking +in tabular environments by reducing the variance of off-distribution transitions. To demonstrate this +failure mode we use the stay inside environment which consists of two rooms separated by a wall +with a small doorway. The agent receives reward for staying in the inside room see Figure 2b. +5.1 +Preference Trajectory Dataset Imbalance and Relearning Failure +The reward model is a function of the dataset D used to train it. One of the simplest ways to change +the preference dataset is to vary the number of timesteps T spent training the sampler between +collecting trajectory fragments. We call the total number of interactions the sampler has with the +environment during reward learning the RL budget. Note the RL budget does not affect the number +of comparisons collected. +Figure 1 shows the learning curves of the sampler and relearner experiments in HalfCheetah. We +find that despite higher RL budget leading to higher sampler returns during reward learning, the +relearners’ performance has the reverse trend; increasing sampler RL budget actually decreases +relearner ground truth return. +We can gain some insight into why this is happening by exploring preference datasets shown in +Figure 2c. First let’s consider the preference dateset produced by one of the runs with the highest- +budget (8M timesteps). We find that the trajectory segments contained in this datset are concentrated +in high ground truth reward regions. On the other hand, when we consider the low-budget dataset (1M +timesteps), the distribution of trajectory segments provides a better coverage across all the ground +truth reward scales within the support. We hypothesize that having an overwhelming proportion of +4 + +high-reward trajectory segments in the preference dataset — and little preference data on trajectories +in the transition from high to low reward — may cause the reward model to effectively over-fit to +the high-reward region. This overfitting leads to poor supervision over randomly initialized policies. +Overall, we believe this could explain the observed relearning failures. +It’s important to note that we did not see a significant increase in relearning failure when increasing +the RL budget in the tabular setting. See Appendix D. +5.2 +Reward Ensembles +1 +5 +Number of Ensemble Members +−500 +0 +500 +1000 +1500 +2000 +Return +Sampler +Relearner +(a) Effect of reward ensembles on sampler and +relearner returns +(b) Ground truth reward +(c) Example no ensemble +(d) Example with ensemble +1 +5 +Number of Ensemble Members +0 +5 +10 +Maximum Reward +(e) Max reward +Figure 2: Ensembles eliminate relearning failures in the stay inside environment. (b) depicts the +ground truth reward in the stay inside environment. (c) shows an example individual learned reward +and (d) with a five member ensemble. Finally, (e) shows the distribution of max learned-reward +across all states. All sub-figures come from the same run which included 20 seeds. +Our tabular experiments provide a concrete, interpretable example of how relearning failures can be +effected by reward model implementation details. In particular, we focus on reward ensembles and +observe that they have drastic effects on relearners but leave the sampler’s performance unchanged. +In the stay inside environment, when using a reward ensemble of size five, all relearners preform at +least as well as their respective samplers, as can be seen in Figure 2a. However, if we use only a single +reward, the relearners behaviour is inconsistent; some relearners do substantially better then their +respective samplers, but almost as many do substantially worse, getting near zero return. Thus, while +adding an ensemble has a minimal effect on the sampler, it changes the behaviour of the relearners. +To understand why this happens we must consider the off-distribution behaviour of our reward models. +In the stay inside environment, the samplers typically stay in the inside half of the environment. +Thus there is often insufficient coverage of the outside half of the environment in our trajectory +dataset. Thus, the reward off-distribution is largely unconstrained by the data. This means that +small changes in the off-distribution behavior of our reward network can become critically important. +Reward models based on neural networks produce spurious high rewards off distribution, see Figure 2. +When these reward delusions are more rewarding than any of the in-distribution transitions, reward +hacking can occur and cause relearning failures. Reward ensembles tends to have lower variance +off distribution than an individual reward network. Thus, any reward delusions tend to have a lower +reward (according to the reward model). This can be directly seen in Figure 2 (c-e). This effect +reduces the chance that the optimal policy will be attracted to one of these spurious rewards during +relearning, which is what we see in Figure 2a. +6 +Limitations and Discussion +Our experiments have a few important limitations. First, they are limited to simple ground truth reward +functions and environments. For example, in Half-Cheetah-v3 [5, 11], the reward function is +essentially a linear in the observation, action and next observation. While these relearning failures also +appear in more complex tasks [15, Section 3.2], it is unclear if it is precisely the same phenomenon +5 + +10 +8 +6 +4 +0 +S2 +0 +-2 +-4 +-60 +-1 +-2that causes them. The design decisions that seem to improve retraining performance in small-scale +experiments, in our case, reward ensembles and less sampler training, may not be the same as those +that address the problem at a larger scale. We leave such explorations to future work. +Overall, we have demonstrated that evaluations of relearning performance can differ substantially +from the results of simply evaluating the sampler agent trained alongside the reward model. We hope +to see future works include relearning evaluation as they appear to hold fruitful insights into the +quality of the learned reward functions. +6 + +Author Contributions +Lev McKinney designed and implemented the tabular experiments and wrote the relevant parts of +the method and experiments sections. In addition, he wrote the introduction, discussion and related +works sections of the paper/appendix. Yawen Duan designed and ran the initial experiments that +displayed reward model relearning failure on continuous control environments, and wrote relevant +sections of the paper. David Krueger provided ideas, guidance and general feedback on experiment +design and analysis. Adam Gleave provided initial ideas of the project, provided high-level and +detailed feedback on experiments and analysis. +Acknowledgments and Disclosure of Funding +This paper was completed as part of an internship at the Center for Human-Compatible Artificial +Intelligence. 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Stiennon, J. Wu, T. B. Brown, A. Radford, D. Amodei, P. F. Christiano, and +G. Irving. Fine-tuning language models from human preferences. CoRR, abs/1909.08593, 2019. +URL http://arxiv.org/abs/1909.08593. +Appendices +A +Training Details and Hyperparameters +A.1 +Reinforcement learning algorithms +In the tabular setting we train the sampler policy using soft Q-Learning [13]. We use soft actor-critic +(SAC) [14] implementations of Stable-Baselines3 [26] in the locomotion control tasks. +Both algorithms are off-policy and use a replay buffer, which ensures their high sample efficiency +compared to on-policy RL algorithms. Note that the learned reward function ˆrφ changes during +training, so we relabel the transitions in the replay buffer after each iteration, similar to PEBBLE +[19]. The main difference between our algorithm and PEBBLE is that we omit the unsupervised +pre-training stage used in PEBBLE. We used the implementations from Imitation Learning Baseline +Implementations [35] to perform the experiments. +A.2 +Continuous Control Experiments +In the tabular setting, all reward networks only take the current state as a one-hot vector. They consist +of a multi-layer perceptron with two hidden layers of size 256 and ReLU activations, similar to those +used in PEBBLE [19]. +Training details +For reward learning experiments, we used the implementations of Preference +Comparisons Algorithm from Imitation Learning Baseline Implementations [35] with a full list of +hyperparameters in Table 1. For the RL component, we used soft actor-critic (SAC) [14] implementa- +tions from Stable-Baselines3 [26] in the locomotion control tasks with a list of hyperparameters in +Table 2. For retraining evaluations, we use the same hyperparameters for SAC to train new agents +against the frozen learned reward models. +10 + +Reward model +The reward model consists of a single multi-layer perceptrons with two +hidden layers of size 256 and LeakyReLU activations with slope 0.01. +The input of the +model consists of the state, action and next state vectors, and the input vector is normalized +by running normalization. +The output the the reward model is normalized by by exponen- +tial moving average. +During relearning experiments, we directly use the raw reward output +from the reward network while being normalzed by a VecNormalize layer in Stable-Baselines3 +(https://stable-baselines3.readthedocs.io/en/master/guide/vec_envs.html#vecenv). +Reward normalization +We compute a normalized version of the learned reward using an Expo- +nential Moving Average to normalize the reward to mean zero and unit standard deviation. This +normalized reward was then used for policy optimization. Note that normalizing the reward does +not change the optimal policy, which is invariant to positive affine transformations. However, it does +simplify the optimization problem. In particular, a normalized reward is a more stable objective for +the critic to learn over time. Additionally, RL hyperparameters can depend on the reward scale (for +example, learning rate should be set inversely proportional to reward scale) – normalizing the learned +reward therefore allows us to use a consistent set of hyperparameters. +Hyperparameter +Value +Segment Length +50 +Total Comparisons +2000 +Number of Iteration +50 +Reward Training Epochs +5 +Query Schedule +constant +Table 1: Reward learning hyperparameters for continuous control experiments +Hyperparameter +Value +Learning Rate +0.0003 +Batch Size +256 +Discount +0.99 +Learning Starts from +10000 +Table 2: SAC hyperparameters for continuous control experiments +A.3 +Tabular Experiments +Similarly to the continuous control experiments we use Imitation’s implementation of preference +comparison [35]. However, we use a tabular soft-q learning algorithm with a replay buffer [13] with +reward relabling [19] to solve the environments. The reward network again uses a similar MLP +architecture to the continuous control setting with a sightly smaller hidden size of 32. Finally, we +normalize the reward functions before ensembling them using a simple running norm over sampled +transitions which is frozen during retraining. Hyperparamaters can be found in Table 3. +Tabular Relearning +When relearning we solve for the soft-optimal policy under the learned reward +function with temperature 0.1 and discount factor 0.99. +B +Environments +Locomotion Control Task +We ran reward learning and relearning on a MuJoCo locomotion +task [33] – HalfCheetah environment from the seals benchmark suite [11], a modification of +HalfCheetah-v3 in the gym environment suite which adds the x-coordinate of the robot’s center of +mass (COM) to the first dimension of the observation space. The ground-truth reward function of +the HalfCheetah environment is a linear combination of the x-velocity of the robot’s COM and a +control cost dependent on the L2 norm of the action vector. Consequently, the reward function in +seals HalfCheetah is a function of the observations, which is not strictly true in the original gym [5] +environment, avoiding a potential confounder. +11 + +Hyper Parameter +Value +Sampler Soft-Q Learning +discount +0.99 +learning rate +5e-2 +replay buffer capacity +∞ +temperature +0.1 +samples from buffer per env sample +10 +initial soft-q value +200 +Reward Learning +trajectory fragment length +30 +total comparison budget +2,500 +RL budget +500,000 +frac. of comparisons from inital random trajs +0.1 +select fragments for comparison +randomly +epochs of training per iteration +1 +number of iterations +100 +query schedule +constant +reward learning rate +1e-3 +Reward Network +reward network hidden layers +[32, 32] +activation function +ReLu +output normilization +Running Norm +Table 3: Tabular Experiment Hyperparamerers +Tabular Environment +We constructed the stay inside environment, which consists of a 20x20 +closed grid of cells. The top "outside” and bottom "inside” halves of the environment are separated +by a wall with a narrow two cell gap in the middle. The reward for each state is shown in Figure 2 (a), +with reward values ranging from +10 to -1. +C +Epic Distance as an Evaluation Metric +As an additional evaluation criterion, we consider using EPIC distance [12] to measure the distance +between learned reward functions and the ground truth reward. EPIC works by canonicalizing +the rewards to be invariant to potential shaping, normalizing them to be invariant to scale, and +then computing the L2 norm of the difference of those functions over a coverage distribution of +transitions. Here we consider two coverage distributions: uniform and expert distribution. The +uniform distribution is uniform over feasible transitions. The expert distribution is the distribution of +a soft-optimal policy with a temperature of 10 to give slightly more coverage. +D +Additional Tabular Experiments +To study the effects of training the sampler for a more time steps, we first consider a simple +environment consisting of a 10x10 grid world. The agent begins in the lower left-hand corner of the +environment and gains a ground-truth reward of 10 for reaching the lower right-hand cell, as seen in +Figure 5. +The performance of the sampler and relearner initially increases with more training timesteps, with +our relearners generalizing well and achieving slightly higher performance than their respective +samplers. However, it quickly plateaus even though we do not see significant reductions in relearner +performance with an increased number of time steps. The EPIC distances of our learned reward +functions from the ground truth reward begin to increase after 400,000 timesteps Figure 4 (b). +Increasing the number of total training timesteps used for DRLHP does seem to degrade the quality +of the reward function according to EPIC distance. However, it does not appear to hurt relearning +performance in the same way in this simple tabular environment. +12 + +10 +3 +10 +2 +10 +1 +(a) Example without ensemble +(b) Example with ensemble +Figure 3: Example on policy distribution +Examples of the on policy distributions of the samplers in the stay inside environment, marginalized +over the entire training run. +200000 +400000 +600000 +800000 +1000000 +DRLHP Training Timesteps +−200 +0 +200 +400 +600 +800 +Return +Agent +Sampler +Relearner +Relearner Minus Sampler +200000 +400000 +600000 +800000 +1000000 +DRLHP Training Timesteps +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +EPIC Distance To Ground Truth Reward +Coverage Distribution +Uniform +Expert +(a) +(b) +Figure 4: Increasing the number of time steps of R.L. training does not seem to significantly effect +relearning failures. +This is a strikingly different effect than we see in HalfCheetah. This may be because in a tabular +setting the sampler either finds the optimal policy induced by the learned reward function every +iteration, so the sampler and relearner have equal performance, or it insufficiently explores the +environment, and reward learning completely fails. This dichotomy leaves little room for the subtle +degradation in relearner performance we see in Figure 1. +13 + +Figure 5: Tiny room environment. The ground-truth reward in the tiny room environment. Note that +the reward only depends on the current state. +14 + +10 +8 +6 +4 +2 +0 \ No newline at end of file diff --git a/fNE_T4oBgHgl3EQf2BxV/content/tmp_files/2301.08338v1.pdf.txt b/fNE_T4oBgHgl3EQf2BxV/content/tmp_files/2301.08338v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..633dfd1ccdd8f814536beffc632ba85c1c88f73a --- /dev/null +++ b/fNE_T4oBgHgl3EQf2BxV/content/tmp_files/2301.08338v1.pdf.txt @@ -0,0 +1,2247 @@ +Local Geometry of Self-similar Sets: Typical Balls, Tangent +Measures and Asymptotic Spectra +Manuel Mor´an 1,2, Marta LLorente3 and Mar´ıa Eugenia Mera1 +1 Departamento de An´alisis Econ´omico y Econom´ıa Cuantitativa. Universidad Complutense de Madrid. Cam- +pus de Somosaguas, 28223 Madrid, Spain. +2 IMI-Institute of Interdisciplinary Mathematics. Universidad Complutense de Madrid. Plaza de Ciencias 3, +28040 Madrid, Spain. +3 Departamento de An´alisis Econ´omico: Econom´ıa Cuantitativa. Universidad Aut´onoma de Madrid, Campus +de Cantoblanco, 28049 Madrid, Spain. +Emails: mmoranca@ucm.es, m.llorente@uam.es, mera@ucm.es +Short Title: Local Geometry of Self-similar Sets +Abstract +We analyse the local geometric structure of self-similar sets with open set condition through +the study of the properties of a distinguished family of spherical neighbourhoods, the typical +balls. We quantify the complexity of the local geometry of self-similar sets, showing that there are +uncountably many classes of spherical neighbourhoods that are not equivalent under similitudes. +We show that, at a tangent level, the uniformity of the Euclidean space is recuperated in the +sense that any typical ball is a tangent measure of the measure ν at ν-a.e. point, where ν is any +self-similar measure. We characterise the spectrum of asymptotic densities of metric measures in +terms of the packing and centred Hausdorff measures. As an example, we compute the spectrum +of asymptotic densities of the Sierpinski gasket. +Keywords: Self-Similar Sets, Hausdorff Measures, Tangent Measures, Density of Measures, Com- +putability of Fractal Measures, Complexity of Topological Spaces, Sierpinski Gasket. +[2020] MSC: 28A78, 28A80, 28A75, 54A05, 54A25 +1 +arXiv:2301.08338v1 [math.DS] 19 Jan 2023 + +1 +Introduction and main results +In order to gauge the vastness of the set of spherical neighbourhoods of a metric space X, it is useful +to consider the quotient spaces SphX/ ≃F, where SphX is the set of spherical neighbourhoods of X +and ≃F is the equivalence class associated with some group F of self-mappings of X : B ≃F B′ ⇔ +B = f(B′) for B, B′ ∈ SphX and some f ∈ F. The regularity of the Euclidean space Rn is made clear +by the fact that if Sn is the set of similarities of Rn, then SphRn/ ≃Sn consists of a unique equivalence +class. +In this paper, we study the local geometry of a self-similar set E ⊂ Rn satisfying the open set +condition (OSC), geometry which is described by the spherical neighbourhoods of E as a metric +subspace of Rn, i.e. by restricted balls of the form B ∩ E, where B is a Euclidean ball. For general +points x, y ∈ E, if B(x, d) denotes the closed Euclidean ball centred at x and with radius d, then +B(x, d) ∩ E and B(y, d) ∩ E are not equivalent by translation, and B(x, d) ∩ E and B(x, d′) ∩ E with +d ̸= d′ are not homothetic-equivalent. Using classical tools of fractal geometry, namely, the s-densities +of metric measures on balls (see Definitions 21 and 22), and Marstrand’s Theorem [1], together with +the results in Sec. 3.2, we are able to prove that, for general self-similar sets with OSC, there are +uncountably many equivalence classes in the quotient spaces SphE/ ≃Sn . This gives account of the +complexity of the purely deterministic self-similar geometry. +In spite of these facts, the literature has established the existence of a strong kind of regularity, on +a tangent level and on average, in the neighbourhoods of a self-similar set. +Recall that a self-similar set is defined as the unique compact set E ⊂ Rn that satisfies the basic +equation of self-similarity +E = ∪m−1 +i=0 fi(E). +(1) +for a given system Ψ = {fi}i∈M , M := {0, 1, . . . , m − 1} of contractive similitudes in Rn. We shall +assume that the system Ψ satisfies the OSC, meaning that there is an open set O ⊂Rn such that +fi(O) ⊂ O for all i ∈ M and fi(O)∩fj(O) = ∅ for i, j ∈ M, i ̸= j. We shall refer to such a set +O as a feasible open set for Ψ. We can assume, without loss of generality, as we shall from now on, +that O∩E ̸= ∅ holds, also called strong open set condition (SOSC) (cf. [2] and [3], see also [4]). If +fi(E) ∩ fj(E) = ∅ for i, j ∈ M, i ̸= j, it is said that the strong separation condition (SSC) holds, in +2 + +which case the OSC is also fulfilled. +We want to understand the local geometry of E through the study of the local behaviour of the +metric s-measures, +Ms⌊E:= +� +µ, Hs⌊E, Hs +Sph⌊E, Cs⌊E, P s⌊E +� +(2) +where s is the similarity dimension of E, dim E, that is, the unique real number s that satisfies +� +i∈M rs +i = 1, ri being the contraction constant of the similarity fi, i ∈ M. Here β⌊E stands for a +measure β restricted to the set E. The measures +Ms := +� +Hs, Hs +Sph, Cs, P s� +(3) +are the s-dimensional Hausdorff measure, spherical Hausdorff measure, centred Hausdorff measure and +packing measure, respectively. Any two measures in Ms⌊E are multiple of each other, moreover, in +the case that s takes the integer value n, they are also multiple of the n-dimensional Lebesgue measure. +Each measure in Ms⌊E highlights different basic geometric properties of subsets of Rn. For α ∈ Ms⌊E, +0 < α(E) < ∞ holds and E is called an s-set (see [5] for further details and Sec. 2.2 for the definitions +of the measures in Ms). We shall present in Sec. 2.1 below the natural probability measure µ. For the +time being, we can see it as the normalised measure, +α +α(E) of any other α ∈ Ms⌊E. +The results in this paper about the regularity of the metric measures are also shared by the +wider class of self-similar measures, MS(E) (see [6] and Sec. +2.1 for a definition). +Whereas the +metric measures, Ms, convey a strong geometric meaning, self-similar measures are an essential tool +in multifractal analysis of logarithmic densities, a topic that has generated a vast amount of literature +for the past 30 years. +1.1 +Scenery flow, tangent distribution and tangent measures +Let ν be a Radon measure on Rn and let x be a point in the support of ν. We can access the local +geometry of ν⌊E around x through the following zooming process: let Tx,t(y) = t(y − x), t > 0, be the +homothety that maps the ball B(x, t−1) onto the unit ball D := B(0, 1). Let νx,t be the probability +measure on D obtained from the normalisation of the restriction to D of the image measure of ν⌊E +under the homothety Tx,t. If M(D) denotes the set of Radon measures on D, then the mapping t → νx,t +can be considered as a measure-valued time series that takes values in the metric space M(D) endowed +3 + +with the weak topology. This time series is called scenery flow of ν around x (cf. [7]). The empirical +distributions Φx,t(ν), t > 0, associated with such “time” series, are probability measures on M(D) (so +they belong to the set M(M(D)) of Radon measures on M(D)). The empirical distribution Φx,t(ν) +gives weight to a set A ⊂ M(D) according to the rate of the time interval [0, t] that the “empirical” +data δνx,t (unit mass at νx,t) stay in A. If the empirical distribution Φx,t(ν) converges to a limit Φx(ν) +as t tends to infinity, then the limiting distribution Φx(ν) is called the tangent distribution of ν at x +(see [8]). +S. Graf [9] proved that if E is a self-similar set with OSC and ν ∈ MS(E), then the limit Φx(ν) exists +ν-a.e. x, and it does not depend on x. Moreover, he constructed an explicit formula for the tangent +distribution. This author gave credit for the first of these results to C. Bandt in [8], and Bandt in +turn gives credit for the same result to S. Graf [10] (indeed a most refreshing case). M. Arbeiter [11], +C. Bandt [10] and A. Py¨or¨al¨a [12] extended these results in different ways. +The uniqueness and +independence of the limit Φx(ν) from x is what M. Gavish, [13], calls, when displayed by a measure, +the uniform scaling scenery property of such a measure. This means that, at a tangent level and in +this sense, the flow scenery recovers the uniformity of the Euclidean space. +Remark 1 There is another way to pass to the limit at the tangent level that leads to tangent measures, +a concept prior to tangent distributions introduced by D. Preiss [14]. There, starting from a measure +ν in the set M(Rn) of Radon measures on Rn, he considers unrestricted zoomings νx,t of ν at x by +homotheties Tx,t as above. Instead of performing an averaging procedure, Preiss considers non-null +and locally finite limits, in the vague topology of M(Rn), of sequences +{cnνx,tn} +with +tn +n→∞ +→ +∞ +and +cn > 0. +Such limit points are called tangent measures of ν at x, and Tan(ν, x) denotes the set of all such limits. +In our approach, following C. Bandt [8], the measures νx,tn are restricted and normalised zoomings, +but the zoomings are through general expanding similitudes, rather than only homotheties. +Let In be the group of isometries of Rn. We may define, in the set M(Rn), the equivalence +relationship +α ∼= β ⇔ there is a g ∈ In and a λ > 0 such that β = λ (g♯(α)), +(4) +4 + +where g♯(α) is the image measure of α under g, i.e. g♯(α)(A) = α(g−1(A)) for α-measurable A ⊂ Rn. +Thus, we identify two measures if they are equal up to an isometry (see, for instance, [10], where +equivalent measures up to isometries are identified in the construction of tangent measures), and we +also identify all measures in the half-straight line {λα : λ > 0, α ∈ M(Rn)} . For α ∈ M(Rn), let �α +denote the equivalence class in M(Rn)/ ∼= to which α belongs, i.e. +�α = {β ∈ M(Rn) : β ∼= α} +(5) +Given a measure ν ∈ M(Rn), we now consider the zoomings νx,tn be of the form (gn)♯ ν⌊B(x,dt−1 +n ) +where gn is a similitude of contraction ratio tn, d ≤ 1, and x ∈ spt(ν) (see (23)). We define the quotient +space � +M(Rn) and the set of tangent equivalence classes of measures, � +Tan(ν, x), by +� +M(Rn) += +{�α : α ∈ M(Rn)} +(6) +� +Tan(ν, x) += +� +�α : there is a sequence cnνx,tn +w +−−−−→ +n→∞ α, with tn → ∞, α ̸= 0 and α ∈ M(Rn) +� +,(7) +where +w→ denotes the weak convergence of measures on M(Rn). +It turns out that, in the course of our research, the case in which the convergence of the magni- +fications occurs in the strong topology of measures in M(Rn) is relevant (see Sec. 1.2 below for a +discussion of this result). We shall write � +Tan +st(ν, x) for the set of equivalence classes, w.r.t. ∼=, of such +strong limits. +Remark 2 In our definition (7) any two zoomings, β = (gn)♯ ν⌊B(x,dt−1 +n ) and β′ = (hn)♯ ν⌊B(x,dt−1 +n ) of +a given spherical neighbourhood B(x, dt−1 +n ) are considered as valid steps in the construction of a tangent +limiting measure α, where gn, hn are different similitudes. This can be considered as the identification +of β and β′ as equivalent zoomings. Notice that β′ = +� +g−1 +n +◦ hn +� +♯ β and that g−1 +n +◦ hn is an isometry. +Thus, the equivalence relationship (4) and the definition in (7) are consistent. +In contrast to the enlightening results obtained in [9], [10] and [11] on the uniform scaling scenery +property of self-similar measures, to the best of our knowledge, the members of Tan(ν, x) for ν ∈ +MS(E) remain unknown. Several natural issues arise here: What is the relationship between Φx(ν) +and Tan(ν, x)? What do the measures in Tan(ν, x) look like? Do they display some uniform property? +As for the first question, see Proposition 1 in [15]. Below, we give a partial answer to the second and +third questions for measures in MS(E) (see (10) and Theorem 12). +5 + +1.2 +Typical balls +A distinguished class of neighbourhoods of E, in terms of which our results are expressed, is the class +of typical balls. +Definition 3 A ball B(x, d) is said to be typical if x ∈ E and B(x, d) ⊂ O, where O is some feasible +open set. We shall write B for the set of typical balls. +The family of typical balls is invariant under the semigroup G generated by Ψ (see Sec. 2), since, +for f ∈ G, it follows from f(O) ⊂ O that f(B) ⊂ B holds. Consider now the set of typical spherical +B-measures, +MS(B) := {α⌊B : B ∈ B, α ∈ MS(E)} . +(8) +It is well known [6] that, for any x ∈ E, the set {f(x) : f ∈ G} is dense in E, so the balls in B are +typical in the sense that, if B ∈ B, then similar copies of B are densely spread over E at small scales +by the action of G. These copies are a countable set of balls. As Theorem 12 shows, the measures in +MS(B) are also typical in a deeper sense since, for any f ∈ G, B ∈ B and α ∈ MS(B), the equality +α⌊f(B)= pff♯(α⌊B) holds for a certain constant pf < 1 associated with f. This means that the images +of typical balls are identical copies, up to the constant pf, to the original ones not only as subsets, but +also from the point of view of any property expressible in terms of self-similar measures. Moreover, +in Theorem 12 it is shown that, for any typical ball B(x, d), for any measure α ∈ MS(E) and for all +points y in a set �E with full α-measure, there is a sequence of balls {B(y, dk)} with dk → 0, a sequence +{fk} of similitudes in G and constants p−1 +fk → ∞, such that +p−1 +fk +� +f −1 +k +� +♯ +� +α⌊(B(y,dk) +� +st +−−−−→ +k→∞ α⌊B(x,d), +(9) +where the convergence in (9) is in the sense of the strong topology of Radon measures. +Theorem 12 also states that, for all x ∈ �E and α ∈ MS(B), +� +MS(B) ⊂ � +Tan +st(α, x) +(10) +holds, where +� +MS(B) = {�α : α ∈ MS(B)} , +(11) +(see (5) for the notation �α). +6 + +The results above imply that the use of general zooming similitudes, grants the strong convergence +of the zoomings to the tangent measures, whereas in the ordinary spaces of tangent measures, where +only homotheties are allowed, convergence can only be ensured in a weak topology sense. See Sec. +3.1.1 below for further details on identifications and topologies of measures. +Remark 4 Putting the results in Sec. 3.3, described in the first paragraph of this section, together +with (10), we see that the self-similar scenery at x ∈ E depends on x on large scales, meaning that +there is a broad variety of balls B(x, d) for varying x that, moreover, also vary with d for fixed x. +Additionally, on a tangent scale, for each α ∈ MS(B) and each x ∈ �E, each typical class of measures +in � +MS(B) is a feasible outcome of the zooming process of α at x, so there is a wide variety of limiting +measures in � +Tan +st(α, x), x ∈ �E. The uniformity of the self-similar setting emerges here in the fact +that the inclusion � +MS(B) ⊂ � +Tan +st(α, x) stands true for any x ∈ �E, so all the points in �E share the +set � +MS(B) of tangent measures. +1.3 +Spectrum of local densities of a self-similar set: the Sierpinski gasket +case +The relevance of the typical balls is stressed by the connection between typical balls and the spectrum +of densities, which in turn determines some basic geometric features of E. +Let α ∈ M(Rn), 0 ≤ s ≤ ∞ and x ∈ Rn. The upper and lower spherical s-densities of α at x are +defined, respectively, by +θ +s +α(x) = lim sup +d→0 +θs +α(x, d), +(12) +θs +α(x) = lim inf +d→0 +θs +α(x, d), +(13) +where the s-density of the ball B(x, d), θs +α(x, d), is given by +θs +α(x, d) = α(B(x, d)) +(2d)s +. +Here the zooming process is summarised in only two scalars, (12) and (13). If θ +s +α(x) = θs +α(x), then +we write θs +α(x) for the common value and call it s-density of α at x. Densities and their connections to +7 + +their underlying measures have been studied extensively in the context of geometric measure theory. A +major contribution from Marstrand (Marstrand’s theorem, [1]) asserts that, in the Euclidean setting, +if the s-density θs +α(x) exists in a set with a finite and positive α-measure with α ∈ M(Rn), then s is +an integer. +The widest class of subsets of Euclidean spaces that are s-sets (i.e. sets with a finite and positive +α-measure) is either the class of self-similar sets that satisfy the OSC, with s being their similarity +dimension (see (2)), or some variations of it, like the Mauldin and Williams graph-directed construc- +tions, cf. [16], and controlled Moran constructions, cf. [17]. Here, we are interested in the case in which +the similarity dimension s is not an integer and, by Marstrand’s theorem described above, θs +α(x) and +θ +s +α(x) do not coincide in subsets with a positive α-measure. This leads to the following definition of +asymptotic spectrum of densities of a given measure α at a point and, more in general, in a subset of +points. +Definition 5 Given a subset A ⊂ Rn, we define the asymptotic spectrum of (non-logarithmic) spher- +ical s-densities, Spec(α, A), for a locally finite measure α by +Spec(α, A) = +� +lim +k→∞ θs +α(x, dk) : x ∈ A and +lim +k→∞ dk = 0 +� +. +(14) +We insert the non-logarithmic epithet above because there is a ample literature on the so-called +multifractal spectrum of logarithmic spherical densities. This literature also focuses on the limiting +behaviour of α on small balls, but the interest is in the upper and lower limits of the quotients +log α(B(x,d)) +log d +when d → 0 (for x ∈ E) and, in particular, in the fractal dimension of both the (α- +null) sets where these limits exist and take particular values [18] and the sets of divergence points +(see [19], [20], [21]) where the limits do not coincide. Much less is known about the behaviour of +non-logarithmic densities, and the research in this paper can be considered a preliminary step in that +direction. +In particular, in Sec. 3, Theorem 14, we present the knowledge to date about the spectrum of +non-logarithmic α-densities, α ∈ Ms⌊E, of self-similar sets E that satisfy the OSC. In particular, we +show that Spec(α, x) is contained in the closed interval +� +α(E) +P s(E), α(E) +Cs(E) +� +for all x in a subset �E of E +with a full α-measure. There arises a natural class of self-similar sets with nice properties, the α-exact +self-similar sets (see notation in 15), which are sets for which the endpoints of such interval belong +8 + +to Spec(α, x), x ∈ �E. Whereas the results for general self-similar sets with OSC presented in Sec. 3 +are of a qualitative nature, in Sec. 4 we shall focus on our prime example of α-exact self-similar set, +the Sierpinski gasket S, and exploit its regularity to accurately approximate the range of values taken +by its spectrum, which is the content of Theorem 26. Moreover, we give a full characterisation of the +spectrum of all the points in S, which is given by the union of two closed intervals of positive length, +namely, +Spec(α, S) = +� +α(S)θs +µ(z0), α(S)θ +s +µ(z0) +� +∪ +� α(S) +P s(S), α(S) +Cs(S) +� +, α ∈ Ms⌊S, +where z0 := (0, 0). Using the numerical approximations of θs +µ(z0), θ +s +µ(z0) obtained in Sec. 4 and of +P s(S) and Cs(S) obtained in [22] and [23], we can also show that these two intervals are disjointed. +In the case that α ∈ {µ, P +s⌊S, Cs⌊S}, we have numerical estimations of these two disjointed intervals. +The Sierpinski gasket is, as far as we know, the first connected self-similar with non-integer dimension +for which the entire spectrum has been computed. +2 +Notation and preliminaries +The self-similar set E given in (1) can be parametrised as E = {π(i) : i ∈ Σ} with parameter space +Σ := M ∞ and geometric projection mapping π : Σ → E given by π(i) = ∩∞ +k=1fi(k)E, where i(k) +denotes the curtailment i1 . . . ik ∈ M k of i = i1i2 · · · ∈ Σ and fi1...ik = fi1 ◦ fi2 ◦ fi3 ◦ ...fik. We adopt +the convention M 0 = ∅ and write M ∗ = ∪∞ +k=0M k for the set of words of finite length. Expressed in +this notation, the semigroup generated by Ψ can be written as G = {fi : i ∈ M ∗} . +For any i ∈ M ∗, we denote by Ei the cylinder sets fi(E), and if i ∈ M 0, then fi(E) := E. The sets +Ei are called k-cylinders if i ∈ M k. We also shorten the notation fi(A) to Ai for a general set A ⊂ Rn. +We write ri := ri1ri2 . . . rik for the contraction ratio of the similitude fi. +Moreover, σ : Σ → Σ shall stand for the shift map given by σ(i1i2i3 . . . ) = i2i3i4 . . . The code +shift can be projected (as a correspondence) onto E, yielding the geometric shift +T (x) := π ◦ σ ◦ π−1(x), +(15) +x ∈ E. The shift orbit of x ∈ E is given by +� +T k(x) : k ∈ N +� +. +Remark 6 Observe that x ∈ T k(A) if and only if fi(x) ∈ A for some i ∈ M k. +9 + +2.1 +Self-similar measures +Let P(Rn) be the space of compactly supported probability Borel measures on Rn, let p = (p0, ..., pm−1) ∈ +Rm be a probability vector and let Mp: P(Rn) → P(Rn) be the Markov operator defined by +Mp(α) = +m−1 +� +i=0 +piα ◦ f −1 +i +, α ∈ P(Rn). +The unique fixed point of the contractive operator Mp is called the self-similar measure µp; that is, +µp = +� +i∈M +piµp ◦ f −1 +i +. +(16) +Moreover, +Mk +p(α) = +� +i∈M k +piα ◦ f −1 +i +w +−→ +k→∞ µp +(17) +for any α ∈ P(Rn), where, for i ∈ M k, pi := pi1 · · · pik. Here Mk +p is the k-th iterate of Mp (see [6] +and [24] for further details). Set +MS(E) := +� +µp : +m−1 +� +i=0 +pi = 1, pi > 0, i = 0, ..., m − 1 +� +. +(18) +For ps := (rs +0, ..., rs +m−1), where s is the similarity dimension of E (recall that ri is the contraction +constant of the similarity fi, i ∈ M), the measure µps is called the natural probability measure on E. +Furthermore, if α ∈ Ms⌊E (see (2) for notation), then +µ := µps = +α +α(E) +(19) +(see [25]). +Notice that, whereas the measures in Ms (see (3) for notation) convey an strong geometrical +meaning, the measures µp in MS(E) do not. +They are concentrated in dense subsets of E, Ep, +whose dimension is given by dim(Ep) = sp := +�m−1 +i=0 +pi log pi +�m−1 +i=0 +pi log ri , but the measure µp is singular w.r.t. the +measures Hsp and P sp (see [26] and [27]). +2.2 +Metric measures +We now briefly recall metric measures. They are the classical tools for analysing the geometric prop- +erties of subsets of Rn. +10 + +The Hausdorff centred measure, Cs(A), of a subset A ⊂ Rn, was defined by Saint Raymond and +Tricot [28] in a two-step process. First, the premeasure Cs +0(A) is defined for any s > 0 by +Cs +0(A) = lim +δ→0 inf +� ∞ +� +i=1 +(2di)s : 2di ≤ δ, i = 1, 2, . . . +� +, +(20) +where the infimum is taken over all coverings, {B(xi, di)}i∈N+ , of A by closed balls B(xi, di) centred +at points xi ∈ A. Then, the centred Hausdorff s-dimensional measure is defined by +Cs(A) = sup {Cs +0(F) : F ⊂ A, F closed} . +The second step in the definition of Cs(A) is due to the lack of monotonicity of Cs +0 (see [29] +and [30, Example 4]). +However, in [30], it was shown that the second step can be omitted when +restricting oneself to self-similar sets with OSC. +With regard to metric measures based on packings, the standard packing measure P s (see [28] +and [31]) is also defined in a two-step process, +P s +0 (A) = lim +δ→0 sup +� ∞ +� +i=1 +(2di)s : 2di ≤ δ, i = 1, 2, . . . +� +, +where the supremum is taken over all packings {B(xi, di)}i∈N+ , with xi ∈ A for all i, and B(xi, di) ∩ +B(xj, dj) = ∅ for i ̸= j. Then, +P s(A) = inf +� ∞ +� +i=1 +P s +0 (Fi) +� +, +where the infimum is taken over all coverings {Fi}i∈N+ of A by closed sets Fi (cf. [32]). In [33], it was +proved that if A is a compact set with P s +0 (A) < ∞, then P s(A) = P s +0 (A), so this simplification applies +to any compact subset of a self-similar set with OSC. +The spherical s-dimensional Hausdorff measure, Hs +Sph(A), is obtained by removing in (20) the +requirement that the balls are centred at points of A. The classical s-dimensional Hausdorff measure, +Hs(A), results if coverings of A by arbitrary subsets, {Ui}i∈ N+ , are considered and 2di is replaced +in (20) with the diameter of Ui, |Ui| (see [34] and [5]). No second step is required for these last two +measures. +The packing and the centred Hausdorff measures have a much simpler expression when dealing +with self-similar sets E that satisfy the OSC as the browse for optimal packings or coverings can be +reduced to the search for optimal density balls within the class of typical balls, B (see Definition 3). +11 + +In particular, for any self-similar E that satisfies the OSC and with similarity dimension s, it is known +(see [36]) that +P s(E) = +� +inf +� +θs +µ(x, d) : B(x, d) ∈ B +��−1 , +(21) +and, Lemma (13) of Sec. 3.2 implies that +Cs(E) = +� +sup +� +θs +µ(x, d) : B(x, d) ∈ B +��−1 . +(22) +3 +Local structure and typical balls +Now we shall study the local structure of a self-similar set E that satisfies the OSC for a feasible +open set O through the study of the scenery flow of α ∈ MS(E) at a.e. x ∈ E, and through the +characterisation of the spectrum of the spherical s-densities of measures in Ms⌊E (Sec. 3.2), a limiting +set that helps to summarise the structure in the neighbourhood of a point (Sec. 1.3). +3.1 +Scenery flow and tangent measures +We start by giving details on the construction of � +Tan(ν, x) for ν ∈ MS(E) and x ∈ E (see 18 for +notation). +3.1.1 +Tangent measures, identifications and topologies. +Recall that the construction of the sets � +Tan(ν, x) and � +Tan +st(ν, x) employs the identification, in the +set M(Rn), of those measures that are equal up to isometries or mutual multiples (see (4), (5), (6) +and (7) for notation). We now examine the construction of the spaces of equivalence classes of tangent +measures above in more detail. +For ν ∈ M(Rn) and x ∈ spt(ν), we first consider sequences {cnνx,tn}∞ +n=0, where for every n ∈ N, +cn > 0, +νx,tn := +1 +ν(B(x, dt−1 +n )) (gx,tn)♯ ν⌊B(x,dt−1 +n ), +(23) +d ≤ 1 and gx,tn is some similarity with expanding ratio tn that maps the ball B(x, t−1 +n ) onto the +ball B(zn, 1), with zn = gx,tn(x), so each νx,tn is a probability measure supported on B(zn, d). Then, +� +Tan(ν, x) and � +Tan +st(ν, x) consist of the equivalence classes of non-null weak and strong limits, re- +spectively, as tn → ∞, of such sequences {cnνx,tn}∞ +n=0 (see (7)). Lemma 8 shows that the elements +12 + +in � +Tan(ν, x) and � +Tan +st(ν, x) do not depend on either the sequence of constants cn or the particu- +lar elements chosen in the equivalence classes � +νx,tn as long as the convergence of these elements is +guaranteed. +Remark 7 The unit ball D does not play any essential role in our definition of tangent measures in +the quotient space � +M(Rn). In the opposite direction (second approach) we may, in a more akin way +to the classical approach, require the similarities gx,tn to map B(x, t−1 +n ) onto B(0, 1), and then define +TanD(ν, x) and Tanst +D(ν, x) as weak and strong limits in M(D), respectively, of sequences of such +measures νx,tn, and � +Tan(ν, x), � +Tan +st(ν, x) as the sets of equivalence classes of measures in TanD(ν, x) +and Tanst +D(ν, x), respectively. +This second method gives spaces of tangent equivalence classes which are particular cases of these in +our primary approach. Are these equivalent methods? In order to answer this question, let a sequence +{cnνx,tn}∞ +n=0, +as in (23), converge to a non-null Radon measure α. By Lemma 8 we may assume +cn = 1 for all n ∈ N+. Since the measures νx,tn are supported on balls B(zn, d) with d ≤ 1 (see +Theorem 12 (i)), the limiting measure α must also be supported on a ball B(z, d) with zn −→ +n→∞ z. Each +measure ν′ +x,tn = (τzn)♯ νx,tn, where τzn(y) = y − zn, is equivalent by translation to νx,tn, and ν′ +x,tn is +supported on D. It is easy to see that νx,tn +w +−−−−→ +n→∞ α implies that ν′ +x,tn +w +−−−−→ +n→∞ α′ = (τz)♯ α, so α′ is +equivalent to α and supported on D. Thus, the second method gives the same space � +Tan(ν, x) than our +primary method. But νx,tn +st +−−−−→ +n→∞ α does not imply that ν′ +x,tn +st +−−−−→ +n→∞ α′, so the second method does not +produce the same space � +Tan +st(ν, x) than our method. +Observe that, if we let ν′ +x,tn = (τz)♯ νx,tn, then νx,tn +st +−−−−→ +n→∞ α does imply ν′ +x,tn +st +−−−−→ +n→∞ α′ = (τz)♯ α. But +now the measure ν′ +x,tn is supported on the ball B(zn −z, d) rather than on D. This observation is useful +because D and all the balls B(zn − z, d) are contained in some ball B(0, R) for R large enough (notice +that zn is a convergent sequence of points), so the convergence νx,tn −→ +n→∞ α (weak or strong) occurs +in M(B(0, R)), and we can see that, if we consider vague convergence of measures, we do not obtain +anything new, since in the Polish space B(0, R) both convergences are equivalent ( [15], Appendix). +Lemma 8 +(i) The sequences {cn}∞ +n=0 in the construction of +� +Tan(ν, x) and � +Tan +st(ν, x) can be taken to be cn = 1, +n = 0, 1, 2, ... +13 + +(ii) Let ν ∈ M(Rn), x ∈ spt(ν) and α ∈ Tan(ν, x). Let {tn}∞ +n=0 ↑ ∞ be such that {νx,tn}∞ +n=0 +w +−−−−→ +n→∞ +α. Assume also that there is a sequence {fn}∞ +n=0 +in the set of isometries In such that +{(fn)♯ νx,tn}∞ +n=0 +w +−−−−→ +n→∞ +α′. Then, there is f ∈ In such that (f)♯α = α′. The same is true if +the +convergence holds in the topology of the strong convergence in M(Rn). +Proof. +(i) By definition, a weak limiting measure α as in (7) is a non-null measure in M(Rn). Therefore, the +sequence of constants {cn} must be bounded above and below by two positive and finite constants. We +can choose a subsequence {cnk}∞ +k=0 that converges to a constant c, and then the sequence cνx,tn must +converge to the weak limit α. This gives νx,tn +w +−−−−→ +n→∞ c−1α, which belongs to the same equivalence class +in � +Tan(ν, x) as α. On the other hand, the non-null weak limits in M(Rn) of sequences {νx,tn}∞ +n=0 are +particular cases of those of sequences {cnνx,tn} . This completes the proof of part (i) for weak limits. +The argument also holds true for strong limits. +(ii) For any n ∈ N+, we can write fn(·) = gn(·) + an, where gn is an orthogonal map and an ∈ Rn. +Recall that νx,tn is supported on B(zn, d), so (gn + an)♯ (νx,tn) is supported on an + B(zn, d) ( [5], +Theorem 1.18), with zn −→ +n→∞ z. This means that, if ν′ +x,tn := (fn)♯ νx,tn converges, in the weak topology +of M(Rn), to some non-null measure α′ in M(Rn), the sequence an must be bounded, and then the +sequence {fn}∞ +n=0 is also bounded in the supremum norm. Therefore, there is a convergent subsequence, +{fnk}∞ +k=0, of {fn}∞ +n=0. Let f := limk→∞ fnk. Since the sequence {(fnk)♯ (νx,tnk )}∞ +k=0 converges to α′, +we have that +α′ = lim +k→∞ fnk♯(νx,tnk ) = f♯α, +(24) +which proves that α′ ∼= α. The second equality in (24) holds true because, for any ϕ in the space +C0(Rn) of continuous, compactly supported functions on Rn and for any ε > 0, there is k0 > 0 such +that for k ≥ k0, we have +∥ϕ ◦ fnk − ϕ ◦ f∥ ≤ ε +2, +���� +� +ϕ ◦ f d(νx,tnk ) − +� +ϕ ◦ f dα +���� ≤ ε +2, +14 + +and then +���� +� +ϕ d +� +fnk ♯(νx,tnk ) +� +− +� +ϕ d(f♯α) +���� +≤ +� +∥ϕ ◦ fnk − ϕ ◦ f ∥ d(νx,tnk ) + +���� +� +ϕ ◦ f d(νx,tnk ) − +� +ϕ ◦ f dα +���� ≤ ε. +If {ν′ +x,tn}∞ +n=0 converges to α′ in the strong topology of M(D), then it also converges in the weak +topology and the argument above applies. +3.1.2 +Scaling properties of typical balls and scenery flow +We need some preliminary lemma and the following definition. +Definition 9 Given a measure α ∈ Ms⌊E, two Euclidean balls B(x, d) and B(x′, d′) are said to be +α-density equivalent if θs +α(x, d) = θs +α(x′, d′). +We start with two elementary scaling properties of typical balls for measures in MS(E) and in +Ms⌊E. +Lemma 10 Let E be a self-similar set generated by the system Ψ = {fi}i∈M of similarities of Rn, +with M = {0, 1, . . . , m − 1} , and similarity dimension s. Let O be a feasible open set (for Ψ) and let +i ∈ M ∗. Then +(i) +µp(fi(A)) = piµp(A), for µp ∈ MS(E) +and µp-measurable A ⊂ O, +(25) +(ii) +µp(f −1 +i +(C)) = p−1 +i µp(C) for µp ∈ MS(E) and µp-measurable C ⊂ Oi, +(26) +(iii) +B(fi(x), rid) is α-density equivalent to B(x, d) for α ∈ Ms⌊E and B(x, d) ⊂ O, +(27) +(iv) +f −1 +i +(B(fi(x), rid)) is α-density equivalent to B(x, d) for α ∈ Ms⌊E and B(x, d) ⊂ Oi. +(28) +Proof. The proof of (25) is trivial from (16) if E satisfies SSC. If SSC does not hold, then +µp(f −1 +j +(fi(A))) ≤ µp(∂O) = 0 for j ̸= i, +15 + +because A ⊂ O and, hence, f −1 +j +(fi(A)) ∩ E ⊂ ∂O, which is known to be a µp- null set (cf. [27]), so +(25) also follows from (16). If we set A = f −1 +i +(C) in (25), we obtain (26) (see also [10]). By (19), we +can apply (25) and (26) to any measure α ∈ Ms⌊E, which easily gives (27) and (28). +Before stating the main theorem of this section, we will see the following lemma. +Lemma 11 +(i) Let g, f : Rn → Rn, α ∈ M(Rn), λ > 0 and A ⊂ Rn be an α-measurable subset. Then, the +following equalities hold true: +• λ (g)♯ (α) = (g)♯ (λα), +• (f ◦ g)♯α = f♯(g)♯(α), and +• (g♯α) ⌊A= g♯(α⌊g−1(A)) +(ii) Let α be a measure on M(Rn), g : Rn → Rn a bijective mapping and β := g♯ (α) . Then, α = +(g−1)♯β. +(iii) If {αk}k∈N is a sequence of measures on M(Rn) and (g)♯ (αk) +st +−−−−→ +k→∞ β, then αk +st +−−−−→ +k→∞ +� +g−1� +♯ (β) . +(iv) Let B(xn, d) := Bn be a sequence of closed balls that converges in the Hausdorff metric to a closed +ball B(x, d) := B, and let α ∈ M(B) with α(∂B) = 0. Then α⌊Bn:= αn +st +−−−−→ +n→∞ α. +Proof. +Parts (i)-(iii) easily follows from the definitions. +Recall that αn +st +−−−−→ +n→∞ α means that αn(A) −→ +n→∞ α(A) for any Borel set A ⊂ Rn. Let α ∈ M(B) and let +K be any compact set contained in the interior U of B. The distance d(K, ∂B) = min {∥x − y∥ : x ∈ K, y ∈ ∂B} +must be a quantity ε > 0 and then K ⊂ B(x, d − ε). The convergence of Bn to B implies that there is +an n0 ∈ N+ such that, for n > n0, ∥x − xn∥ ≤ ε. Then, if z ∈ K, +∥z − xn∥ ≤ ∥z − x∥ + ∥x − xn∥ ≤ d, +which shows that K ⊂ B ∩ Bn for n > n0. Then, for such values of n, we have +αn (K) = α(Bn ∩ K) = α(K) +We now prove that αn +st +−−−−→ +n→∞ α also holds in the σ-field B(B) of Borel subsets of B. Let +A := +� +A ⊂ B : A is α-measurable and +lim +n→∞ αn(A) = α(A) +� +. +16 + +(Notice that any α-measurable set is also αn-measurable for all n ∈ N+). It is easy to check that +B ∈ A, that B − A := Ac ∈ A if A ∈ A, and that A is closed under a finite union of its members or, in +short, that A is a field. Let Fk be a sequence of members of A. In order to show that ∪k∈N+Fk ∈ A, we +first write ∪k∈N+Fk = ∪k∈N+Gk, where Gk = ∪k +i=1Fi. This shows that ∪k∈N+Fk can be expressed as a +countable union of the increasing sequence Gk of members of A. Furthermore, ∪k∈N+Fk = ∪k∈N+Hk, +where Hk = (Gk − Gk−1) with G0 = ∅. Now, each Hk ∈ A and Hk ∩ Hk′ = ∅ for k ̸= k′. Then, using +that each αn is a measure, we have +lim +n→∞ αn +� � +k∈N+ +Hk +� += +� +k∈N+ +lim +n→∞ αn (Hk) = +� +k∈N+ +α (Hk) = α +� � +k∈N+ +Hk +� +. +This completes the proof that A is a σ-field. Notice that any closed set K ⊂ B can be written as the +union of the α and αn-null set K ∩ ∂B and of the set K − ∂B, which belongs to A as a countable +union of compact sets K ∩ B(x, d − n−1) ⊂ U. Thus, the class K of closed subsets of B is contained in +A. We know that the σ-fields generated by K and by A satisfy B(B) = σ(K) ⊂ σ(A) = A. This gives +the strong convergence of αn to α on B(B). +We can now go to the scenery flow of measures in MS(E). +Theorem 12 Let E be a self-similar set generated by the system Ψ = {fi}i∈M of similarities on +Rn, with M = {0, 1, . . . , m − 1} and similarity dimension s. Let O be a feasible open set (for Ψ) and +µp ∈ MS(E). Then, for any µp-measurable set B ⊂ O and i ∈ M ∗, the following statements hold +true. +(i) +µp⌊Bi= pi (fi)♯ (µp⌊B) . +(ii) +µp⌊B= p−1 +i +� +f −1 +i +� +♯ (µp⌊Bi) . +(29) +(iii) There is a subset �E ⊂ E with µp( �E) = 1 such that, if x ∈ E and B(x, d) ⊂ O, then for any y ∈ �E, +there is a sequence {ij}j∈N+ with ij ∈ M ∗ and a sequence of balls +� +B(y, drij) +� +j∈N+ such that +p−1 +ij +� +f −1 +ij +� +♯ +� +µp⌊B(y,drij ) +� +st +−−−→ +j→∞ µp⌊B(x,d) +(iv) For any x ∈ �E, +� +MS(B) ⊂ � +Tan +st(µp, x), +17 + +where MS(B) is defined in (8). +Proof. In order to show (i), let µp ∈ MS(E), i ∈ M ∗ and let B ⊂ O and A ⊂ Rn be µp-measurable +sets. Then, +(pi(fi)♯ (µp⌊B)) (A) = pi (µp⌊B) +� +f −1 +i +(A +� +) += piµp(f −1 +i +(A ∩ Bi)) = µp⌊Bi(A), +where the third equality follows from (26) and (i) is proved. Analogously, (ii) follows from (25). +Now, let +�E = +� +y ∈ E : +� +T k(y) : k ∈ N+� +is dense in E +� +(30) +(see (15) in Sec. 2 the definition of T ). It is well known (cf. [37]) that the set �E has a full µp-measure. +Let x ∈ E, B(x, d) ⊂ O, y ∈ �E and {xj}j∈N+ such that limj→∞ xj = x (in the Euclidean metric) with +xj ∈ T kj(y) for every j ∈ N+. We may also assume that B(xj, d) ⊂ O for every j ∈ N+. We shorten +B(xj, d) to Bj and B(x, d) to B. Since limj→∞ xj = x, it follows that +� +Bj� +j∈N converges to B in the +Hausdorff metric. Also, µp(∂B) = 0 because µp ∈ MS(E) (cf. [35]). Then, Lemma 11 (iv) implies +that +µp⌊Bj +st +−−−→ +j→∞ µp⌊B. +(31) +Now, notice that, since xj ∈ T kj(y) for each j ∈ N, there is ij ∈ M kj such that fij(xj) = y (see +Remark 6). Then, f −1 +ij (B(y, drij)) = Bj. By (29) applied to Bj and ij ∈ M ∗, we see that +µp⌊Bj= p−1 +ij +� +f −1 +ij +� +♯ +� +µp⌊B(y,drij ) +� +, +(32) +which concludes the proof of (iii). +Observe that, in the terminology of Sec. 3.1.1, the right hand term in (32) is, ctjνy,tj for ν = µp, +tj = r−1 +ij +and ctj = p−1 +ij µp(B(y, drij)) (recall that νy,tj was a normalised blowup and notice also that +we may assume, rescaling E if necessary, that all typical balls have a radius d ≤ 1). So, (31) and (7) +give � +µp⌊B ∈ � +Tan +st(µp, x) and part (iv) is proved. +3.2 +Asymptotic spectra and measure-exact self-similar sets +We shall write Im(θs +α, B) to designate the set +Im(θs +α, B) := {θs +α(x, d) : B(x, d) ∈ B} +18 + +(see notation in Definition 3), which plays a relevant role in the geometric analysis of E (see (21) and +the lemma below). +Lemma 13 Let E be a self-similar set generated by the system of similarities of Rn, Ψ = {fi}i∈M , +with M = {0, 1, . . . , m − 1} , and similarity dimension s. If E satisfies the OSC, then +(i) +Cs(E) = +� +sup +� +θs +µ(x, d) : B(x, d) ∈ BO +��−1 , +where BO := {B(x, d) ∈ B : B(x, d) ⊂ O} and O is any feasible open set for Ψ. +(ii) +Cs(E) = +� +sup Im(θs +µ, B) +�−1 . +Proof. It is known that for a general self-similar set that satisfies the OSC (see [36] and [30]), +Cs(E) = +� +sup{θs +µ(x, d) : x ∈ E and d > 0} +�−1 +(33) +holds. Let O be any feasible open set. Then, it is enough to show that +sup +(x,d)∈E×R+ θs +µ(x, d) ≤ +sup +B(x,d)∈BO +θs +µ(x, d). +Should this not be the case, there would exist (x0, d0) ∈ E × R+ such that B(x0, d0) /∈ BO and +θs +µ(x0, d0) > +sup +B(x,d)∈BO +θs +µ(x, d). +In order to show that this contradicts (33), take x∗ ∈ E∩O such that there is i ∈ M ∗ with fi(x∗) = x∗. +Let ρ1 := min {∥x∗ − z∥ : z ∈ ∂O} . Observe that, if we take ρ2 > 0 so that B(x0, d0) ⊂ B(x∗, ρ2) and +k ∈ N+, satisfying that rk +i ρ2 < ρ1, then +f k +i (B(x0, d0)) ⊂ f k +i (B(x∗, ρ2)) = B(x∗, rk +i ρ2) ⊂ O, +which, using that f k +i (B(x0, d0) ∩ S) ⊂ f k +i (B(x0, d0)) ∩ S, raises the contradiction +θs +µ(x0, d0) ≤ r−ks +i +µ(f k +i (B(x0, d0))) +(2d0)s += µ(B(f k +i (x0), rk +i d0)) +(2d0rk +i )s +≤ +sup +B(x,d)∈BO +θs +µ(x, d). +Part (ii) is trivial from (i). +In the next theorem, we shall establish the relationships between the pointwise and global spectra, +the set Im(θs +α, B) and its extreme values α(E) (P s(E))−1 and α(E) (Cs(E))−1 . +19 + +Theorem 14 Let E ⊂ Rn be a self-similar set that satisfies the SOSC with feasible open set O and +similarity dimension s, and let α ∈ Ms⌊E. Then, the following statements hold true. +(i) For x ∈ E, it holds that +Spec(α, x) = +� +θs +α(x), θ +s +α(x) +� +(see (13) and (12) for notation) +Spec(α, E) ⊂ [κ1, κ2] +with 0 < κ1 ≤ κ2 < ∞. +(ii) There is a subset �E ⊂ E with µ( �E) = 1 such that, for any y ∈ �E, +Spec(α, y) = Spec(α, �E) = Spec(α, O ∩ E). +(iii) +� α(E) +P s(E), α(E) +Cs(E) +� +⊂ Im(θs +α, B) ⊂ Spec(α, O∩E) ⊂ +� α(E) +P s(E), α(E) +Cs(E) +� +. +Proof. That θs +α(x) and θ +s +α(x) belong to and are the extreme values of Spec(α, x) follows from the +definitions. That all the intermediate values in between also belong to Spec(α, x) is a consequence +of the continuousness of θs +α(x, d), with respect to d. This last property follows from the fact that the +α-measure of the boundary of Euclidean balls is always null [35] for any measure α ∈ MS(E), which +proves the first assertion of (i). The second assertion is well known [6]. +In order to prove (ii), let �E be the full µ-measure subset of points of E that have a dense geometric +shift orbit in E (see (30)) and let y ∈ �E. The inclusions Spec(α, y) ⊂ Spec(α, �E) ⊂ Spec(α, O ∩ E) are +trivial as �E ⊂ O. This follows from the fact that, if y /∈ O, then T (y) ∩ O = ∅ because fi(O) ⊂ O for +any i ∈ M, and repeating the same argument, we see that T k(y) could not be dense in E. +The corresponding equalities would follow if we prove Spec(α, O ∩ E) ⊂ Spec(α, y). This holds true +because, if z = limk→∞ θs +α(x, dk) for x ∈ O∩E and dk −→ +k→∞ 0, since B(x, dk) ∈ B for any sufficiently +large k, we can apply Theorem 12 (iii) to see that, for such values of k, θs +α(x, dk) ∈ Spec(α, y) and, +hence, limk→∞ θs +α(x, dk) ∈ Spec(α, y) easily follows from (i). This ends the proof of (ii). +Finally, the first inclusion in (iii) for α = µ follows from the continuousness of the function θs +µ(x, d) on +Rn × R+ since +1 +P s(E) ≤ θs +µ(x, d) ≤ +1 +Cs(E) +20 + +holds if B(x, d) ∈ B as a straightforward consequence of (21) and (22). The arguments given in the +proof of (ii) applied to µ show that, if B(x, d) ∈ B, then θs +µ(x, d) ∈ Spec(µ, O ∩ E), which gives the +next inclusion in (iii). The last inclusion follows from the observation that Spec(µ, O∩E) consists of +limiting values of sequences with terms in Im(θs +µ, B), whose extreme values are +1 +P s(E) and +1 +Cs(E). Using +(19), we get that θs +α(x, d) = α(E)θs +µ(x, d), and (iii) follows for any α ∈ Ms⌊E. +Of note is the case in which the extreme values of θs +α(x, d) are attained on B. In this case, we have +the following result. +Corollary 15 Let α ∈ {µ, P s⌊E, Cs⌊E}. Under the hypotheses of Theorem 14, if there are two balls +B(x1, d1) and B(x2, d2), both in B, such that +θs +µ(x1, d1) = inf +� +θs +µ(x, d) : B(x, d) ∈ B +� +(34) +and +θs +µ(x2, d2) = sup +� +θs +µ(x, d) : B(x, d) ∈ B +� +, +(35) +the inclusions in Theorem 14 (iii) can be replaced with equalities. +Proof. The first inclusion in Theorem 14 (iii), together with (21), (22), (34) and (35), implies that +Im(θs +µ, B) = +� +1 +P s(E), +1 +Cs(E) +� +, +which, in turn, gives that Im(θs +α, B) = Spec(α, O∩E). +Corollary 15 motivates the introduction of the class of α-exact self-similar sets with special prop- +erties. +Definition 16 We say that the self-similar set E is α-exact if there exists B ∈ Cα such that +µ(B) +|B|s = sup +�µ(B) +|B|s : B ∈ Cα +� +if α ∈ {Cs⌊E, Hs⌊E, HSph⌊E} , and +µ(B) +|B|s = inf +�µ(B) +|B|s : B ∈ Cα +� +, +if α = P s⌊E, where Cα is what we call “the relevant class of sets” for the measure α, which is defined +as +21 + +• Cα := B if α ∈ {P s⌊E, Cs⌊E} , +• CHs⌊E := {B ⊂ Rn : B is a convex set} and +• CHs +Sph⌊E := {B ⊂ Rn : B is a closed ball}. +One nice property that α-exact self-similar sets have is that they possess optimal coverings or +packings, that is, almost-coverings (i.e. coverings for α-almost all points in E) or packings whose +s-volume gives the exact value of the corresponding α-measure, whilst if α-exactness is not fulfilled, +we can only hope to find coverings or packings with s-volume arbitrarily close to the corresponding +α-measure. +Example 17 Self-similar sets E with the strong separation condition are an example of α-exact self- +similar sets. See [38] for α ∈ +� +P s⌊E, Hs⌊E, Hs +Sph⌊E +� +and [30] for α = Cs. +Example 18 The Sierpinski gasket S is an example of a set where the strong separation condition +does not hold, and that is a P s⌊S-exact (see [22]) and Cs⌊S-exact (see [23]) set. +In [39], it is shown a class of self-similar sets E with OSC in the line whose members can be non- +Hs⌊E-exact (and, consequently, non-Hs +Sph⌊E-exact since these two measures coincide in the line), and +the authors find conditions under which they are Hs⌊E-exact. +Example 19 Self-similar sets E with OSC in the line, with similarity dimension s, and that admit +an open interval as a feasible open set, are an example of P s⌊E-exact self-similar sets [40]. +3.3 +Complexity of the local structure of self-similar sets +We now show how these results allow us to explore the complexity of the local geometric structure of +self-similar sets that satisfy the OSC condition. +First, we need to properly define the equivalence classes of restricted balls. Notice that different +Euclidean balls, even if they share the centre, can produce the same restricted balls. This motivates +the following definitions that are valid for general subsets of Rn. +Definition 20 Given a subset A ⊂ Rn, the spherical diameter of A is defined by +|A|Sph = inf {2d : A = A ∩ B(x, d) for some x ∈ A} +22 + +Definition 21 Given a subset A ⊂ Rn, we say that the restricted ball B(x, d) ∩ A is proper and write +B(x, d) ∩ A ∈ P(A) if x ∈ A and 2d = |B(x, d) ∩ A|Sph . +Definition 22 Given a measure α on Rn and an α-measurable s-set A ⊂ Rn, we define the α-spherical +s-density of A by +θs +Sph(α)(A) = +α(A) +� +|A|Sph +�s . +Definition 23 Given a subset A ⊂ Rn and two restricted balls B(x, d)∩A, B(x′, d′)∩A both in P(A), +we say that they are similarity-equivalent and write B(x, d) ∩ A ≃Sn B(x′, d′) ∩ A if there is an f ∈ Sn +such that +B(x′, d′) ∩ A = f(B(x, d) ∩ A). +Lemma 24 Let A ⊂ Rn and B(x, d) ∩ A ∈ P(A). +(i) If f ∈ Sn has similarity constant rf, then f(B(x, d)) ∩ f(A) ∈ P(f(A)) and |f(B(x, d)) ∩ f(A)|p = +rfd. +(ii) Let α ∈ Ms and let A be an α-measurable s-set. If B(x, d) ∩ A ≃Sn B(x′, d′) ∩ A, then +θs +Sph(α)(B(x, d) ∩ A) = θs +Sph(α)(B(x′, d′) ∩ A) +Proof. Let A ⊂ Rn, B(x, d) ∩ A ∈ P(A). In order to show (i), assume that f(B(x, d)) ∩ f(A) is not +proper. Then, there is a ball B(y, ρ) such that +B(y, ρ) ∩ f(A) = f(B(x, d)) ∩ f(A) +with y ∈ f(A) and ρ < rfd. Then B(f −1(y), r−1 +f ρ) ∩ A = B(x, d) ∩ A with f −1(y) ∈ A and +r−1 +f ρ < d, in contradiction with |B(x, d) ∩ A|Sph = 2d. Therefore, f(B(x, d)) ∩ f(A) ∈ P(f(A)) +and |f(B(x, d)) ∩ f(A)|Sph = 2rfd. +Part (ii) is now trivial since α ∈ Ms and, hence, +α(B(x′, d′) ∩ A) = α(f(B(x, d) ∩ A)) = rs +fα(B(x, d) ∩ A) +and, by (i), +� +|B(x′, d′) ∩ A|Sph +�s += +� +|f(B(x, d) ∩ f(A))|Sph +�s += rs +f(2d)s = rs +f +� +|B(x, d) ∩ A|Sph +�s +. +23 + +Now we can proceed to state our result for the complexity of the local geometry of self-similar sets +with OSC. +Corollary 25 Under the assumptions of Theorem 14, assume that s is a non-integer real number. +Then, there is an uncountable number of equivalence classes in the quotient space SphE/ ≃Sn . +Proof. By Lemma 24 (ii), we know that all restricted balls in an equivalence class of SphE/ ≃Sn +share the same µ-spherical s-density, which allows us to naturally define a mapping θs +µ : SphE/ ≃Sn→ +Im(θs +µ, B). This implies that the inverse +� +θs +µ +�−1 : Im(θs +µ, B)→SphE/ ≃Sn of such mapping is an injective +correspondence. Using Marstrand’s Theorem, parts (ii) and (iii) of Theorem 14 and that µ( �E) = 1 > 0, +it follows that either Cs(E) < P s(E) or s is an integer (notice that from the definitions in Sec. 2.2 +it is easy to see that Cs(E) ≤ P s(E)). This, together with Theorem 14 (iii), means that Im(θs +µ, B) +contains an interval with uncountably many points and the proof is completed. +4 +The spectrum of the Sierpinski gasket +In this section, we shall apply the results obtained in Theorem 14 to fully characterise the asymptotic +spectra of the Sierpinski gasket S. +Recall that the Sierpinski gasket or Sierpinski triangle is a special case of a self-similar set generated +by a system Ψ = {f0,f1,f2} of three contracting similitudes of the plane, with contraction ratios +ri := 1/2, i ∈ M, given by +f0(x, y) = 1 +2(x, y), +f1(x, y) = 1 +2(x, y) + (1 +2, 0) +and +f2(x, y) = 1 +2(x, y) + 1 +2(1 +2, +√ +3 +2 ). +(36) +We shall denote by zi the fixed point of each fi, i = 0, 1, 2 that is, z0 = (0, 0), z1 = (1, 0) and +z2 = ( 1 +2, +√ +3 +2 ), and by T the equilateral triangle with vertexes zi, i ∈ M. +It is well known that S is a connected set that satisfies the OSC and has similarity dimension +s = log 3 +log 2. +Thanks to previous work on the packing and Hausdorff centred measures of the Sierpinski gasket (cf. +[22] and [23]), we know that S is both P s⌊S and Cs⌊S-exact, and we have fairly precise approximations +of the values of P s(S) and Cs(S). +24 + +4.1 +Theoretical results +Theorem 26 Let S be the Sierpinski gasket, �S = +� +y ∈ S : +� +T k(y) : k ∈ N +� +is dense in S +� +, B be the +collection of typical balls, R be a feasible open set for S, and α ∈ Ms⌊S. Then, the following statements +hold true. +(i) +Spec(α, y) = Spec(α, �S) = Spec(α, R ∩ S) = Im(θs +α, B) = +� α(S) +P s(S), α(S) +Cs(S) +� +, y ∈ �S. +(37) +(ii) Spec(α, S) is given by the union of two closed intervals of positive length: +Spec(α, S) = +� +θs +α(z0), θ +s +α(z0) +� +∪ +� α(S) +P s(S), α(S) +Cs(S) +� +, +(38) +where z0 = (0, 0). Furthermore, +θs +α(z0) = min +� +θs +α(z0, d) : 1 +2 ≤ d ≤ 1 +� +(39) +and +θ +s +α(z0) = max +� +θs +α(z0, d) : 1 +2 ≤ d ≤ 1 +� +. +(40) +Proof. Our previous work guarantees that S is a P s-exact (see [22]) and Cs-exact (see [23]) set. Then, +the four equalities in (i) follow as a consequence of Theorem 14 and Corollary 15. +In order to prove (38), let Ri, i ∈ {0, 1, 2} be the three open rhombi composed of the topological +interior of the union of the triangle T and its reflection across the edge of T opposite the point zi, +i ∈ M (see R2 in Fig. 1). Using that +S = {z0, z1, z2} ∪ (S ∩ ∪2 +i=0Ri), +(41) +we obtain +Spec(α, S) = Spec(α, S ∩ ∪2 +i=0Ri) ∪ +� +∪2 +i=0 Spec(α, zi) +� += += +� α(S) +P s(S), α(S) +Cs(S) +� +∪ Spec(α, z0), +where the last equality follows from (37), (41) and the fact that, by symmetry, Spec(α, zi) must be +identical for i ∈ {0, 1, 2} . +25 + +Figure 1: A feasible open set. +An open rhombus R2 that is a feasible open set for S. +Observe now that, if d ≤ 1/2, then B(z0, d) ∩ S = B(z0, d) ∩ f0(S). Hence, using that α is an +s-dimensional metric measure +θs +α(z0, d) = α(B(z0, d) ∩ f0(S)) +(2d)s += α(f0(B(z0, 2d) ∩ S)) +(2d)s += α(B(z0, 2d) ∩ S)) +(4d)s += θs +α(z0, 2d). +If 2d ≤ 1/2, we can repeat the argument k times until 1/2 ≤ 2kd ≤ 1 and θs +α(z0, d) = θs +α(z0, 2kd). This +shows that +min {θs +α(z0, d) : 0 ≤ d ≤ 1} = min +� +θs +α(z0, d) : 1 +2 ≤ d ≤ 1 +� += θs +α(z0), +where the last equality can easily be checked and, analogously, (40) holds. +Remark 27 Notice that part (i) of Theorem 26 shows that there is a set of full α-measure whose +points exhibit a strongly regular behaviour, whereas part (ii) underlines the special local behaviour of the +vertexes as the most isolated points in S. However, the set of exceptional points does not consist only of +the vertexes as there might be other exceptional points, all of them belonging to the set ∪2 +i=0 (Ri ∩ S)− �S. +The pointwise α-density spectrum of such points is contained in +� +α(S) +P s(S), α(S) +Cs(S) +� +. The detection and +characterisation of the behaviour of these points remains an open issue. +26 + +22 +T +Z1 +R24.2 +Numerical results +Following the structure of the algorithms developed in [22, 23, 41, 42] for the numerical estimation of +the metric measures of self-similar sets, the construction of the computational algorithm used in this +work in order to approximate the values of θs +µ(z0) and θ +s +µ(z0) relies upon the discrete approximations +of both the Sierpinski gasket and its invariant measure µ. Recall that any two measures in Ms⌊S are +mutually multiple of each other (see (19)), so we can obtain Spec(α, S) from Spec(µ, S) if we know +α(S). +The Sierpinski gasket, as the attractor of Ψ = {f0, f1, f2} (see (36)), is the unique non-empty +compact set that admits the self-similar decomposition S = F(S), where F is the Hutchinson operator +defined, for A ⊂ R2, by +F(A) := f0(A) ∪ f1(A) ∪ f2(A). +It is well-known that, for any non-empty compact subset A ⊂ R2, S can be built with an arbitrary +level of detail by increasing the iterations k in F k(A), where F k = F ◦F...◦F is the k-th iterate of the +contracting operator F. This is because F k(A) +k→∞ +→ +S in the Hausdorff metric (cf. [6]). Furthermore, +if A ⊂ S, then F k(A) ⊂ S for any k ∈ N+. In particular, if we take A1 := {z0, z1, z2} as the initial +compact set, we obtain the set +Ak := F k−1(A1) ⊂ S, k ≥ 2, +(42) +which approximates S at the iteration k of our algorithm. +The relation between the Markov operator and the natural probability measure µps given in (17), +with s = log 3 +log 2 and pi = rs +i = 3−k, and (19) leads to the following relation: +Mk +ps(α) = 1 +3k +� +i∈M k +α ◦ f −1 +i +w→ µ, +α ∈ P(R2). +(43) +If we consider µ1 := 1 +3(δz0 + δz1 + δz2) as an initial measure α in (43), where δx is a unit mass at +x, then +µk := Mk−1 +ps (µ1) = +1 +3k−1 +� +i∈M k−1 +µ1 ◦ f −1 +i += 1 +3k +� +i∈M k−1 +� +δfi(z0) + δfi(z1) + δfi(z2) +� +(44) +is a probability measure supported on Ak ⊂ S and µk +w→ µ. +The discrete measure µk is the approximation of the invariant measure µ that our algorithm takes at +iteration k. +27 + +Lemmas 28 and 30 (Lemma 28 is proved in [23]), provide precise relationships between the measures +µk and µ. +Lemma 28 +(i) Let {Si : i ∈ I ⊂ M k}, k ∈ N+, be a collection of k-cylinder sets of S. Then, +µ +�� +i∈I +Si +� +≤ µk +�� +i∈I +Si +� +(ii) Let A ⊂ S, k ∈ N+, and let I = {i ∈ M k : Si ∩ A ̸= ∅}. Then, +µk(A) ≤ µ +�� +i∈I +Si +� +Remark 29 The comparisons between the measures µ and µk on collections of cylinders and sets +given in the lemma above are passed to enlarged and reduced balls in part (i) of the next lemma. Since +our algorithms compute only µk-densities of balls with centres in Ak (see (42)) and with some point of +Ak in their boundaries, in part (ii) of this lemma we approximate the µ-measure of a ball centred at x +with the µk-measure of a ball with its same centre and with a point of Ak at its boundary. +In order to obtain more accurate estimates of θs +µ(z0) and θ +s +µ(z0) (as we also do in [22] and [23] for the +estimation of P s(S) and Cs(S)), it is necessary to consider open balls when searching balls of minimal +µk-density (see (46)), whereas in the search of balls with maximal µk-density, the approximating balls +must be taken to be closed balls (see (47)). In the definition of θs +µ(·) and θ +s +µ(·), the use of open or +closed balls has no relevance because the µ-measure of the boundary of any ball is null. However, in the +case of densities of the discrete measures µk, the values obtained in one or the other case do actually +matter, mainly if k is not large. +From now on, we shall use the notation ˚ +B(x, d) := {y ∈ R2 : |x − y| < d} and ˚θs +α for the s-density +of α defined using open balls. +Lemma 30 Let k > 0, x ∈ R2, and 2−k < d ≤ maxi∈M ∥zi − x∥ . Then, +(i) µk(B(x, d − 2−k)) ≤ µ(B(x, d)) ≤ µk(˚ +B(x, d + 2−k)) +(ii) If B(x, d) ∩ Ak ̸= ∅, then there are points yk and zk in Ak such that +µk +� +˚ +B(x, dyk) +� +≤ µ(B(x, d)) ≤ µk(B(x, dzk)), +28 + +where dyk := |yk − x| , dzk := |zk − x| , and {dyk, dzk} ∈ [d − 2−k, d + 2−k]. +Proof. +(i) Let +Hk := {i ∈ M k : B(x, d − 2−k) ∩ Si ̸= ∅} +For any i ∈ Hk, Si ⊂ B(x, d) holds, so ∪i∈HkSi ⊂ B(x, d). Using Lemma 28 (ii), we have +µk(B(x, d − 2−k)) ≤ µ(∪i∈HkSi) ≤ µ(B(x, d)). +Let +Gk := {i ∈ M k : Si ⊂ ˚ +B(x, d + 2−k)}. +Then, ˚ +B(x, d) ∩ S ⊂ ∪i∈GkSi and ∪i∈GkSi ⊂ ˚ +B(x, d + 2−k). Using Lemma 28 (i), we get +µ(B(x, d)) = µ(˚ +B(x, d) ∩ S) ≤ µ(∪i∈GkSi) ≤ µk(∪i∈GkSi) ≤ µk(˚ +B(x, d + 2−k)) +(ii) Let d∗ = maxi∈M ∥zi − x∥ . If S ⊂ B(x, d), then d = d∗ and µ(B(x, d∗)) = 1 = µk(B(x, d∗)) > +µk((˚ +B(x, d∗)), so property (ii) holds for dyk = dzk = d∗. Let us now assume that S ⊈ B(x, d). We +prove first that +Fk := {i ∈ M k : ∂B(x, d) ∩ Si ̸= ∅} ̸= ∅. +(45) +If Fk = ∅, then +∪i∈M kSi ⊂ ˚ +B(x, d) ∪ (B(x, d))c. +We know that (∪i∈M kSi) ∩ ˚ +B(x, d) ̸= ∅ because B(x, d) ∩ Ak ̸= ∅ and Fk = ∅, and we also know +that (∪i∈M kSi) ∩ (B(x, d))c ̸= ∅ because S ⊈ B(x, d) and Fk = ∅. This contradicts that ∪i∈M kSi is +a connected set, and (45) must hold. +Using (i), we have that +µ(B(x, d)) ≤ µk(B(x, d + 2−k)) = µk(B(x, dzk)), +where zk satisfies dzk = ∥zk − x∥ with +dzk = max{∥y − x∥ : y ∈ Ak ∩ B(x, d + 2−k)}. +The inequality dzk ≤ d+2−k is obvious, and dzk ≥ d−2−k follows because Fk ̸= ∅ and each k-cylinder +Si, i ∈ M k contains some point in Ak. +29 + +Using the first inequality in (i), we have +µ(B(x, d)) ≥ µk(B(x, d − 2−k)) = µk(˚ +B(x, dyk)), +where yk satisfies dyk = ∥yk − x∥ with +dyk = min{∥y − x∥ : y ∈ Ak ∩ +� +B(x, d − 2−k) +�c}. +The inequality dyk ≥ d − 2−k is obvious, and dyk ≤ d + 2−k follows because Fk ̸= ∅. +Theorem 26 allows us to characterise Spec(α, S) for α ∈ {µ, P s⌊S, Cs⌊S} through only four +numbers, namely, θs +µ(z0), θ +s +µ(z0), P s(S) and Cs(S). Thanks to previous numerical work that uses the +measures µk and the sets Ak (see (44) and (42)) as approximations of µ and S, respectively, we have +estimates given by our algorithms Pk of P s(S) (see [22]) and Ck of Cs(S) (see [23]) and precise error +bounds for such estimates. We show in Theorem 31 below how to obtain estimates ξk of θs +µ(z0) and +ξk of θ +s +µ(z0), that such estimates converge to the real values, and we give accurate bounds for them, +that is θs +µ(z0) ∈ [ξinf +k , ξsup +k +] and θ +s +µ(z0) ∈ [ξ +inf +k , ξ +sup +k +] (see the definition of ξk, ξk and of the intervals +[ξinf +k , ξsup +k +] and [ξ +inf +k , ξ +sup +k +] in Theorem 31). This allows us to implement an algorithm along the lines of +those developed for the estimation of Cs(S) and P s(S) (see [22,23]). +Theorem 31 For k > 1, let +ξk := min +� +˚θs +µk(z0, d) : d = |x − z0| , x ∈ Ak, d ∈ [1 +2 − 2−k, 1] +� +(46) +and +ξk := max +� +θs +µk(z0, d) : d = |x − z0| , x ∈ Ak, d ∈ [1 +2 − 2−k, 1] +� +(47) +be the estimates of θs +µ(z0) and θ +s +µ(z0), respectively. Let dk be such that ˚θs +µk(z0, dk) = ξk, and let Dk be +such that θs +µk(z0, Dk) = ξk. +Then, +{θs +µ(z0), ξk} ∈ [ξinf +k , ξsup +k +], +(48) +and +{θ +s +µ(z0), ξk} ∈ [ξ +inf +k , ξ +sup +k +], +(49) +where +ξinf +k += Kkξk, +Kk = (1 − 21−k)s, +ξsup +k += µk(˚ +B(z0, dk + 2−k)) +(2dk)s +, +(50) +30 + +ξ +inf +k += µk(B(z0, Dk − 2−k)) +(2Dk)s +, +Kk = (1 + 21−k)s, +ξ +sup +k += Kkξk. +(51) +Proof. That ξk ∈ [ξinf +k , ξsup +k +] and ξk ∈ [ξ +inf +k , ξ +sup +k +] is obvious from the definitions. +We prove first that θs +µ(z0) ∈ [ξinf +k , ξsup +k +]. Using Lemma 30 (i) and (39), we obtain +θs +µ(z0) ≤ µ(B(z0, dk)) +(2dk)s +≤ µk(˚ +B(z0, dk + 2−k)) +(2dk)s += ξsup +k +. +Let d ∈ [ 1 +2, 1] be such that θs +µ(z0) = µ(B(z0,d)) +(2d)s +. Lemma 30 (ii) guarantees the existence of yk ∈ Ak such +that µ(B(z0, d)) ≥ µk(˚ +B(z0, dyk)), where dyk := |yk − z0| ∈ [d − 2−k, d + 2−k] ⊂ [ 1 +2 − 2−k, 1]. This, +together with (46) and the inequality d ≥ 1 +2 gives +θs +µ(z0) = µ(B(z0, d)) +(2d)s +≥ µk(˚ +B(z0, dyk)) +(2d)s += +�dyk +d +�s µk(˚ +B(z0, dyk)) +(2dyk)s +≥ +�dyk +d +�s +ξk ≥ +�d − 2−k +d +�s +ξk ≥ ξinf +k . +The proof that θ +s +µ(z0) ∈ [ξ +inf +k , ξ +sup +k +] is analogous. Using Lemma 30 (i) and (40), we obtain +θ +s +µ(z0) ≥ µ(B(z0, Dk)) +(2Dk)s +≥ µk(B(z0, Dk − 2−k)) +(2Dk)s += ξ +inf +k . +Let D ∈ [ 1 +2, 1] be such that θ +s +µ(z0) = µ(B(z0,D)) +(2D)s +. Lemma 30 (ii) guarantees the existence of zk ∈ Ak +such that µ(B(z0, D)) ≤ µk(B(z0, dzk)), where dzk := |zk − z0| ∈ [D − 2−k, D + 2−k] ⊂ [ 1 +2 − 2−k, 1]. +This, together with (47) and the inequality D ≥ 1 +2 gives +θ +s +µ(z0) = µ(B(z0, D)) +(2D)s +≤ µk(B(z0, dzk)) +(2D)s += +�dzk +D +�s µk(B(z0, dzk)) +(2dzk)s +≤ +�dzk +D +�s +ξk +≤ +�D + 2−k +D +�s +ξk ≤ ξ +sup +k +. +We present in Table 1 the estimates ξk, and ξk of θs +µ(z0) and θ +s +µ(z0) (see (48) and (49) for +definitions), respectively, and the corresponding lower and upper bounds in the 100% confidence inter- +vals [ξinf +k , ξsup +k +], [ξ +inf +k , ξ +sup +k +] (see (51),(49)) obtained by our algorithm for k = 14 (see the definition these +values in (46), (47), (50) and (51)). We also provide the radii, dk and Dk, of the µk-optimal balls. +See in Fig. 2a the graph of the function θs +µ14(z0, d) as a function of d ∈ [ε, 1], and in Fig. 2b the +points (g(d), θs +µ14(z0, d)), where g(d) := ε + +ε−1 +log(ε)(log(d) − log(ε)) and ε = 0.05. This is a suitable +logarithmic scale, [10], which allows us to see the periodicity of this function at such a scale. +31 + +(a) Values of θs +µ14(z0, d) for d ∈ [ε, 1] and ε = 0.05. +(b) Values of (g(d), θs +µ14(z0, d)), where g(d) := ε + +ε−1 +log(ε)(log(d) − log(ε)) and ε = 0.05. +Figure 2: Densities at z0 +We present in Table 2 the estimates Pk of P s(S) and Ck of Cs(S) obtained by our algorithms for +k = 14. The lower and upper bounds of P s(S) are denoted by P inf +k +and P sup +k +, respectively, and the +bounds of Cs(S) are denoted by Cinf +k +and Csup +k +, respectively. These results were computed in [22] and +[23]), respectively. Recall that (P s(S))−1 and (Cs(S))−1 are the µ-densities of the balls of minimum +and maximum µ-density in the set of typical balls. The estimates Pk and Ck are obtained by replacing +S with Ak and µ with µk. Again, we have used open balls in the estimation of the density of the +ball of minimum µk-density, and closed balls for the density of the ball of maximum µk-density. The +centre and radius of the open ball of minimum µk-density are denoted by x∗ +k and dk, respectively, +and the centre and radius corresponding to the closed ball of maximum µk-density are denoted by +y∗ +k and Dk. The table also contains the corresponding optimal µk-densities and their bounds. The +upper bound P sup +k +:= KP +k Pk of P s(S) is slightly improved here with respect to the one given in [22]. +Here KP +k := (1 − 25−k +√ +3 )−s instead of the value KP +k = (1 − 26−k +√ +3 )−s used in [22]. This gives the value +P sup +14 += 1.671292 given in Table 2 instead of the value P sup +14 += 1.668305 given in Table 1 in [22]. +The results of the following corollary are based on the estimates of Tables 1 and 2. +Corollary 32 Let S be the Sierpinski gasket. +(i) For any α ∈ Ms⌊S, Spec(α, S) is the union of two closed disjointed intervals. +32 + +"(z +00.36 +μ14 +0.35 +0.34 +0.33 +0.32 +0.31 +0.3 +0.29 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +d0.36 +μ14 +0.35 +0.34 +0.33 +0.32 +0.31 +0.3 +0.29 +0.2 +0 +0.1 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +g(d)(ii) +Spec(µ, S) ∼ [0.2997, 0.3567] ∪ [0.5994, 0.9951] +[0.2998, 0.3566] ∪ [0.5999, 0.9944] ⊂ Spec(µ, S) ⊂ [0.2996, 0.3568] ∪ [0.5983, 0.9970] +(iii) +Spec(P s⌊S, S) ∼ [0.5, 0.5951] ∪ [1, 1.6602] +[0.5010, 0.5945] ∪ [1, 1.6578] ⊂ Spec(P s⌊S, S) ⊂ [0.4995, 0.5963] ∪ [1, 1.6662] +(iv) +Spec(Cs⌊S, S) ∼ [0.3012, 0.3584] ∪ [0.6023, 1] +[0.3015, 0.3577] ∪ [0.6032, 1] ⊂ Spec(Cs⌊S, S) ⊂ [0.3005, 0.3588] ∪ [0.6002, 1] +Proof. We know (see (38) in Theorem 14) that +Spec(µ, S) = +� +θs +µ(z0), θ +s +µ(z0) +� +∪ +� +1 +P s(S), +1 +Cs(S) +� +, +(52) +and that (see (19)) +Spec(α, S) = α(S)Spec(µ, S), +α ∈ Ms⌊S. +(53) +The two intervals in Spec(α, S), α ∈ Ms⌊S are disjointed if θ +s +µ(z0) < +1 +P s(S). Such a condition holds +(see Theorem 31, and Tables 1 and 2) because +θ +s +µ(z0) ≤ ξ +sup +14 < 0.3568 +and +1 +P s(S) ≥ +1 +P sup +14 +> 0.5983. +Using (52), Theorem 31 and (53), we have that +Spec(µ, S) ∼ [ξ14, ξ14] ∪ +� 1 +P14 +, +1 +C14 +� +, +� +ξsup +14 , ξ +inf +14 +� +∪ +� 1 +P inf +14 +, +1 +Csup +14 +� +⊂ Spec(µ, S) ⊂ +� +ξinf +14 , ξ +sup +14 +� +∪ +� +1 +P sup +14 +, +1 +Cinf +14 +� +, +Spec(P s⌊S, S) ∼ [P14ξµ14, P14ξµ14] ∪ +� +1, P14 +C14 +� +, +� +P sup +14 ξsup +14 , P inf +14 ξ +inf +14 +� +∪ +� +1, P inf +14 +Csup +14 +� +⊂ Spec(P s⌊S, S) ⊂ +� +P inf +14 ξinf +14 , P sup +14 ξ +sup +14 +� +∪ +� +1, P sup +14 +Cinf +14 +� +, +33 + +(and, analogously, for Spec(Cs⌊S, S)), and the proof is completed using the corresponding estimates +of Tables 1 and 2. +ξ14 +[ξinf +14 , ξsup +14 ] +d14 +0.299714 +[0.299656,0.299763] +0.642272 +ξ14 +[ξ +inf +14 , ξ +sup +14 ] +D14 +0.356687 +[0.356645,0.356756] +0.913663 +Table 1: Extreme densities at z0 +Estimates of θs +µ(z0) and θ +s +µ(z0), bounds and radii, dk and Dk, of the µk-optimal balls for k = 14. +x∗ +14 +d14 +P14 +[P inf +14 , P sup +14 ] +(P14)−1 = ˚θs +µ14(x∗ +14, d14) +� +(P sup +14 )−1 , +� +P inf +14 +�−1� +(0.5,0) +0.160543 +1.668305 +[1.667178, 1.671292] +0.599411 +[0.598339, 0.599816] +y∗ +14 +D14 +C14 +[Cinf +14 , Csup +14 ] +(C14)−1 = θs +µ14(y∗ +14, D14) +� +(Csup +14 )−1 , +� +Cinf +14 +�−1� +� +5 +16, +√ +3 +16 +� +0.145957 +1.004903 +[1.003109,1.005611] +0.995121 +[0.994420, 0.996901] +Table 2: Packing and Centred measure estimates of S +Centres and radii of the balls ˚ +B(x∗ +14, d14) and B(y∗ +14, D14) of minimum and maximum µ14-densities, +estimates P14 and C14 of P s(S) and Cs(S), and bounds. The last two columns in the table are the +µ14-densities of the optimal balls (inverses of P s(S) and Cs(S)) and their bounds. +Acknowledgement 33 This work was supported by the Universidad Conmplutense de Madrid and +the Banco de Santander (PR108/20-14). +References +[1] J. M. Marstrand, The (φ, s) regular subsets of n space, Trans. Am. Math. Soc. 113 (1964) 369-392. +[2] S. P. Lalley, The packing and covering functions of some self-similar fractals, Indiana Univ. Math. +J. 37(3) (1988) 699-710. +34 + +[3] A. Schief, Separation Properties for Self-Similar Sets, Proc. Amer. Math. Soc. 122(1) (1994) +111-115. +[4] M. Mor´an, Dynamical boundary of a self-similar set, Fundam. Math. 160 (1999) 1-14. +[5] P. Mattila, Geometry of sets and measures in Euclidean Spaces (Cambridge University Press, +1995). +[6] J. E. Hutchinson, Fractals and self-similarity, Ind. J. Math. 30 (1984) 713-747. +[7] T. Bedford, A. M. Fisher and M. 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Mor´an, Rate of convergence: the packing and centered Hausdorff +measures of totally disconnected self-similar sets, Chaos, Solitons & Fractals, 98 (2017) 220-232. +37 + diff --git a/fNE_T4oBgHgl3EQf2BxV/content/tmp_files/load_file.txt b/fNE_T4oBgHgl3EQf2BxV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2a171994a92c832680717fe470f437e7f287890 --- /dev/null +++ b/fNE_T4oBgHgl3EQf2BxV/content/tmp_files/load_file.txt @@ -0,0 +1,926 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf,len=925 +page_content='Local Geometry of Self-similar Sets: Typical Balls, Tangent Measures and Asymptotic Spectra Manuel Mor´an 1,2, Marta LLorente3 and Mar´ıa Eugenia Mera1 1 Departamento de An´alisis Econ´omico y Econom´ıa Cuantitativa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Universidad Complutense de Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Cam- pus de Somosaguas, 28223 Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 2 IMI-Institute of Interdisciplinary Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Universidad Complutense de Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Plaza de Ciencias 3, 28040 Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3 Departamento de An´alisis Econ´omico: Econom´ıa Cuantitativa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Universidad Aut´onoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Emails: mmoranca@ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='es, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='llorente@uam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='es, mera@ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='es Short Title: Local Geometry of Self-similar Sets Abstract We analyse the local geometric structure of self-similar sets with open set condition through the study of the properties of a distinguished family of spherical neighbourhoods, the typical balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We quantify the complexity of the local geometry of self-similar sets, showing that there are uncountably many classes of spherical neighbourhoods that are not equivalent under similitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We show that, at a tangent level, the uniformity of the Euclidean space is recuperated in the sense that any typical ball is a tangent measure of the measure ν at ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' point, where ν is any self-similar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We characterise the spectrum of asymptotic densities of metric measures in terms of the packing and centred Hausdorff measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' As an example, we compute the spectrum of asymptotic densities of the Sierpinski gasket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Keywords: Self-Similar Sets, Hausdorff Measures, Tangent Measures, Density of Measures, Com- putability of Fractal Measures, Complexity of Topological Spaces, Sierpinski Gasket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' [2020] MSC: 28A78, 28A80, 28A75, 54A05, 54A25 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='08338v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='DS] 19 Jan 2023 1 Introduction and main results In order to gauge the vastness of the set of spherical neighbourhoods of a metric space X, it is useful to consider the quotient spaces SphX/ ≃F, where SphX is the set of spherical neighbourhoods of X and ≃F is the equivalence class associated with some group F of self-mappings of X : B ≃F B′ ⇔ B = f(B′) for B, B′ ∈ SphX and some f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The regularity of the Euclidean space Rn is made clear by the fact that if Sn is the set of similarities of Rn, then SphRn/ ≃Sn consists of a unique equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In this paper, we study the local geometry of a self-similar set E ⊂ Rn satisfying the open set condition (OSC), geometry which is described by the spherical neighbourhoods of E as a metric subspace of Rn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' by restricted balls of the form B ∩ E, where B is a Euclidean ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' For general points x, y ∈ E, if B(x, d) denotes the closed Euclidean ball centred at x and with radius d, then B(x, d) ∩ E and B(y, d) ∩ E are not equivalent by translation, and B(x, d) ∩ E and B(x, d′) ∩ E with d ̸= d′ are not homothetic-equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Using classical tools of fractal geometry, namely, the s-densities of metric measures on balls (see Definitions 21 and 22), and Marstrand’s Theorem [1], together with the results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2, we are able to prove that, for general self-similar sets with OSC, there are uncountably many equivalence classes in the quotient spaces SphE/ ≃Sn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This gives account of the complexity of the purely deterministic self-similar geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In spite of these facts, the literature has established the existence of a strong kind of regularity, on a tangent level and on average, in the neighbourhoods of a self-similar set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Recall that a self-similar set is defined as the unique compact set E ⊂ Rn that satisfies the basic equation of self-similarity E = ∪m−1 i=0 fi(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (1) for a given system Ψ = {fi}i∈M , M := {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' , m − 1} of contractive similitudes in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We shall assume that the system Ψ satisfies the OSC, meaning that there is an open set O ⊂Rn such that fi(O) ⊂ O for all i ∈ M and fi(O)∩fj(O) = ∅ for i, j ∈ M, i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We shall refer to such a set O as a feasible open set for Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We can assume, without loss of generality, as we shall from now on, that O∩E ̸= ∅ holds, also called strong open set condition (SOSC) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' [2] and [3], see also [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' If fi(E) ∩ fj(E) = ∅ for i, j ∈ M, i ̸= j, it is said that the strong separation condition (SSC) holds, in 2 which case the OSC is also fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We want to understand the local geometry of E through the study of the local behaviour of the metric s-measures, Ms⌊E:= � µ, Hs⌊E, Hs Sph⌊E, Cs⌊E, P s⌊E � (2) where s is the similarity dimension of E, dim E, that is, the unique real number s that satisfies � i∈M rs i = 1, ri being the contraction constant of the similarity fi, i ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Here β⌊E stands for a measure β restricted to the set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The measures Ms := � Hs, Hs Sph, Cs, P s� (3) are the s-dimensional Hausdorff measure, spherical Hausdorff measure, centred Hausdorff measure and packing measure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Any two measures in Ms⌊E are multiple of each other, moreover, in the case that s takes the integer value n, they are also multiple of the n-dimensional Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Each measure in Ms⌊E highlights different basic geometric properties of subsets of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' For α ∈ Ms⌊E, 0 < α(E) < ∞ holds and E is called an s-set (see [5] for further details and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2 for the definitions of the measures in Ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We shall present in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1 below the natural probability measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' For the time being, we can see it as the normalised measure, α α(E) of any other α ∈ Ms⌊E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The results in this paper about the regularity of the metric measures are also shared by the wider class of self-similar measures, MS(E) (see [6] and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1 for a definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Whereas the metric measures, Ms, convey a strong geometric meaning, self-similar measures are an essential tool in multifractal analysis of logarithmic densities, a topic that has generated a vast amount of literature for the past 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1 Scenery flow, tangent distribution and tangent measures Let ν be a Radon measure on Rn and let x be a point in the support of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We can access the local geometry of ν⌊E around x through the following zooming process: let Tx,t(y) = t(y − x), t > 0, be the homothety that maps the ball B(x, t−1) onto the unit ball D := B(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let νx,t be the probability measure on D obtained from the normalisation of the restriction to D of the image measure of ν⌊E under the homothety Tx,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' If M(D) denotes the set of Radon measures on D, then the mapping t → νx,t can be considered as a measure-valued time series that takes values in the metric space M(D) endowed 3 with the weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This time series is called scenery flow of ν around x (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The empirical distributions Φx,t(ν), t > 0, associated with such “time” series, are probability measures on M(D) (so they belong to the set M(M(D)) of Radon measures on M(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The empirical distribution Φx,t(ν) gives weight to a set A ⊂ M(D) according to the rate of the time interval [0, t] that the “empirical” data δνx,t (unit mass at νx,t) stay in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' If the empirical distribution Φx,t(ν) converges to a limit Φx(ν) as t tends to infinity, then the limiting distribution Φx(ν) is called the tangent distribution of ν at x (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Graf [9] proved that if E is a self-similar set with OSC and ν ∈ MS(E), then the limit Φx(ν) exists ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' x, and it does not depend on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Moreover, he constructed an explicit formula for the tangent distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This author gave credit for the first of these results to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Bandt in [8], and Bandt in turn gives credit for the same result to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Graf [10] (indeed a most refreshing case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Arbeiter [11], C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Bandt [10] and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Py¨or¨al¨a [12] extended these results in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The uniqueness and independence of the limit Φx(ν) from x is what M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Gavish, [13], calls, when displayed by a measure, the uniform scaling scenery property of such a measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This means that, at a tangent level and in this sense, the flow scenery recovers the uniformity of the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Remark 1 There is another way to pass to the limit at the tangent level that leads to tangent measures, a concept prior to tangent distributions introduced by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Preiss [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' There, starting from a measure ν in the set M(Rn) of Radon measures on Rn, he considers unrestricted zoomings νx,t of ν at x by homotheties Tx,t as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Instead of performing an averaging procedure, Preiss considers non-null and locally finite limits, in the vague topology of M(Rn), of sequences {cnνx,tn} with tn n→∞ → ∞ and cn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Such limit points are called tangent measures of ν at x, and Tan(ν, x) denotes the set of all such limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In our approach, following C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Bandt [8], the measures νx,tn are restricted and normalised zoomings, but the zoomings are through general expanding similitudes, rather than only homotheties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let In be the group of isometries of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We may define, in the set M(Rn), the equivalence relationship α ∼= β ⇔ there is a g ∈ In and a λ > 0 such that β = λ (g♯(α)), (4) 4 where g♯(α) is the image measure of α under g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' g♯(α)(A) = α(g−1(A)) for α-measurable A ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Thus, we identify two measures if they are equal up to an isometry (see, for instance, [10], where equivalent measures up to isometries are identified in the construction of tangent measures), and we also identify all measures in the half-straight line {λα : λ > 0, α ∈ M(Rn)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' For α ∈ M(Rn), let �α denote the equivalence class in M(Rn)/ ∼= to which α belongs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' �α = {β ∈ M(Rn) : β ∼= α} (5) Given a measure ν ∈ M(Rn), we now consider the zoomings νx,tn be of the form (gn)♯ ν⌊B(x,dt−1 n ) where gn is a similitude of contraction ratio tn, d ≤ 1, and x ∈ spt(ν) (see (23)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We define the quotient space � M(Rn) and the set of tangent equivalence classes of measures, � Tan(ν, x), by � M(Rn) = {�α : α ∈ M(Rn)} (6) � Tan(ν, x) = � �α : there is a sequence cnνx,tn w −−−−→ n→∞ α, with tn → ∞, α ̸= 0 and α ∈ M(Rn) � ,(7) where w→ denotes the weak convergence of measures on M(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' It turns out that, in the course of our research, the case in which the convergence of the magni- fications occurs in the strong topology of measures in M(Rn) is relevant (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2 below for a discussion of this result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We shall write � Tan st(ν, x) for the set of equivalence classes, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' ∼=, of such strong limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Remark 2 In our definition (7) any two zoomings, β = (gn)♯ ν⌊B(x,dt−1 n ) and β′ = (hn)♯ ν⌊B(x,dt−1 n ) of a given spherical neighbourhood B(x, dt−1 n ) are considered as valid steps in the construction of a tangent limiting measure α, where gn, hn are different similitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This can be considered as the identification of β and β′ as equivalent zoomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Notice that β′ = � g−1 n hn � ♯ β and that g−1 n hn is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Thus, the equivalence relationship (4) and the definition in (7) are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In contrast to the enlightening results obtained in [9], [10] and [11] on the uniform scaling scenery property of self-similar measures, to the best of our knowledge, the members of Tan(ν, x) for ν ∈ MS(E) remain unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Several natural issues arise here: What is the relationship between Φx(ν) and Tan(ν, x)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' What do the measures in Tan(ν, x) look like?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Do they display some uniform property?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' As for the first question, see Proposition 1 in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Below, we give a partial answer to the second and third questions for measures in MS(E) (see (10) and Theorem 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2 Typical balls A distinguished class of neighbourhoods of E, in terms of which our results are expressed, is the class of typical balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Definition 3 A ball B(x, d) is said to be typical if x ∈ E and B(x, d) ⊂ O, where O is some feasible open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We shall write B for the set of typical balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The family of typical balls is invariant under the semigroup G generated by Ψ (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 2), since, for f ∈ G, it follows from f(O) ⊂ O that f(B) ⊂ B holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Consider now the set of typical spherical B-measures, MS(B) := {α⌊B : B ∈ B, α ∈ MS(E)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (8) It is well known [6] that, for any x ∈ E, the set {f(x) : f ∈ G} is dense in E, so the balls in B are typical in the sense that, if B ∈ B, then similar copies of B are densely spread over E at small scales by the action of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' These copies are a countable set of balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' As Theorem 12 shows, the measures in MS(B) are also typical in a deeper sense since, for any f ∈ G, B ∈ B and α ∈ MS(B), the equality α⌊f(B)= pff♯(α⌊B) holds for a certain constant pf < 1 associated with f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This means that the images of typical balls are identical copies, up to the constant pf, to the original ones not only as subsets, but also from the point of view of any property expressible in terms of self-similar measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Moreover, in Theorem 12 it is shown that, for any typical ball B(x, d), for any measure α ∈ MS(E) and for all points y in a set �E with full α-measure, there is a sequence of balls {B(y, dk)} with dk → 0, a sequence {fk} of similitudes in G and constants p−1 fk → ∞, such that p−1 fk � f −1 k � ♯ � α⌊(B(y,dk) � st −−−−→ k→∞ α⌊B(x,d), (9) where the convergence in (9) is in the sense of the strong topology of Radon measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Theorem 12 also states that, for all x ∈ �E and α ∈ MS(B), � MS(B) ⊂ � Tan st(α, x) (10) holds, where � MS(B) = {�α : α ∈ MS(B)} , (11) (see (5) for the notation �α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 6 The results above imply that the use of general zooming similitudes, grants the strong convergence of the zoomings to the tangent measures, whereas in the ordinary spaces of tangent measures, where only homotheties are allowed, convergence can only be ensured in a weak topology sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1 below for further details on identifications and topologies of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Remark 4 Putting the results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3, described in the first paragraph of this section, together with (10), we see that the self-similar scenery at x ∈ E depends on x on large scales, meaning that there is a broad variety of balls B(x, d) for varying x that, moreover, also vary with d for fixed x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Additionally, on a tangent scale, for each α ∈ MS(B) and each x ∈ �E, each typical class of measures in � MS(B) is a feasible outcome of the zooming process of α at x, so there is a wide variety of limiting measures in � Tan st(α, x), x ∈ �E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The uniformity of the self-similar setting emerges here in the fact that the inclusion � MS(B) ⊂ � Tan st(α, x) stands true for any x ∈ �E, so all the points in �E share the set � MS(B) of tangent measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3 Spectrum of local densities of a self-similar set: the Sierpinski gasket case The relevance of the typical balls is stressed by the connection between typical balls and the spectrum of densities, which in turn determines some basic geometric features of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let α ∈ M(Rn), 0 ≤ s ≤ ∞ and x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The upper and lower spherical s-densities of α at x are defined, respectively, by θ s α(x) = lim sup d→0 θs α(x, d), (12) θs α(x) = lim inf d→0 θs α(x, d), (13) where the s-density of the ball B(x, d), θs α(x, d), is given by θs α(x, d) = α(B(x, d)) (2d)s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Here the zooming process is summarised in only two scalars, (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' If θ s α(x) = θs α(x), then we write θs α(x) for the common value and call it s-density of α at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Densities and their connections to 7 their underlying measures have been studied extensively in the context of geometric measure theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' A major contribution from Marstrand (Marstrand’s theorem, [1]) asserts that, in the Euclidean setting, if the s-density θs α(x) exists in a set with a finite and positive α-measure with α ∈ M(Rn), then s is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The widest class of subsets of Euclidean spaces that are s-sets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' sets with a finite and positive α-measure) is either the class of self-similar sets that satisfy the OSC, with s being their similarity dimension (see (2)), or some variations of it, like the Mauldin and Williams graph-directed construc- tions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' [16], and controlled Moran constructions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Here, we are interested in the case in which the similarity dimension s is not an integer and, by Marstrand’s theorem described above, θs α(x) and θ s α(x) do not coincide in subsets with a positive α-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This leads to the following definition of asymptotic spectrum of densities of a given measure α at a point and, more in general, in a subset of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Definition 5 Given a subset A ⊂ Rn, we define the asymptotic spectrum of (non-logarithmic) spher- ical s-densities, Spec(α, A), for a locally finite measure α by Spec(α, A) = � lim k→∞ θs α(x, dk) : x ∈ A and lim k→∞ dk = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (14) We insert the non-logarithmic epithet above because there is a ample literature on the so-called multifractal spectrum of logarithmic spherical densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This literature also focuses on the limiting behaviour of α on small balls, but the interest is in the upper and lower limits of the quotients log α(B(x,d)) log d when d → 0 (for x ∈ E) and, in particular, in the fractal dimension of both the (α- null) sets where these limits exist and take particular values [18] and the sets of divergence points (see [19], [20], [21]) where the limits do not coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Much less is known about the behaviour of non-logarithmic densities, and the research in this paper can be considered a preliminary step in that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In particular, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3, Theorem 14, we present the knowledge to date about the spectrum of non-logarithmic α-densities, α ∈ Ms⌊E, of self-similar sets E that satisfy the OSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In particular, we show that Spec(α, x) is contained in the closed interval � α(E) P s(E), α(E) Cs(E) � for all x in a subset �E of E with a full α-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' There arises a natural class of self-similar sets with nice properties, the α-exact self-similar sets (see notation in 15), which are sets for which the endpoints of such interval belong 8 to Spec(α, x), x ∈ �E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Whereas the results for general self-similar sets with OSC presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3 are of a qualitative nature, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 4 we shall focus on our prime example of α-exact self-similar set, the Sierpinski gasket S, and exploit its regularity to accurately approximate the range of values taken by its spectrum, which is the content of Theorem 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Moreover, we give a full characterisation of the spectrum of all the points in S, which is given by the union of two closed intervals of positive length, namely, Spec(α, S) = � α(S)θs µ(z0), α(S)θ s µ(z0) � ∪ � α(S) P s(S), α(S) Cs(S) � , α ∈ Ms⌊S, where z0 := (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Using the numerical approximations of θs µ(z0), θ s µ(z0) obtained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 4 and of P s(S) and Cs(S) obtained in [22] and [23], we can also show that these two intervals are disjointed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In the case that α ∈ {µ, P s⌊S, Cs⌊S}, we have numerical estimations of these two disjointed intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The Sierpinski gasket is, as far as we know, the first connected self-similar with non-integer dimension for which the entire spectrum has been computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 2 Notation and preliminaries The self-similar set E given in (1) can be parametrised as E = {π(i) : i ∈ Σ} with parameter space Σ := M ∞ and geometric projection mapping π : Σ → E given by π(i) = ∩∞ k=1fi(k)E, where i(k) denotes the curtailment i1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' ik ∈ M k of i = i1i2 · · · ∈ Σ and fi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='ik = fi1 ◦ fi2 ◦ fi3 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='fik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We adopt the convention M 0 = ∅ and write M ∗ = ∪∞ k=0M k for the set of words of finite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Expressed in this notation, the semigroup generated by Ψ can be written as G = {fi : i ∈ M ∗} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' For any i ∈ M ∗, we denote by Ei the cylinder sets fi(E), and if i ∈ M 0, then fi(E) := E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The sets Ei are called k-cylinders if i ∈ M k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We also shorten the notation fi(A) to Ai for a general set A ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We write ri := ri1ri2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' rik for the contraction ratio of the similitude fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Moreover, σ : Σ → Σ shall stand for the shift map given by σ(i1i2i3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' ) = i2i3i4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The code shift can be projected (as a correspondence) onto E, yielding the geometric shift T (x) := π ◦ σ ◦ π−1(x), (15) x ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The shift orbit of x ∈ E is given by � T k(x) : k ∈ N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Remark 6 Observe that x ∈ T k(A) if and only if fi(x) ∈ A for some i ∈ M k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1 Self-similar measures Let P(Rn) be the space of compactly supported probability Borel measures on Rn, let p = (p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=', pm−1) ∈ Rm be a probability vector and let Mp: P(Rn) → P(Rn) be the Markov operator defined by Mp(α) = m−1 � i=0 piα ◦ f −1 i , α ∈ P(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The unique fixed point of the contractive operator Mp is called the self-similar measure µp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' that is, µp = � i∈M piµp ◦ f −1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (16) Moreover, Mk p(α) = � i∈M k piα ◦ f −1 i w −→ k→∞ µp (17) for any α ∈ P(Rn), where, for i ∈ M k, pi := pi1 · · · pik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Here Mk p is the k-th iterate of Mp (see [6] and [24] for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Set MS(E) := � µp : m−1 � i=0 pi = 1, pi > 0, i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=', m − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (18) For ps := (rs 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=', rs m−1), where s is the similarity dimension of E (recall that ri is the contraction constant of the similarity fi, i ∈ M), the measure µps is called the natural probability measure on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Furthermore, if α ∈ Ms⌊E (see (2) for notation), then µ := µps = α α(E) (19) (see [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Notice that, whereas the measures in Ms (see (3) for notation) convey an strong geometrical meaning, the measures µp in MS(E) do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' They are concentrated in dense subsets of E, Ep, whose dimension is given by dim(Ep) = sp := �m−1 i=0 pi log pi �m−1 i=0 pi log ri , but the measure µp is singular w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' the measures Hsp and P sp (see [26] and [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2 Metric measures We now briefly recall metric measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' They are the classical tools for analysing the geometric prop- erties of subsets of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 10 The Hausdorff centred measure, Cs(A), of a subset A ⊂ Rn, was defined by Saint Raymond and Tricot [28] in a two-step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' First, the premeasure Cs 0(A) is defined for any s > 0 by Cs 0(A) = lim δ→0 inf � ∞ � i=1 (2di)s : 2di ≤ δ, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' � , (20) where the infimum is taken over all coverings, {B(xi, di)}i∈N+ , of A by closed balls B(xi, di) centred at points xi ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, the centred Hausdorff s-dimensional measure is defined by Cs(A) = sup {Cs 0(F) : F ⊂ A, F closed} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The second step in the definition of Cs(A) is due to the lack of monotonicity of Cs 0 (see [29] and [30, Example 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' However, in [30], it was shown that the second step can be omitted when restricting oneself to self-similar sets with OSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' With regard to metric measures based on packings, the standard packing measure P s (see [28] and [31]) is also defined in a two-step process, P s 0 (A) = lim δ→0 sup � ∞ � i=1 (2di)s : 2di ≤ δ, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' � , where the supremum is taken over all packings {B(xi, di)}i∈N+ , with xi ∈ A for all i, and B(xi, di) ∩ B(xj, dj) = ∅ for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, P s(A) = inf � ∞ � i=1 P s 0 (Fi) � , where the infimum is taken over all coverings {Fi}i∈N+ of A by closed sets Fi (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In [33], it was proved that if A is a compact set with P s 0 (A) < ∞, then P s(A) = P s 0 (A), so this simplification applies to any compact subset of a self-similar set with OSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The spherical s-dimensional Hausdorff measure, Hs Sph(A), is obtained by removing in (20) the requirement that the balls are centred at points of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The classical s-dimensional Hausdorff measure, Hs(A), results if coverings of A by arbitrary subsets, {Ui}i∈ N+ , are considered and 2di is replaced in (20) with the diameter of Ui, |Ui| (see [34] and [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' No second step is required for these last two measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The packing and the centred Hausdorff measures have a much simpler expression when dealing with self-similar sets E that satisfy the OSC as the browse for optimal packings or coverings can be reduced to the search for optimal density balls within the class of typical balls, B (see Definition 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 11 In particular, for any self-similar E that satisfies the OSC and with similarity dimension s, it is known (see [36]) that P s(E) = � inf � θs µ(x, d) : B(x, d) ∈ B ��−1 , (21) and, Lemma (13) of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2 implies that Cs(E) = � sup � θs µ(x, d) : B(x, d) ∈ B ��−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (22) 3 Local structure and typical balls Now we shall study the local structure of a self-similar set E that satisfies the OSC for a feasible open set O through the study of the scenery flow of α ∈ MS(E) at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' x ∈ E, and through the characterisation of the spectrum of the spherical s-densities of measures in Ms⌊E (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2), a limiting set that helps to summarise the structure in the neighbourhood of a point (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1 Scenery flow and tangent measures We start by giving details on the construction of � Tan(ν, x) for ν ∈ MS(E) and x ∈ E (see 18 for notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1 Tangent measures, identifications and topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Recall that the construction of the sets � Tan(ν, x) and � Tan st(ν, x) employs the identification, in the set M(Rn), of those measures that are equal up to isometries or mutual multiples (see (4), (5), (6) and (7) for notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We now examine the construction of the spaces of equivalence classes of tangent measures above in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' For ν ∈ M(Rn) and x ∈ spt(ν), we first consider sequences {cnνx,tn}∞ n=0, where for every n ∈ N, cn > 0, νx,tn := 1 ν(B(x, dt−1 n )) (gx,tn)♯ ν⌊B(x,dt−1 n ), (23) d ≤ 1 and gx,tn is some similarity with expanding ratio tn that maps the ball B(x, t−1 n ) onto the ball B(zn, 1), with zn = gx,tn(x), so each νx,tn is a probability measure supported on B(zn, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, � Tan(ν, x) and � Tan st(ν, x) consist of the equivalence classes of non-null weak and strong limits, re- spectively, as tn → ∞, of such sequences {cnνx,tn}∞ n=0 (see (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Lemma 8 shows that the elements 12 in � Tan(ν, x) and � Tan st(ν, x) do not depend on either the sequence of constants cn or the particu- lar elements chosen in the equivalence classes � νx,tn as long as the convergence of these elements is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Remark 7 The unit ball D does not play any essential role in our definition of tangent measures in the quotient space � M(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In the opposite direction (second approach) we may, in a more akin way to the classical approach, require the similarities gx,tn to map B(x, t−1 n ) onto B(0, 1), and then define TanD(ν, x) and Tanst D(ν, x) as weak and strong limits in M(D), respectively, of sequences of such measures νx,tn, and � Tan(ν, x), � Tan st(ν, x) as the sets of equivalence classes of measures in TanD(ν, x) and Tanst D(ν, x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This second method gives spaces of tangent equivalence classes which are particular cases of these in our primary approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Are these equivalent methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In order to answer this question, let a sequence {cnνx,tn}∞ n=0, as in (23), converge to a non-null Radon measure α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' By Lemma 8 we may assume cn = 1 for all n ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Since the measures νx,tn are supported on balls B(zn, d) with d ≤ 1 (see Theorem 12 (i)), the limiting measure α must also be supported on a ball B(z, d) with zn −→ n→∞ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Each measure ν′ x,tn = (τzn)♯ νx,tn, where τzn(y) = y − zn, is equivalent by translation to νx,tn, and ν′ x,tn is supported on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' It is easy to see that νx,tn w −−−−→ n→∞ α implies that ν′ x,tn w −−−−→ n→∞ α′ = (τz)♯ α, so α′ is equivalent to α and supported on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Thus, the second method gives the same space � Tan(ν, x) than our primary method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' But νx,tn st −−−−→ n→∞ α does not imply that ν′ x,tn st −−−−→ n→∞ α′, so the second method does not produce the same space � Tan st(ν, x) than our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Observe that, if we let ν′ x,tn = (τz)♯ νx,tn, then νx,tn st −−−−→ n→∞ α does imply ν′ x,tn st −−−−→ n→∞ α′ = (τz)♯ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' But now the measure ν′ x,tn is supported on the ball B(zn −z, d) rather than on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This observation is useful because D and all the balls B(zn − z, d) are contained in some ball B(0, R) for R large enough (notice that zn is a convergent sequence of points), so the convergence νx,tn −→ n→∞ α (weak or strong) occurs in M(B(0, R)), and we can see that, if we consider vague convergence of measures, we do not obtain anything new, since in the Polish space B(0, R) both convergences are equivalent ( [15], Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Lemma 8 (i) The sequences {cn}∞ n=0 in the construction of � Tan(ν, x) and � Tan st(ν, x) can be taken to be cn = 1, n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 13 (ii) Let ν ∈ M(Rn), x ∈ spt(ν) and α ∈ Tan(ν, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let {tn}∞ n=0 ↑ ∞ be such that {νx,tn}∞ n=0 w −−−−→ n→∞ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Assume also that there is a sequence {fn}∞ n=0 in the set of isometries In such that {(fn)♯ νx,tn}∞ n=0 w −−−−→ n→∞ α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, there is f ∈ In such that (f)♯α = α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The same is true if the convergence holds in the topology of the strong convergence in M(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (i) By definition, a weak limiting measure α as in (7) is a non-null measure in M(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Therefore, the sequence of constants {cn} must be bounded above and below by two positive and finite constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We can choose a subsequence {cnk}∞ k=0 that converges to a constant c, and then the sequence cνx,tn must converge to the weak limit α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This gives νx,tn w −−−−→ n→∞ c−1α, which belongs to the same equivalence class in � Tan(ν, x) as α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' On the other hand, the non-null weak limits in M(Rn) of sequences {νx,tn}∞ n=0 are particular cases of those of sequences {cnνx,tn} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This completes the proof of part (i) for weak limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The argument also holds true for strong limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (ii) For any n ∈ N+, we can write fn(·) = gn(·) + an, where gn is an orthogonal map and an ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Recall that νx,tn is supported on B(zn, d), so (gn + an)♯ (νx,tn) is supported on an + B(zn, d) ( [5], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='18), with zn −→ n→∞ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This means that, if ν′ x,tn := (fn)♯ νx,tn converges, in the weak topology of M(Rn), to some non-null measure α′ in M(Rn), the sequence an must be bounded, and then the sequence {fn}∞ n=0 is also bounded in the supremum norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Therefore, there is a convergent subsequence, {fnk}∞ k=0, of {fn}∞ n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let f := limk→∞ fnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Since the sequence {(fnk)♯ (νx,tnk )}∞ k=0 converges to α′, we have that α′ = lim k→∞ fnk♯(νx,tnk ) = f♯α, (24) which proves that α′ ∼= α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The second equality in (24) holds true because, for any ϕ in the space C0(Rn) of continuous, compactly supported functions on Rn and for any ε > 0, there is k0 > 0 such that for k ≥ k0, we have ∥ϕ ◦ fnk − ϕ ◦ f∥ ≤ ε 2, ���� � ϕ ◦ f d(νx,tnk ) − � ϕ ◦ f dα ���� ≤ ε 2, 14 and then ���� � ϕ d � fnk ♯(νx,tnk ) � − � ϕ d(f♯α) ���� ≤ � ∥ϕ ◦ fnk − ϕ ◦ f ∥ d(νx,tnk ) + ���� � ϕ ◦ f d(νx,tnk ) − � ϕ ◦ f dα ���� ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' If {ν′ x,tn}∞ n=0 converges to α′ in the strong topology of M(D), then it also converges in the weak topology and the argument above applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2 Scaling properties of typical balls and scenery flow We need some preliminary lemma and the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Definition 9 Given a measure α ∈ Ms⌊E, two Euclidean balls B(x, d) and B(x′, d′) are said to be α-density equivalent if θs α(x, d) = θs α(x′, d′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We start with two elementary scaling properties of typical balls for measures in MS(E) and in Ms⌊E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Lemma 10 Let E be a self-similar set generated by the system Ψ = {fi}i∈M of similarities of Rn, with M = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' , m − 1} , and similarity dimension s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let O be a feasible open set (for Ψ) and let i ∈ M ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then (i) µp(fi(A)) = piµp(A), for µp ∈ MS(E) and µp-measurable A ⊂ O, (25) (ii) µp(f −1 i (C)) = p−1 i µp(C) for µp ∈ MS(E) and µp-measurable C ⊂ Oi, (26) (iii) B(fi(x), rid) is α-density equivalent to B(x, d) for α ∈ Ms⌊E and B(x, d) ⊂ O, (27) (iv) f −1 i (B(fi(x), rid)) is α-density equivalent to B(x, d) for α ∈ Ms⌊E and B(x, d) ⊂ Oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (28) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The proof of (25) is trivial from (16) if E satisfies SSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' If SSC does not hold, then µp(f −1 j (fi(A))) ≤ µp(∂O) = 0 for j ̸= i, 15 because A ⊂ O and, hence, f −1 j (fi(A)) ∩ E ⊂ ∂O, which is known to be a µp- null set (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' [27]), so (25) also follows from (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' If we set A = f −1 i (C) in (25), we obtain (26) (see also [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' By (19), we can apply (25) and (26) to any measure α ∈ Ms⌊E, which easily gives (27) and (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Before stating the main theorem of this section, we will see the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Lemma 11 (i) Let g, f : Rn → Rn, α ∈ M(Rn), λ > 0 and A ⊂ Rn be an α-measurable subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, the following equalities hold true: λ (g)♯ (α) = (g)♯ (λα), (f ◦ g)♯α = f♯(g)♯(α), and (g♯α) ⌊A= g♯(α⌊g−1(A)) (ii) Let α be a measure on M(Rn), g : Rn → Rn a bijective mapping and β := g♯ (α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, α = (g−1)♯β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (iii) If {αk}k∈N is a sequence of measures on M(Rn) and (g)♯ (αk) st −−−−→ k→∞ β, then αk st −−−−→ k→∞ � g−1� ♯ (β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (iv) Let B(xn, d) := Bn be a sequence of closed balls that converges in the Hausdorff metric to a closed ball B(x, d) := B, and let α ∈ M(B) with α(∂B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then α⌊Bn:= αn st −−−−→ n→∞ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Parts (i)-(iii) easily follows from the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Recall that αn st −−−−→ n→∞ α means that αn(A) −→ n→∞ α(A) for any Borel set A ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let α ∈ M(B) and let K be any compact set contained in the interior U of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The distance d(K, ∂B) = min {∥x − y∥ : x ∈ K, y ∈ ∂B} must be a quantity ε > 0 and then K ⊂ B(x, d − ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The convergence of Bn to B implies that there is an n0 ∈ N+ such that, for n > n0, ∥x − xn∥ ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, if z ∈ K, ∥z − xn∥ ≤ ∥z − x∥ + ∥x − xn∥ ≤ d, which shows that K ⊂ B ∩ Bn for n > n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, for such values of n, we have αn (K) = α(Bn ∩ K) = α(K) We now prove that αn st −−−−→ n→∞ α also holds in the σ-field B(B) of Borel subsets of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let A := � A ⊂ B : A is α-measurable and lim n→∞ αn(A) = α(A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 16 (Notice that any α-measurable set is also αn-measurable for all n ∈ N+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' It is easy to check that B ∈ A, that B − A := Ac ∈ A if A ∈ A, and that A is closed under a finite union of its members or, in short, that A is a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let Fk be a sequence of members of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In order to show that ∪k∈N+Fk ∈ A, we first write ∪k∈N+Fk = ∪k∈N+Gk, where Gk = ∪k i=1Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This shows that ∪k∈N+Fk can be expressed as a countable union of the increasing sequence Gk of members of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Furthermore, ∪k∈N+Fk = ∪k∈N+Hk, where Hk = (Gk − Gk−1) with G0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Now, each Hk ∈ A and Hk ∩ Hk′ = ∅ for k ̸= k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, using that each αn is a measure, we have lim n→∞ αn � � k∈N+ Hk � = � k∈N+ lim n→∞ αn (Hk) = � k∈N+ α (Hk) = α � � k∈N+ Hk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This completes the proof that A is a σ-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Notice that any closed set K ⊂ B can be written as the union of the α and αn-null set K ∩ ∂B and of the set K − ∂B, which belongs to A as a countable union of compact sets K ∩ B(x, d − n−1) ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Thus, the class K of closed subsets of B is contained in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We know that the σ-fields generated by K and by A satisfy B(B) = σ(K) ⊂ σ(A) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This gives the strong convergence of αn to α on B(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We can now go to the scenery flow of measures in MS(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Theorem 12 Let E be a self-similar set generated by the system Ψ = {fi}i∈M of similarities on Rn, with M = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' , m − 1} and similarity dimension s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let O be a feasible open set (for Ψ) and µp ∈ MS(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, for any µp-measurable set B ⊂ O and i ∈ M ∗, the following statements hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (i) µp⌊Bi= pi (fi)♯ (µp⌊B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (ii) µp⌊B= p−1 i � f −1 i � ♯ (µp⌊Bi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (29) (iii) There is a subset �E ⊂ E with µp( �E) = 1 such that, if x ∈ E and B(x, d) ⊂ O, then for any y ∈ �E, there is a sequence {ij}j∈N+ with ij ∈ M ∗ and a sequence of balls � B(y, drij) � j∈N+ such that p−1 ij � f −1 ij � ♯ � µp⌊B(y,drij ) � st −−−→ j→∞ µp⌊B(x,d) (iv) For any x ∈ �E, � MS(B) ⊂ � Tan st(µp, x), 17 where MS(B) is defined in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In order to show (i), let µp ∈ MS(E), i ∈ M ∗ and let B ⊂ O and A ⊂ Rn be µp-measurable sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, (pi(fi)♯ (µp⌊B)) (A) = pi (µp⌊B) � f −1 i (A � ) = piµp(f −1 i (A ∩ Bi)) = µp⌊Bi(A), where the third equality follows from (26) and (i) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Analogously, (ii) follows from (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Now, let �E = � y ∈ E : � T k(y) : k ∈ N+� is dense in E � (30) (see (15) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 2 the definition of T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' It is well known (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' [37]) that the set �E has a full µp-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let x ∈ E, B(x, d) ⊂ O, y ∈ �E and {xj}j∈N+ such that limj→∞ xj = x (in the Euclidean metric) with xj ∈ T kj(y) for every j ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We may also assume that B(xj, d) ⊂ O for every j ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We shorten B(xj, d) to Bj and B(x, d) to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Since limj→∞ xj = x, it follows that � Bj� j∈N converges to B in the Hausdorff metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Also, µp(∂B) = 0 because µp ∈ MS(E) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, Lemma 11 (iv) implies that µp⌊Bj st −−−→ j→∞ µp⌊B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (31) Now, notice that, since xj ∈ T kj(y) for each j ∈ N, there is ij ∈ M kj such that fij(xj) = y (see Remark 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, f −1 ij (B(y, drij)) = Bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' By (29) applied to Bj and ij ∈ M ∗, we see that µp⌊Bj= p−1 ij � f −1 ij � ♯ � µp⌊B(y,drij ) � , (32) which concludes the proof of (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Observe that, in the terminology of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1, the right hand term in (32) is, ctjνy,tj for ν = µp, tj = r−1 ij and ctj = p−1 ij µp(B(y, drij)) (recall that νy,tj was a normalised blowup and notice also that we may assume, rescaling E if necessary, that all typical balls have a radius d ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' So, (31) and (7) give � µp⌊B ∈ � Tan st(µp, x) and part (iv) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2 Asymptotic spectra and measure-exact self-similar sets We shall write Im(θs α, B) to designate the set Im(θs α, B) := {θs α(x, d) : B(x, d) ∈ B} 18 (see notation in Definition 3), which plays a relevant role in the geometric analysis of E (see (21) and the lemma below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Lemma 13 Let E be a self-similar set generated by the system of similarities of Rn, Ψ = {fi}i∈M , with M = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' , m − 1} , and similarity dimension s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' If E satisfies the OSC, then (i) Cs(E) = � sup � θs µ(x, d) : B(x, d) ∈ BO ��−1 , where BO := {B(x, d) ∈ B : B(x, d) ⊂ O} and O is any feasible open set for Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (ii) Cs(E) = � sup Im(θs µ, B) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' It is known that for a general self-similar set that satisfies the OSC (see [36] and [30]), Cs(E) = � sup{θs µ(x, d) : x ∈ E and d > 0} �−1 (33) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let O be any feasible open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, it is enough to show that sup (x,d)∈E×R+ θs µ(x, d) ≤ sup B(x,d)∈BO θs µ(x, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Should this not be the case, there would exist (x0, d0) ∈ E × R+ such that B(x0, d0) /∈ BO and θs µ(x0, d0) > sup B(x,d)∈BO θs µ(x, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In order to show that this contradicts (33), take x∗ ∈ E∩O such that there is i ∈ M ∗ with fi(x∗) = x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let ρ1 := min {∥x∗ − z∥ : z ∈ ∂O} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Observe that, if we take ρ2 > 0 so that B(x0, d0) ⊂ B(x∗, ρ2) and k ∈ N+, satisfying that rk i ρ2 < ρ1, then f k i (B(x0, d0)) ⊂ f k i (B(x∗, ρ2)) = B(x∗, rk i ρ2) ⊂ O, which, using that f k i (B(x0, d0) ∩ S) ⊂ f k i (B(x0, d0)) ∩ S, raises the contradiction θs µ(x0, d0) ≤ r−ks i µ(f k i (B(x0, d0))) (2d0)s = µ(B(f k i (x0), rk i d0)) (2d0rk i )s ≤ sup B(x,d)∈BO θs µ(x, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Part (ii) is trivial from (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In the next theorem, we shall establish the relationships between the pointwise and global spectra, the set Im(θs α, B) and its extreme values α(E) (P s(E))−1 and α(E) (Cs(E))−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 19 Theorem 14 Let E ⊂ Rn be a self-similar set that satisfies the SOSC with feasible open set O and similarity dimension s, and let α ∈ Ms⌊E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, the following statements hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (i) For x ∈ E, it holds that Spec(α, x) = � θs α(x), θ s α(x) � (see (13) and (12) for notation) Spec(α, E) ⊂ [κ1, κ2] with 0 < κ1 ≤ κ2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (ii) There is a subset �E ⊂ E with µ( �E) = 1 such that, for any y ∈ �E, Spec(α, y) = Spec(α, �E) = Spec(α, O ∩ E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (iii) � α(E) P s(E), α(E) Cs(E) � ⊂ Im(θs α, B) ⊂ Spec(α, O∩E) ⊂ � α(E) P s(E), α(E) Cs(E) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' That θs α(x) and θ s α(x) belong to and are the extreme values of Spec(α, x) follows from the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' That all the intermediate values in between also belong to Spec(α, x) is a consequence of the continuousness of θs α(x, d), with respect to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This last property follows from the fact that the α-measure of the boundary of Euclidean balls is always null [35] for any measure α ∈ MS(E), which proves the first assertion of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The second assertion is well known [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In order to prove (ii), let �E be the full µ-measure subset of points of E that have a dense geometric shift orbit in E (see (30)) and let y ∈ �E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The inclusions Spec(α, y) ⊂ Spec(α, �E) ⊂ Spec(α, O ∩ E) are trivial as �E ⊂ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This follows from the fact that, if y /∈ O, then T (y) ∩ O = ∅ because fi(O) ⊂ O for any i ∈ M, and repeating the same argument, we see that T k(y) could not be dense in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The corresponding equalities would follow if we prove Spec(α, O ∩ E) ⊂ Spec(α, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This holds true because, if z = limk→∞ θs α(x, dk) for x ∈ O∩E and dk −→ k→∞ 0, since B(x, dk) ∈ B for any sufficiently large k, we can apply Theorem 12 (iii) to see that, for such values of k, θs α(x, dk) ∈ Spec(α, y) and, hence, limk→∞ θs α(x, dk) ∈ Spec(α, y) easily follows from (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This ends the proof of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Finally, the first inclusion in (iii) for α = µ follows from the continuousness of the function θs µ(x, d) on Rn × R+ since 1 P s(E) ≤ θs µ(x, d) ≤ 1 Cs(E) 20 holds if B(x, d) ∈ B as a straightforward consequence of (21) and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The arguments given in the proof of (ii) applied to µ show that, if B(x, d) ∈ B, then θs µ(x, d) ∈ Spec(µ, O ∩ E), which gives the next inclusion in (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The last inclusion follows from the observation that Spec(µ, O∩E) consists of limiting values of sequences with terms in Im(θs µ, B), whose extreme values are 1 P s(E) and 1 Cs(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Using (19), we get that θs α(x, d) = α(E)θs µ(x, d), and (iii) follows for any α ∈ Ms⌊E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Of note is the case in which the extreme values of θs α(x, d) are attained on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In this case, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Corollary 15 Let α ∈ {µ, P s⌊E, Cs⌊E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Under the hypotheses of Theorem 14, if there are two balls B(x1, d1) and B(x2, d2), both in B, such that θs µ(x1, d1) = inf � θs µ(x, d) : B(x, d) ∈ B � (34) and θs µ(x2, d2) = sup � θs µ(x, d) : B(x, d) ∈ B � , (35) the inclusions in Theorem 14 (iii) can be replaced with equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The first inclusion in Theorem 14 (iii), together with (21), (22), (34) and (35), implies that Im(θs µ, B) = � 1 P s(E), 1 Cs(E) � , which, in turn, gives that Im(θs α, B) = Spec(α, O∩E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Corollary 15 motivates the introduction of the class of α-exact self-similar sets with special prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Definition 16 We say that the self-similar set E is α-exact if there exists B ∈ Cα such that µ(B) |B|s = sup �µ(B) |B|s : B ∈ Cα � if α ∈ {Cs⌊E, Hs⌊E, HSph⌊E} , and µ(B) |B|s = inf �µ(B) |B|s : B ∈ Cα � , if α = P s⌊E, where Cα is what we call “the relevant class of sets” for the measure α, which is defined as 21 Cα := B if α ∈ {P s⌊E, Cs⌊E} , CHs⌊E := {B ⊂ Rn : B is a convex set} and CHs Sph⌊E := {B ⊂ Rn : B is a closed ball}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' One nice property that α-exact self-similar sets have is that they possess optimal coverings or packings, that is, almost-coverings (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' coverings for α-almost all points in E) or packings whose s-volume gives the exact value of the corresponding α-measure, whilst if α-exactness is not fulfilled, we can only hope to find coverings or packings with s-volume arbitrarily close to the corresponding α-measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Example 17 Self-similar sets E with the strong separation condition are an example of α-exact self- similar sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' See [38] for α ∈ � P s⌊E, Hs⌊E, Hs Sph⌊E � and [30] for α = Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Example 18 The Sierpinski gasket S is an example of a set where the strong separation condition does not hold, and that is a P s⌊S-exact (see [22]) and Cs⌊S-exact (see [23]) set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In [39], it is shown a class of self-similar sets E with OSC in the line whose members can be non- Hs⌊E-exact (and, consequently, non-Hs Sph⌊E-exact since these two measures coincide in the line), and the authors find conditions under which they are Hs⌊E-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Example 19 Self-similar sets E with OSC in the line, with similarity dimension s, and that admit an open interval as a feasible open set, are an example of P s⌊E-exact self-similar sets [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3 Complexity of the local structure of self-similar sets We now show how these results allow us to explore the complexity of the local geometric structure of self-similar sets that satisfy the OSC condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' First, we need to properly define the equivalence classes of restricted balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Notice that different Euclidean balls, even if they share the centre, can produce the same restricted balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This motivates the following definitions that are valid for general subsets of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Definition 20 Given a subset A ⊂ Rn, the spherical diameter of A is defined by |A|Sph = inf {2d : A = A ∩ B(x, d) for some x ∈ A} 22 Definition 21 Given a subset A ⊂ Rn, we say that the restricted ball B(x, d) ∩ A is proper and write B(x, d) ∩ A ∈ P(A) if x ∈ A and 2d = |B(x, d) ∩ A|Sph .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Definition 22 Given a measure α on Rn and an α-measurable s-set A ⊂ Rn, we define the α-spherical s-density of A by θs Sph(α)(A) = α(A) � |A|Sph �s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Definition 23 Given a subset A ⊂ Rn and two restricted balls B(x, d)∩A, B(x′, d′)∩A both in P(A), we say that they are similarity-equivalent and write B(x, d) ∩ A ≃Sn B(x′, d′) ∩ A if there is an f ∈ Sn such that B(x′, d′) ∩ A = f(B(x, d) ∩ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Lemma 24 Let A ⊂ Rn and B(x, d) ∩ A ∈ P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (i) If f ∈ Sn has similarity constant rf, then f(B(x, d)) ∩ f(A) ∈ P(f(A)) and |f(B(x, d)) ∩ f(A)|p = rfd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (ii) Let α ∈ Ms and let A be an α-measurable s-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' If B(x, d) ∩ A ≃Sn B(x′, d′) ∩ A, then θs Sph(α)(B(x, d) ∩ A) = θs Sph(α)(B(x′, d′) ∩ A) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let A ⊂ Rn, B(x, d) ∩ A ∈ P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In order to show (i), assume that f(B(x, d)) ∩ f(A) is not proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, there is a ball B(y, ρ) such that B(y, ρ) ∩ f(A) = f(B(x, d)) ∩ f(A) with y ∈ f(A) and ρ < rfd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then B(f −1(y), r−1 f ρ) ∩ A = B(x, d) ∩ A with f −1(y) ∈ A and r−1 f ρ < d, in contradiction with |B(x, d) ∩ A|Sph = 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Therefore, f(B(x, d)) ∩ f(A) ∈ P(f(A)) and |f(B(x, d)) ∩ f(A)|Sph = 2rfd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Part (ii) is now trivial since α ∈ Ms and, hence, α(B(x′, d′) ∩ A) = α(f(B(x, d) ∩ A)) = rs fα(B(x, d) ∩ A) and, by (i), � |B(x′, d′) ∩ A|Sph �s = � |f(B(x, d) ∩ f(A))|Sph �s = rs f(2d)s = rs f � |B(x, d) ∩ A|Sph �s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 23 Now we can proceed to state our result for the complexity of the local geometry of self-similar sets with OSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Corollary 25 Under the assumptions of Theorem 14, assume that s is a non-integer real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, there is an uncountable number of equivalence classes in the quotient space SphE/ ≃Sn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' By Lemma 24 (ii), we know that all restricted balls in an equivalence class of SphE/ ≃Sn share the same µ-spherical s-density, which allows us to naturally define a mapping θs µ : SphE/ ≃Sn→ Im(θs µ, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This implies that the inverse � θs µ �−1 : Im(θs µ, B)→SphE/ ≃Sn of such mapping is an injective correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Using Marstrand’s Theorem, parts (ii) and (iii) of Theorem 14 and that µ( �E) = 1 > 0, it follows that either Cs(E) < P s(E) or s is an integer (notice that from the definitions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2 it is easy to see that Cs(E) ≤ P s(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This, together with Theorem 14 (iii), means that Im(θs µ, B) contains an interval with uncountably many points and the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 4 The spectrum of the Sierpinski gasket In this section, we shall apply the results obtained in Theorem 14 to fully characterise the asymptotic spectra of the Sierpinski gasket S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Recall that the Sierpinski gasket or Sierpinski triangle is a special case of a self-similar set generated by a system Ψ = {f0,f1,f2} of three contracting similitudes of the plane, with contraction ratios ri := 1/2, i ∈ M, given by f0(x, y) = 1 2(x, y), f1(x, y) = 1 2(x, y) + (1 2, 0) and f2(x, y) = 1 2(x, y) + 1 2(1 2, √ 3 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (36) We shall denote by zi the fixed point of each fi, i = 0, 1, 2 that is, z0 = (0, 0), z1 = (1, 0) and z2 = ( 1 2, √ 3 2 ), and by T the equilateral triangle with vertexes zi, i ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' It is well known that S is a connected set that satisfies the OSC and has similarity dimension s = log 3 log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Thanks to previous work on the packing and Hausdorff centred measures of the Sierpinski gasket (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' [22] and [23]), we know that S is both P s⌊S and Cs⌊S-exact, and we have fairly precise approximations of the values of P s(S) and Cs(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1 Theoretical results Theorem 26 Let S be the Sierpinski gasket, �S = � y ∈ S : � T k(y) : k ∈ N � is dense in S � , B be the collection of typical balls, R be a feasible open set for S, and α ∈ Ms⌊S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, the following statements hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (i) Spec(α, y) = Spec(α, �S) = Spec(α, R ∩ S) = Im(θs α, B) = � α(S) P s(S), α(S) Cs(S) � , y ∈ �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (37) (ii) Spec(α, S) is given by the union of two closed intervals of positive length: Spec(α, S) = � θs α(z0), θ s α(z0) � ∪ � α(S) P s(S), α(S) Cs(S) � , (38) where z0 = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Furthermore, θs α(z0) = min � θs α(z0, d) : 1 2 ≤ d ≤ 1 � (39) and θ s α(z0) = max � θs α(z0, d) : 1 2 ≤ d ≤ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (40) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Our previous work guarantees that S is a P s-exact (see [22]) and Cs-exact (see [23]) set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, the four equalities in (i) follow as a consequence of Theorem 14 and Corollary 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In order to prove (38), let Ri, i ∈ {0, 1, 2} be the three open rhombi composed of the topological interior of the union of the triangle T and its reflection across the edge of T opposite the point zi, i ∈ M (see R2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Using that S = {z0, z1, z2} ∪ (S ∩ ∪2 i=0Ri), (41) we obtain Spec(α, S) = Spec(α, S ∩ ∪2 i=0Ri) ∪ � ∪2 i=0 Spec(α, zi) � = = � α(S) P s(S), α(S) Cs(S) � ∪ Spec(α, z0), where the last equality follows from (37), (41) and the fact that, by symmetry, Spec(α, zi) must be identical for i ∈ {0, 1, 2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 25 Figure 1: A feasible open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' An open rhombus R2 that is a feasible open set for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Observe now that, if d ≤ 1/2, then B(z0, d) ∩ S = B(z0, d) ∩ f0(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Hence, using that α is an s-dimensional metric measure θs α(z0, d) = α(B(z0, d) ∩ f0(S)) (2d)s = α(f0(B(z0, 2d) ∩ S)) (2d)s = α(B(z0, 2d) ∩ S)) (4d)s = θs α(z0, 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' If 2d ≤ 1/2, we can repeat the argument k times until 1/2 ≤ 2kd ≤ 1 and θs α(z0, d) = θs α(z0, 2kd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This shows that min {θs α(z0, d) : 0 ≤ d ≤ 1} = min � θs α(z0, d) : 1 2 ≤ d ≤ 1 � = θs α(z0), where the last equality can easily be checked and, analogously, (40) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Remark 27 Notice that part (i) of Theorem 26 shows that there is a set of full α-measure whose points exhibit a strongly regular behaviour, whereas part (ii) underlines the special local behaviour of the vertexes as the most isolated points in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' However, the set of exceptional points does not consist only of the vertexes as there might be other exceptional points, all of them belonging to the set ∪2 i=0 (Ri ∩ S)− �S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The pointwise α-density spectrum of such points is contained in � α(S) P s(S), α(S) Cs(S) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The detection and characterisation of the behaviour of these points remains an open issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 26 22 T Z1 R24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2 Numerical results Following the structure of the algorithms developed in [22, 23, 41, 42] for the numerical estimation of the metric measures of self-similar sets, the construction of the computational algorithm used in this work in order to approximate the values of θs µ(z0) and θ s µ(z0) relies upon the discrete approximations of both the Sierpinski gasket and its invariant measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Recall that any two measures in Ms⌊S are mutually multiple of each other (see (19)), so we can obtain Spec(α, S) from Spec(µ, S) if we know α(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The Sierpinski gasket, as the attractor of Ψ = {f0, f1, f2} (see (36)), is the unique non-empty compact set that admits the self-similar decomposition S = F(S), where F is the Hutchinson operator defined, for A ⊂ R2, by F(A) := f0(A) ∪ f1(A) ∪ f2(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' It is well-known that, for any non-empty compact subset A ⊂ R2, S can be built with an arbitrary level of detail by increasing the iterations k in F k(A), where F k = F ◦F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='◦F is the k-th iterate of the contracting operator F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This is because F k(A) k→∞ → S in the Hausdorff metric (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Furthermore, if A ⊂ S, then F k(A) ⊂ S for any k ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In particular, if we take A1 := {z0, z1, z2} as the initial compact set, we obtain the set Ak := F k−1(A1) ⊂ S, k ≥ 2, (42) which approximates S at the iteration k of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The relation between the Markov operator and the natural probability measure µps given in (17), with s = log 3 log 2 and pi = rs i = 3−k, and (19) leads to the following relation: Mk ps(α) = 1 3k � i∈M k α ◦ f −1 i w→ µ, α ∈ P(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (43) If we consider µ1 := 1 3(δz0 + δz1 + δz2) as an initial measure α in (43), where δx is a unit mass at x, then µk := Mk−1 ps (µ1) = 1 3k−1 � i∈M k−1 µ1 ◦ f −1 i = 1 3k � i∈M k−1 � δfi(z0) + δfi(z1) + δfi(z2) � (44) is a probability measure supported on Ak ⊂ S and µk w→ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The discrete measure µk is the approximation of the invariant measure µ that our algorithm takes at iteration k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 27 Lemmas 28 and 30 (Lemma 28 is proved in [23]), provide precise relationships between the measures µk and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Lemma 28 (i) Let {Si : i ∈ I ⊂ M k}, k ∈ N+, be a collection of k-cylinder sets of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, µ �� i∈I Si � ≤ µk �� i∈I Si � (ii) Let A ⊂ S, k ∈ N+, and let I = {i ∈ M k : Si ∩ A ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, µk(A) ≤ µ �� i∈I Si � Remark 29 The comparisons between the measures µ and µk on collections of cylinders and sets given in the lemma above are passed to enlarged and reduced balls in part (i) of the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Since our algorithms compute only µk-densities of balls with centres in Ak (see (42)) and with some point of Ak in their boundaries, in part (ii) of this lemma we approximate the µ-measure of a ball centred at x with the µk-measure of a ball with its same centre and with a point of Ak at its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In order to obtain more accurate estimates of θs µ(z0) and θ s µ(z0) (as we also do in [22] and [23] for the estimation of P s(S) and Cs(S)), it is necessary to consider open balls when searching balls of minimal µk-density (see (46)), whereas in the search of balls with maximal µk-density, the approximating balls must be taken to be closed balls (see (47)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' In the definition of θs µ(·) and θ s µ(·), the use of open or closed balls has no relevance because the µ-measure of the boundary of any ball is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' However, in the case of densities of the discrete measures µk, the values obtained in one or the other case do actually matter, mainly if k is not large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' From now on, we shall use the notation ˚ B(x, d) := {y ∈ R2 : |x − y| < d} and ˚θs α for the s-density of α defined using open balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Lemma 30 Let k > 0, x ∈ R2, and 2−k < d ≤ maxi∈M ∥zi − x∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, (i) µk(B(x, d − 2−k)) ≤ µ(B(x, d)) ≤ µk(˚ B(x, d + 2−k)) (ii) If B(x, d) ∩ Ak ̸= ∅, then there are points yk and zk in Ak such that µk � ˚ B(x, dyk) � ≤ µ(B(x, d)) ≤ µk(B(x, dzk)), 28 where dyk := |yk − x| , dzk := |zk − x| , and {dyk, dzk} ∈ [d − 2−k, d + 2−k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (i) Let Hk := {i ∈ M k : B(x, d − 2−k) ∩ Si ̸= ∅} For any i ∈ Hk, Si ⊂ B(x, d) holds, so ∪i∈HkSi ⊂ B(x, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Using Lemma 28 (ii), we have µk(B(x, d − 2−k)) ≤ µ(∪i∈HkSi) ≤ µ(B(x, d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let Gk := {i ∈ M k : Si ⊂ ˚ B(x, d + 2−k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, ˚ B(x, d) ∩ S ⊂ ∪i∈GkSi and ∪i∈GkSi ⊂ ˚ B(x, d + 2−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Using Lemma 28 (i), we get µ(B(x, d)) = µ(˚ B(x, d) ∩ S) ≤ µ(∪i∈GkSi) ≤ µk(∪i∈GkSi) ≤ µk(˚ B(x, d + 2−k)) (ii) Let d∗ = maxi∈M ∥zi − x∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' If S ⊂ B(x, d), then d = d∗ and µ(B(x, d∗)) = 1 = µk(B(x, d∗)) > µk((˚ B(x, d∗)), so property (ii) holds for dyk = dzk = d∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let us now assume that S ⊈ B(x, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We prove first that Fk := {i ∈ M k : ∂B(x, d) ∩ Si ̸= ∅} ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (45) If Fk = ∅, then ∪i∈M kSi ⊂ ˚ B(x, d) ∪ (B(x, d))c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We know that (∪i∈M kSi) ∩ ˚ B(x, d) ̸= ∅ because B(x, d) ∩ Ak ̸= ∅ and Fk = ∅, and we also know that (∪i∈M kSi) ∩ (B(x, d))c ̸= ∅ because S ⊈ B(x, d) and Fk = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This contradicts that ∪i∈M kSi is a connected set, and (45) must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Using (i), we have that µ(B(x, d)) ≤ µk(B(x, d + 2−k)) = µk(B(x, dzk)), where zk satisfies dzk = ∥zk − x∥ with dzk = max{∥y − x∥ : y ∈ Ak ∩ B(x, d + 2−k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The inequality dzk ≤ d+2−k is obvious, and dzk ≥ d−2−k follows because Fk ̸= ∅ and each k-cylinder Si, i ∈ M k contains some point in Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 29 Using the first inequality in (i), we have µ(B(x, d)) ≥ µk(B(x, d − 2−k)) = µk(˚ B(x, dyk)), where yk satisfies dyk = ∥yk − x∥ with dyk = min{∥y − x∥ : y ∈ Ak ∩ � B(x, d − 2−k) �c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The inequality dyk ≥ d − 2−k is obvious, and dyk ≤ d + 2−k follows because Fk ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Theorem 26 allows us to characterise Spec(α, S) for α ∈ {µ, P s⌊S, Cs⌊S} through only four numbers, namely, θs µ(z0), θ s µ(z0), P s(S) and Cs(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Thanks to previous numerical work that uses the measures µk and the sets Ak (see (44) and (42)) as approximations of µ and S, respectively, we have estimates given by our algorithms Pk of P s(S) (see [22]) and Ck of Cs(S) (see [23]) and precise error bounds for such estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We show in Theorem 31 below how to obtain estimates ξk of θs µ(z0) and ξk of θ s µ(z0), that such estimates converge to the real values, and we give accurate bounds for them, that is θs µ(z0) ∈ [ξinf k , ξsup k ] and θ s µ(z0) ∈ [ξ inf k , ξ sup k ] (see the definition of ξk, ξk and of the intervals [ξinf k , ξsup k ] and [ξ inf k , ξ sup k ] in Theorem 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This allows us to implement an algorithm along the lines of those developed for the estimation of Cs(S) and P s(S) (see [22,23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Theorem 31 For k > 1, let ξk := min � ˚θs µk(z0, d) : d = |x − z0| , x ∈ Ak, d ∈ [1 2 − 2−k, 1] � (46) and ξk := max � θs µk(z0, d) : d = |x − z0| , x ∈ Ak, d ∈ [1 2 − 2−k, 1] � (47) be the estimates of θs µ(z0) and θ s µ(z0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let dk be such that ˚θs µk(z0, dk) = ξk, and let Dk be such that θs µk(z0, Dk) = ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Then, {θs µ(z0), ξk} ∈ [ξinf k , ξsup k ], (48) and {θ s µ(z0), ξk} ∈ [ξ inf k , ξ sup k ], (49) where ξinf k = Kkξk, Kk = (1 − 21−k)s, ξsup k = µk(˚ B(z0, dk + 2−k)) (2dk)s , (50) 30 ξ inf k = µk(B(z0, Dk − 2−k)) (2Dk)s , Kk = (1 + 21−k)s, ξ sup k = Kkξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (51) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' That ξk ∈ [ξinf k , ξsup k ] and ξk ∈ [ξ inf k , ξ sup k ] is obvious from the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We prove first that θs µ(z0) ∈ [ξinf k , ξsup k ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Using Lemma 30 (i) and (39), we obtain θs µ(z0) ≤ µ(B(z0, dk)) (2dk)s ≤ µk(˚ B(z0, dk + 2−k)) (2dk)s = ξsup k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let d ∈ [ 1 2, 1] be such that θs µ(z0) = µ(B(z0,d)) (2d)s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Lemma 30 (ii) guarantees the existence of yk ∈ Ak such that µ(B(z0, d)) ≥ µk(˚ B(z0, dyk)), where dyk := |yk − z0| ∈ [d − 2−k, d + 2−k] ⊂ [ 1 2 − 2−k, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This, together with (46) and the inequality d ≥ 1 2 gives θs µ(z0) = µ(B(z0, d)) (2d)s ≥ µk(˚ B(z0, dyk)) (2d)s = �dyk d �s µk(˚ B(z0, dyk)) (2dyk)s ≥ �dyk d �s ξk ≥ �d − 2−k d �s ξk ≥ ξinf k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The proof that θ s µ(z0) ∈ [ξ inf k , ξ sup k ] is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Using Lemma 30 (i) and (40), we obtain θ s µ(z0) ≥ µ(B(z0, Dk)) (2Dk)s ≥ µk(B(z0, Dk − 2−k)) (2Dk)s = ξ inf k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Let D ∈ [ 1 2, 1] be such that θ s µ(z0) = µ(B(z0,D)) (2D)s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Lemma 30 (ii) guarantees the existence of zk ∈ Ak such that µ(B(z0, D)) ≤ µk(B(z0, dzk)), where dzk := |zk − z0| ∈ [D − 2−k, D + 2−k] ⊂ [ 1 2 − 2−k, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This, together with (47) and the inequality D ≥ 1 2 gives θ s µ(z0) = µ(B(z0, D)) (2D)s ≤ µk(B(z0, dzk)) (2D)s = �dzk D �s µk(B(z0, dzk)) (2dzk)s ≤ �dzk D �s ξk ≤ �D + 2−k D �s ξk ≤ ξ sup k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We present in Table 1 the estimates ξk, and ξk of θs µ(z0) and θ s µ(z0) (see (48) and (49) for definitions), respectively, and the corresponding lower and upper bounds in the 100% confidence inter- vals [ξinf k , ξsup k ], [ξ inf k , ξ sup k ] (see (51),(49)) obtained by our algorithm for k = 14 (see the definition these values in (46), (47), (50) and (51)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We also provide the radii, dk and Dk, of the µk-optimal balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' See in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 2a the graph of the function θs µ14(z0, d) as a function of d ∈ [ε, 1], and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 2b the points (g(d), θs µ14(z0, d)), where g(d) := ε + ε−1 log(ε)(log(d) − log(ε)) and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This is a suitable logarithmic scale, [10], which allows us to see the periodicity of this function at such a scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 31 (a) Values of θs µ14(z0, d) for d ∈ [ε, 1] and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (b) Values of (g(d), θs µ14(z0, d)), where g(d) := ε + ε−1 log(ε)(log(d) − log(ε)) and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Figure 2: Densities at z0 We present in Table 2 the estimates Pk of P s(S) and Ck of Cs(S) obtained by our algorithms for k = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The lower and upper bounds of P s(S) are denoted by P inf k and P sup k , respectively, and the bounds of Cs(S) are denoted by Cinf k and Csup k , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' These results were computed in [22] and [23]), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Recall that (P s(S))−1 and (Cs(S))−1 are the µ-densities of the balls of minimum and maximum µ-density in the set of typical balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The estimates Pk and Ck are obtained by replacing S with Ak and µ with µk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Again, we have used open balls in the estimation of the density of the ball of minimum µk-density, and closed balls for the density of the ball of maximum µk-density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The centre and radius of the open ball of minimum µk-density are denoted by x∗ k and dk, respectively, and the centre and radius corresponding to the closed ball of maximum µk-density are denoted by y∗ k and Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The table also contains the corresponding optimal µk-densities and their bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The upper bound P sup k := KP k Pk of P s(S) is slightly improved here with respect to the one given in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Here KP k := (1 − 25−k √ 3 )−s instead of the value KP k = (1 − 26−k √ 3 )−s used in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' This gives the value P sup 14 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='671292 given in Table 2 instead of the value P sup 14 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='668305 given in Table 1 in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The results of the following corollary are based on the estimates of Tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Corollary 32 Let S be the Sierpinski gasket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (i) For any α ∈ Ms⌊S, Spec(α, S) is the union of two closed disjointed intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 32 "(z 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='36 μ14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='29 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='9 1 d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='36 μ14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='9 1 g(d)(ii) Spec(µ, S) ∼ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2997, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3567] ∪ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='5994, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='9951] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2998, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3566] ∪ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='5999, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='9944] ⊂ Spec(µ, S) ⊂ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='2996, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3568] ∪ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='5983, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='9970] (iii) Spec(P s⌊S, S) ∼ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='5951] ∪ [1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='6602] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='5010, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='5945] ∪ [1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='6578] ⊂ Spec(P s⌊S, S) ⊂ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='4995, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='5963] ∪ [1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='6662] (iv) Spec(Cs⌊S, S) ∼ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3012, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3584] ∪ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='6023, 1] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3577] ∪ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='6032, 1] ⊂ Spec(Cs⌊S, S) ⊂ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3588] ∪ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='6002, 1] Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' We know (see (38) in Theorem 14) that Spec(µ, S) = � θs µ(z0), θ s µ(z0) � ∪ � 1 P s(S), 1 Cs(S) � , (52) and that (see (19)) Spec(α, S) = α(S)Spec(µ, S), α ∈ Ms⌊S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' (53) The two intervals in Spec(α, S), α ∈ Ms⌊S are disjointed if θ s µ(z0) < 1 P s(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Such a condition holds (see Theorem 31, and Tables 1 and 2) because θ s µ(z0) ≤ ξ sup 14 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='3568 and 1 P s(S) ≥ 1 P sup 14 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='5983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Using (52),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Theorem 31 and (53),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' we have that Spec(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' S) ∼ [ξ14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' ξ14] ∪ � 1 P14 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 1 C14 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' � ξsup 14 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' ξ inf 14 � ∪ � 1 P inf 14 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 1 Csup 14 � ⊂ Spec(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' S) ⊂ � ξinf 14 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' ξ sup 14 � ∪ � 1 P sup 14 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 1 Cinf 14 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Spec(P s⌊S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' S) ∼ [P14ξµ14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' P14ξµ14] ∪ � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' P14 C14 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' � P sup 14 ξsup 14 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' P inf 14 ξ inf 14 � ∪ � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' P inf 14 Csup 14 � ⊂ Spec(P s⌊S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' S) ⊂ � P inf 14 ξinf 14 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' P sup 14 ξ sup 14 � ∪ � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' P sup 14 Cinf 14 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 33 (and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' analogously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' for Spec(Cs⌊S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' S)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' and the proof is completed using the corresponding estimates of Tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' ξ14 [ξinf 14 , ξsup 14 ] d14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='299714 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='299656,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='299763] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='642272 ξ14 [ξ inf 14 , ξ sup 14 ] D14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='356687 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='356645,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='356756] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='913663 Table 1: Extreme densities at z0 Estimates of θs µ(z0) and θ s µ(z0), bounds and radii, dk and Dk, of the µk-optimal balls for k = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' x∗ 14 d14 P14 [P inf 14 , P sup 14 ] (P14)−1 = ˚θs µ14(x∗ 14, d14) � (P sup 14 )−1 , � P inf 14 �−1� (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='5,0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='160543 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='668305 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='667178, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='671292] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='599411 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='598339, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='599816] y∗ 14 D14 C14 [Cinf 14 , Csup 14 ] (C14)−1 = θs µ14(y∗ 14, D14) � (Csup 14 )−1 , � Cinf 14 �−1� � 5 16, √ 3 16 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='145957 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='004903 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='003109,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='005611] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='995121 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='994420, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content='996901] Table 2: Packing and Centred measure estimates of S Centres and radii of the balls ˚ B(x∗ 14, d14) and B(y∗ 14, D14) of minimum and maximum µ14-densities, estimates P14 and C14 of P s(S) and Cs(S), and bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' The last two columns in the table are the µ14-densities of the optimal balls (inverses of P s(S) and Cs(S)) and their bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' Acknowledgement 33 This work was supported by the Universidad Conmplutense de Madrid and the Banco de Santander (PR108/20-14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' References [1] 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convergence: the packing and centered Hausdorff measures of totally disconnected self-similar sets, Chaos, Solitons & Fractals, 98 (2017) 220-232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} +page_content=' 37' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE_T4oBgHgl3EQf2BxV/content/2301.08338v1.pdf'} diff --git a/hNE4T4oBgHgl3EQfrQ2w/content/tmp_files/2301.05207v1.pdf.txt b/hNE4T4oBgHgl3EQfrQ2w/content/tmp_files/2301.05207v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c245196e2a10ea7cc4ec81583489c6c248714a86 --- /dev/null +++ b/hNE4T4oBgHgl3EQfrQ2w/content/tmp_files/2301.05207v1.pdf.txt @@ -0,0 +1,945 @@ +arXiv:2301.05207v1 [math.CO] 12 Jan 2023 +INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS +KAREN GUNDERSON, KAREN MEAGHER, JOY MORRIS, AND VENKATA RAGHU TEJ PANTANGI +Abstract. In this article, we study the order and structure of the largest induced forests in +some families of graphs. First we prove a variation of the ratio bound that gives an upper +bound on the order of the largest induced forest in a graph. Next we define a canonical +induced forest to be a forest that is formed by adding a vertex to a coclique and give +several examples of graphs where the maximal forest is a canonical induced forest. These +examples are all distance-regular graphs with the property that the Delsarte-Hoffman ratio +bound for cocliques holds with equality. We conclude with some examples of related +graphs where there are induced forests that are larger than a canonical forest. +1. Introduction +In this paper we study both the cardinality and structure of the largest sets of vertices +inducing forests in some distance-regular graphs. For a graph G, let τ(G) be the maximum +number of vertices inducing a forest in G. The quantity τ(G) is called the acyclic number +of G. Letting α(G) denote the independence number of G, the order of the largest coclique, +it is clear that for any non-empty graph, τ(G) ≥ α(G) + 1 as adding any vertex to an +independent set will induce a forest. The main results of this article are to give bounds on +τ(G) for certain distance-regular graphs and to identify graphs in which every maximum +induced forest can be obtained by adding a single vertex to an independent set. +A number of other graph parameters and special kinds of vertex subsets bear some re- +lationship to this acyclic number τ(G). An induced forest in a graph is complementary to +a set of vertices whose removal induces an acyclic graph and this is sometimes known as +a ‘decycling set’ of a graph, or a ‘feedback vertex set’. Recall that a graph is k-degenerate +if and only if every subgraph has a vertex of valency at most k. The notion of degeneracy +arises in colouring problems and in the study of ‘cores’ of graphs, related to their con- +nectivity properties. A graph is empty if and only if it is 0-degenerate, while a graph is a +forest if and only if it is 1-degenerate. Thus, the largest coclique in a graph is the largest +set of vertices that induce a 0-degenerate subgraph, while the largest induced forest can be +thought of as the largest set of vertices inducing a 1-degenerate subgraph. +Date: January 13, 2023. +2010 Mathematics Subject Classification. Primary: 05C69, Secondary: 05C35, 05C25. +Key words and phrases. induced forests, distance-regular graphs, acyclic number, 1-degenerate subgraphs. +The first author was supported by Natural Science and Engineering Research Council of Canada (grant +RGPIN-2016-05949). +The second author was supported by Natural Science and Engineering Research Council of Canada (grant +RGPIN-03952-2018). +The third and fourth authors were supported by the Natural Science and Engineering Research Council of +Canada (grant RGPIN-2017-04905). +The authors are all indebted to the support of the Pacific Institute for Mathematical Sciences (PIMS), through +the establishment of the Collaborative Research Group on Movement and Symmetry in Graphs which funded this +work. +1 + +2 +K. GUNDERSON, K. MEAGHER, J. MORRIS, AND VENKATA RAGHU TEJ PANTANGI +Alon, Kahn, Seymour [2] showed that τ(G) ≥ � +v∈V 2/(d(v) + 1), where d(v) denotes +the valency of v. In fact, this is a special case of the general bound they prove for k- +degenerate induced subgraphs. In the case of a d-regular graph on n vertices, this implies +that τ(G) ≥ 2n/(d + 1). This bound is tight when (d + 1) | n for a graph consisting +of disjoint copies of Kd+1. Bondy, Hopkins and Staton [8] showed that if d = 3 and G +is connected (so that the previous tight examples do not apply), then τ(G) ≥ 5n−2 +8 +(here +n is the number of vertices). They also provided examples where their bound is tight. +Further refinements have been given for regular graphs of large girth [20, 21, 23]. Bau, +Wormald, and Zhou [6] showed that for random 3-regular graphs, asymptotically almost +surely, τ(G) = n−⌈(n+2)/4⌉ = ⌊(3n−2)/4⌋and gave bounds for random r-regular graphs in +general. Alon, Mubayi and Thomas [4] gave bounds on τ(G) in terms of the independence +number and the maximum valency. +The largest induced forests and smallest decycling sets in specific families of graphs +have been well-studied in the literature, for example: planar graphs [1], bipartite graphs [3, +10], hypercubes [5, 14, 27] and binomial random graphs [22]. Related work has concerned +the largest induced trees [13, 15, 24, 25, 26] and the largest induced matchings [9, 11]. +One of the most well known results in extremal graph theory is the Erd˝os-Ko-Rado +theorem ([12]). +Theorem 1.1 (Erd˝os, Ko, Rado). Let n > 2k and let F = {F1, F2, . . ., Fm} be an intersect- +ing family of k-sets from [n]. Then, +|F | ≤ +�n − 1 +k − 1 +� +, +with equality if and only if F consists of all k-sets containing a fixed element x ∈ [n]. +This celebrated result can be interpreted as a characterization of the cardinality and +structure of independent sets of maximum possible cardinality in the Kneser graph. The +Kneser graph K(n, k) is defined for any n, k ∈ Z+ to be the graph whose vertices are all of +the k-sets from [n] = {1, 2, . . ., n} with two vertices A, B adjacent if and only if A∩B = ∅. If +n < 2k, then K(n, k) has no edges, so we assume that n ≥ 2k. Translated into the setting of +the Kneser graphs, the Erd˝os-Ko-Rado Theorem states that for n > 2k, α(K(n, k)) = +�n−1 +k−1 +� +and any coclique of this order consists of all k-sets that contain a common element. +Similar such characterizations were made for maximum cocliques in many other fami- +lies of graphs. We refer to [17] for a through survey of such results. +The main results of this article characterize the largest induced forests in some distance- +regular graphs. The graphs we consider are distance-regular graphs for which the charac- +terization of maximum independent sets is known. Throughout this article, we will refer to +induced forests of maximum possible order as maximum induced forests. Let G = (V, E) +be a graph and S be a coclique of V. As noted previously, for any v ∈ V \ S , the set S ∪ {v} +induces a forest, so that τ(G) ≥ α(G) + 1. A natural next step is to find graphs in which +every maximum induced forest can be constructed by adding a vertex to an independent +set. +Definition 1.2. Let G = (V, E) be a graph and let F ⊂ V induce a forest. We say that F is +a canonical induced forest if there is a vertex v ∈ F such that F \ {v} is an independent set. +Often we refer to this as just a canonical forest in G. +The following result, simply known as the Delsarte-Hoffman ratio bound, is a spectral +graph theoretic method that has been used to characterize the maximum cocliques in many +families of graphs. + +INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS +3 +Theorem 1.3. (see [18, Theorem 3.2]) Let G be a k-regular graph on n vertices and let λ +be the smallest eigenvalue of the adjacency matrix of G. Then we have +α(G) ≤ n(−λ) +k − λ . +This result is an application of the Cauchy Interlacing Theorem (see [18, Theorem 2.1]). +Applying the same technique, we will show the following spectral upper bound for the +order of an induced forest in a regular graph. +Theorem 1.4. Let G be a k-regular graph on n vertices and let λ be the smallest eigenvalue +of the adjacency matrix of G. Then +τ(G) ≤ n(2 − λ) + +� +n2(2 − λ)2 − 8n(k − λ) +2(k − λ) +< −nλ +k − λ + +2n +k − λ. +An edge-counting argument provides an alternative bound on the order of an induced +forest in a regular graph that is sometimes better than the spectral bound (see discussion +after Lemma 4.1). +Theorem 1.5. Let G be a k-regular graph on n vertices. Let f be the number of vertices +and c the number of connected components in an induced forest of G. Then +f ≤ nk − 2c +2k − 2 ≤ nk − 2 +2k − 2. +The first summand of the right-hand side of the inequality in Theorem 1.4 is equal +to the Delsarte-Hoffman ratio bound on the independence number α(G). It is natural to +investigate the orders of forests in regular graphs for which the Delsarte-Hoffman ratio +bound is tight. Below is a list of five families of such graphs in which the maximum forest +is formed by adding a single vertex to a coclique. +Theorem 1.6. In the following graphs, every maximum forest is a canonical forest: +(1) the Kneser graph K(n, k), for every k ≥ 2 and n ≥ 2k3; +(2) the q-Kneser graph Kq(n, k), for k ≥ 2, n > 3k − 2 and q sufficiently large; +(3) the non-collinearity graph on points in a generalized quadrangle with parameters +(s, t) and s > 3; +(4) Xm,n = ⊗mKn with m ≥ 2 and n > 2m(m − 1); +(5) the complement of the block graph of an orthogonal array with parameters m, n +with n > 1 + 2m(m − 1); +We were able to make a few refinements in some subfamilies of the graphs mentioned +in the above result. These can be found in Theorems 3.4, 3.6, and 3.9. +We prove Theorems 1.4 and 1.5 in Section 2. In Section 3, we prove the results of +Theorem 1.6, characterizing induced forests in some other families of graphs. In Section 4, +we produce an infinite family of graphs with “large” maximum forests. +2. Upper bounds +We begin this section by proving Theorem 1.5. +Proof of Theorem 1.5. Let G = (V, E) be a k-regular graph on n vertices, and let F be an +induced forest of G with f vertices and c connected components. +Now, F has f − c edges. Since each of the f vertices of F has k incident edges and +each of the f − c edges of F is counted twice in the valency of vertices of F, there are + +4 +K. GUNDERSON, K. MEAGHER, J. MORRIS, AND VENKATA RAGHU TEJ PANTANGI +fk − 2(f − c) = f(k − 2) + 2c edges of G that join vertices of F to vertices that are not in F. +In total, this makes f(k − 1) + c edges of G that are incident with at least one vertex of F. +Clearly, the number of edges of G that are incident with at least one vertex of F cannot +exceed the total number of edges of G, which by the Handshaking Lemma is nk/2. So +f(k − 1) + c ≤ nk/2. +Rearranging this inequality produces the given result, which is maximized when c = 1. +□ +We next work toward the proof of Theorem 1.4. Let G = (V, E) be a k-regular graph +on n vertices. Let k = λ1 ≥ λ2 · · · ≥ λn be the eigenvalues of its adjacency matrix. The +following result from [18] gives algebraic bounds for induced subgraphs. We include the +proof for completeness. +Theorem 2.1. [18, Theorem 3.5] Let G be a k-regular graph on n vertices and suppose +that G has an induced subgraph G′ with n′ vertices and m′ edges. Then +λ2 ≥ 2m′n − k(n′)2 +n′(n − n′) +≥ λn. +Proof. Consider the partition π = {G′, G′} of the vertex set. The corresponding quotient +matrix is +� +2m′ +n′ +k − 2m′ +n′ +n′k−2m′ +n−n′ +k − n′k−2m′ +n−n′ +� +. +The eigenvalues of this matrix are k and 2m′ +n′ − n′k−2m′ +n−n′ += 2m′n−n′2k +n′(n−n′) . The result follows by +Cauchy’s Interlacing Theorem (see [18, Theorem 2.1]). +□ +We are now ready to prove Theorem 1.4. +Proof of Theorem 1.4. Let F be an induced forest in G on f vertices with c connected +components. Since F has exactly f − c edges and f vertices, using the above result, we +have +2(f − c)n − f 2k +f(n − f) +≥ λn, +and thus +(k − λn)f 2 + n(λn − 2)f + 2cn ≤ 0. +As c ≥ 1, we have (k − λn)f 2 + n(λn − 2)f + 2n ≤ 0, and thus f ≤ +n(2−λn)+√ +n2(2−λn)2−8n(k−λn) +2(k−λn) +. +□ +We now use Theorem 1.4 to find the acyclic number of a small graph. +Example 1. Consider the complement P′(9) of the Paley graph on 9 vertices. The vertex +set of this graph is the field F9 of size 9; and two elements a, b ∈ F9 are adjacent if and +only if a − b is not a quadratic residue in F9. We identify F9 � F3[x]/⟨x2 + 1⟩ and the set of +quadratic residues is S = {0, 1, 2, x, 2x}. The induced subgraph F3 ∪ {x + 1, x + 2} is a +path on 5 vertices, in P′(9). This construction implies that τ (P′(9)) ≥ 5. It is well-known +that P′(9) is a strongly-regular graph whose specturm is (4, 1, −2). Using Theorem 1.4, +we have τ(P′(9)) < 6. We note that Theorem 1.5 gives us the same upper bound. Thus we +have τ(P′(9)) = 5. +We were not able to extend this to other Paley graphs. In Section 4, we present some +observations (on the acyclic number) stemming from computations on small order Paley +graphs. + +INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS +5 +3. Graphs whose maximum induced forests are canonical. +In this section, we characterize maximum induced forests in some families of regular +graphs. In particular, we will prove Theorem 1.6 using a counting method for each graph. +Let G be a regular graph. We recall that the order τ(G) of a maximum induced forest +satisfies τ(G) ≥ α(G) + 1. To show that every maximum induced forest in G is canonical, +it suffices to show that |F| < α(G) + 1 for every non-canonical induced forest F. Note that +an induced forest F in G is not canonical if and only if F contains either a copy of P4 (a +path with 4 vertices) or a copy of P2 + P2 (the disjoint union of two edges) as an induced +subgraph. We now find an upper bound on the order of an induced forest F that does not +contain either a P4 or a P2 + P2. +Given a pair (a, b) of adjacent vertices in G, by N(a, b), we denote the set of vertices in +G that are not adjacent to either of a or b; and by η(a, b), we denote |N(a, b)|. We denote +the maximum such value by +η(G) = max {η(a, b) | a, b ∈ G and a ∼ b} . +Lemma 3.1. If F is a non-canonical forest in a graph G, then |F| ≤ 2 + 2η(G). +Proof. First assume that F contains a path on four vertices; call this subgraph P. Since +F is a forest, every v ∈ F \ P is adjacent to at most one vertex of P. Therefore, every +v ∈ F \ P is non-adjacent to at least one leaf and the neighbour of that leaf in P. Suppose +that P is made up of vertices {a, b, c, d} with a ∼ b, b ∼ c and c ∼ d. Then we see that +F ⊂ N(a, b) ∪ N(c, d) ∪ {b, c}, completing the proof in this case. +Next assume that F does not contain a path on four vertices but has an induced subgraph +Q that is isomorphic to P2 + P2. Let Q be made up of vertices {a, b, c, d} such that a ∼ b +and c ∼ d. Since F is a forest that does not contain a path on four vertices, every v ∈ F \ Q +is adjacent to at most one vertex of Q, so is non-adjacent to a pair of adjacent vertices of +Q. Thus we have F ⊆ N(a, b) ∪ N(c, d). +□ +This lemma is particularly applicable to strongly-regular graphs since the value of +η(α, β) is the same for all pairs (α, β) of adjacent vertices. We now recall that given +n, k, a, c ∈ N, a strongly-regular graph with parameters (n, k : a, c) is a k-regular graph +on n vertices such that (i) every pair of adjacent vertices have exactly a neighbours in com- +mon; and (ii) every pair of non-adjacent vertices have exactly c neighbours in common. +Using inclusion-exclusion on the parameters of a strongly-regular graph to get the value of +η(α, β) yields the following result. +Corollary 3.2. Let G be a strongly-regular graph with parameters (n, k : a, c). If +1 + 2(n − 2k + a) < α(G), +then every maximum induced forest is a canonical induced forest. +This corollary can be used to prove that for n ≥ 17 the maximum forests in K(n, 2) are +canonical (we omit this proof, since Theorem 3.4 gives a stronger result). +In the following subsections, we apply Lemma 3.1 to show that maximum induced +forests in some families of graphs must be canonical. +3.1. Kneser Graphs. In this section we consider the Kneser graphs K(n, k) with n ≥ 2k. +The graph K(2k, k) consists of exactly 1 +2 +�2k +k +� +disjoint edges and is itself a forest, so we will +only consider n > 2k. It is well known from the Erd˝os-Ko-Rado Theorem [12] that the +order of a maximum coclique in K(n, k) is +�n−1 +k−1 +� +and that the Delsarte-Hoffman ratio bound + +6 +K. GUNDERSON, K. MEAGHER, J. MORRIS, AND VENKATA RAGHU TEJ PANTANGI +holds with equality. Thus a canonical forest has order +�n−1 +k−1 +� ++ 1. We will show for n large +relative to k that this is the largest possible induced forest. +Theorem 3.3. For every k ≥ 2 and n ≥ 2k3, we have +τ(K(n, k)) = +�n − 1 +k − 1 +� ++ 1. +Moreover, every maximum induced forest is a canonical induced forest. +Proof. Let γ and δ be a pair of adjacent vertices in K(n, k). Elementary counting arguments +(overcounting sets whose intersection with γ or δ has cardinality greater than 1) show that +there are at most k2�n−2 +k−2 +� +k-subsets of [n] intersecting both γ and δ. Thus in this case, we +have η(K(n, k)) ≤ k2�n−2 +k−2 +� +. +By Lemma 3.1, if F is a non-canonical induced forest, then |F| ≤ 2 + 2k2�n−2 +k−2 +� +. In the +case n ≥ 2k3, we have +2 + 2k2 +�n − 2 +k − 2 +� +< 1 + +�n − 1 +k − 1 +� +. +Therefore non-canonicalinduced forests are smaller than the canonical induced forests. +□ +We consider one special case of Kneser graphs, in which the same sort of counting can +be done more precisely. +Theorem 3.4. For n ≥ 5 +τ(K(n, 2)) = max{n, 7}. +If n > 7, every maximum induced forest in K(n, 2) is canonical. +Proof. For any n, a canonical forest in K(n, 2) has order n. Further, τ(K(n, 2)) ≥ 7 for any +n ≥ 5, this is seen by taking the following vertex set: +{{1, 2}, {3, 4}, {1, 3}, {2, 4}, {1, 4}, {2, 3}, {1, 5}}. +Recall that any non-canonical forest contains either a copy of P4 or a copy of P2 + P2. +Assume F is an induced forest in K(n, 2). If F contains a copy of P4, then the vertices +of this P4 must be the sets {a, b}, {c, d}, {a, e}, {b, c} for some a, b, c, d, e. Any other vertex +in F is adjacent to at most one of these vertices. There are only 6 other vertices in K(n, 2) +that are nonadjacent to any of the pairs of adjacent vertices on this path, and exactly 3 of +these vertices are nonadjacent to 3 vertices of the path ({a, c}, {b, d}, and {b, e}). So any +such F contains at most 7 vertices. +Similarly, if F contains a copy of P2 + P2, then the vertices of this subgraph must be +the sets {a, b}, {c, d}, {a, c}, {b, d} for some a, b, c, d. If F has no P4, then F cannot include +any vertex of the form {a, e}, {b, e}, {c, e}, or {d, e} (for any e � {a, b, c, d}). Since any vertex +in F must be nonadjacent to at least 2 of the vertices of the P2 + P2, this implies that the +elements of the 2-set defining the vertex must lie entirely in {a, b, c, d}, so there are only 2 +other vertices that can be added: {a, d} and {b, c}. So any such F contains at most 6 vertices. +Therefore any induced forest that is not canonical contains no more than 7 vertices and +the result follows. +□ + +INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS +7 +3.2. q-Kneser Graphs. The next family we consider is the q-Kneser graphs. Let n, k be +positive integers with n ≥ 2k, and q be a power of a prime. The vertex set of the graph +Kq(n, k) is the set of all k-dimensional subspaces of Fn +q; two vertices are adjacent if and only +if they intersect trivially. It is well known that the cardinality of a coclique in this graph +is +�n−1 +k−1 +� +q, and that the Delsarte-Hoffman ratio bound holds with equality (see [16] or [17, +Chapter 9] for notation and details). The canonical induced forests have 1 + +�n−1 +k−1 +� +q vertices. +We obtain the following characterization of maximum induced forests in q-Kneser graphs. +Theorem 3.5. For k ≥ 2, n > 3k − 2 and q sufficiently large, we have +τ(Kq(n, k)) = +�n − 1 +k − 1 +� +q ++ 1. +Moreover, every maximum induced forest is canonical. +Proof. Let γ and δ be two adjacent vertices in K(n, k)q. If ω is a k-subspace intersecting +non-trivially with both γ and δ, then it contains a subspace of the form ⟨x⟩ + ⟨y⟩, where +x ∈ γ \{0} and y ∈ δ\{0}. A subspace of the form ⟨x⟩+⟨y⟩ can be chosen in +�k +1 +�2 +q ways. It is +a well known fact that there are +�n−2 +k−2 +� +q subspaces of dimension k, which contain a specific +2-dimensional subspace. Thus we have η(Kq(n, k)) ≤ +�k +1 +�2 +q +�n−2 +k−2 +� +q. +If F is a non-canonical induced forest, then by Lemma 3.1, we have +|F| ≤ 2 + 2 +�k +1 +�2 +q +�n − 2 +k − 2 +� +q +. +We will now show that, provided n > 3k − 2 and q sufficiently large, this upper bound +is smaller than 1 + +�n−1 +k−1 +� +q. Since +�n−1 +k−1 +� +q = +�n−1 +1 +� +q +�k−1 +1 +� +q +�n−2 +k−2 +� +q, we have +1 + +�n − 1 +k − 1 +� +q +− 2 − 2 +�k +1 +�2 +q +�n − 2 +k − 2 +� +q += +�n − 2 +k − 2 +� +q + +�n−1 +1 +� +q − 2 +�k−1 +1 +� +q +�k +1 +�2 +q +�k−1 +1 +� +q + − 1. +Expanding the q-binomial coefficients gives that +�n − 1 +1 +� +q +− 2 +�k − 1 +1 +� +q +�k +1 +�2 +q += qn−1 − 1 +q − 1 +− 2 +�qk−1 − 1 +q − 1 +� �qk − 1 +q − 1 +�2 +, +and, provided that n − 2 > 3k − 4, this is a monic polynomial of degree n − 2 and hence +positive for a sufficiently large q. So for n > 3k−2 and q sufficiently large, the order of any +forest is bounded above by 1 + +�n−1 +k−1 +� +q, and this bound is met by only canonical forests. +□ +As in the case of Kneser graphs, we consider the special case of strongly-regular q- +Kneser graphs with k = 2, in which the same sort of counting can be done more precisely. +In particular, the following result gives a complete characterization of maximum forests in +Kq(n, 2) provided n ≥ 4. +Theorem 3.6. For n ≥ 4 +τ(Kq(n, 2)) = max + +�n − 1 +1 +� +q ++ 1, 8 + . +If (n, q) � (4, 2), then every maximum induced forest in K(n, 2) is canonical. + +8 +K. GUNDERSON, K. MEAGHER, J. MORRIS, AND VENKATA RAGHU TEJ PANTANGI +Proof. Let F be a non-canonical forest in Kq(n, 2). Then F contains either a copy of P4 (a +path with 4 vertices) or a copy of P2 + P2 (the disjoint union of two edges) as an induced +subgraph. +First assume that F has four vertices {X, Y, V, W} inducing a path, with X ∼ Y, Y ∼ V, +and V ∼ W. From the discussion prior to Lemma 3.1, we have F ⊂ N(X, Y) ∪ N(V, W) ∪ +{Y, V}. As Kq(n, 2) is an strongly-regulargraph, the graph induced by N(X, Y) is isomorphic +to the graph induced by N(V, W). We now have |F| ≤ 2 + 2τ(N(V, W)), where τ(N(V, W)) +is the order of a maximum forest induced in the graph N(V, W). Similarly, if F has four +vertices {X, Y, V, W} inducing a disjoint union of two edges, with X ∼ Y and V ∼ W, then +|F| ≤ 2τ(N(V, W)). Therefore the order of a non-canonical forest is bounded above by +2 + 2τ(N(V, W)). We will now try to look at the structure of N(V, W). +As V and W are adjacent, V and W are disjoint 2-subpaces of F4 +q, and thus any U ∈ +N(V, W) is completely determined by U ∩ V and U ∩ W. Let U1, U2 ∈ N(V, W) intersect +non-trivially. Let e1, e2 ∈ V and f1, f2 ∈ W be such that ei ∈ Ui ∩ V and fi ∈ Ui ∩ W. As +U1 and U2 intersect non-trivially, there are a, b, c, d ∈ Fq such that ae1 + b f1 = ce2 + d f2. +This can be rewritten as ae1 − ce2 = d f2 − b f1. As V and W are disjoint, we must have +ae1 = ce2 and d f2 = b f1. We have now concluded that for U1, U2 ∈ N(V, W), U1 ∼ U2 if +and only if U1 ∩ V � U2 ∩ V and U1 ∩ W � U2 ∩ W. This shows that the subgraph induced +by N(V, W) is the two fold tensor product of the complete graph Kq+1. We dealt with these +graphs in Subsection 3.4. Using the notation in Subsection 3.4, we have N(V, W) � X2,q+1. +In Theorem 3.8 we will show that provided q ≥ 3, we have τ(X2,q+1) = q + 2. Therefore +if q ≥ 3, the order of a non-canonical forest is bounded above by 2 +2τ(N(V, W)) = 2q +6. +The order of the largest canonical forest is α(Kq(n, 2))+1 = +�n−1 +1 +� +q +1. Elementary algebra +shows that +�n−1 +1 +� +q + 1 > 2q + 6 for all n ≥ 4 and q ≥ 3. Therefore provided q ≥ 3, every +maximum forest in Kq(n, 2) is canonical. +We now shift our attention to q = 2. If V, W are two adjacent vertices in K2(n, 2), then +we have seen that N(V, W) � X2,3 = K3 ⊗ K3. The algebraic bound Theorem 1.4 shows that +τ(X2,3) < 6. We can check either by hand or computer that X2,3 has an induced path with 5 +vertices, and therefore τ(X2,3) = 5. Thus the order of a non-canonical forest F is bounded +above by 2+2τ(N(V, W)) = 12. For n > 4, we have α(K2(n, 2))+1 = 2n−1 > 12. Therefore +provided n > 4, every maximum forest in K2(n, 2) is canonical. +We are now left with the case of K2(4, 2). With the help of a computer algebra system +such as Sage ([28]), we can show that τ(K2(4, 2)) = α(K2(4, 2)) + 1 = 8. It can also be +shown that there are paths on 8 vertices in K2(4, 2). Thus not all maximum forests are +canonical in this case. +□ +3.3. Non-collinearity Graphs of Generalized Quadrangles. The next family we con- +sider is the family of non-collinearity graphs on generalized quadrangles. Let G be a gen- +eralized quadrangle with parameters s, t. By XG, we denote the graph whose vertices are +the points of G, in which two points are adjacent if and only if they are not collinear. It +is well known that XG is a strongly-regular graph with +� +s2t, −s, t +� +as the set of distinct +eigenvalues (see [17, Section 5.6]). By the Delsarte-Hoffman ratio bound for cocliques +(Theorem 1.3), +α(XG) ≤ (s + 1)(st + 1)s +s2t + s += s + 1. +The set of all points on a line form a coclique, so this bound is tight. We obtain the +following characterization of maximum induced forests in XG. + +INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS +9 +Theorem 3.7. Let G be a generalized quadrangle with parameters (s, t) and let XG be the +non-collinearity graph on points in G. Suppose that s > 3, then, +τ(XG) = s + 2. +Moreover, every maximum induced forest in XG is canonical. +Proof. Consider an induced forest F which contains a path P on 4 vertices as an induced +subgraph. Let {A, B, C, D} be the vertices inducing P, with A ∼ B, B ∼ C, and C ∼ D. As +F is a forest, any V ∈ F \ P must be non-adjacent to at least three vertices in {A, B, C, D}. +Suppose V is non-adjacent to each vertex in {A,C, D}. In other words, V is collinear with +every point in {A,C, D}. Thus V must lie on both the lines −−→ +AC and −−→ +AD. This implies that +V = A, which is contrary to our assumption V ∈ F \ P. By the same argument, V cannot +be simultaneously non-adjacent to every vertex in {A, B, D}. Thus V must be adjacent to +one of A or D. If V is adjacent to A, then as F is a forest, V must be collinear to every +point in {B,C, D}. Similarly, if V is adjacent to D, then V must be collinear to every point +in {A, B,C}. As G is a generalized quadrangle, given a line L and a point P not on L, there +is a unique point on L, that is collinear with P. Let Q1 be the unique point on −−→ +BD that is +collinear with C, and let Q2 be the unique point on −−→ +AC that is collinear with B. We can +now conclude that F ⊆ {Q1, A, B, C, D, Q2}. We now claim that Q1 and Q2 are non- +collinear. Let us assume the contrary, then we see that {Q1, C, Q2} form a triangle (not in +the graph) in G. This is impossible as a generalized quadrangle cannot contain a triangle, +and therefore Q1 and Q2 are not collinear. Thus S := {Q1, A, B, C, D, Q2} induces a +cycle on 6 vertices. As F ⊂ {Q1, A, B, C, D, Q2} is a forest, we must have |F| ≤ 5. +Now consider an induced forest F that contains a copy of P2 + P2. Let {P, Q} and {R, S } +be two edges in distinct connected components of the forest. The points P, R, Q, S form +vertices of a quadrilateral in G. Suppose that |F| > 4, then any V ∈ F \ {P, R, Q, S } +must be non-adjacent to at least one point in both {P, Q} and {R, S }. Without loss of +generality, let V be non-adjacent with R and Q. We claim that V must be on the line −−→ +RQ. +Assuming the contrary implies the existence of the triangle VRQ in G, which is absurd as +G is a generalized quadrangle. Again since G is a generalized quadrangle, R is the unique +point on −−→ +RQ collinear with P; and Q is the unique point on −−→ +RQ collinear with S . Therefore +V ∈ −−→ +RQ, must be simultaneously non-collinear with both P and S . Now the set {R, P, V, S } +induces a path on four vertices in F. By the argument in the above paragraph, existence of +such a path implies that |F| ≤ 5. +From the previous two paragraphs, we can conclude that the size of a non-canonical +forest is at most 5. Since when s > 3 any canonical forest has s + 2 ≥ 6 vertices, we have +shown that if s > 3, the only maximum forests in XG are the canonical ones. +□ +3.4. Tensor powers of complete graphs. We next consider a family of graphs in the +Hamming scheme. Consider the complete graph on n vertices, Kn. By Xm,n, we denote +the m-fold tensor product ⊗mKn. This is the mth graph in the Hamming Scheme H(m, n). +The vertex set can be considered as sequences of length m with entries from the additive +group Zn, with two sequences adjacent if and only if they differ at every coordinate. This +is an (n − 1)m-regular graph whose smallest eigenvalue is −(n − 1)m−1. Application of the +Delsarte-Hoffman ratio bound (Theorem 1.3) shows that α(Xm,n) ≤ nm−1. This bound is +met by the subset of sequences whose first coordinate is 0. +If m = 1, then Xm,n = Kn and any maximum forest is an edge which is a canonical +maximum forest. Also, if n = 1 then Xm,n is simply K1, so trivially any maximum forest is +canonical. + +10 +K. GUNDERSON, K. MEAGHER, J. MORRIS, AND VENKATA RAGHU TEJ PANTANGI +We obtain the following characterization of maximum induced forests in Xm,n. +Theorem 3.8. Let m, n be positive integers with m ≥ 2 and n > 2m(m − 1). Then +τ(Xm,n) = nm−1 + 1, +and every maximum induced forest in Xm,n is canonical. +Proof. As before, we investigate the orders of non-canonical forests. A simple counting +argument shows that η = m(m−1)nm−2 and therefore by Lemma 3.1 a non-canonical forest +has order at most 2 + 2m(m − 1)nm−2. Thus canonical forests are the largest, provided that +2 + 2m(m − 1)nm−2 < nm−1 + 1, +or, equivalently, +1 < nm−2(n − 2m(m − 1)). +If m ≥ 2, then the above equation holds whenever n > 2m(m − 1). +□ +As in the case of the q-Kneser graphs and the Kneser graphs, we consider the spe- +cial case of strongly regular tensor powers of complete graphs, in which the same sort of +counting can be done more precisely. In particular, the following result gives a complete +characterization of maximum forests in X2,n provided n ≥ 4. +Theorem 3.9. Given n ≥ 3, we have τ(X2,n) = max({5, n + 1}). Moreover when n ≥ 4, +every maximum induced forest is canonical. +Proof. Firstly given an edge {A, B}, we observe that |N(A, B)| = 2, where N(A, B) is the +set of vertices that are not adjacent to either A or B. Suppose that F is a forest, with +an induced subgraph P � P4. Suppose that P is made up of vertices {A, B,C, D} with +A ∼ B, B ∼ C and C ∼ D. Then from the discussion prior to Lemma 3.1, we know that +F ⊂ N(A, B) ∪ N(C, D) ∪ {B,C}. Suppose that X and Y are vertices such that N(A, B) = +{D, X} and N(C, D) = {A, Y}. Without loss of generality, we may assume that A = (a, b), +B = (c, d), D = (a, d), and C = (e, b), for some a, b, c, d, e ∈ Zn (not necessarily distinct) +such that a � c, b � d, a � e, and e � c. This forces X = (c, b) and Y = (e, d). Therefore X +is adjecent to Y, and thus ⊂ N(A, B) ∪ N(C, D) ∪ {B,C} is a 6 cycle. Therefore |F| ≤ 5. +If G is a forest with an induced copy of P2 + P2, then a similar argument shows that +|G| ≤ 5 (by adding a vertex that induces the same P5 that arises if we start with a P4 as +above). This completes the proof. +□ +3.5. Orthogonal Array Graphs. We finally consider a family of strongly-regular graphs +associated with orthogonal arrays. Let m and n be positive integers with m < n + 1. An +orthogonal array with parameters (m, n) is an m × n2 array with entries in Zn with the +property that every 2 × n2 array consists of all n2 possible pairs. Given an orthogonal array +O with parameters (m, n), by XO, we denote the graph on columns of O, where two columns +are adjacent if and only if there are no rows in which they have the same entry. We note +that the graph XO is the complement of the block graph of the orthogonal array O. It is +well known, see for example [17, Theorem 5.5.1], that this is a strongly-regular graph with +valency m(n − 1) and least eigenvalue m − n − 1. Application of the Delsarte-Hoffman ratio +bound (Theorem 1.3) shows that α(XO) ≤ n. This bound is met by the set of columns of O +whose first entry is 1. +Theorem 3.10. Let m, n be positive integers with n > 1 + 2m(m − 1) and let O be an +orthogonal array with parameters (m, n). Then +τ(XO) = n + 1. + +INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS +11 +Moreover, every maximum induced forest in Xm,n is canonical. +Proof. We now apply Lemma 3.1 to characterize the maximum independent sets in XO. We +note that η(XO) is the number of common neighbours of two non-adjacent vertices in the +complement of XO. By [17, Theorem 5.5.1], we see that η(XO) = m(m−1). By Lemma 3.1, +if F is a non-canonical induced forest, we have |F| ≤ 2 + 2m(m − 1). +We can now conclude that if α(XO) + 1 = n + 1 > 2 + 2m(m − 1), then every maximum +induced forest is canonical. +□ +4. Kneser graphs with non-canonical maximum forests +As noted in Subsection 3.1, K(2k, k) is a forest, so all of these graphs have non-canonical +maximum forests. The logical next family of Kneser graphs to consider are the graphs +K(2k + 1, k), these graphs also have non-canonical maximum forests. +Lemma 4.1. If k > 3, the graph K(2k + 1, k) has a forest of order +�2k +k +� ++ 2k − 2. +hence the maximum forests are not canonical. +Proof. Let F1 be the set of all vertices in K(2k+1, k) that do not contain the element 2k+1; +F1 is a set of 1 +2 +�2k +k +� += +�2k−1 +k +� +disjoint edges. +For i = 1, . . ., 2k − 2, define xi = {i, i + 1, . . ., i + k − 3} with the entries taken modulo +2k −1. Define the set F2 of vertices of the form γi = xi ∪{2k, 2k +1} with i = 1, . . ., 2k −2. +Clearly F2 is a coclique and any vertex in F2 is adjacent to at most one vertex in any edge +of F1 (specifically, the vertex that does not contain 2k). Further, vertices γi and γ j, have +exactly one common neighbour in F1 if j = i + 1, and no common neighbours otherwise. +Thus F1 ∪ F2 forms a forest of order +�2k +k +� ++ 2k − 2. +□ +The eigenvalue bound from Theorem 1.4 in this case is +τ(K(2k + 1, k)) < +�2k+1 +k +�� k +k−1 +� +�k+1 +k +� ++ +� k +k−1 +� + +2 +�2k+1 +k +� +�k+1 +k +� ++ +� k +k−1 +� = k + 2 +k +� 2k +k − 1 +� +. +This bound is larger than the forest given in Lemma 4.1. We can do better using the +bound produced by Theorem 1.5, which is +�2k+1 +k +��k+1 +k +� +− 2 +2 +�k+1 +k +� +− 2 += k + 1 +2k +�2k + 1 +k +� +− 1 +k +but this is still significantly larger than the forest our construction produces. +The final case to consider is K(7, 3), and in this case Theorem 1.5 tells us that an induced +forest has order at most +4 +6 +�7 +3 +� +− 1 +3 = 2(35) − 1 +3 += 23 +which can be achieved by the forest consisting of all triples from {1, . . ., 6} along with +{1, 2, 7}, {1, 3, 7} and {2, 3, 7}. + +12 +K. GUNDERSON, K. MEAGHER, J. MORRIS, AND VENKATA RAGHU TEJ PANTANGI +5. Further Work +It would be interesting to have more examples of graphs G with α(G) very close to +τ(G). We suspect that a characterization of the graphs with τ(G) = α(G)+1 is unlikely, but +perhaps we can find properties of a graph that would imply these two values are close. In +a sense, any such graph would have large independent sets that are uniformly connected to +the vertices in its complement. Specifically, any two adjacent vertices outside of the large +independent set would have to be adjacent to at least one common vertex in the independent +set, and non-adjacent vertices to at least two. This may lead to some structure conditions +on a graph that imply that τ(G) = α(G) + 1. We also suspect that focusing the search on +strongly-regular graph may produce more interesting examples. +All the examples of graphs we considered in this paper are graphs whose maximum +independent sets have been characterized. Maximum independent sets in Paley graph on +a square number vertices were characterized by Blokhius [7]. We will now discuss some +computational results we obtained regarding induced forests in these graphs. Let q be a +power of an odd prime. Let Fq and Fq2 be a fields of cardinality q. By P(q2), we denote the +Paley graph on q2 vertices. The vertex set for P(q2) is F2 +q, and two vertices are adjacent if +and only if their difference is a quadratic residue in the Fq2. It is well-known that the Paley +graph is self-complementary. In this regard, we could consider the complement P′(q2) of +the P(q2). We do so because the maximum independent sets in the complement have the +following natural characterization. +Theorem 5.1. (Blokhius [7]) Let q be a power of a prime and S be the set of non-zero +squares in Fq2, then α(P′(q2)) = q and the set {sFq + e : s ∈ S and e ∈ Fq2} is the set of all +independent sets of size q. +So the size of any canonical forest in P′(q2) is q + 1. We will now use Theorem 1.4 +to obtain an upper bound on the acyclic number. P′(q2) is strongly-regular graph and its +spectrum is well known to be ( q2−1 +2 , q−1 +2 , − q+1 +2 ) (see [17, Section 5.8]). Using Theorem 1.4, +we have +τ(P(q2) < q2(q2 + 5) +q2 + q +< q + 4 +In Example 1, we concluded that τ(P′(9)) = 5. From the discussion above the size of a +canonical forest in P′(9) is 4 and thus in this case, maximum forests are not canonical. We +will now consider two more Paley graphs of small order. +Example 2. Consider the graph P′(25), by Theorem 1.4, a forest cannot have more than +8 vertices. We have F25 � F5[x]/⟨x2 + x + 1⟩, and the set of quadratic residues is S = +{0} ∪ {a, ax, a +� +x + 1 +� +| a ∈ F∗ +5}. The induced subgraph F5 ∪ {x + 2, x + 4} is a forest (in +fact, a tree) of order 7 formed by adding two vertices to a maximum independent set. Since +canonical forests have order 6, this cannot be maximum forests. A computational search +indicates that 7 is the order of a maximum forest in this graph. +Example 3. Consider the complement of Paley graph on 49 vertices. Again Theorem 1.4 +implies a forest can have no more than 10 vertices. We have F49 � F7[x]/⟨x2 + 1⟩. The +set of quadratic residues is S = {0} ∪ {a, ax, a +� +x + 1 +� +, a +� +x − 1 +� +| a ∈ F∗ +7}. The induced +subgraph F7 ∪ {x + 2, x + 5} is a forest (in fact, a tree) of order 9 formed by adding two +vertices to a maximum independent set. Again computations indicate that 9 is the order of +a maximum forest in this graph, and canonical forests have order 8. + +INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS +13 +In Example 1 and the above examples, maximum induced forests which are in fact trees +were obtained by adding two vertices to a maximum independent set. Using Blokhius’s +characterization (Theorem 5.1) of maximum independent sets, we used Sage [28] to search +if similar constructions were possible in bigger Paley graphs. We checked for all prime +powers 7 < q ≤ 67 that adding two vertices to a maximum independent set in P′(q2), will +not result in a forest. So the examples we found may be anomalies occurring for small +values of q. We make the following conjecture. +Conjecture 5.2. For q > 7 a prime power, τ(P(q2)) = q + 1. +Paley graphs on q vertices can be defined whenever q is a prime power with q ≡ 1 +(mod 4). Let P′(q) denote the graph on the field Fq, in which two vertices are adjacent +if and only if their difference is not a quadratic residue in Fq. When q is an even power, +we conjectured above that τ(P′(q)) = √q + 1. It is natural to ask the question of what +happens when p is not an even power of a prime. Applying Theorem 1.4, we can show that +τ(P′(q)) < √q + 4. In this case, the order of the maximum independent sets is not known +in general, but it is bounded by √q, and can be significantly smaller. For instance, when q +is a prime, [19] shows that α(P′(q)) < +�q +2 +1. From our computer searches it seems even +in this case τ(P′(q)) is close to √q, so sometimes the induced forests are much larger than +α(P′(q)). Further, τ(P′(q)) seems to be non-decreasing with q, which is not the case for the +size of an independent set, and close to the eigenvalue bound. This may just be the case for +small values of q, so more computational results would be helpful. A key missing result is +a construction of an induced forest of size close to √q. Forests are bipartite graphs, and +so existence of an induced forest of size √q in P′(q) implies the existence of independent +sets of size at least √q/2. When q is not an ever power of a prime, there are no known +constructions of such large independent sets in P′(q). +References +[1] Jin Akiyama and Mamoru Watanabe. Maximum induced forests of planar graphs. Graphs Combin., +3(1):201–202, 1987. +[2] N. Alon, J. Kahn, and P. D. Seymour. Large induced degenerate subgraphs. Graphs Combin., 3(3):203–211, +1987. +[3] Noga Alon. Problems and results in extremal combinatorics. I. volume 273, pages 31–53. 2003. Euro- +Comb’01 (Barcelona). +[4] Noga Alon, Dhruv Mubayi, and Robin Thomas. Large induced forests in sparse graphs. Journal of Graph +Theory, 38(3):113–123, 2001. +[5] Sheng Bau, Lowell W. Beineke, Genmin Du, Zhishan Liu, and Robert C. Vandell. Decycling cubes and +grids. Util. Math., 59:129–137, 2001. +[6] Sheng Bau, Nicholas C. Wormald, and Sanming Zhou. Decycling numbers of random regular graphs. vol- +ume 21, pages 397–413. 2002. Random structures and algorithms (Poznan, 2001). +[7] Aart Blokhuis. On subsets of gf (q2) with square differences. In Indagationes Mathematicae (Proceedings), +volume 87, pages 369–372. Elsevier, 1984. +[8] J Adrian Bondy, Glenn Hopkins, and William Staton. Lower bounds for induced forests in cubic graphs. +Canadian mathematical bulletin, 30(2):193–199, 1987. +[9] Kathie Cameron. Induced matchings. Discrete Applied Mathematics, 24(1-3):97–102, 1989. +[10] David Conlon, Jacob Fox, and Benny Sudakov. Short proofs of some extremal results. Combin. Probab. +Comput., 23(1):8–28, 2014. +[11] Oliver Cooley, Nemanja Draganic, Mihyun Kang, and Benny Sudakov. Large induced matchings in random +graphs. SIAM Journal on Discrete Mathematics, 35(1):267–280, 2021. +[12] Paul Erd˝os, Chao Ko, and Richard Rado. Intersection theorems for systems of finite sets. Quart. J. Math. +Oxford Ser.(2), 12:313–320, 1961. +[13] Paul Erd˝os, Michael Saks, and Vera T S`os. Maximum induced trees in graphs. Journal of Combinatorial +Theory, Series B, 41(1):61–79, 1986. + +14 +K. GUNDERSON, K. MEAGHER, J. MORRIS, AND VENKATA RAGHU TEJ PANTANGI +[14] Riccardo Focardi, Flaminia L. Luccio, and David Peleg. Feedback vertex set in hypercubes. Inform. Process. +Lett., 76(1-2):1–5, 2000. +[15] Jacob Fox, Po-Shen Loh, and Benny Sudakov. Large induced trees in kr-free graphs. Journal of Combina- +torial Theory, Series B, 99(2):494–501, 2009. +[16] P´eter Frankl and Richard M Wilson. The Erd˝os-Ko-Rado theorem for vector spaces. Journal of Combinato- +rial Theory, Series A, 43(2):228–236, 1986. +[17] Chris Godsil and Karen Meagher. Erd˜os–Ko–Rado Theorems: Algebraic Approaches. Cambridge Studies +in Advanced Mathematics. Cambridge University Press, 2015. +[18] Willem H Haemers. Interlacing eigenvalues and graphs. Linear Algebra and its applications, 226:593–616, +1995. +[19] Brandon Hanson and Giorgis Petridis. Refined estimates concerning sumsets contained in the roots of unity. +Proceedings of the London Mathematical Society, 122(3):353–358, 2021. +[20] Carlos Hoppen and Nicholas Wormald. Induced forests in regular graphs with large girth. Combinatorics, +Probability and Computing, 17(3):389–410, 2008. +[21] Tom Kelly and Chun-Hung Liu. Size of the largest induced forest in subcubic graphs of girth at least four +and five. Journal of Graph Theory, 89(4):457–478, 2018. +[22] Maria Krivoshapko and Maksim Zhukovskii. Maximum induced forests in random graphs. Discrete Applied +Mathematics, 305:211–213, 2021. +[23] Jiping Liu and Cheng Zhao. A new bound on the feedback vertex sets in cubic graphs. Discrete Mathematics, +148(1-3):119–131, 1996. +[24] Jiˇr´ı Matouˇsek and Robert ˇS´amal. Induced trees in triangle-free graphs. The Electronic Journal of Combina- +torics, pages R41–R41, 2008. +[25] Zbigniew Palka and Andrzej Ruci´nski. On the order of the largest induced tree in a random graph. Discrete +Applied Mathematics, 15(1):75–83, 1986. +[26] Florian Pfender. Rooted induced trees in triangle-free graphs. Journal of Graph Theory, 64(3):206–209, +2010. +[27] David A. Pike. Decycling hypercubes. Graphs Combin., 19(4):547–550, 2003. +[28] W. A. Stein et al. Sage Mathematics Software (Version 8.6). The Sage Development Team, 2018. +http://www.sagemath.org. +Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada +Email address: Karen.Gunderson@umanitoba.ca +Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada +Email address: karen.meagher@uregina.ca +Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta T1K +3M4, Canada +Email address: joy.morris@uleth.ca +Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta T1K +3M4, Canada +Email address: raghu.pantangi@uleth.ca + diff --git a/hNE4T4oBgHgl3EQfrQ2w/content/tmp_files/load_file.txt b/hNE4T4oBgHgl3EQfrQ2w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b8239d8142fd94d9e179dbb22a9be2db039b87e --- /dev/null +++ b/hNE4T4oBgHgl3EQfrQ2w/content/tmp_files/load_file.txt @@ -0,0 +1,693 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf,len=692 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='05207v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='CO] 12 Jan 2023 INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS KAREN GUNDERSON, KAREN MEAGHER, JOY MORRIS, AND VENKATA RAGHU TEJ PANTANGI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In this article, we study the order and structure of the largest induced forests in some families of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' First we prove a variation of the ratio bound that gives an upper bound on the order of the largest induced forest in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Next we define a canonical induced forest to be a forest that is formed by adding a vertex to a coclique and give several examples of graphs where the maximal forest is a canonical induced forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' These examples are all distance-regular graphs with the property that the Delsarte-Hoffman ratio bound for cocliques holds with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We conclude with some examples of related graphs where there are induced forests that are larger than a canonical forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Introduction In this paper we study both the cardinality and structure of the largest sets of vertices inducing forests in some distance-regular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' For a graph G, let τ(G) be the maximum number of vertices inducing a forest in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The quantity τ(G) is called the acyclic number of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Letting α(G) denote the independence number of G, the order of the largest coclique, it is clear that for any non-empty graph, τ(G) ≥ α(G) + 1 as adding any vertex to an independent set will induce a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The main results of this article are to give bounds on τ(G) for certain distance-regular graphs and to identify graphs in which every maximum induced forest can be obtained by adding a single vertex to an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' A number of other graph parameters and special kinds of vertex subsets bear some re- lationship to this acyclic number τ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' An induced forest in a graph is complementary to a set of vertices whose removal induces an acyclic graph and this is sometimes known as a ‘decycling set’ of a graph, or a ‘feedback vertex set’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Recall that a graph is k-degenerate if and only if every subgraph has a vertex of valency at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The notion of degeneracy arises in colouring problems and in the study of ‘cores’ of graphs, related to their con- nectivity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' A graph is empty if and only if it is 0-degenerate, while a graph is a forest if and only if it is 1-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus, the largest coclique in a graph is the largest set of vertices that induce a 0-degenerate subgraph, while the largest induced forest can be thought of as the largest set of vertices inducing a 1-degenerate subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Primary: 05C69, Secondary: 05C35, 05C25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' induced forests, distance-regular graphs, acyclic number, 1-degenerate subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The first author was supported by Natural Science and Engineering Research Council of Canada (grant RGPIN-2016-05949).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The second author was supported by Natural Science and Engineering Research Council of Canada (grant RGPIN-03952-2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The third and fourth authors were supported by the Natural Science and Engineering Research Council of Canada (grant RGPIN-2017-04905).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The authors are all indebted to the support of the Pacific Institute for Mathematical Sciences (PIMS), through the establishment of the Collaborative Research Group on Movement and Symmetry in Graphs which funded this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' 1 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' GUNDERSON, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' MEAGHER, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' MORRIS, AND VENKATA RAGHU TEJ PANTANGI Alon, Kahn, Seymour [2] showed that τ(G) ≥ � v∈V 2/(d(v) + 1), where d(v) denotes the valency of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In fact, this is a special case of the general bound they prove for k- degenerate induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In the case of a d-regular graph on n vertices, this implies that τ(G) ≥ 2n/(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This bound is tight when (d + 1) | n for a graph consisting of disjoint copies of Kd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Bondy, Hopkins and Staton [8] showed that if d = 3 and G is connected (so that the previous tight examples do not apply), then τ(G) ≥ 5n−2 8 (here n is the number of vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' They also provided examples where their bound is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Further refinements have been given for regular graphs of large girth [20, 21, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Bau, Wormald, and Zhou [6] showed that for random 3-regular graphs, asymptotically almost surely, τ(G) = n−⌈(n+2)/4⌉ = ⌊(3n−2)/4⌋and gave bounds for random r-regular graphs in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Alon, Mubayi and Thomas [4] gave bounds on τ(G) in terms of the independence number and the maximum valency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The largest induced forests and smallest decycling sets in specific families of graphs have been well-studied in the literature, for example: planar graphs [1], bipartite graphs [3, 10], hypercubes [5, 14, 27] and binomial random graphs [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Related work has concerned the largest induced trees [13, 15, 24, 25, 26] and the largest induced matchings [9, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' One of the most well known results in extremal graph theory is the Erd˝os-Ko-Rado theorem ([12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1 (Erd˝os, Ko, Rado).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let n > 2k and let F = {F1, F2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=', Fm} be an intersect- ing family of k-sets from [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Then, |F | ≤ �n − 1 k − 1 � , with equality if and only if F consists of all k-sets containing a fixed element x ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This celebrated result can be interpreted as a characterization of the cardinality and structure of independent sets of maximum possible cardinality in the Kneser graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The Kneser graph K(n, k) is defined for any n, k ∈ Z+ to be the graph whose vertices are all of the k-sets from [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=', n} with two vertices A, B adjacent if and only if A∩B = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If n < 2k, then K(n, k) has no edges, so we assume that n ≥ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Translated into the setting of the Kneser graphs, the Erd˝os-Ko-Rado Theorem states that for n > 2k, α(K(n, k)) = �n−1 k−1 � and any coclique of this order consists of all k-sets that contain a common element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Similar such characterizations were made for maximum cocliques in many other fami- lies of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We refer to [17] for a through survey of such results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The main results of this article characterize the largest induced forests in some distance- regular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The graphs we consider are distance-regular graphs for which the charac- terization of maximum independent sets is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Throughout this article, we will refer to induced forests of maximum possible order as maximum induced forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let G = (V, E) be a graph and S be a coclique of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' As noted previously, for any v ∈ V \\ S , the set S ∪ {v} induces a forest, so that τ(G) ≥ α(G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' A natural next step is to find graphs in which every maximum induced forest can be constructed by adding a vertex to an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let G = (V, E) be a graph and let F ⊂ V induce a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We say that F is a canonical induced forest if there is a vertex v ∈ F such that F \\ {v} is an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Often we refer to this as just a canonical forest in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The following result, simply known as the Delsarte-Hoffman ratio bound, is a spectral graph theoretic method that has been used to characterize the maximum cocliques in many families of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' (see [18, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='2]) Let G be a k-regular graph on n vertices and let λ be the smallest eigenvalue of the adjacency matrix of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Then we have α(G) ≤ n(−λ) k − λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This result is an application of the Cauchy Interlacing Theorem (see [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Applying the same technique, we will show the following spectral upper bound for the order of an induced forest in a regular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let G be a k-regular graph on n vertices and let λ be the smallest eigenvalue of the adjacency matrix of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Then τ(G) ≤ n(2 − λ) + � n2(2 − λ)2 − 8n(k − λ) 2(k − λ) < −nλ k − λ + 2n k − λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' An edge-counting argument provides an alternative bound on the order of an induced forest in a regular graph that is sometimes better than the spectral bound (see discussion after Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let G be a k-regular graph on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let f be the number of vertices and c the number of connected components in an induced forest of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Then f ≤ nk − 2c 2k − 2 ≤ nk − 2 2k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The first summand of the right-hand side of the inequality in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4 is equal to the Delsarte-Hoffman ratio bound on the independence number α(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' It is natural to investigate the orders of forests in regular graphs for which the Delsarte-Hoffman ratio bound is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Below is a list of five families of such graphs in which the maximum forest is formed by adding a single vertex to a coclique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In the following graphs, every maximum forest is a canonical forest: (1) the Kneser graph K(n, k), for every k ≥ 2 and n ≥ 2k3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' (2) the q-Kneser graph Kq(n, k), for k ≥ 2, n > 3k − 2 and q sufficiently large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' (3) the non-collinearity graph on points in a generalized quadrangle with parameters (s, t) and s > 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' (4) Xm,n = ⊗mKn with m ≥ 2 and n > 2m(m − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' (5) the complement of the block graph of an orthogonal array with parameters m, n with n > 1 + 2m(m − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We were able to make a few refinements in some subfamilies of the graphs mentioned in the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' These can be found in Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='6, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='5 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In Section 3, we prove the results of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='6, characterizing induced forests in some other families of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In Section 4, we produce an infinite family of graphs with “large” maximum forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Upper bounds We begin this section by proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let G = (V, E) be a k-regular graph on n vertices, and let F be an induced forest of G with f vertices and c connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Now, F has f − c edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Since each of the f vertices of F has k incident edges and each of the f − c edges of F is counted twice in the valency of vertices of F, there are 4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' GUNDERSON, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' MEAGHER, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' MORRIS, AND VENKATA RAGHU TEJ PANTANGI fk − 2(f − c) = f(k − 2) + 2c edges of G that join vertices of F to vertices that are not in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In total, this makes f(k − 1) + c edges of G that are incident with at least one vertex of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Clearly, the number of edges of G that are incident with at least one vertex of F cannot exceed the total number of edges of G, which by the Handshaking Lemma is nk/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' So f(k − 1) + c ≤ nk/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Rearranging this inequality produces the given result, which is maximized when c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ We next work toward the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let G = (V, E) be a k-regular graph on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let k = λ1 ≥ λ2 · · · ≥ λn be the eigenvalues of its adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The following result from [18] gives algebraic bounds for induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We include the proof for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' [18, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='5] Let G be a k-regular graph on n vertices and suppose that G has an induced subgraph G′ with n′ vertices and m′ edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Then λ2 ≥ 2m′n − k(n′)2 n′(n − n′) ≥ λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Consider the partition π = {G′, G′} of the vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The corresponding quotient matrix is � 2m′ n′ k − 2m′ n′ n′k−2m′ n−n′ k − n′k−2m′ n−n′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The eigenvalues of this matrix are k and 2m′ n′ − n′k−2m′ n−n′ = 2m′n−n′2k n′(n−n′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The result follows by Cauchy’s Interlacing Theorem (see [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ We are now ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let F be an induced forest in G on f vertices with c connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Since F has exactly f − c edges and f vertices, using the above result, we have 2(f − c)n − f 2k f(n − f) ≥ λn, and thus (k − λn)f 2 + n(λn − 2)f + 2cn ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' As c ≥ 1, we have (k − λn)f 2 + n(λn − 2)f + 2n ≤ 0, and thus f ≤ n(2−λn)+√ n2(2−λn)2−8n(k−λn) 2(k−λn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ We now use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4 to find the acyclic number of a small graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Consider the complement P′(9) of the Paley graph on 9 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The vertex set of this graph is the field F9 of size 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' and two elements a, b ∈ F9 are adjacent if and only if a − b is not a quadratic residue in F9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We identify F9 � F3[x]/⟨x2 + 1⟩ and the set of quadratic residues is S = {0, 1, 2, x, 2x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The induced subgraph F3 ∪ {x + 1, x + 2} is a path on 5 vertices, in P′(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This construction implies that τ (P′(9)) ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' It is well-known that P′(9) is a strongly-regular graph whose specturm is (4, 1, −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4, we have τ(P′(9)) < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='5 gives us the same upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus we have τ(P′(9)) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We were not able to extend this to other Paley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In Section 4, we present some observations (on the acyclic number) stemming from computations on small order Paley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Graphs whose maximum induced forests are canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In this section, we characterize maximum induced forests in some families of regular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In particular, we will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='6 using a counting method for each graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let G be a regular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We recall that the order τ(G) of a maximum induced forest satisfies τ(G) ≥ α(G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' To show that every maximum induced forest in G is canonical, it suffices to show that |F| < α(G) + 1 for every non-canonical induced forest F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Note that an induced forest F in G is not canonical if and only if F contains either a copy of P4 (a path with 4 vertices) or a copy of P2 + P2 (the disjoint union of two edges) as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We now find an upper bound on the order of an induced forest F that does not contain either a P4 or a P2 + P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Given a pair (a, b) of adjacent vertices in G, by N(a, b), we denote the set of vertices in G that are not adjacent to either of a or b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' and by η(a, b), we denote |N(a, b)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We denote the maximum such value by η(G) = max {η(a, b) | a, b ∈ G and a ∼ b} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If F is a non-canonical forest in a graph G, then |F| ≤ 2 + 2η(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' First assume that F contains a path on four vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' call this subgraph P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Since F is a forest, every v ∈ F \\ P is adjacent to at most one vertex of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Therefore, every v ∈ F \\ P is non-adjacent to at least one leaf and the neighbour of that leaf in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Suppose that P is made up of vertices {a, b, c, d} with a ∼ b, b ∼ c and c ∼ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Then we see that F ⊂ N(a, b) ∪ N(c, d) ∪ {b, c}, completing the proof in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Next assume that F does not contain a path on four vertices but has an induced subgraph Q that is isomorphic to P2 + P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let Q be made up of vertices {a, b, c, d} such that a ∼ b and c ∼ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Since F is a forest that does not contain a path on four vertices, every v ∈ F \\ Q is adjacent to at most one vertex of Q, so is non-adjacent to a pair of adjacent vertices of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus we have F ⊆ N(a, b) ∪ N(c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ This lemma is particularly applicable to strongly-regular graphs since the value of η(α, β) is the same for all pairs (α, β) of adjacent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We now recall that given n, k, a, c ∈ N, a strongly-regular graph with parameters (n, k : a, c) is a k-regular graph on n vertices such that (i) every pair of adjacent vertices have exactly a neighbours in com- mon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' and (ii) every pair of non-adjacent vertices have exactly c neighbours in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Using inclusion-exclusion on the parameters of a strongly-regular graph to get the value of η(α, β) yields the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let G be a strongly-regular graph with parameters (n, k : a, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If 1 + 2(n − 2k + a) < α(G), then every maximum induced forest is a canonical induced forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This corollary can be used to prove that for n ≥ 17 the maximum forests in K(n, 2) are canonical (we omit this proof, since Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4 gives a stronger result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In the following subsections, we apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1 to show that maximum induced forests in some families of graphs must be canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Kneser Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In this section we consider the Kneser graphs K(n, k) with n ≥ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The graph K(2k, k) consists of exactly 1 2 �2k k � disjoint edges and is itself a forest, so we will only consider n > 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' It is well known from the Erd˝os-Ko-Rado Theorem [12] that the order of a maximum coclique in K(n, k) is �n−1 k−1 � and that the Delsarte-Hoffman ratio bound 6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' GUNDERSON, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' MEAGHER, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' MORRIS, AND VENKATA RAGHU TEJ PANTANGI holds with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus a canonical forest has order �n−1 k−1 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We will show for n large relative to k that this is the largest possible induced forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' For every k ≥ 2 and n ≥ 2k3, we have τ(K(n, k)) = �n − 1 k − 1 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Moreover, every maximum induced forest is a canonical induced forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let γ and δ be a pair of adjacent vertices in K(n, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Elementary counting arguments (overcounting sets whose intersection with γ or δ has cardinality greater than 1) show that there are at most k2�n−2 k−2 � k-subsets of [n] intersecting both γ and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus in this case, we have η(K(n, k)) ≤ k2�n−2 k−2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1, if F is a non-canonical induced forest, then |F| ≤ 2 + 2k2�n−2 k−2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In the case n ≥ 2k3, we have 2 + 2k2 �n − 2 k − 2 � < 1 + �n − 1 k − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Therefore non-canonicalinduced forests are smaller than the canonical induced forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ We consider one special case of Kneser graphs, in which the same sort of counting can be done more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' For n ≥ 5 τ(K(n, 2)) = max{n, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If n > 7, every maximum induced forest in K(n, 2) is canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' For any n, a canonical forest in K(n, 2) has order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Further, τ(K(n, 2)) ≥ 7 for any n ≥ 5, this is seen by taking the following vertex set: {{1, 2}, {3, 4}, {1, 3}, {2, 4}, {1, 4}, {2, 3}, {1, 5}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Recall that any non-canonical forest contains either a copy of P4 or a copy of P2 + P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Assume F is an induced forest in K(n, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If F contains a copy of P4, then the vertices of this P4 must be the sets {a, b}, {c, d}, {a, e}, {b, c} for some a, b, c, d, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Any other vertex in F is adjacent to at most one of these vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' There are only 6 other vertices in K(n, 2) that are nonadjacent to any of the pairs of adjacent vertices on this path, and exactly 3 of these vertices are nonadjacent to 3 vertices of the path ({a, c}, {b, d}, and {b, e}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' So any such F contains at most 7 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Similarly, if F contains a copy of P2 + P2, then the vertices of this subgraph must be the sets {a, b}, {c, d}, {a, c}, {b, d} for some a, b, c, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If F has no P4, then F cannot include any vertex of the form {a, e}, {b, e}, {c, e}, or {d, e} (for any e � {a, b, c, d}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Since any vertex in F must be nonadjacent to at least 2 of the vertices of the P2 + P2, this implies that the elements of the 2-set defining the vertex must lie entirely in {a, b, c, d}, so there are only 2 other vertices that can be added: {a, d} and {b, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' So any such F contains at most 6 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Therefore any induced forest that is not canonical contains no more than 7 vertices and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' q-Kneser Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The next family we consider is the q-Kneser graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let n, k be positive integers with n ≥ 2k, and q be a power of a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The vertex set of the graph Kq(n, k) is the set of all k-dimensional subspaces of Fn q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' two vertices are adjacent if and only if they intersect trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' It is well known that the cardinality of a coclique in this graph is �n−1 k−1 � q, and that the Delsarte-Hoffman ratio bound holds with equality (see [16] or [17, Chapter 9] for notation and details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The canonical induced forests have 1 + �n−1 k−1 � q vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We obtain the following characterization of maximum induced forests in q-Kneser graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' For k ≥ 2, n > 3k − 2 and q sufficiently large, we have τ(Kq(n, k)) = �n − 1 k − 1 � q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Moreover, every maximum induced forest is canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let γ and δ be two adjacent vertices in K(n, k)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If ω is a k-subspace intersecting non-trivially with both γ and δ, then it contains a subspace of the form ⟨x⟩ + ⟨y⟩, where x ∈ γ \\{0} and y ∈ δ\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' A subspace of the form ⟨x⟩+⟨y⟩ can be chosen in �k 1 �2 q ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' It is a well known fact that there are �n−2 k−2 � q subspaces of dimension k, which contain a specific 2-dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus we have η(Kq(n, k)) ≤ �k 1 �2 q �n−2 k−2 � q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If F is a non-canonical induced forest, then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1, we have |F| ≤ 2 + 2 �k 1 �2 q �n − 2 k − 2 � q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We will now show that, provided n > 3k − 2 and q sufficiently large, this upper bound is smaller than 1 + �n−1 k−1 � q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Since �n−1 k−1 � q = �n−1 1 � q �k−1 1 � q �n−2 k−2 � q, we have 1 + �n − 1 k − 1 � q − 2 − 2 �k 1 �2 q �n − 2 k − 2 � q = �n − 2 k − 2 � q \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed �n−1 1 � q − 2 �k−1 1 � q �k 1 �2 q �k−1 1 � q \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Expanding the q-binomial coefficients gives that �n − 1 1 � q − 2 �k − 1 1 � q �k 1 �2 q = qn−1 − 1 q − 1 − 2 �qk−1 − 1 q − 1 � �qk − 1 q − 1 �2 , and, provided that n − 2 > 3k − 4, this is a monic polynomial of degree n − 2 and hence positive for a sufficiently large q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' So for n > 3k−2 and q sufficiently large, the order of any forest is bounded above by 1 + �n−1 k−1 � q, and this bound is met by only canonical forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ As in the case of Kneser graphs, we consider the special case of strongly-regular q- Kneser graphs with k = 2, in which the same sort of counting can be done more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In particular, the following result gives a complete characterization of maximum forests in Kq(n, 2) provided n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' For n ≥ 4 τ(Kq(n, 2)) = max \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 �n − 1 1 � q + 1, 8 \uf8fc\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If (n, q) � (4, 2), then every maximum induced forest in K(n, 2) is canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' 8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' GUNDERSON, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' MEAGHER, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' MORRIS, AND VENKATA RAGHU TEJ PANTANGI Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let F be a non-canonical forest in Kq(n, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Then F contains either a copy of P4 (a path with 4 vertices) or a copy of P2 + P2 (the disjoint union of two edges) as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' First assume that F has four vertices {X, Y, V, W} inducing a path, with X ∼ Y, Y ∼ V, and V ∼ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' From the discussion prior to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1, we have F ⊂ N(X, Y) ∪ N(V, W) ∪ {Y, V}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' As Kq(n, 2) is an strongly-regulargraph, the graph induced by N(X, Y) is isomorphic to the graph induced by N(V, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We now have |F| ≤ 2 + 2τ(N(V, W)), where τ(N(V, W)) is the order of a maximum forest induced in the graph N(V, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Similarly, if F has four vertices {X, Y, V, W} inducing a disjoint union of two edges, with X ∼ Y and V ∼ W, then |F| ≤ 2τ(N(V, W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Therefore the order of a non-canonical forest is bounded above by 2 + 2τ(N(V, W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We will now try to look at the structure of N(V, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' As V and W are adjacent, V and W are disjoint 2-subpaces of F4 q, and thus any U ∈ N(V, W) is completely determined by U ∩ V and U ∩ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let U1, U2 ∈ N(V, W) intersect non-trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let e1, e2 ∈ V and f1, f2 ∈ W be such that ei ∈ Ui ∩ V and fi ∈ Ui ∩ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' As U1 and U2 intersect non-trivially, there are a, b, c, d ∈ Fq such that ae1 + b f1 = ce2 + d f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This can be rewritten as ae1 − ce2 = d f2 − b f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' As V and W are disjoint, we must have ae1 = ce2 and d f2 = b f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We have now concluded that for U1, U2 ∈ N(V, W), U1 ∼ U2 if and only if U1 ∩ V � U2 ∩ V and U1 ∩ W � U2 ∩ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This shows that the subgraph induced by N(V, W) is the two fold tensor product of the complete graph Kq+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We dealt with these graphs in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Using the notation in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4, we have N(V, W) � X2,q+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='8 we will show that provided q ≥ 3, we have τ(X2,q+1) = q + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Therefore if q ≥ 3, the order of a non-canonical forest is bounded above by 2 +2τ(N(V, W)) = 2q +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The order of the largest canonical forest is α(Kq(n, 2))+1 = �n−1 1 � q +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Elementary algebra shows that �n−1 1 � q + 1 > 2q + 6 for all n ≥ 4 and q ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Therefore provided q ≥ 3, every maximum forest in Kq(n, 2) is canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We now shift our attention to q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If V, W are two adjacent vertices in K2(n, 2), then we have seen that N(V, W) � X2,3 = K3 ⊗ K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The algebraic bound Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4 shows that τ(X2,3) < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We can check either by hand or computer that X2,3 has an induced path with 5 vertices, and therefore τ(X2,3) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus the order of a non-canonical forest F is bounded above by 2+2τ(N(V, W)) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' For n > 4, we have α(K2(n, 2))+1 = 2n−1 > 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Therefore provided n > 4, every maximum forest in K2(n, 2) is canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We are now left with the case of K2(4, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' With the help of a computer algebra system such as Sage ([28]), we can show that τ(K2(4, 2)) = α(K2(4, 2)) + 1 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' It can also be shown that there are paths on 8 vertices in K2(4, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus not all maximum forests are canonical in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Non-collinearity Graphs of Generalized Quadrangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The next family we con- sider is the family of non-collinearity graphs on generalized quadrangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let G be a gen- eralized quadrangle with parameters s, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' By XG, we denote the graph whose vertices are the points of G, in which two points are adjacent if and only if they are not collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' It is well known that XG is a strongly-regular graph with � s2t, −s, t � as the set of distinct eigenvalues (see [17, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' By the Delsarte-Hoffman ratio bound for cocliques (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='3), α(XG) ≤ (s + 1)(st + 1)s s2t + s = s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The set of all points on a line form a coclique, so this bound is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We obtain the following characterization of maximum induced forests in XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS 9 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let G be a generalized quadrangle with parameters (s, t) and let XG be the non-collinearity graph on points in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Suppose that s > 3, then, τ(XG) = s + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Moreover, every maximum induced forest in XG is canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Consider an induced forest F which contains a path P on 4 vertices as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let {A, B, C, D} be the vertices inducing P, with A ∼ B, B ∼ C, and C ∼ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' As F is a forest, any V ∈ F \\ P must be non-adjacent to at least three vertices in {A, B, C, D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Suppose V is non-adjacent to each vertex in {A,C, D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In other words, V is collinear with every point in {A,C, D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus V must lie on both the lines −−→ AC and −−→ AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This implies that V = A, which is contrary to our assumption V ∈ F \\ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' By the same argument, V cannot be simultaneously non-adjacent to every vertex in {A, B, D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus V must be adjacent to one of A or D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If V is adjacent to A, then as F is a forest, V must be collinear to every point in {B,C, D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Similarly, if V is adjacent to D, then V must be collinear to every point in {A, B,C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' As G is a generalized quadrangle, given a line L and a point P not on L, there is a unique point on L, that is collinear with P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let Q1 be the unique point on −−→ BD that is collinear with C, and let Q2 be the unique point on −−→ AC that is collinear with B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We can now conclude that F ⊆ {Q1, A, B, C, D, Q2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We now claim that Q1 and Q2 are non- collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let us assume the contrary, then we see that {Q1, C, Q2} form a triangle (not in the graph) in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This is impossible as a generalized quadrangle cannot contain a triangle, and therefore Q1 and Q2 are not collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus S := {Q1, A, B, C, D, Q2} induces a cycle on 6 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' As F ⊂ {Q1, A, B, C, D, Q2} is a forest, we must have |F| ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Now consider an induced forest F that contains a copy of P2 + P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let {P, Q} and {R, S } be two edges in distinct connected components of the forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The points P, R, Q, S form vertices of a quadrilateral in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Suppose that |F| > 4, then any V ∈ F \\ {P, R, Q, S } must be non-adjacent to at least one point in both {P, Q} and {R, S }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Without loss of generality, let V be non-adjacent with R and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We claim that V must be on the line −−→ RQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Assuming the contrary implies the existence of the triangle VRQ in G, which is absurd as G is a generalized quadrangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Again since G is a generalized quadrangle, R is the unique point on −−→ RQ collinear with P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' and Q is the unique point on −−→ RQ collinear with S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Therefore V ∈ −−→ RQ, must be simultaneously non-collinear with both P and S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Now the set {R, P, V, S } induces a path on four vertices in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' By the argument in the above paragraph, existence of such a path implies that |F| ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' From the previous two paragraphs, we can conclude that the size of a non-canonical forest is at most 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Since when s > 3 any canonical forest has s + 2 ≥ 6 vertices, we have shown that if s > 3, the only maximum forests in XG are the canonical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Tensor powers of complete graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We next consider a family of graphs in the Hamming scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Consider the complete graph on n vertices, Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' By Xm,n, we denote the m-fold tensor product ⊗mKn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This is the mth graph in the Hamming Scheme H(m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The vertex set can be considered as sequences of length m with entries from the additive group Zn, with two sequences adjacent if and only if they differ at every coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This is an (n − 1)m-regular graph whose smallest eigenvalue is −(n − 1)m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Application of the Delsarte-Hoffman ratio bound (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='3) shows that α(Xm,n) ≤ nm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This bound is met by the subset of sequences whose first coordinate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If m = 1, then Xm,n = Kn and any maximum forest is an edge which is a canonical maximum forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Also, if n = 1 then Xm,n is simply K1, so trivially any maximum forest is canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' GUNDERSON, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' MEAGHER, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' MORRIS, AND VENKATA RAGHU TEJ PANTANGI We obtain the following characterization of maximum induced forests in Xm,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let m, n be positive integers with m ≥ 2 and n > 2m(m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Then τ(Xm,n) = nm−1 + 1, and every maximum induced forest in Xm,n is canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' As before, we investigate the orders of non-canonical forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' A simple counting argument shows that η = m(m−1)nm−2 and therefore by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1 a non-canonical forest has order at most 2 + 2m(m − 1)nm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus canonical forests are the largest, provided that 2 + 2m(m − 1)nm−2 < nm−1 + 1, or, equivalently, 1 < nm−2(n − 2m(m − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If m ≥ 2, then the above equation holds whenever n > 2m(m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ As in the case of the q-Kneser graphs and the Kneser graphs, we consider the spe- cial case of strongly regular tensor powers of complete graphs, in which the same sort of counting can be done more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In particular, the following result gives a complete characterization of maximum forests in X2,n provided n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Given n ≥ 3, we have τ(X2,n) = max({5, n + 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Moreover when n ≥ 4, every maximum induced forest is canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Firstly given an edge {A, B}, we observe that |N(A, B)| = 2, where N(A, B) is the set of vertices that are not adjacent to either A or B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Suppose that F is a forest, with an induced subgraph P � P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Suppose that P is made up of vertices {A, B,C, D} with A ∼ B, B ∼ C and C ∼ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Then from the discussion prior to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1, we know that F ⊂ N(A, B) ∪ N(C, D) ∪ {B,C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Suppose that X and Y are vertices such that N(A, B) = {D, X} and N(C, D) = {A, Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Without loss of generality, we may assume that A = (a, b), B = (c, d), D = (a, d), and C = (e, b), for some a, b, c, d, e ∈ Zn (not necessarily distinct) such that a � c, b � d, a � e, and e � c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This forces X = (c, b) and Y = (e, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Therefore X is adjecent to Y, and thus ⊂ N(A, B) ∪ N(C, D) ∪ {B,C} is a 6 cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Therefore |F| ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If G is a forest with an induced copy of P2 + P2, then a similar argument shows that |G| ≤ 5 (by adding a vertex that induces the same P5 that arises if we start with a P4 as above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Orthogonal Array Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We finally consider a family of strongly-regular graphs associated with orthogonal arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let m and n be positive integers with m < n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' An orthogonal array with parameters (m, n) is an m × n2 array with entries in Zn with the property that every 2 × n2 array consists of all n2 possible pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Given an orthogonal array O with parameters (m, n), by XO, we denote the graph on columns of O, where two columns are adjacent if and only if there are no rows in which they have the same entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We note that the graph XO is the complement of the block graph of the orthogonal array O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' It is well known, see for example [17, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1], that this is a strongly-regular graph with valency m(n − 1) and least eigenvalue m − n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Application of the Delsarte-Hoffman ratio bound (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='3) shows that α(XO) ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This bound is met by the set of columns of O whose first entry is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let m, n be positive integers with n > 1 + 2m(m − 1) and let O be an orthogonal array with parameters (m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Then τ(XO) = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS 11 Moreover, every maximum induced forest in Xm,n is canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We now apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1 to characterize the maximum independent sets in XO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We note that η(XO) is the number of common neighbours of two non-adjacent vertices in the complement of XO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' By [17, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1], we see that η(XO) = m(m−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1, if F is a non-canonical induced forest, we have |F| ≤ 2 + 2m(m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We can now conclude that if α(XO) + 1 = n + 1 > 2 + 2m(m − 1), then every maximum induced forest is canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Kneser graphs with non-canonical maximum forests As noted in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1, K(2k, k) is a forest, so all of these graphs have non-canonical maximum forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The logical next family of Kneser graphs to consider are the graphs K(2k + 1, k), these graphs also have non-canonical maximum forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' If k > 3, the graph K(2k + 1, k) has a forest of order �2k k � + 2k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' hence the maximum forests are not canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let F1 be the set of all vertices in K(2k+1, k) that do not contain the element 2k+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' F1 is a set of 1 2 �2k k � = �2k−1 k � disjoint edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=', 2k − 2, define xi = {i, i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=', i + k − 3} with the entries taken modulo 2k −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Define the set F2 of vertices of the form γi = xi ∪{2k, 2k +1} with i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=', 2k −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Clearly F2 is a coclique and any vertex in F2 is adjacent to at most one vertex in any edge of F1 (specifically, the vertex that does not contain 2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Further, vertices γi and γ j, have exactly one common neighbour in F1 if j = i + 1, and no common neighbours otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Thus F1 ∪ F2 forms a forest of order �2k k � + 2k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' □ The eigenvalue bound from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4 in this case is τ(K(2k + 1, k)) < �2k+1 k �� k k−1 � �k+1 k � + � k k−1 � + 2 �2k+1 k � �k+1 k � + � k k−1 � = k + 2 k � 2k k − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This bound is larger than the forest given in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We can do better using the bound produced by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='5, which is �2k+1 k ��k+1 k � − 2 2 �k+1 k � − 2 = k + 1 2k �2k + 1 k � − 1 k but this is still significantly larger than the forest our construction produces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The final case to consider is K(7, 3), and in this case Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='5 tells us that an induced forest has order at most 4 6 �7 3 � − 1 3 = 2(35) − 1 3 = 23 which can be achieved by the forest consisting of all triples from {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=', 6} along with {1, 2, 7}, {1, 3, 7} and {2, 3, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' 12 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' GUNDERSON, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' MEAGHER, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' MORRIS, AND VENKATA RAGHU TEJ PANTANGI 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Further Work It would be interesting to have more examples of graphs G with α(G) very close to τ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We suspect that a characterization of the graphs with τ(G) = α(G)+1 is unlikely, but perhaps we can find properties of a graph that would imply these two values are close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In a sense, any such graph would have large independent sets that are uniformly connected to the vertices in its complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Specifically, any two adjacent vertices outside of the large independent set would have to be adjacent to at least one common vertex in the independent set, and non-adjacent vertices to at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This may lead to some structure conditions on a graph that imply that τ(G) = α(G) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We also suspect that focusing the search on strongly-regular graph may produce more interesting examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' All the examples of graphs we considered in this paper are graphs whose maximum independent sets have been characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Maximum independent sets in Paley graph on a square number vertices were characterized by Blokhius [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We will now discuss some computational results we obtained regarding induced forests in these graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let q be a power of an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let Fq and Fq2 be a fields of cardinality q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' By P(q2), we denote the Paley graph on q2 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The vertex set for P(q2) is F2 q, and two vertices are adjacent if and only if their difference is a quadratic residue in the Fq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' It is well-known that the Paley graph is self-complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In this regard, we could consider the complement P′(q2) of the P(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We do so because the maximum independent sets in the complement have the following natural characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' (Blokhius [7]) Let q be a power of a prime and S be the set of non-zero squares in Fq2, then α(P′(q2)) = q and the set {sFq + e : s ∈ S and e ∈ Fq2} is the set of all independent sets of size q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' So the size of any canonical forest in P′(q2) is q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We will now use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4 to obtain an upper bound on the acyclic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' P′(q2) is strongly-regular graph and its spectrum is well known to be ( q2−1 2 , q−1 2 , − q+1 2 ) (see [17, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4, we have τ(P(q2) < q2(q2 + 5) q2 + q < q + 4 In Example 1, we concluded that τ(P′(9)) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' From the discussion above the size of a canonical forest in P′(9) is 4 and thus in this case, maximum forests are not canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We will now consider two more Paley graphs of small order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Consider the graph P′(25), by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4, a forest cannot have more than 8 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We have F25 � F5[x]/⟨x2 + x + 1⟩, and the set of quadratic residues is S = {0} ∪ {a, ax, a � x + 1 � | a ∈ F∗ 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The induced subgraph F5 ∪ {x + 2, x + 4} is a forest (in fact, a tree) of order 7 formed by adding two vertices to a maximum independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Since canonical forests have order 6, this cannot be maximum forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' A computational search indicates that 7 is the order of a maximum forest in this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Consider the complement of Paley graph on 49 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Again Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4 implies a forest can have no more than 10 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We have F49 � F7[x]/⟨x2 + 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The set of quadratic residues is S = {0} ∪ {a, ax, a � x + 1 � , a � x − 1 � | a ∈ F∗ 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' The induced subgraph F7 ∪ {x + 2, x + 5} is a forest (in fact, a tree) of order 9 formed by adding two vertices to a maximum independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Again computations indicate that 9 is the order of a maximum forest in this graph, and canonical forests have order 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' INDUCED FORESTS IN SOME DISTANCE-REGULAR GRAPHS 13 In Example 1 and the above examples, maximum induced forests which are in fact trees were obtained by adding two vertices to a maximum independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Using Blokhius’s characterization (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='1) of maximum independent sets, we used Sage [28] to search if similar constructions were possible in bigger Paley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We checked for all prime powers 7 < q ≤ 67 that adding two vertices to a maximum independent set in P′(q2), will not result in a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' So the examples we found may be anomalies occurring for small values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' We make the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' For q > 7 a prime power, τ(P(q2)) = q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Paley graphs on q vertices can be defined whenever q is a prime power with q ≡ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Let P′(q) denote the graph on the field Fq, in which two vertices are adjacent if and only if their difference is not a quadratic residue in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' When q is an even power, we conjectured above that τ(P′(q)) = √q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' It is natural to ask the question of what happens when p is not an even power of a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content='4, we can show that τ(P′(q)) < √q + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' In this case, the order of the maximum independent sets is not known in general, but it is bounded by √q, and can be significantly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' For instance, when q is a prime, [19] shows that α(P′(q)) < �q 2 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' From our computer searches it seems even in this case τ(P′(q)) is close to √q, so sometimes the induced forests are much larger than α(P′(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Further, τ(P′(q)) seems to be non-decreasing with q, which is not the case for the size of an independent set, and close to the eigenvalue bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' This may just be the case for small values of q, so more computational results would be helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' A key missing result is a construction of an induced forest of size close to √q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Forests are bipartite graphs, and so existence of an induced forest of size √q in P′(q) implies the existence of independent sets of size at least √q/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' When q is not an ever power of a prime, there are no known constructions of such large independent sets in P′(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' References [1] Jin Akiyama and Mamoru Watanabe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Maximum induced forests of planar graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Graphs Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=', 3(1):201–202, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Alon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Kahn, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Seymour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Large induced degenerate subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Graphs Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=', 3(3):203–211, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' [3] Noga Alon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Problems and results in extremal combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' volume 273, pages 31–53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Euro- Comb’01 (Barcelona).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' [4] Noga Alon, Dhruv Mubayi, and Robin Thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Large induced forests in sparse graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Journal of Graph Theory, 38(3):113–123, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' [5] Sheng Bau, Lowell W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Beineke, Genmin Du, Zhishan Liu, and Robert C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Vandell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Decycling cubes and grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Util.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=', 59:129–137, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' [6] Sheng Bau, Nicholas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Wormald, and Sanming Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Decycling numbers of random regular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' vol- ume 21, pages 397–413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' Random structures and algorithms (Poznan, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNE4T4oBgHgl3EQfrQ2w/content/2301.05207v1.pdf'} +page_content=' [7] Aart Blokhuis.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 1 Boosting Automated Patch Correctness Prediction via Pre-trained Language Model Quanjun Zhang, Chunrong Fang*, Weisong Sun, Yan Liu, Tieke He*, Xiaodong Hao, Zhenyu Chen Abstract—Automated program repair (APR) aims to fix software bugs automatically without human debugging efforts and plays a crucial role in software development and maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Despite the recent significant progress in the number of fixed bugs, APR is still challenged by a long-standing overfitting problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', the generated patch is plausible but overfitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Various techniques have thus been proposed to address the overfitting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Among them, leveraging deep learning approaches to predict patch correctness automatically is emerging along with the available large-scale patch benchmarks recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' However, existing learning-based techniques mainly rely on manually-designed code features, which can be extremely costly and challenging to construct in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this paper, we propose APPT, a pre-trained model-based automated patch correctness assessment technique, which treats the source code as a sequence of tokens without extra overhead to design a mass of features from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In particular, APPT adopts a pre-trained model as the encoder stack, followed by an LSTM stack and a deep learning classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Although our idea is general and can be built on various existing pre-trained models, we have implemented APPT based on the BERT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We conduct an extensive experiment on 1,183 Defects4J patches and the experimental results show that APPT achieves prediction accuracy of 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% and recall of 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3%, outperforming the state-of-the-art technique CACHE by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Our additional investigation on 49,694 real-world patches shows that APPT achieves the optimum performance (exceeding 99% in five common metrics for assessing patch classification techniques) compared with existing representation learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also prove that adopting advanced pre-trained models can further provide substantial advancement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', GraphCodeBERT-based APPT improves BERT-based APPT by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% in precision and recall, respectively), highlighting the generalizability of APPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Index Terms—Automated Program Repair, Patch Correctness, Pre-trained Model !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 1 INTRODUCTION S Oftware bugs are inevitable in modern software systems and result in fatal consequences, such as costing trillions of dollars in financial loss and affecting billions of people around the world [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' It is incredibly time-consuming and labor-intensive for developers to fix such bugs due to the increasing size and complexity of modern software systems [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Automated program repair (APR) aims to fix revealed software bugs without human intervention auto- matically and has attracted massive attention from both academia and industry in the past decades [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Despite an emerging research area, a variety of APR techniques have been proposed and continuously achieved promising results in terms of the number of fixed bugs in the literature [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' However, it is fundamentally difficult to achieve high precision for generated patches due to the weak program specifications [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Existing APR techniques usually leverage the developer-written test cases as the criteria to assess the correctness of the generated patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In fact, a generated patch passing the available test cases may not generalize to other potential test cases, leading to a long-standing challenge of APR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', the overfitting issue) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, when a bug is detected in functionality, a patch can be sim- ply generated by deleting the functionality and the available Quanjun Zhang, Chunrong Fang, Weisong Sun, Yan Liu, Tieke He, Xiaodong Hao and Zhenyu Chen are with the State Key Laboratory for Novel Software Technology, Nanjing University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' E-mail: quanjun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='zhang@smail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='cn, fangchunrong@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='cn, weisongsun@smail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='cn, MF21320104@smail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='cn, hetieke@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='cn, MF21320054@smail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='cn, zychen@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='cn Chunrong Fang and Tieke He are the corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Manuscript received xxx xxx, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' revised xxx xxx, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' test cases usually fail to exercise the deleted functionality [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this case, developers need to consume tremendous time and effort to filter the overfitting patches, resulting in a negative debugging performance when APR techniques are applied in practice [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Thus, various automated patch correctness assessment (APCA) techniques have been proposed to determine whether a generated patch is indeed correct or not [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' According to extracted features, the traditional APCA tech- niques can be categorized into two groups: static and dy- namic ones [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Static techniques tend to analyze the code changed patterns or code similarity based on the syntactic and semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [14] define a set of generic forbidden transformations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', the above- mentioned functionality deleting) for the buggy program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In contrast, dynamic techniques usually execute the plausible patches against extra test cases generated by automated test generation tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Evosuite [15] and Randoop [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [17] generate new test cases and de- termine patch correctness based on the behavior similarity of the test case executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' However, the static techniques may suffer from prediction precision problems, while it is pretty time-consuming for dynamic techniques to generate additional test cases and execute all patched programs [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Recently, inspired by large-scale patch benchmarks being released [6], [7], some learning-based APCA techniques have been proposed to assess patch correctness by em- bedding buggy and patched code snippets [12], [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [20] leverage the abstract syntax tree (AST) path to represent the patch and build a deep learning classifier to predict the correctness of the patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='12453v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='SE] 29 Jan 2023 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 2 Similarly, He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [18] extract code features at the AST level statically and train a probabilistic model to perform patch prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' However, despite outstanding prediction results, existing learning-based APCA techniques mainly employ complex code-aware features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', AST path in [20]) or manually-designed code features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', 202 code features in [18]), which are costly to conduct and extract in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, we propose, APPT, the first Automated Pre- trained model-based Patch correcTness assessment tech- nique, which employs the pre-training and fine-tuning to address the above limitation of prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We first adopt the large pre-trained model as the encoder stack to extract code representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We then employ bidirectional LSTM layers to capture rich dependency information between the buggy and patched code snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Finally, we build a deep learning classifier to predict whether the patch is overfitting or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' APPT treats only the source code tokens as the input and automatically extracts code features using a well- trained encoder stack, getting rid of the need for code-aware features and manually-designed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Although APPT is conceptually general and can be built on various pre-trained models, we have implemented APPT as a practical APCA tool based on the BERT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Our experimental results on 1,183 Defects4J patches indicate that APPT improves the state-of-the-art technique CACHE by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% accuracy, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2% precision, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% recall, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% F1-score and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We conduct an additional investigation on 49,694 real-world patches from five different patch benchmarks and the results show that APPT exceeds 99% in accuracy, precision, re- call, F1-score and AUC metrics, outperforming the existing representation learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also adopt different pre-trained models to further investigate the generalization ability of APPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The results demonstrate that APPT with advanced pre-trained models can enhance the prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, precision and recall of APPT can be improved by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% when equipped with GraphCodeBERT, which are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2% and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2% higher than the state-of-the-art technique CACHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' To sum up, we make the following major contributions: New Direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' This paper opens a new direction for patch correctness prediction to directly utilize large pre- trained models by pre-training and fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Com- pared with existing learning-based APCA techniques, our approach does not need any additional efforts to design and extract complex code features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Novel Technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We propose APPT, a BERT-based APCA technique that leverages the pre-training and classifier to predict patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' To the best of our knowledge, we are the first to exploit fine-tuning the pre-trained model for assessing patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Extensive Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We conduct various empirical studies to investigate and evaluate APPT on diverse patch benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The results show that APPT achieves significantly better overall performance than existing learning-based and traditional APCA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Available Artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We release the relevant materials (including source code, patches and results) used in the experiments for replication and future research1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' All artifacts relevant to this work can be found at anonymouswebsite, accessed August 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' repair strategy test suite correct patch plausible patch developer generated patch overfitting patch suspicious code fault localization deployment Localization Phase Repair Phase buggy program Verification Phase Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 1: Overview of APR 2 BACKGROUND 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 Automated Program Repair APR techniques’ primary objective is to identify and fix program bugs automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 1 illustrates the workflow of the typical APR technique, which is usually composed of three steps: (1) the localization phrase utilizes off-the- shelf fault localization techniques to recognize the suspi- cious code elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', statements or methods) [21], [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' (2) the repair phrase then modifies these elements based on a set of transformation rules to generate various new program variants, also called candidate patches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' (3) the ver- ification phrase adopts the original test cases as the oracle to check whether candidate patches execute as expected or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Specifically, a candidate patch passing the original test cases is called a plausible patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' A plausible patch that is semantically equivalent to the developer patch denotes a correct patch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' otherwise, it is an overfitting patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' It is fundamentally challenging to ensure the correctness of the plausible patches due to the weak specification of the program behavior in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Existing studies have demon- strated that manually identifying the overfitting patches is time-consuming and may harm the debugging performance of developers [10], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Thus, various techniques have been proposed to validate patch correctness automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Ac- cording to whether the dynamic execution or machine learn- ing is required [13], we categorize them into three main cat- egories: static-based techniques, dynamic-based techniques and learning-based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Static-based APCA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' These techniques aim to prioritize correct patches over overfitting ones by static code features, such as code-deleting program transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Dynamic-based APCA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' These techniques aim to filter out overfitting patches by executing extra test cases, which are generated based on fixed or patched programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' According to whether the correct patches are required, these techniques can be further categorized into dynamic with oracle-based ones and dynamic without oracle-based ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Learning-based APCA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' These techniques aim to predict the correctness of plausible patches enhanced by machine learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' They usually extract the manually-designed code features and then adopt a classi- fier to perform patch prediction [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Some techniques are IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 3 proposed to adopt code embedding techniques to extract code features automatically [20], which are also denoted as representation learning-based APCA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Recently, an increasing number of research efforts have attempted to use machine learning techniques to learn from existing patch benchmarks for predicting potential patch correctness, achieving promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, we adopt the large pre-trained model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', BERT) to encode plausible patches and train a deep learning classifier to predict patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Compared to existing techniques, our paper is the first work to predict patch correctness by pre-training and fine-tuning the pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 Pre-trained Model Recently, Pre-trained language models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', BERT) have significantly improved performance across a wide range of natural language processing (NLP) tasks, such as machine translation and text classification [24]–[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Typically, the models are pre-trained to derive generic language represen- tations by self-supervised training on large-scale unlabeled data and then are transferred to benefit multiple down- stream tasks by fine-tuning on limited data annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Existing pre-trained models usually adopt the encoder- decoder architectures, where an encoder encodes an input sequence as a fixed-length vector representation, and a decoder generates an output sequence based on the in- put representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Encoder-only models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', BERT [24]) usually pre-train a bidirectional transformer in which each token can attend to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Encoder-only models are good at understanding tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', code search), but their bidirectionality nature requires an additional decoder for generation tasks, where this decoder initializes from scratch and cannot benefit from the pre-training tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Decoder-only models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', GPT [25]) are pre-trained using unidirectional language modeling that only allows tokens to attend to the previous tokens and itself to predict the next token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Decoder-only models are good at auto-regressive tasks like code completion, but the unidirectional framework is sub- optimal for understanding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Encoder-decoder models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', T5 [26]) often make use of denoising pre-training objec- tives that corrupt the source input and require the decoder to recover them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Compared to encoder-only and decoder- only models that favor understanding and auto-regressive tasks, encoder-decoder models can support generation tasks like code summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, we treat the patch correctness assessment as a binary classification task and we consider encoder-only models to get embeddings of code snippets according to existing work [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Inspired by the success of pre-trained models in NLP, many recent attempts have been adopted to boost numer- ous code-related tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', code summarization and code search) with pre-trained models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', GraphCodeBERT) [28], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Despite the promising results, little work aims to explore the capabilities of pre-trained models in sup- porting patch correctness assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, BERT is selected to exploit pre-trained models for automated patch correctness assessment, as it has been widely adopted in various code-related tasks and is quite effective for classi- fication tasks [28], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Two advanced BERT-style models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', CodeBERT and GraphCodeBERT) are also selected to investigate the generalization ability of APPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 3 APPROACH Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 2 presents the overall framework of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Generally, APPT accepts a buggy program and a plausible patch that passes the available test cases as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' APPT ex- tracts the buggy code snippet and its corresponding patched code snippet, and adopts four strategies to truncate the code tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' APPT then uses the pre-trained BERT model for embedding the truncated tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' After obtaining the representations for the buggy and patched code snippets, APPT uses four pre-defined functions for integrating the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Finally, APPT adopts a deep learning clas- sifier to return the final result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', correct or overfitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 Code Extraction Given a buggy program, existing APR tools may return a plausible patch p (if it exists) that passes all available test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Code extraction phrase aims to take the returned patch and the buggy program as the inputs, and output the corresponding buggy and patched code tokens (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Specifically, we get the buggy and patched code snippets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Cb and Cp) by parsing the patch file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Firstly, we select removed and added lines as the buggy and patched lines, marked with “+” and ‘-’, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Secondly, to keep the context information about the plausible patch, we keep unchanged lines (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', without +” and ‘- in the beginning) as part of each code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Finally, the buggy (or patched) code snippet are made up by the buggy (patched) lines and common context part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We treat the buggy (or patched) code snippet as se- quences of tokens and utilize a subword tokenization method to address out-of-vocabulary (OOV) problem by further breakdowning identifiers into their subtokens [30] when tokenizing the code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, we keep the original tokenization vocabulary instead of building a new vocabulary using byte pair encoding (BPE) algorithm as we want APPT to inherit the natural language understanding ability and start learning prediction from a good initial point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' After the buggy (or patched) code tokens are extracted, we attempt to take them as the inputs into the token em- bedding phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' However, pre-trained models are usually limited to a particular input length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, BERT can only take input sequences up to 512 tokens in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We further truncate the inputs whose length is longer than 512 after tokenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Following existing work [31], we use different methods to truncate the method pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' head-only: keep the first 512 tokens in Cb and Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' tail-only: keep the last 512 tokens in Cb and Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' mid-only: select 512 tokens in the middle of in Cb and Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' hybrid: select the first 256 and the last 256 tokens in Cb and Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In our experiment, we use the head-only method to truncate the code tokens by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also discuss the impact of different truncation methods in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Finally, the buggy and patched code tokens (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Tb and Tp) are extracted based on Cb and Tp to fit the maximum length limit of BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX 2022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Plausible Patch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Buggy Code Vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Patched Code Vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='con × ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='mix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Code Change ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Correct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Overfitting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Token ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Truncation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Head-Tail ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Mid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Tail ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='(b) Token Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='(a) Code Extraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='(c) Patch Classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Encoder Stack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Self-Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Feed-Forward ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Layer-Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Encoder 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Encoder 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Encoder 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Word Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Truncated Tokens ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Softmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='LSTM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='LSTM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='LSTM Stack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Integration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Patched Lines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Buggy Lines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Context Part ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Patched Code Snippet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Buggy Code Snippet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Buggy Tokens ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Patched Tokens ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Tokenizer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Code ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Tokenization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Code Snippets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Parsed Patch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 2: Overview of APPT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 Token Embedding Token Embedding phrase takes the buggy (or patched) code tokens (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Tb or Tp) as input and embeds it into the buggy (or patched) vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Eb or Ep) as output (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' APPT implements a stack of twelve layers of encoder blocks to extract the hidden state of the code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Each encoder block consists of three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The first part is a multi-head self-attention layer to learn long-range dependencies in the input code tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The second part is a simple, position-wise fully connected feed- forward neural network, which can linearly transform the token embedding for better feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The third part is a residual connection around each component, followed by a layer normalization to ensure the stability of code token embeddings distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In particular, the self-attention mechanism computes the representation of each code token by considering the position relationship between the code tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The self- attention mechanism mainly relies on three main vectors, query Q, key K, and value V , by mapping a query and a set of key-value pairs to an output vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We employ a scaled dot-product self-attention to calculate the attention scores of each token by taking the dot product between all of the query vectors and key vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The attention scores are then normalized to probabilities using the softmax function to get the attention weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Finally, the value vectors can be updated by taking a dot product between the value vectors and the attention weight vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The self-attention operation is computed using three matrices Q, K and V as follows: Attention(Q, K, V ) = softmax �QKT √dk � V (1) To capture richer semantic meanings of the input code tokens, we further use a multi-head mechanism to real- ize the self-attention, which allows the model to jointly attend the information from different code representation subspaces at different positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For d-dimension Q, K, and V , we split those vectors into h heads where each head has d/h-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' After all of the self-attention operations, each head will then be concatenated back again to feed into a fully-connected feed-forward neural network including two linear transformations with a ReLU activation in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The multi-head mechanism can be summarized by the fol- lowing equation: MultiHead(Q, K, V ) = Concat (head1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' , headh) W O (2) where headi = Attention(QW Q i , KW Q i , V W Q i ) and W O is used to linearly project to the expected dimension after concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Therefore, the encoder stack can take an input code snippet and output a real-valued vector for each code token within the code snippet based on the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3 Patch Classification After the embedding vectors of the buggy and patched code snippets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Eb and Ep) are extracted by the encoder stack, patch classification phrase first integrates the two vectors into a single input vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Econ) and then adopts a deep learn- ing classifier to predict the patch correctness automatically (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 Representations Integration Given two vectors Eb and Ep with n dimensions repre- senting the buggy and patched code snippets, respectively, we integrate the two vectors into one code changed vector for patch classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In detail, we leverage different ap- proaches to integrate them to characterize the differences between Eb and Ep from diverse aspects, such as an vector- wise concatenation operation Econ, element-wise addition operation Eadd , element-wise subtraction operation Esub, Hadamard product Epro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also attempt to capture crossed features between the two vectors by concatenating the above integrated vectors Emix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The integration approaches are selected due to their promising results in previous studies [12], [32], which are listed as follows: (1) Econ is a concatenation operation between Eb and Ep on vector-wise level with 2n dimension (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Econ = Eb � Ep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' (2) Eadd is an addition operation between Eb and Ep on element-wise level with n dimensions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Eadd = Eb+ Ep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' (3) Esub is a subtraction operation between Eb and Ep on element-wise level with n dimensions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Esub = Eb− Ep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' (4) Epro is a Hadamard product operation between Eb and Ep on element-wise level with n dimensions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Esub = Eb ⊙ Ep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' (5) Emix is a concatenation over Econ, Eadd, Esub and Epro on vector-wise level with 5n dimension (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Emix = Econ � Eadd � Esub � Esub).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 1902IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 LSTM Stack After the embedding vector (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Econ) of the changed code tokens is extracted, APPT aims to determine the given patch’s correctness based on a deep learning classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' To extract more hidden code change features, we further feed the code changed vector into a Long Short-Term Memory (LSTM) stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The LSTM stack has two bidirectional LSTM layers, the output of which is a new state generated by concatenating the hidden states from both directions at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' LSTM is a specialized recurrent neural network (RNN) for modeling long-term dependencies of sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' A common LSTM gate unit is composed of a cell, an input gate, an output gate and a forget gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Thanks to the gated mechanism, LSTM is well-suited to extract the contextual semantic features containing token sequential dependencies and has been widely used in various kinds of tasks, such as vulnerability detection [33], fault localization [34], and automated program repair [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In APPT, the LSTM stack computes a mapping from an input code changed vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', xT ) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Econ) to an output vector z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', zT ) by calculating the network gate unit activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We implement the gated mechanism by leveraging the input gates and forget gates to control the propagation of cell states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Specifically, when updating the cell state, the input gates decide what new information from the current input to be included in the cell states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Equation 3), and forget gates decide what information to be excluded from the cell states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Equation 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Based on new and forgetting information, cell states as the memory of the LSTM unit can be updated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Equation 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The output gate then determines the value for the next hidden state by point-wise multiplication of the output gate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Finally, the value of the current cell state passed through tanh function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Equation 7), by which the output of LSTM stack is calculated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Equation 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' it = sigmoid (Wixxt + Wihht−1 + bi) (3) ft = sigmoid (Wfxxt + Wfhht−1 + bf) (4) ct = ft ⊙ ct−1 + it ⊙ tanh (Wgxxt + Wghht−1 + bg) (5) ot = sigmoid (Woxxt + Wohht−1 + bo) (6) ht = ot ⊙ tanh (ct) (7) zt = Wzhht + bz (8) where the W terms denote weight matrices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Wix is the matrix of weights from the input gate to the input), the b terms denote bias vectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', bi is the input gate bias vector) and ⊙ denotes element-wise multiplication of the vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3 Classifier After the computation of all LSTM iterations, the embedding vectors of changed code tokens are further fed to a designed deep learning classifier to predict the patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The classifier is composed of two fully connected layers followed by a binary predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In APPT, we apply a standard softmax function to obtain the probability distribution over correct- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' A patch is labeled as correct if its probability of being correct is larger than that of being incorrect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' otherwise, it is considered overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In particular, for patch p, z denotes its output of the last iteration in the LSTM stack, which is further linearly trans- formed into a real number as Equation 9, where W ∈ Rd×1, b ∈ R, and n denotes the number of class (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', correct and overfitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We then leverage softmax function to normalize the output of patch p as Equation 10, where s denotes the correct or overfitting probability of patch p predicted by the model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' yi = Wzi + bi ∀i ∈ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' n (9) s (yi) = exp {yi} �n i=1 exp {yj} (10) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4 Training To train the network, we calculate the loss to update the neural weights based on its predicted result and ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We use the cross-entropy loss, which has been widely used in some classification tasks and patch prediction stud- ies [20], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In particular, gi ∈ {0, 1} denotes whether the i-th patch is correct or overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The cross-entropy loss compares a target gi with a prediction s in a logarithmic and hence exponential fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The objective function is computed in Equation 11, which is minimized constantly in the training to update the parameters in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' L = � i −[gi · log(s) + (1 − gi) · log(1 − s)] (11) We employ the dropout technique to improve the robust- ness of APPT and the Adam approach [37] to optimize the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 4 EXPERIMENT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 Research Questions The empirical study is conducted to answer the following research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' RQ1: How does APPT perform compared with existing state-of-the-art representation learning-based APCA techniques?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' RQ2: How does APPT perform compared with existing state-of-the-art traditional and learning-based APCA techniques?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' RQ3: To what extent do the different choices affect the overall effectiveness of APPT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1: To what extent do the token truncation choices affect the overall effectiveness of APPT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2: To what extent do the vector concatenation choices affect the overall effectiveness of APPT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 6 TABLE 1: APR tools in small benchmark Category APR Tools Heuristic-based jGenProg [41], jKali [41], jMutRepair [41], SimFix [42], ARJA [38], GenProg-A [38], Kali-A [38], RSRepair-A [38], CapGen [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Constraint-based DynaMoth [44], Nopol [45], ACS [46], Cardumen [47], JAID [48], SketchFix [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Template-based kPAR [50], FixMiner [51], AVATAR [52], TBar [5], SOFix [53], HDRepair [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Learning-based SequenceR [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3: To what extent do the pre-trained model choices affect the overall effectiveness of APPT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' RQ1 aims to compare APPT with 16 representation learning techniques to explore to what extent APPT outperforms these techniques, including three classifiers multiplied (de- cision tree, logistic regression, and naive Bayes) by five rep- resentation methods (BERT, code2vec, code2seq, Doc2Vec, and CC2Vec) from Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [12], and the most recent technique CACHE from Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' RQ2 is designed to investigate the effectiveness of APPT by comparing it with both dynamic and static techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The latest learning- based APCA technique, ODS, is also evaluated in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' RQ3 focuses on impact analysis of APPT, which is further refined into three sub-RQs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In detail, RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 explores how the four token truncation choices affect the effectiveness of APPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 explores how the five vector concatenation methods affect the effectiveness of APPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3 replaces BERT with advanced CodeBERT and GraphCodeBERT to investigate the impact of the pre-trained models on the effectiveness of APPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 Dataset With the rapid development of APR research in the last decades, a broad range of repair techniques has been pro- posed [38]–[40], resulting in a growing number of patches across many benchmarks being released [7], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The large- scale patch benchmarks enable deep learning-based pre- diction techniques to learn the distribution of correct and overfitting patches for patch correctness assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this study, we adopt two patch datasets based on the recent studies [12], [13], [20], a small one containing 1,183 Defects4J labeled patches and a large one containing 50,794 real-world labeled patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' On the small dataset, we mainly focus on the released patches from Defects4J [56], which is the most widely- adopted benchmark in APR research [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We select the benchmarks released by two recent large-scale studies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [13] and Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Specifically, the first benchmark [13] includes the labeled patches provided by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [7], Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [17] and Defects4J developers [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The second benchmark [12] includes the labeled patches from Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [7] and also considers the patches generated by some well-known APR tools that are not included in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [7] to better explore the overfitting problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', JAID [48], SketchFix [49], CapGen [43], SOFix [53] and SequenceR [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' To avoid the data leakage issue in the two benchmarks, a filtering process is also conducted to discard duplicate patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In particular, given a patch whose all the blank spaces are removed, the left text information is compared with that from the other patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' If two patches are iden- tical concerning their text information, they are considered TABLE 2: Datasets used in our experiment Datasets Subjects # Correct # Overfitting Total Small Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [12] 468 532 1,000 Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [13] 248 654 902 Our Study 532 648 1,183 Large ManySStuBs4J [57] 51,433 0 51,433 RepairThemAll [6] 900 63,393 64,293 Our Study 25,589 24,105 49,694 duplicates, resulting in 1,183 patches in our small dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The patches are generated by 22 distinct APR tools, which can be divided into four categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', heuristic-based, constraint-based, template-based, and learning-based tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The detailed information on these covered APR tools is presented in Table 1, where the first column lists the four repair technique categories and the second column list the corresponding repair techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' On the large dataset, we further consider a variety of patches generated from other benchmarks, to evaluate the generality of APPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Recently, existing studies demonstrate that APR techniques may overfit Defects4J in terms of repairability [6], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Thus, some other benchmarks have been conducted to evaluate the performance of APR tech- niques, such as Bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='jar [58], IntroclassJava [59], BEARS [60] and QuixBugs [61], providing substantial patches on the large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, we consider a large patch dataset released by a recent study [20] to investigate the generality of APPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The large patch dataset includes the labeled patches provided from RepairThemAll framework [6] and ManySStuBs4J [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In particular, RepairThemAll framework [6] contains 64,293 patches using 11 Java test- suite-based repair tools and 2,141 bugs from five diverse benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' However, there exists an imbalanced dataset issue as over 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6%2 (63,393/64,293) of generated patches are actually labeled as incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Recent studies have re- vealed that a well-balanced dataset is essential when investi- gating deep learning-based prediction techniques [12], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' To compensate the lack of correct patches, the large patch dataset then includes ManySStuBs4J [57], which provides simple bug-fix changes mined from 1,000 popular open- source Java projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The bug-fix changes are correct fix attempts of real-world bugs and thus are considered cor- rect patches in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Finally, a large balanced patch dataset is built from the RepairThemAll framework and ManySStuBs4J by discarding duplicate patches and filtering the ones from small student-written programming assignments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', IntroClassJava).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The dataset involves all 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The RepairThemAll Framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='com/ program-repair/RepairThemAll, accessed August 2022 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 7 TABLE 3: Compared APCA techniques in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' with Oracle Required without Oracle Required Dynamic-based Evosuite [15], Randoop [16], DiffTGen [62], Daikon [63] PATCH-SIM [17], E-PATCH-SIM [17], R-Opad [63], E-Opad [63] Static-based � ssFix [64], CapGen [43], Anti-patterns [14], S3 [65] Learning-based � ODS [18], Random Forest [12], Embedding learning [12], CACHE [20], Our proposed APPT denotes the representation learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' available patches generated on RepairThemAll framework and ManySStuBs4J, resulting in 49,694 patches after dedu- plication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Statistics on the two datasets are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Table 2 has two main rows representing the two datasets, each of which has three sub-rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The first and second sub-rows list the two sources in the corresponding dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The third column lists the filtered patches used in our experiment from the two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also present the number of correct, overfitting and total patches in the last three columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3 Baselines Various APCA techniques have been proposed in the litera- ture to validate patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Following existing studies [17], [20], we attempt to select state-of-the-art techniques designed for Java language as Java is the most targeted language in APR community [7] and the existing patches of real-world bugs are usually available in Java language [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We first consider the recent empirical study by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [13] to identify existing APCA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We then select recent advanced studies [12], [20] that are not included in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In general, following existing work [13], [20], the existing APCA techniques can be categorized into static, dynamic and learning-based APCA techniques according to whether test execution is needed or deep learning techniques are adopted (mentioned in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Meanwhile, according to whether the ground-truth patch is required, they can be further categorized into two categories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', with or without oracle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Particularly, similar to our proposed method APPT, CHCHE and embedding learning techniques adopt repre- sentation models to embed changed code and a deep learn- ing classier to predict patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Such techniques can be further considered as representation learning APCA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The details of the selected APCA techniques are illus- trated in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The first column lists three APCA cate- gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The second and third columns list whether the oracle information is equipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also list the representation learning techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', APPT) in the light gray box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We summarize the selected techniques as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 Dynamic-based APCA Techniques Dynamic-based techniques are designed to distinguish cor- rect patches from overfitting patches based on the outcome or the execution traces of the original or generated test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Simple Test Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The overfitting issue is preva- lent in the repair process due to the weak adequacy of existing test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Thus, researchers use test case generation tools to generate extra test cases based on the fixed program and check whether or not the generated patches that pass the original test cases can pass the extra test cases [23], [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, we adopt Evosuite [15] and Randoop [16] as the test case generation tools, as they have been widely investigated in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' DiffTGen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [62] identify overfitting patches by executing test cases generated by an external test generator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Evosuite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Different from simple test generation gener- ating test cases randomly, DiffTGen generates test cases to uncover the syntactic differences between the patched and buggy program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' A plausible patch is regarded as overfitting if the output of the patched program is not the same as that of the correct program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' DiffTGen needs a human- written patch as a reference and requires providing human- amenable testing information for the developers to provide oracles the generated test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Daikon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Daikon is a dynamic-based technique based on the program invariant with oracle information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [63] adopt the program invariant to explore the differences between an overfitting and a correct patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' A patch is considered correct if its inferred invariant is identical to that of the ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' If there exists a different comparison, the patch is considered overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' PATCH-SIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [17] consider the execution traces of the passing tests on the buggy and patched pro- grams are likely to be similar, while the execution traces of failing tests on the buggy and patched programs are likely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Based on the concept, they approximate the correctness of a patch based on the execution trace without the oracle information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' PATCH-SIM adopts Randoop to generate additional test cases to collect dynamic execution information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, we also replace Randoop with Evosuite to comprehensively explore the impact of test generation techniques (denoted as E-PATCH-SIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Opad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [67] adopt fuzzing testing to generate new test cases and employ two test oracles (crash and memory-safety) to enhance the validity checking of patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The original implementation of Opad is not designed for Java language and uses American Fuzz Lop (AFL) as the fuzzing technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, following recent studies [13], [20], we replace AFL with Randoop and Evosuite to generate new test cases on the Java programs and denote them as R-Opad and E-Opad, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 Static-based APCA Techniques Static-based techniques usually adopt static analysis tools to extract some designed static features and then check patch correctness based on such features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 8 ssFix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' ssFix [64] is a static-based technique that utilizes token-based syntax representation to generate patches with a higher probability of correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' ssFix first performs a syntactic code search to find code snippets from a codebase that is syntax-related to the context of a bug to generate correct patches, and then prioritizes the patches based on the modification types and the modification sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' CapGen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [43] propose three aspects of context information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', genealogy contexts, variable contexts and dependency contexts) embedded in an AST node and its surrounding codes to prioritize correct patches over overfit- ting ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, following recent studies [13], [20], we extract the three context information as static features to investigate patch correctness assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Anti-patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [14] define a set of rules that essentially capture disallowed modifications to the buggy program, and a patch is overfitting if it falls into the rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' A recent study [13] has shown that the manually-defined anti-patterns may have false positives for correct patches, resulting in destructive effects in patch correctness predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Le et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [65] assume that a correct patch is often syntactically and semantically close to a buggy code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Thus, they adopt six syntactic features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', AST differenc- ing, cosine similarity and locality of variables and constants) and semantic features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', model counting, output coverage and anti-patterns) to measure the distance between a candi- date patch and the buggy code snippet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3 Learning-based APCA Techniques Learning-based techniques can predict whether a plausible patch is correct or not based on machine learning tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' ODS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [18] first extract 202 code features at the abstract syntax tree level and then use supervised learning to learn a probabilistic model automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The results show that ODS can achieve better prediction performance than the dynamic-based technique PATCH-SIM with a faster speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' CACHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [20] propose a context-aware APCA technique CACHE by taking both the changed code snippet and the correlated unchanged code snippet into considera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' CACHE first parses the patched code snippet into AST representation and then utilizes the AST path technique to capture the structure information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Random Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [13] investigate the effective- ness of adopting deep learning models to predict patch cor- rectness based on eight static features (two from ssFix, three from S3, and three from CapGen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' To integrate the static features, six widely-used classification models (including Random Forest, Decision Table, J48, Naive Bayes, Logistic Regression, and SMO) are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The results demonstrate that Random Forest can achieve both superior precision and recall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, following existing work [20], we also adopt Random Forest to predict the patch correctness based on the integrated static features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Embedding Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [12] propose to leverage representation learning techniques to produce embedding for buggy and patched code snippets and then adopt su- pervised learning classifies to predict patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In particular, nine representation learning APCA techniques are evaluated, involving three embedding techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', CC2vec, BERT and Doc2Vec) and three classifiers (logistic regression, decision tree and naive bayes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4 Model Selection To the best of our knowledge, APPT is the first automated patch correctness prediction technique by fine-tuning the existing pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this paper, we adopt BERT as the encoder stack due to its powerful performance in previous work [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Specifically, BERT is pre-trained on large amounts of text data with two self-supervised goals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', masked language modeling (MLM) and next sentence prediction (NSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' MLM aims to let the model predict the masked words by masking 15% of words in each sentence randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' NSP aims to further improve the model’s ability to understand the rela- tionship between two sentences by letting the model predict whether the given sentence pair is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The model then can be fine-tuned to adapt to some specific downstream tasks and has achieved remarkable state-of-the-art results on a variety of natural language processing tasks, such as question answering and language inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' There exist two model architectures at different sizes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', BERTbase and BERTlarge [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The former has 12 layers and 12 attention heads, and the embedding size is 768, while the latter has a double layer number and 16 attention heads, and the embedding size is changed to 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this paper, we do not modify the vocabulary size and use the pre-trained BERTbase as the fine-tuning starting point instead of starting from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this paper, APPT is conceptually and practically gen- eralizable to various pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also select CodeBERT and GraphCodeBERT as the encoder stack to evaluate the scalability of APPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' CodeBERT and Graph- CodeBERT share the same model architecture as BERT, while utilizing paired natural language and programming language to pre-train the model to support code-related tasks (mentioned in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5 Evaluation Metrics We evaluate the prediction performance of various APCA approaches by accuracy, precision, recall, F1-score and AUC metrics, which have been widely adopted in patch correct- ness assessment research and other classification tasks [12], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Given the number of true positives (TPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' a TP refers to an overfitting patch that is identified as overfitting),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' false positives (FPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' a FP refers to a correct patch that is identified as overfitting),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' false negatives (FNs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' a FN refers to an over- fitting patch is identified as correct) and true negatives (TNs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' a TN refers to a correct patch that is identified as correct),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' the metrics are defined as follows: Accuracy: the proportion of correctly reported (whether the patch is correct or not) patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Accuracy measures the probability that the prediction of APCA techniques is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Accuracy = TP + TN TP + FP + FN + TN (12) Precision: the proportion of real overfitting patches over the reported overfitting patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Precision measures how IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 9 much we can trust the APCA techniques when it predicts a patch as overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Precision = TP TP + FP (13) Recall: the proportion of reported overfitting patches over all the real overfitting patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Recall measures the ability of the APCA techniques to find all the overfitting patches in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Recall = TP TP + FN (14) F1-score: twice the multiplication of precision and recall divided by the sum of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' F1-score measures the trade-off between precision and recall by taking their harmonic mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' F1-score = 2 ∗ Precision ∗ Recall Precision + Recall (15) AUC: the entire two-dimensional area underneath the entire receiver operating characteristic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' AUC mea- sures the probability that the classifier will rank a randomly chosen overfitting patch higher than that of a randomly chosen correct patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The higher the AUC, the better the APCA techniques is at predicting real overfitting patches as overfitting and real correct patches as correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' AUC = � I �Poverfitting , Pcorrect � M × N I �Poverfitting, Pcorrect � = � � � 1, Poverfitting > Pcorrect 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5, Poverfitting = Pcorrect 0, Poverfitting < Pcorrect (16) where M and N denote the number of overfitting and correct patches, while Poverfitting and Pcorrect denote the pre- diction probability for the overfitting and correct patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6 Implementation Details All of our approaches are built based on PyTorch frame- work3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We use the Hugging Face4 implementation version of BERT in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Considering previous work recom- mendation [26], [40], we utilize “bert-base-uncased” (refer to BERTbase) as the initial point, as the base version is quite lightweight to employ in practice with comparable effec- tiveness compared against the large version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' There exist 12 layers of transformer blocks and 12 self-attention heads in the “bert-base-uncased” model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The optimizer is Adam [37] with 5e − 5 learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The batch size is 16 and dropout rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We train for most 50 epochs and the max length of the input is set to 512 due to model limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' All the training and evaluation of our methods are conducted on one Ubuntu 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3 server with two Tesla V100-SXM2 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 5 RESULTS AND ANALYSIS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 RQ1: Comparing with Representation Learning- based APCA Techniques 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 Experimental Design As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3, APPT, CACHE and embedding learning techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', techniques within the light gray 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='org/, accessed August 2022 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Hugging Face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='co/, accessed August 2022 box in Table 3) can be categorized as representation learning APCA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this section, we aim to explore the per- formance of APPT when compared with these representa- tion learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In particular, embedding learning techniques [12] mainly adopt embedding models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', BERT, Doc2Vec, and CC2Vec) to embed buggy and patched code fragments, and then train classification models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Deci- sion Tree, Logistic Regression, and Naive Bayes) to predict patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Following previous study [20], we also consider two additional embedding models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', code2vec and code2seq) in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Meanwhile, CACHE can also be considered as a representation learning technique, which incorporates the context information in embedding code changes, and trains a deep learning classifier to predict the patch correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In total, 16 representation learning techniques are con- sidered in our experiment, involving five embedding tech- niques multiplied by three classification models, and one context-aware representation learning technique CACHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Following the previous study [12], we perform a 5-fold cross-validation on both the small and large datasets for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 Results Comparison results against the existing representation learning techniques are presented in Table 4 to Table 5 for the both small and large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The first column lists the three classifiers and the second column lists the five embed- ding approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The remaining columns list the detailed values of accuracy, precision, recall, F1-score and AUC met- rics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We present the most recent representation learning work CACHE and our APPT in the bottom part of Table 4 and Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' It can be observed that APPT achieves the best performance under each experimental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' On the small dataset, APPT is around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% higher than the state-of-the-art technique CACHE in terms of all metrics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% for accuracy, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% for precision, 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% for recall, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% for F1-score, and 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% for AUC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Compared with all representation learning tech- niques, APPT achieves the best performance in terms of accuracy, precision, F1-score and AUC metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In particular, the values of APPT on the accuracy and precision metrics are 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% and 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7%, respectively, while the optimal values of all other techniques are 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% and 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' This suggests that APPT can generally achieve the most accurate predic- tions, and the patches identified as overfitting by APPT are of high confidence to be overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Regarding recall, the values of CC2vec and code2vec can sometimes exceed those of APPT since they tend to classify most patches as overfit- ting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', CC2vec with Naive Bayes classifies 1,051 out of 1,183 patches as overfitting and thus achieves a high recall of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' However, these techniques achieve relatively low precision (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', CC2vec with Naive Bayes classifier has only 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2% for recall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' On the contrary, APPT can achieve a high recall exceeding 81% while maintaining a high precision of 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' On the large dataset, we can find APPT achieves over 99% for the five metrics, outperforming all existing ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, APPT reaches 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% in terms of AUC, which is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% higher than the second highest value IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 10 TABLE 4: Effectiveness of APPT compared with representation learning-based APCA techniques on the small dataset Classifier Embedding Accuracy Precision Recall F1-score AUC Decision Tree BERT 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% CC2vec 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% code2vec 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% code2seq 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% Doc2Vec 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% Logistic Regression BERT 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% CC2vec 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% code2vec 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2% code2seq 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% Doc2Vec 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% Na¨ıve Bayes BERT 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% CC2vec 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% code2seq 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% Doc2Vec 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% CACHE 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% APPT 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% 80.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% obtained from the most recent technique CACHE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' This suggests that APPT is more capable of dis- tinguishing correct and overfitting patches than CACHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Besides, the improvement against CACHE for accuracy, precision, recall and F1-score metrics achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also find that the perfor- mance achieved on the large dataset is commonly higher than that achieved on the small dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, the average value among the five metrics increases from 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='06% to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='26%, resulting in a 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% improvement rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Based on our analysis on the two datasets, the possible reason for this improvement is that bugs on the large dataset are usually simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We observe that all ManySStuBs4J patches on the large dataset are single-line operations, while patches on the small dataset usually cross multiple lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', more than 40% of Defects4J developer patches are multiple line patches [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' It is easy for the neural networks to learn the correctness distribution of such simple code changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Meanwhile, the difference in patch scale between the two datasets may be the second reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We find there exist 49,694 patches on the large dataset, which is 42 times larger than that of the small dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The amount of training data is often the single most dominant factor that determines the performance of the neural networks [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' More available patches benefit the neural networks to learn diverse code changes better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Answer to RQ1: Overall, our analysis on representation learning techniques reveals that (1) APPT can outper- form a state-of-the-art representation learning technique CACHE under all metrics and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' (2) on the small dataset, APPT achieves 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% for accuracy and 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% for AUC, which surpass CACHE by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' (3) on the large dataset, APPT exceeds 99% on all metrics, yet none of existing representation learning techniques achieves that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 RQ2: Comparing with Traditional and Learning- based APCA Techniques 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 Experimental Design In this section, we aim to further compare the proposed method APPT with the existing APCA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We select the remaining techniques mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3 (except IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 11 TABLE 6: Effectiveness of APPT compared with the traditional and learning-based APCA technique Category APCA Accuracy Precision Recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' F1-score Dynamic-based w-oracle Evosuite 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% Randoop 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2% DiffTGen 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% Daikon 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% wo-oracle R-Opad 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% E-Opad 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% PATCH-SIM 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% E-PATCH-SIM 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% Static-based Anti-patterns 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% S3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% ssFix 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% CapGen 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% Learning-based Random Forest 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% ODS 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% APPT 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% representation learning techniques discussed in RQ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In total, 14 APCA techniques are considered in the experiment, involving four static techniques (Anti-patterns, ssFix, Cap- Gen and S3), eight dynamic techniques (Evosuite, Randoop, DiffTGen, Daikon, R-Opad, E-Opad, PATCH-SIM and E- PATCH-SIM) and two learning techniques (Random Forest and ODS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' As it is time-consuming to run all the techniques (espe- cially for dynamic and learning ones), following the existing work [20], we reuse the released results from the recent work [13], [18], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We collect the detailed results of all selected APCA techniques from Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [20], which are concluded based on 902 patches (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [13] in Table 2) and a 10-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' To fairly compare with all the state-of-the-art techniques, we perform our experiment in the same experimental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 Results The experiment results are listed in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The first two columns list the selected techniques and their corresponding categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The remaining columns list the detailed values of accuracy, precision, recall and F1-score metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Compared with traditional dynamic-based and static- based APCA techniques, we can find that APPT reaches 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4%, 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% and 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% in terms of accuracy, recall and F1-score, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Specifically, APPT achieves the best overall performance with the three metrics, and none of the previous techniques exceeds 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' As for precision, more than 91% of patches reported by APPT are indeed overfit- ting patches, which is better than all static-based techniques and three dynamic-based techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Daikon, PATCH- SIM, and E-PATCH-SIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Although some dynamic ones have higher precision values, it is time-consuming to gen- erate additional test cases and collect run-time information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' More importantly, the recall of these techniques is usually low (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3% for R-Opad), or the ground-truth oracle is needed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Evosuite and Randoop techniques), limiting the application of such techniques in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Compared with learning-based techniques, we find that APPT still performs better than a state-of-the-art technique ODS with respect to all four metrics (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% for accuracy, 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4% for precision, 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% for recall, 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% for F1-score, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Overall, the improvement against Random Forest and ODS reaches 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5%∼17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1%∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Considering that it is expen- sive for ODS to extract hundreds of manually-designed code features at AST level, our approach simply adopting the pre-trained model to encode a sequence of tokens is even more promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also highlight this direction of integrating code-aware features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', code edits and AST representation) with pre-trained models for patch correct- ness assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Answer to RQ2: Overall, our comparison results reveal that, (1) APPT can achieve remarkable performance com- pared to exiting static-based techniques with a high re- call reaching 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' (2) APPT can achieve higher pre- cision than a state-of-the-art dynamic-based technique PATCH-SIM by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' (3) compared with existing learning- based techniques, APPT can achieve the best performance among all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3 RQ3: The Impact Analysis 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 Experimental Design To further explore how different fine-tuning choices affect the prediction performance of pre-trained models, we first consider and replace the head-only token truncation with other truncation methods, such as hybrid, mid-only and tail- only token truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We then adopt different methods to merge the buggy method vector and patched method vec- tor, such as concatenate, additional, subtraction, and prod- uct operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also mix the above-mentioned merged vectors as an additional concatenation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Recently, following the BERT model architecture, researchers use some code-related pre-trained tasks to capture the semantic connection between natural language and programming language, so as to further adapt these pre-training models for programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Thus, we replace the BERT with two advanced models pre-trained with the programming language, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', CodeBERT [28] and GraphCodeBERT [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 Results: The Impact of Token Truncation Choice Table 7 presents the prediction results under different trun- cation choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The first column lists the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 12 TABLE 7: Effectiveness of APPT with different truncation choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Dataset Truncation Accuracy Precision Recall F1-score AUC small APPThybrid 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='04% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='67% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='34% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='92% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='43% APPThead 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='72% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='84% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='17% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='76% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='55% APPTmid 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='48% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='27% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='41% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='85% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='34% APPTtail 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='20% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='00% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='40% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='38% 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='45% large APPThybrid 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='13% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='09% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='13% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='11% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='86% APPThead 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='04% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='17% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='86% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='01% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='54% APPTmid 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='36% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='62% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='17% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='35% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='18% APPTtail 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='85% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='28% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='30% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='77% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='49% TABLE 8: Effectiveness of APPT with different concatenation choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Dataset Truncation Accuracy Precision Recall F1-score AUC small APPTconcat 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='04% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='67% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='34% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='92% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='43% APPTaddition 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='83% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='24% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='12% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='83% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='44% APPTsubtraction 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='38% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='42% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='27% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='72% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='59% APPTproduct 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='27% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='37% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='32% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='81% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='46% APPTmix 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='90% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='21% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='18% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='64% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='46 large APPTconcat 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='13% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='09% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='13% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='11% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='86% APPTaddition 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='96% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='80% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='07% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='93% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='81% APPTsubtraction 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='31% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='14% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='29% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='17% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='46% APPTproduct 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='82% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='88% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='69% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='78% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='78% APPTmix 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='10% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='99% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='17% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='08% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='79% second column lists the four truncation choices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', head- only, mid-only, tail-only and hybrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The remaining columns list the detailed values of accuracy, precision, recall and F1- score and AUC metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' On the small dataset, we can find that the head-only approach achieves the optimum performance for accuracy (79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='72%), precision (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='84%), recall (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='84%) and F1-score (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='76%), while the hybrid approach achieves the optimum AUC score (83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='43%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The mid-only approach, considering the middle tokens in the buggy and patched methods, achieves the third-best performance for all metrics, followed by the tail-only approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Similar performance can be ob- served on the large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, the head-only and hybrid approaches have the best performance in all metrics, while the mid-only and tail-only ones are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The results demonstrate that the head-only approach extracting the beginning code tokens is effective in distinguishing the buggy and patched code snippets for the pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3 RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 Results: The Impact of The Vector Concate- nation Choice Table 8 presents the prediction results under different con- catenation choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The first column lists the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The second column lists the five concatenation choices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', concat, addition, subtraction, product and mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The remain- ing columns list the detailed values of accuracy, precision, recall and F1-score and AUC metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' On the small dataset, although conceptually simple, APPTconcat can obtain 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='04%, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='67%, 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='34%, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='92% and 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='43% for accuracy, precision, recall, F1-score and AUC metrics, four of which are highest among all investigated concatenation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' APPTproduct has the highest recall score (96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='32%), while it performs worse than APPTconcat by 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='77%, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='30%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='11% and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='97% for the other four metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' APPTaddition and APPTsubtraction perform the ad- dition and subtraction operation for buggy and patched vectors, and have similar performance for all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Mean- while, a mixed method APPTmix that applies these different comparison functions to represent the changed embedding vector can achieve better results than APPTconcat, which is also consistent with the existing study results [12], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Such results indicate that the pre-trained model can better capture the code change information by integrating differ- ent concatenation ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' On the large dataset, APPTconcat achieves the best performance in accuracy, F1-score and AUC metrics, while APPTsubtraction and APPTmix perform best in precision and recall respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The difference in performance is similar as the methods have relatively high metric values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, all metric values are higher than 99% for APPTconcat and APPTmix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4 RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3 Results: The Impact of Pre-trained Model Choice Table 9 demonstrates the predicted performance of three pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The first column lists the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The second column lists the three models , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', BERT, CodeBERT, and GraphCodeBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The remaining columns list the detailed values of accuracy, precision, recall and F1- score and AUC metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Generally speaking, all of the adopted models achieve a higher performance than state-of-the-art technique CACHE on all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, on the small dataset, BERT, CodeBERT and GraphCodeBERT reach 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9%, 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3%, and 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% with respect to the F1-score, which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9%, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3%, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% higher than CACHE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' A similar im- provement can also be observed on the large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' This demonstrates the model choice may not impact the per- formance dramatically, and pre-trained models can consis- tently achieve state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Specifically, to compare the performance of different pre- trained models, we can observe that both CodeBERT and GraphCodeBert achieve a better value for all metrics on the IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 13 TABLE 9: Effectiveness of APPT with different pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Dataset Model Accuracy Precision Recall F1-score AUC small APPTbert 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='04% (↑ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='67% (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='34% (↑ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='92% (↑ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='34% (↑ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1) APPTcodebert 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='49% (↑ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='10% (↑ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='6) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='73% (↑ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='35% (↑ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='32% (↑ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0) APPTgraphcodebert 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='83% (↑ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='68% (↑ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='63% (↑ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='47% (↑ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='79% (↑ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5) large APPTbert 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='13% (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='09% (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='13% (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='11% (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='86% (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0) APPTcodebert 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='57% (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='71% (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='40% (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='55% (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='89% (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0) APPTgraphcodebert 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='61% (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='61% (↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='7) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='59% (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='4) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='60% (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='90% (↑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0) ↑ denotes performance improvement against state-of-the-art technique CACHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' small dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' This superior performance also generalizes to large datasets, where CodeBERT and GraphCodeBert have better or competitive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', AUC) performance on the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' One possible explanation for this is that BERT is designed for natural language processing tasks, while CodeBERT and GraphCodeBERT regard a source code as a sequence of tokens or graph representation and then pre- train models on source code to support code-related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' This indicates that although pre-trained models in NLP can achieve state-of-the-art performance for assessing patch correctness, the adoption of pre-trained models targeting source code can further boost the improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Answer to RQ3: The performance under different choices demonstrates that: (1) the beginning code tokens can rep- resent the buggy and patched code snippets well for the pre-trained model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' (2) the concat of buggy and patched vectors is better than other methods to distinguish the changed code snippets, while the integration of differ- ent concatenation ways can achieve optimum results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' (3) advanced pre-trained models can provide a stable even better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 6 DISCUSSION 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 Threats to Validity To facilitate the replication and verification of our exper- iments, we have made the relevant materials (including source code, trained models, and patch data) available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Despite that, our study still faces some threats to validity, listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The first threat to validity lies in the patch benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We focus on the Defects4J database with reproducible real faults and collect 1,183 patches generated by existing APR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' However, the patch benchmark may not consider all available APR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' To address this, following the latest work [20], we include the 22 APR tools covering four cate- gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' It should be worth noting that although the learning- based category contains only SequenceR, it contains 73 patches, which is the largest number for a single APR tool [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also mitigate the potential bias by using multiple evaluation metrics to exhaustively assess the APCA tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Further, we adopt another large benchmark contain- ing 49,694 real-world patches to evaluate the generalization ability of the studied techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Overall, to the best of our knowledge, the used patch benchmarks are the largest set explored in the literature on patch correctness assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The second threat to validity is that the performance of APPT may not generalize to other pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We select BERT in our experiment due to its powerful perfor- mance in recent code-related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' However, it is unclear whether the conclusions in our experiment (discussed in Section 5) can be maintained when using other pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We have mitigated the potential threat by using CodeBERT and GraphCodeBERT to demonstrate the per- formance of APPT under different pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The investigated pre-trained models include both code-related ones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', CodeBERT) and natural language-specific ones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', BERT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also rely on two diverse patch benchmarks to ensure the generality of the experimental conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The last threat to validity is the implementation of the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In our work, we compare APPT against a wide range of APCA techniques with different categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Imple- menting these baselines may introduce a potential threat to the internal validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' To mitigate this threat, following the recent work [20], we conduct the experiment under the same setting and reuse the released results from the original work [12], [13], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Further, we carefully check the reused results and publicly release all our materials for further verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 Comparison with BATS In our work, following some recent APCA work [12], [13], 30 related APCA techniques with different categories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', 16 representation learning-based ones, 9 dynamic-based ones, 4 static-based ones and 2 learning-based ones) are compared in our experiment (discussed in Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' To the best of our knowledge, the selected baselines are the largest set on patch correctness prediction in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' However, there may exist other possible techniques that could have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, the recent BATS [19] predicts patch correctness based on the similarity of failing test cases, which can be complementary to the state-of-the-art APCA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We do not include BATS in our experiment (discussed in Section 5) because it requires historical test cases as the search space for searching similar cases, which are not available in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We then perform an additional evaluation by assessing APPT on the dataset provided in BATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' However, BATS fails to assess some plausible patches as it considers only historical test cases with the similarity which are higher than a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, BATS with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8 threshold value is able to predict only 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='9% (114/1278) of the plausible patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Thus, we compare APPT against BATS with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0 threshold value, which can perform prediction for all patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We also compare APPT against BATS with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8 threshold value, as it achieves the best recall, F1-score and AUC performance among all threshold values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The results are presented in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The first column lists APPT and BATS (with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0 and IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 14 TABLE 10: Comparison with a state-of-the-art learning- based APCA technique BATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' APCA #Patch Accurancy Preciosn Recall F1-score BATS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0) 1278 (1278) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='50% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='81% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='82% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='94% BATS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8) 114 (1278) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='54% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='16% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='21% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='18% APPA 1278 (1278) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='05% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='39% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='38% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='88% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8 threshold values, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The second column lists the number of predicted patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Each cell is represented as x(y), where x is the number of patches predicted by APPT and BATS and y is the total number of patches in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The remaining columns list the detailed performance under the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We can find APPT achieves 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='39%∼85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='05%, improving the metrics by 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='56%∼34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='58% when compared with BATS (threshold is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' When the threshold of BATS is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8, APPT can still improve the metrics by 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='40% on average while predicting 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1% more plausible patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Overall, the results demonstrate that APPT per- forms better than BATS in terms of the number of predicted patches and the prediction metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 7 IMPLICATION AND GUIDELINE Based on the observations in our experiment, we can sum- marize the following essential practical guidelines for future patch correctness assessment studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Simple features can work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Our study demonstrates that APPT, representing source code as a sequence of tokens, performs even better than the existing learning techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', CACHE) considering complex code-aware characteris- tics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', abstract syntax tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Also, the token sequences can already outperform manually-designed static features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', the line number) and time-consuming dynamic features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', code coverage) in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Such observations indi- cate that simple features, such as code sequences, should not be just ignored and a systematic study to explore the impact of different code representations is needed in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In fact, they should be considered and even integrated with different features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', data flow graph) to design more advanced patch correctness assessment techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The quality of the training dataset is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We can find that APPT achieves 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='5% precision in Table 4 while the precision is decreased by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='8% in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Similar performance can also be observed in Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The results show that more training data cannot always lead to better performance for patch correctness assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' It is crucial to automatically select the most informative training set that represents the whole patch benchmarks to optimize the prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, it is interesting to explore how the number of patches is distributed across fix patterns and how to select balanced patches for each fix pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Future work can also be conducted to investigate training data selection approaches targeting specific bug benchmarks under prediction or even specific bug types under prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Pre-trained model-based APCA techniques require more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Our results show that the BERT-based APPT performs even better than the state-of-the-art APCA tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Also, the CodeBERT-based and GraphCodeBERT- based APPT can further enhance the prediction effective- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Such observation motivates future researchers to in- vestigate more advanced APCA techniques by employing different pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, it is interesting to propose domain-specific pre-trained models by designing repair-related pre-training tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Meanwhile, thorough eval- uations are recommended to explore how different features, such as bug types and fix patterns, influence the perfor- mance of pre-trained models in patch correctness prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 8 RELATED WORK In this paper, we adopt pre-trained language models to predict patch correctness generated by off-the-shelf auto- mated program repair tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Our work is related to auto- mated program repair, patch correctness assessment and pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We have introduced the existing work about patch correctness assessment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Thus, in this section, we focus on and discuss the existing work on automated program repair techniques (Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1) and pre- trained models (Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 Automated Program Repair Over the past decade, researchers have proposed a variety of techniques to generate patches based on different hypothe- ses [1], [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Following recent work [2], [7], [11], we cate- gorize them into four main categories: heuristic-based [38], [41], [70], constraint-based [44], [45], [71], template-based [5], [51], [52] and learning-based repair techniques [35], [39], [40], [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Heuristic-based repair techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' These techniques usu- ally use a heuristic algorithm to find a valid patch by iteratively exploring a search space of syntactic program modifications [38], [41], [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Among them, GenProg [70] proposed in the early days has been considered a seminal work in this field, which uses genetic programming to search for correct repairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' GenProg represents candidate re- pairs as sequences of edits to source code and evaluate them by the execution results of test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Those candidates that pass more test cases are considered to have a higher fitness and are iteratively applied to produce new candidates based on mutation and crossover operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The recent SimFix technique [42] utilizes code change operations from existing patches across different projects and similar code snippets within the buggy project to build two search spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Then, the intersection of the above two search spaces is further used to search the final patch using basic heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Constraint-based repair techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' These techniques mainly focus on repairing conditional statements, which can repair more than half of the bugs repaired by existing APR approaches [44], [45], [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In detail, these techniques transform the patch generation into a constraint-solving problem, and use a solver to obtain a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, Nopol [45] relies on an SMT solver to solve the condition synthesis problem after identifying potential locations of patches by angelic fix localization and collecting test execution traces of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Among them, ACS [46] refining the ranking of ingredients for condition synthesis is considered one of the most advanced constraint-based repair techniques [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Template-based repair techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' These techniques gener- ate patches by designing pre-defined fix patterns to mutate IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 15 buggy code snippets with the retrieved donor code [5], [51], [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [5] revisit the repair performance of repair patterns using a systematic study that evaluates the effectiveness of a variety of fix patterns summarized from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Among them, the recent PraPR technique [73] is able to generate plausible and correct patches for 148 and 43 real bugs, respectively, which is the largest number of bugs reported as fixed for Defects4J when published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Learning-based repair techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' These techniques at- tempt to fix bugs enhanced by machine learning techniques [30], [35], [39], [74]–[76] and are getting increasing attention recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, Tufano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [75] extensively evalu- ate the ability of neural machine translation techniques to generate patches from bug-fixes commits in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [35] adopt a tree-based RNN encoder-decoder model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', DLFix) to learn code contexts and transformations from pre- vious bug fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Lutellier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [39] propose a new context- aware NMT architecture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', CoCoNut) that represents the buggy source code and its surrounding context separately, to automatically fix bugs in multiple programming lan- guages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In our experiment, we select 22 representative APR tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', SimFix, ACS, and SEQUENCER) from the four cate- gories, representing state-of-the-art techniques in the corre- sponding category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Then we evaluate APPT on the plausible patches (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', passing the original test cases) generated by these APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 Pre-trained Model Our approach is inspired by the application of pre-trained models in NLP and code-related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this section, we first introduce the existing studies about pre-trained models in NLP (Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1) and SE (Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We then discuss the application of pre-trained models to some code-related tasks in SE (Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='1 Pre-trained Model in NLP Recent work has demonstrated substantial gains on many NLP tasks and benchmarks by pre-training on a large corpus of text followed by fine-tuning on a specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For exam- ple, Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [24] propose a new language representation model BERT to pre-train deep bidirectional representations from the unlabeled text by jointly conditioning on both left and right contexts in all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' To explore the landscape of transfer learning techniques for NLP, Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [26] propose a text-to-text transfer transformer T5 by introducing a unified framework that converts all text-based language problems into a text-to-text format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [25] pro- pose an autoregressive language model GPT-3 without any gradient updates or fine-tuning, with tasks and few-shot demonstrations specified purely via text interaction with the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, we choose BERT to encode a given plau- sible patch into a fixed-length representation vector as the input of the deep learning classifier, due to the powerful performance of BERT in previous work [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2 Pre-trained Model in SE Inspired by the application of pre-trained models in NLP, many researchers apply the pre-trained model to code- related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Instead of designing new network architec- tures, SE researchers usually adopt existing architectures in NLP and design some code-aware pre-training tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', code-AST prediction and bimodal dual generation) to learn representations of the source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Then the pre- trained models are further fine-tuned to some diversified code-related tasks such as code-code (clone detection, de- fect detection, cloze test, code completion, code refinement, and code-to-code translation), text-code (natural language code search, text-to-code generation), and code-text (code summarization) scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [28] present a bimodal pre- trained model (CodeBERT) for natural language and pro- gramming languages by masked language modeling and replaced token detection to support code search and code documentation generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [29] present the first pre-trained model (GraphCodeBERT) that leverages code structure to learn code representation to improve code understanding tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', code search, clone detection, code translation, and code refinement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [27] present UniXcoder, a unified cross-modal pre-trained model for programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' UniXcoder utilizes mask attention matrices with prefix adapters to control the behavior of the model and leverages cross-modal contents such as AST and code comment to enhance code representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In contrast to most studies pre-training a large-scale model from scratch costly, we attempt to boost patch correctness assessment on top of the existing pre-trained language model fine-tuning paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' In this work, to further explore the generalization ability of APPT, we select other BERT-like models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', CodeBERT and GraphCodeBERT) as the encoder stack due to their powerful performance in the code-related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='3 Applications of Pre-trained Model in SE In addition to the above-mentioned typical code-related tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', automatic bug-fixing, injection of code mutants, generation of asserts in tests and code summarization in [78]), researchers have also applied pre-trained models to some other domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', code completion, and program repair) in SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' For example, Cinisell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [77] evaluate the performance of the BERT model in the task of code completion at different granularity levels, including single tokens, one or multiple entire statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' The results show that the model achieves promising results superior to state-of-the-art n-gram mod- els, and the model learns better on some specific datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', Android) when code abstraction is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Ciborowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [79] apply BERT to the bug localization problem with the goal of improved retrieval quality, especially on bug reports where straightforward textual similarity would not suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Recently, Salza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [80] investigate how transfer learning can be applied to code search by pre-training and fine-tuning a BERT-based model on combinations of natural language and source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Mashhadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' [81] propose a novel pre-trained model-based APR technique by fine- tuning CodeBERT on the ManySStuBs4J benchmark and find the approach generates fix codes for different types of bugs with comparable effectiveness and efficacy compared with state-of-the-art APR techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' XXX, XXX 2022 16 Although there exist some SE tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', code review and bug localization) benefitting from pre-trained models, in this work, we perform the first application of pre-trained models to predict the generated patch correctness in auto- mated program repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' 9 CONCLUSION In this work, we present APPT, a novel automated patch correctness prediction technique based on the pre-training model and classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We first adopt the off-the-shelf pre- trained model as the encoder stack and LSTM stack to enhance the dependency relationships among the buggy and patched code snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Then we build a deep learning classifier by two fully connected layers and a standard softmax function to predict whether the patch is overfitting or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We conduct experiments on both patch datasets and show that APPT significantly outperforms state-of-the-art learning-based and traditional APCA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We fur- ther demonstrate that APPT is generalizable to various pre- trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' Based on these observations, some impli- cations and guidelines on improving the existing learning- based techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=', the usage of simple features and pre- trained models) are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' We highlight the direction of applying pre-trained models to predict patch correctness automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idFMT4oBgHgl3EQf4zEN/content/2301.12453v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work is supported partially by the 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mode 100644 index 0000000000000000000000000000000000000000..145d0f82562c5eb00674f75c88d356802e641662 --- /dev/null +++ b/jdAzT4oBgHgl3EQf4_6S/content/tmp_files/load_file.txt @@ -0,0 +1,764 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf,len=763 +page_content='Kagome qubit ice Alejandro Lopez-Bezanilla1, Jack Raymond2, Kelly Boothby2, Juan Carrasquilla3,4, Cristiano Nisoli1,∗ and Andrew D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' King2† 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 2D-Wave Systems, Burnaby, British Columbia, Canada, V5G 4M9, Canada 3Vector Institute, University of Toronto, Toronto, Ontario, M5G 1M1, Canada and 4Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada (Dated: January 6, 2023) Abstract Topological phases of spin liquids with constrained disorder can host a kinetics of fractionalized excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' However, spin-liquid phases with distinct kinetic regimes have proven difficult to observe experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Here we present a realization of kagome spin ice in the superconducting qubits of a quantum annealer, and use it to demonstrate a field-induced kinetic crossover between spin-liquid phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Employing fine control over local magnetic fields, we show evidence of both the Ice-I phase and an unconventional field-induced Ice-II phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In the latter, a charge-ordered yet spin-disordered topological phase, the kinetics proceeds via pair creation and annihilation of strongly correlated, charge conserving, fractionalized excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' As these kinetic regimes have resisted characterization in other artificial spin ice realizations, our results demonstrate the utility of quantum-driven kinetics in advancing the study of topological phases of spin liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='01853v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='stat-mech] 4 Jan 2023 INTRODUCTION Dynamics in crystals typically proceeds via motion of topological defects such as dislo- cation gliding [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' One might expect the kinetics of disordered systems to be naturally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' But in spin liquids, where disorder is present but constrained, kinetics often also proceeds through defects or excitations are endowed with a conserved topological charge [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' For instance, frustrated spin systems, such as pyrochlore [3, 4] or square [5–7] spin ices, remain disordered at low temperature, leading to a Pauling residual entropy, and their disorder is constrained by the so-called ice rule [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' There, kinetics consists of creation/annihilation and walks of localized violations of the ice rule, in the form of emergent magnetic monopoles [9] that conserve a topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Among spin ices, the kagome ice model [10–18] has been widely studied because it mim- ics a remarkable variety of natural and artificial systems, from rare-earth pyrochlores [3], to nanomagnetic fabrications [19], gravitationally trapped colloids [15], and many other sys- tems [20–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Kagome spin ice can in principle manifest various unusual phases [13–15], but the large energy scales of artificial implementations pose an experimental challenge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' thor- ough measurements of these phases and the physical conditions driving the phase-to-phase transition are scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Here we present a kagome qubit ice realized in a superconducting quantum annealer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Using this experimental platform, we study its field-induced spin-liquid phases and quantum- activated kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' We experimentally establish that topological constraints affecting the dynamics proceeds via charge-conserving fractionalized excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Using thousands of programmable external magnetic fields, we detune the system from its more common ice- rule-obeying “Ice-I” phase into a field-induced “Ice-II” phase, which exhibits charge order while remaining spin-disordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' RESULTS Kagome spin ice Kagome spin ice consists of magnetic dipoles as classical binary Ising spins arranged along the edges of a hexagonal lattice and therefore on the sites of a kagome lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' They point from one triangular “ice vertex” (kagome plaquette) to another (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' We can 2 thus introduce the notion of a magnetic charge for a vertex, defined as the number of spins pointing toward the vertex minus those pointing away from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Because of the odd coordination, a vertex can host only nonzero, odd charges q = −3, −1, 1, 3 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' The simplest magnetic kagome model includes interactions only among spins impinging on the same vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Since not all pairs of spins at a vertex can simultaneously assume an energy-minimizing head-to-tail configuration, the system is frustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' The ground state is therefore an extensively degenerate ensemble of disordered spins obeying the (pseudo-) ice rule: frustration is minimized when each vertex has two spins pointing in and one pointing out, or vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' This ice manifold is often called the Ice-I phase, and can be thought as a spin liquid forming an overall neutral plasma of disordered ±1 magnetic charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In the Ice-II phase [13, 14, 29], disordered spins still still obey the ice rule but charges are ordered in an ionic lattice [30–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Kagome qubit ice In this work, we realize kagome spin ice in a quantum annealer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Its superconducting flux qubits are described by the transverse field Ising Hamiltonian HQ = −Γ � i ˆσx i + J � � i hiˆσz i + � ij Jijˆσz i ˆσz j � , (1) where ˆσx and ˆσz are Pauli matrices on the qubits, J is an energy prefactor on the classi- cal Ising Hamiltonian, hi are per-qubit programmable longitudinal fields [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' ], and Jij are programmable two-qubit couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Γ is a transverse field entangling the Pauli matrices and thus controls quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Kagome spin ice can be mapped to a classical Ising model [34], and therefore to the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Consider alternating A and B vertices pointing up (△) and down (▽) respectively in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' We assign an Ising spin value si = +1 if it points into the A vertex, and si = −1 if it points into the B vertex (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' (Compare with the Then, standard kagome ice corresponds to the Hamiltonian HI = J � ⟨i,j⟩ sisj + � i hisi (2) where each nearest-neighbor spin is coupled antiferromagnetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 3 We then embed the kagome lattice in the graph of available two-body couplers, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 1c, by modifying an embedding of a Z2 lattice gauge theory into the transverse-field Ising model [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Each kagome site is represented by a ferromagnetic three-qubit chain, and nearest-neighbor chains are coupled antiferromagnetically with two physical couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' (Three qubits are needed for each kagome lattice site because it is not possible to directly couple two arbitrarily-chosen qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=') We use h and J (with no index) to denote the total field on a three-qubit chain and the total coupling between two neighboring chains,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' obtaining the kagome qubit ice (KQI) Hamiltonian HKQI = −˜Γ � i ˜σx i + J � h � i ˜σz i + J � ⟨i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='j⟩ ˜σz i ˜σz j � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' (3) where ˜Γ = Γ3/J2 FM is an effective transverse field on the three-qubit chains for a ferromag- netic chain coupling JFM [36],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' ˜σi denotes a logical moment represented by a three-qubit chain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' and indices i and j are also over three-qubit chains,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' rather than individual qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Phases When Γ = ˜Γ = 0 and h = 0, the extensively degenerate ground state manifold of HKQI corresponds to that of HI, which is the commonly seen Ice-I phase [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' But we can go beyond this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In nanoscopic realizations, another phase of lower entropy is possible [30, 31, 33, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In such systems, it is driven by the long range nature of the dipolar interactions [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' It still has disordered ice-rule obeying spins, but with charges ordered in an ionic lattice where A and B vertices have opposite charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' While the spins remain disordered, though at lower entropy [13, 14, 38], their disorder is topologically constrained: it can be mapped to a dimer cover model [12, 38] and considered a case of classical topological order [2, 29, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' This is often called the Ice-II phase, and its topological nature should show a topologically protected kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' (Note also that Ice-II can also be considered a broken symmetry phase with unsaturated order parameter in the context of magnetic fragmentation [18, 41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=') Indeed, the kinetics in the Ice-I phase is not gapped: It is possible to flip a single spin— or indeed an extensive number of single spins—without violating the ice rule and thus without creating an excitation (see also Supplementary Informations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Thus, the system can kinetically explore the phase from within the local low energy manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Instead, in the Ice-II phase any individual spin flip disrupts the charge balance, thus 4 creating an excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Therefore [2] the kinetics of the Ice-II phase must proceed either via pair creation, motion, and annihilation of gapped excitations, or else via cooperative, ungapped flips of entire loops of head-to-tail spins which do not alter the charge distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Such kinetics was never probed in previous realizations of kagome ice because the Ice-II phase has proved very hard to reach [30, 31, 33, 37] (see Supplementary Informations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Fortunately, the quantum annealer offers another route: we can induce it by the field h, acting on σz, and then we can study field-induced Ice-II kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' If we define a staggered charge qs on a vertex such that qs = −q for A vertices and qs = q for B vertices, then the field h determines the vertex energies ε−3, ε−1, ε+1, ε+3 for vertices with qs = −3, −1, 1, and 3 respectively, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 2 (see also SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' For 0 < h/J < 4, ε+1 has the lowest energy, leading to the charge-ordered, spin-disordered Ice-II phase as the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Within this window, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 2 shows a regime crossover at h/J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' The lowest excitations are charge-order violations upsetting the ionic crystals of charges when 0 < h/J < 2, and ice rule violations when 2 < h/J < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' The two types of excitations are degenerate at h/J = 2 where the excitation gap is highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Then, for h/J > 4, the ground state degeneracy vanishes, replaced by an ordered state in which all A and B vertices have charge −3 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' To estimate pseudo-equilibrium properties of the kagome qubit ice in these different phases, we begin with a random spin state and repeatedly expose the system to quantum fluctuations as described by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' (1), by cycling the transverse field Γ on and off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' An appropriate magnitude of transverse field drives the kinetics of this kagome qubit ice without erasing the state memory, as previously demonstrated in square ice [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' After each exposure, we read out a classical spin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' This leads to a sequence of states amenable to statistics (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 3 summarizes experimental results for varying h/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 3a shows real-space samples, represented as vertex charges, for increasing values of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' At h = 0 we see the expected disordered charge plasma of the Ice-I phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Increasing h first leads to ionic ordering of the charge ( the Ice-II phase) eventually giving way to a polarized state in which the longitudinal field overcomes the ice rule, forming ionic crystals of ±3 charge, and all spins have value si = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 3b shows the corresponding result in reciprocal space via the Fourier transform of the spins defined as S(q) ∝ � ij eiq(ri−rj) (⟨sisj⟩ − ⟨si⟩⟨sj⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Our sign convention for the spins 5 leads to the appearance of peaks only in the Ice-II phase and its proximity, and the formation of pinch points in the topologically protected region with h/J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 3c, cuts of the Fourier transforms through the high-symmetry points in the extended Brillouin zone clearly show growing peaks at K in the proximity of the Ice-II phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' These peaks correspond to the expected logarithmic divergence of the dipolar correlations [12] (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 5 in ref [12], obtained from a dimer model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' They, and the pinch points, follow therefore from the topological properties induced on the phase by the charge ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' From an implementation point of view, S(q) reveals a highly symmetric system in which the multi-qubit embedding of kagome spins preserves isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' This is an important advance over previous work [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 3d plots the charge order parameter, defined as one third the average staggered charge of a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' The two two broad plateaus at ±1/3 correspond to the Ice-II phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 3e confirms the high ice-rule obedience throughout the Ice-I and Ice-II phases, which breaks down at h/J > |4| where, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 2, the lowest energy vertex no longer obeys the ice rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Topologically protected quasi-classical kinetics These measurements validate the annealer’s effectiveness as an experimental platform for probing phases of the Ising kagome spin ice system near a low-temperature thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Because consecutive output states are separated dynamically by a relatively short exposure to a relatively weak transverse field Γ (compared to J), we can also probe the quasi-classical kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' As mentioned above, in the Ice-II ground state a single spin flip always corresponds to fractionalized excitations, as either violations of the Ice-II charge-order constraint, or violations of the kagome ice rule (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' We can define a topological charge (or t-charge) as qt = q + 1, qt = q − 1 for A and B vertices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In the Ice-II charge-ordered ground state, the topological charge is zero on all vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Instead, excitations of the Ice-II phase are topologically charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Their t-charge is conserved: flipping a spin creates a pair of fractional excitations of t-charges ±2 and zero net t-charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Further flips can separate the t-charges, which can then annihilated when meeting other, opposite ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' This situation of paired fractional excitations is very reminiscent of square and pyrochlore ice [7, 43], although here the topological charge is not the magnetic charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 6 To probe the thermal and quantum-activated kinetics of the Ice-I and Ice-II phases, we compare QA output samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Between consecutive samples, the qubits are exposed to the a transverse field for 1 µs, and at the same time J is dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' This protocol is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Since the system is in a thermal bath at 12 mK, this allows both quantum and thermal fluctuations to drive dynamics [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In agreement with the description above, our results show a kinetics of fractionalized excitations, that can be created and annihilated in pairs of opposite topological charge, and more rarely a kinetics consisting of flips of entire loops of spins—which can always be construed mathematically as creations followed by annihilation of topologically charged pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 4b shows two representative samples from each of h/J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5, 4, correspond- ing roughly to the boundaries and the middle of the field-induced Ice-II phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Ice-rule and charge-order violations are shown as triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Between the two samples, we highlight the spins that flip during the exposure to fluctuations, as well as the motion of fractional excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' At h/J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5 the charge order is fragile and we are close to the Ice-I phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' We see many excitations popping up erratically, and they are charge order violations, due to their small energy cost (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' At h/J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5 we see far fewer excitations, and the kinetics consists of their wandering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' We also see flipping of closed loops of spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' One fractional excitation escapes off the boundary, one appears from the boundary, and one moves to another location through a chain of flipped spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' This picture is consistent with the large energy gap shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 2, which suppresses pair creation of excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' At h/J = 4, we again see a regime in which excitations can appear at low cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' these cheap excitations are now ice-rule violations, in contrast to the charge-order violations seen near the Ice-I phase, consistent with the energetics (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' To quantify the creation/annihilation and motion of fractional excitations, we consider the subgraph of the honeycomb lattice whose edges correspond to flipped spins (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 4c– d) between consecutive states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' We measure the degrees (valencies) of honeycomb sites in this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' A closed loop of flipped spins results in only degree-two honeycomb sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Conversely, an open chain of flipped spins will have degree two in the interior, and degree one on the ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' This can involve the motion of a fractional excitation, with or without 7 creation/annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In general, degree-two spins correspond to motion of excitations, while degree-one spins correspond to creation/annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 4e shows that the system is overall most active around h/J = 0 and h/J = 4, which corresponds to points of degeneracy (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 2) where excitations are cheapest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' The plot of the relative frequency of excitation motion over pair creation/annihilation shows a maximum around h/J = 2, the point of maximum gap: where excitations are most expensive, kinetics consists mostly of their random walk, much like monopoles in square or pyrochlore spin ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' The non-monotonicity of the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 3e shows that in kagome qubit ice, by tuning the gap of the phase, the topological protection of the kinetics can be controlled, from a hard to distinguish soup of excitations at h/J = 0, 4, to a clear picture of creation/annihilation and motion of fractionalized excitations around the value h/J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' DISCUSSION We have realized kagome qubit spin ice in 2742 superconducting flux qubits of a quantum annealing processor and explored its field-induced spin-liquid ice phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' We have studied the quantum-activated, topologically protected kinetics of the Ice-II phase and shown that it proceeds via creation/annihilation and propagation of charge-conserving fractionalized excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' We emphasize that quantum fluctuations are used here only to drive kinetics, but can be employed in the future to study entangled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Furthermore, the kagome antiferromagnet in a transverse field Γ has a rich ground-state phase diagram [44] arising from high-order perturbations in Γ, which may be probed in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Our results demonstrate that quantum annealers are capable of implementing exotic programmable phases of frustrated spin sliquids, whose gap and topologically-protected kinetic regimes can be finely tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 8 a A A A A B B B B A A A A B B B B c A B B B Ising spin (flux qubit) FM AFM b + \xad A B Ising +1 + \xad B A Ising –1 +3 +1 \xad1 \xad3 Ice\xadrule obeying Ice\xadrule breaking 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5 mm d FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 1: Kagome qubit ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' A, Kagome spin ice consists of magnetic dipoles on the edges of a hexagonal lattice, which point in or out of triangular plaquettes (vertices) of the dual kagome lattice (gray lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' B, Each vertex in a given configuration has a nonzero charge: ±1-charged vertices satisfy the kagome ice rule;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' ±3-charged vertices do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Denoting triangles pointing up and down by A and B respectively, one can map dipoles to Ising spins according to whether not the dipole points into an A triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' C, In the kagome qubit ice, each kagome site is realized using a ferromagnetically-coupled three-qubit chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Sites impinging on the same triangular ice vertex are coupled antiferromagnetically, leading to geometric frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' D, Optical image of the superconducting quantum annealing processor in a sample holder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 2742 qubits are used to realize a 913-spin kagome ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 9 0 1 2 3 4 5 6 −4 −2 0 2 4 Ice\xadII, first excitation is charge violation Ice\xadII, first excitation is ice rule violation Ordered phase gap maximized h/J Energies ε−3 ε+3 ε−1 ε+1 Ice\xadI FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 2: Ice vertex energies (normalized to J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In the Ice-I phase (h = 0), the ice-rule vertex states ε+1 and ε−1 are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Detuning h leads energetic preference towards the (staggered) +1-charged configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Within the ice region 0 ≤ h/J ≤ 4, the energy gap is maximized at h/J = 2, where charge-imbalance excitations are degenerate with ice-rule excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 10 2π π 0 −π −2π 2π π 0 −π −2π qx (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=') qy 2π π 0 −π −2π 2π π 0 −π −2π qx (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=') qy 2π π 0 −π −2π 2π π 0 −π −2π qx (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=') qy 2π π 0 −π −2π 2π π 0 −π −2π qx (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=') qy 0 2 4 intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' units) Γ K Γ′ h/J = 0 h/J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5 h/J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5 h/J = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5 −4 0 4 −1 0 1 Charge order parameter h/J −4 0 4 0 1 Ice rule obedience h/J Γ K Γ′ Γ 0 2 4 6 S(q) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5 h/J a) b) c) d) e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 3: Field-induced charge phases and qubit ice structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' A, Charge states for varying external field h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' At h = 0, vertices have no energetic preference between −1 and +1 charge (light blue and red respectively), leading to charge disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' As h increases, A and B vertices energetically favor −1 and +1 charge respectively, leading to long-range order in the staggered charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Eventually h polarizes the sites, leading to a preponderance of −3 and +3 charged vertices (dark blue and red respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' B, Fourier transforms S(q) calculated from QA experimental output, with Brillouin zone in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' C, Cuts of S(q) for varying h/J through high-symmetry points Γ, K, and Γ′ (shown in b) show the effect of the longitudinal field on peak height at K and pinch-point width at Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' D, Charge order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' E, Proportion of vertices obeying the ice rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 4: Kinetics and field-induced topological protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' A, Within the quantum annealer, the kinetics is driven by a reverse anneal protocol wherein the qubits (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' (1)) are exposed to quantum fluctuations (Γ) and thermal fluctuations (T/J ) for a duration of 1 µs between projected classical output states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' B, Quantum annealer output samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' For each h/J, two consecutive states are shown, along with the spin-flip difference between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' For the states, up and down triangles denote charge-order violations and ice-rule violations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' For the spin-flip differences, crosses denote excitations in state i, and circles denote excitations in state i + 1.' metadata={'source': 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D-Wave, without whom this work would not be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' The work of ALB and CN was carried out under the auspices of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' DoE through the Los Alamos National Laboratory, operated by Triad National Security, LLC (Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 892333218NCA000001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' JC acknowledges support from the Natural Sciences and Engineering Research Council (NSERC), the Shared Hier- archical Academic Research Computing Network (SHARCNET), Compute Canada, Google Quantum Research Award, and the Canadian Institute for Advanced Research (CIFAR) AI chair program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 15 Competing interests The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Author Contributions J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' first proposed the idea of a Kagome embedding in a D-Wave QA to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='. A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' conceived the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' contributed to the design of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' realized the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' performed supporting mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' performed QA experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' performed data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' provided the theoretical framework for experiment design and result interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' drafted the manuscript with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' All authors contributed to the final version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Methods The QA processor used in this work was a D-Wave Advantage QPU (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 1d) housed in Burnaby, BC, Canada, operating at T = 12 mK and accessed remotely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' The QPU contains 5627 operable superconducting flux qubits of which we used 2739 to implement our kagome qubit spin ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' The architecture is discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In quantum annealing, the Hamiltonian (1) in the Main Text is controlled by an annealing parameter s ranging from 0 to 1: HQ(s) = −Γ(s) � i σx i + J (s) � � i hiσz i + � ij Jijσz i σz j � , (4) where Γ(0) ≫ J (0) and Γ(1) ≈ 0 ≪ J (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Thus a typical “forward anneal”, in which s is ramped linearly for the duration of anneal time ta (s = t/ta) begins in an easily-prepared superposition ground state and ends in a low-energy state of a classical Ising Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' For simulating spin systems, it has proven useful [7, 46] to employ a “quantum evolution Monte Carlo” method, in which a chain of classical samples S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Sk is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' To generate Si, the system is initialized in state S0 at the end of the anneal (s = 1), then “reverse annealed” back to some intermediate s∗, paused at s∗ to allow equilibration for some time tp, then quickly quenched back to 16 s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Although this method can be used to estimate observables from a transverse field Ising model at s∗ [46], here we just use quantum fluctuations as a driver of mixing dynamics between low energy state in the kagome ice system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In this work we generate chains of k = 128 samples, starting with a random initial state S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' To estimate equilibrium properties (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 3) we use tp = 256 µs and discard the first 64 samples of each chain (and the random initial state) as Monte Carlo burn-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' For dynamics inquiries (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 4) we use tp = 1 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In both cases we interrogate Hamiltonian (4) using s∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='32, which was chosen to give an appropriate amount of mixing in one microsecond (smaller s leads to faster mixing since both Γ/J and T/J are larger [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' When statistical quantities are estimated, we take the average of 200 repeated experimental iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' each iteration includes a call to the QPU for each value of h probed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Graph embedding The qubits in the QA processor are intercoupled in a “Pegasus” layout [47], in which a qubit is coupled to up to 15 other qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' From these available couplers we select a geometry that represents a kagome graph using three qubits per kagome spin as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' We show the full embedded lattice in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' The kagome embedding does not require the use of all qubits, and it is possible to embed a defect-free lattice with no site vacancies, despite the existence of some inoperable qubits (empty circle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Since ferromagnetic chains are sometimes broken, they are majority-voted to provide an unambiguous mapping from the qubit system to the kagome system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' We run all experiments presented herein with Jij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='9 for AFM couplers and Jij = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5 for FM couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' This choice of ferromagnetic coupling is sufficient to guarantee that chains are almost never broken in QPU output, despite the frustration in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Disorder suppression In this application we perform many experiments on a single programmed lattice, whose classical ground state is highly degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Under such conditions it is appropriate to refine the general-purpose QA calibration by exploiting symmetries in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' For example, when h = 0 each qubit should have average magnetization ⟨si⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Thus we 17 tune per-qubit flux offsets to balance qubits at zero for h = 0, then use the same flux offsets when h ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In this experiment, we are not interested in probing boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Rather, we want to simulate the thermodynamic limit of an infinite system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' In an infinite system, for any fixed h, the correlation of two neighboring kagome sites ⟨sisj⟩ is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Thus we fine-tune the AFM couplers to promote this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Since the three-qubit FM chains are almost never broken, we do not fine-tune the FM couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Similarly, for any fixed h ̸= 0, the magnetization of each qubit should be equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' we fine-tune the per-qubit fields hi to promote this property (maintaining the property that the average 1 N � i hi does not change from the nominal value h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' These calibration refinements are performed before collecting the analyzed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 7 shows an example of this refinement for J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='9, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='6, with the magnetizations and correlations achieved, and the programmed values that achieve them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 18 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5 1 0 2 4 6 8 10 s Energy scale (GHz) Γ(s) J (s) a) b) 0 s∗ 1 relaxation: tp reverse anneal: tq(1 − s∗) readout quench: tq(1 − s∗) readout/wait: tw Time Annealing parameter s FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 5: Quantum annealing schedule and protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' a, Transverse field Γ(s) and Ising energy scale J (s) as a function of annealing parameter s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Note that the total coupling between two three- qubit chains is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='8J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' b, Quantum evolution Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' A sequence of classical readout states is generated by repeated exposure to quantum fluctuations and thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 19 a) b) c) d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 6: Embedding of the kagome lattice into the qubit graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' a–b, Each kagome site is represented by three qubits, coupled together ferromagnetically in a chain (Jij = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' The entire qubit graph and embedding are shown in a with green and orange lines representing FM and AFM couplers respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' b shows a detailed zoom, with operable and inoperable qubits represented by filled and empty circles respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' c–d, The embedding shown in a realizes a 729-site kagome lattice, which can be viewed as Ising spins (c), or magnetic dipoles (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='32 0 200 400 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' qubit magnetization Occurrences −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='33 0 200 400 600 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' coupler spin-spin correlation Occurrences 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='91 0 100 200 300 Adjusted coupling Occurrences 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='2 0 200 400 600 800 Adjusted longitudinal eld Occurrences 2e-5 0 2e-5 0 100 200 300 Qubit ux-bias o\x1bset (h = 0) Occurrences a) b) c) d) e) boundary bulk FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 7: Suppressed disorder with fine-tuned Hamiltonian terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Example data are shown for nominal AFM coupling values of J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='9 and local fields of h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' a–b, Over 100 iterations, tightly-concentrated average qubit magnetizations and spin-spin correlations of coupled pairs in- dicate a balanced degenerate ice system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' c–e, This is achieved by small adjustements of couplers (c), adjustment of fields (d), and qubits are balanced using flux-bias offsets at h = 0 that are also used at nonzero h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' Note the two modes in d, where boundary spins are assigned roughly half the field of bulk spins, in accordance with their degree in the graph, to achieve similar magnetizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQf4_6S/content/2301.01853v1.pdf'} diff --git a/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf b/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..18e9a414e362cc28750d5da1233d86027eb6ca3d --- /dev/null +++ b/jdE0T4oBgHgl3EQf7QI8/content/2301.02773v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7973ec63d2fba6a42e51d131b3343ebcf1553eaa3557287b1f2c318df9fd5b68 +size 507391 diff --git a/k9FIT4oBgHgl3EQfriuF/content/tmp_files/2301.11332v1.pdf.txt b/k9FIT4oBgHgl3EQfriuF/content/tmp_files/2301.11332v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3aea89dae4ff7289250678bf8819d7c7174f081 --- /dev/null +++ b/k9FIT4oBgHgl3EQfriuF/content/tmp_files/2301.11332v1.pdf.txt @@ -0,0 +1,1178 @@ +Springer Nature 2021 LATEX template +Locating γ-Ray Sources on the Celestial +Sphere via Modal Clustering +Anna Montin1†, Alessandra R. Brazzale1*† and Giovanna +Menardi1† +1*Department of Statistical Sciences, University of Padova, Via +C. Battisti 241, Padova, 35121, (PD), Italy. +*Corresponding author(s). E-mail(s): +alessandra.brazzale@unipd.it; +Contributing authors: anna.montin@studenti.unipd.it; +giovanna.menardi@unipd.it; +†These authors contributed equally to this work. +Abstract +Searching for as yet undetected γ-ray sources is a major target of the +Fermi LAT Collaboration. We present an algorithm capable of identify- +ing such type of sources by non-parametrically clustering the directions +of arrival of the high-energy photons detected by the telescope onboard +the Fermi spacecraft. In particular, the sources will be identified using +a von Mises-Fisher kernel estimate of the photon count density on the +unit sphere via an adjustment of the mean-shift algorithm to account +for the directional nature of data. This choice entails a number of desir- +able benefits. It allows us to by-pass the difficulties inherent on the +borders of any projection of the photon directions onto a 2-dimensional +plane, while guaranteeing high flexibility. The smoothing parameter will +be chosen adaptively, by combining scientific input with optimal selec- +tion guidelines, as known from the literature. Using statistical tools +from hypothesis testing and classification, we furthermore present an +automatic way to skim off sound candidate sources from the γ-ray +emitting diffuse background and to quantify their significance. The algo- +rithm was calibrated on simulated data provided by the Fermi LAT +Collaboration and will be illustrated on a real Fermi LAT case-study. +Keywords: directional data, kernel density estimator, man-shift algorithm, +tree-based classification +1 +arXiv:2301.11332v1 [astro-ph.IM] 25 Jan 2023 + +Springer Nature 2021 LATEX template +2 +Locating γ-Ray Sources via Modal Clustering +1 Motivation and rationale +1.1 High-energy astrophysics +The past three decades have been a golden era for Astronomy. Pioneering +technology has driven remarkable acceleration in the rate of detection and +characterization of celestial objects, and new space missions will have more and +better quality data to help find and characterize these objects. Discoveries in +this field are of utmost relevance as they contain a wealth of information about +the history of the Universe, and impact on the understanding of our Galaxy +and our own Solar system. An important example is high-energy astrophysics, +which acts at the interface between particle physics and astronomy to study +the multitude of extreme phenomena which inhabit the Cosmos. To date, the +observation of γ-ray photons, that is, of quanta of light in the highest energy +range, has provided the basis for a large number of astronomical discoveries. γ- +rays are usually generated from accelerated charged particles, such as electrons +or protons, boosted by extreme celestial objects such as supermassive black +holes, supernova remnants, pulsars and active galactic nuclei, to name a few. +The study of these γ-ray emitting sources improves our understanding of high- +energy astrophysical phenomena, and might even resolve the mystery of the +fundamental nature of dark matter. +The Fermi Gamma-ray Space Telescope is an international and multi- +agency space mission launched in June 2008 which studies the Cosmos in +the energy range 10 keV – 300 GeV. The primary instrument onboard the +Fermi spacecraft is the Large Area Telescope (LAT), a wide field-of-view pair- +conversion telescope which was designed to perform an all-sky survey aimed +at discovering and locating high-energy emitting sources. The standard proce- +dure of the Fermi LAT Collaboration for point-like source detection relies on +so-called single-source models (Hobson et al., 2009, par. 7.4), which require the +sky map to be split into small regions. The presence of a possible new source is +assessed on a pixel-by-pixel basis: Poisson regression is used to model the num- +ber of photons associated with each pixel and likelihood ratio tests assess the +significance of the source (Mattox et al., 1996). See also van Dyk et al. (2001) +for a Bayesian treatment with appplication to low-count X-ray data collected +by the Chandra X-Ray Observatory. Conversely, variable-source-number mod- +els address the problem from a more global perspective, as they simultaneously +identify and locate all possible sources in a given sky map (Hobson et al., 2009, +par. 7.3). Since point-like sources present themselves as spatially concentrated +photon emissions, the problem can naturally be recast as a clustering prob- +lem. Recent examples of variable-source-number modelling of X-ray and γ-ray +photon count data using finite and infinite mixtures are Jones et al. (2015), +Costantin et al. (2020), Costantin et al. (2020), Sottosanti et al. (2021) and +Meyer et al. (2021). +The data provided by the Fermi LAT Collaboration typically consist of an +event list which gives the direction in the sky of each detected photon together +with additional information, the primary one being its energy content and + +Springer Nature 2021 LATEX template +Locating γ-Ray Sources via Modal Clustering +3 +Fig. 1 Left: Polar coordinates as recorded by the LAT (Image credit: Mardia and Jupp, +2000). φ is the longitude and θ is the co-latitude. Center and right: Fermi-LAT γ-ray photon +count maps for a 5-year observation period. Center: in polar coordinates. Right: in Galactic +coordinates. Yellow: region of size (l, b) ∈ [95◦, 135◦] × [−40◦, −10◦] analyzed in Section 5.2. +Green: photon counts used to train the post-processing classifier. Red: Galactic plane. +the so-called event type which expresses the quality of the measurement. This +information is used to determine the number of the emitting extra-galactic +sources, measure their intensities, and assign to them the corresponding indi- +vidual photon counts. A major challenge of trying and detecting high-energy +phenomena from astronomical data is to separate the signal of the putative +emitting source from noise. The Fermi LAT data, in particular, are char- +acterized by two types of noise: (i) measurement error associated with the +components of the LAT (tracker, calorimeter etc.) and (ii) the diffuse γ-ray +background which spreads over the entire area observed by the telescope. +The former is expressed through the LAT’s point spread function (Ackermann +et al., 2013), which is typically included into the model. Different phenom- +ena contribute to the residual γ-ray background (Acero et al., 2016). Broadly +speaking, its origins can be brought under two headings: galactic interstel- +lar emission (GIE), that is, the interaction of galactic cosmic rays with gas +and radiation fields, and a residual all-sky emission. The latter is commonly +called the isotropic diffuse gamma-ray background (IGRB), and includes the +γ-ray emission from faint unresolved sources and any residual galactic emis- +sion which is approximately isotropic. Costantin et al. (2020) translate the +simulation-based background model developed by Acero et al. (2016) into a +workable parametric formulation, while Sottosanti et al. (2021) reconstruct it +via a flexible Bayesian nonparametric model based on B-splines. +A further challenge of analysing the Fermi LAT data refers to the geometry +of the problem. As the distance to the emitting source is not given, the data +points are placed on the celestial sphere with Earth at its center and unit +radius, as shown in the middle panel of Figure 1. Directions are expressed in +Galactic coordinates, that is longitude l and latitude b, which place the origin +of the Cartesian system in the center of our galaxy — the Milky Way — and +align the x-axis with the Galactic plane (right panel of Figure 1). This is the +situation considered by Jones et al. (2015), Costantin et al. (2020), Sottosanti +et al. (2021) and Meyer et al. (2021). Instead of projecting data onto a 2- +dimensional map, we may rather express directions in 3 dimensions through +polar coordinates, that is, co-latitude (θ) and longitude (φ) in geographical +terms; see the left panel of Figure 1. These can easily be back-transformed to + +44 +4.Springer Nature 2021 LATEX template +4 +Locating γ-Ray Sources via Modal Clustering +Cartesian coordinates x = [cos θ, sin θ cos φ, sin θ sin φ]⊤ on the unit sphere, as +done by Costantin et al. (2020). A thorough treatment of directional data can +be found in Mardia and Jupp (2000). +1.2 The statistical state of the art +The discovery of celestial objects is an intrinsically interdisciplinary field which +combines both, statistical and astrophysical methodology. Statistical learning, +by which we mean the ability of discovering patterns and regularities in the +data, plays a central role in knowledge discovery. This also includes allocating +objects to a pre-assigned or unknown number of groups according to a set of +observed attributes or features, which is a natural activity of any science. A +major distinction is made depending on whether the groups are defined, and +known a priori, or need be detected using the data. Clustering, or unsuper- +vised learning, considers the latter situation. A surge of techniques has been +proposed over the years, which differ significantly in their definition of what +a cluster is and how to identify it (Hennig et al., 2015). A precise statisti- +cal notion of what a “group” is, is provided by the density-based approach. +Here, the clusters are associated with some specific features of the probability +distribution which is assumed to underlie the data. This idea has been devel- +oped into two distinct directions. The model-based or parametric approach +represents the probability distribution of the data as a mixture of parametric +distributions. A cluster is associated with each component of the mixture and +the observations are allocated to the cluster with maximal density among the +components. Standard accounts are the seminal works of Fraley and Raftery +(1998, 2002). A less widespread density-based clustering formulation is referred +to as modal or nonparametric clustering and dates back to Carmichael et al. +(1968). Here, the underlying density is reconstructed from the data using suit- +able nonparametric density estimators, and clusters are associated with the +domain of attraction of their modes. The rather scattered theory is reviewed +in Menardi (2016). Chac´on (2015) provides some new insight into the theoreti- +cal foundations of modal clustering making use of Morse theory (Milnor et al., +1969). +In this paper, we advocate the use of nonparametric, or modal clustering +for γ-ray source detection using a von Mises-Fisher kernel on the unit sphere. +This choice entails a number of desirable benefits. It allows us to by-pass +the difficulties inherent on the borders of any 2-dimensional projection of the +photon directions. But, it also guarantees high flexibility and adaptability, +while posing on a sound theoretical ground. The sources will be identified +via an adjustment of the mean-shift algorithm to account for the directional +nature of the Fermi LAT data. The issue of selecting the smoothing parameter +is addressed adaptively, by combining scientific input with optimal selection +guidelines, as known from the literature. Using known results from hypothesis +testing and classification, we furthermore present an automatic way to pinpoint +sound candidate sources and to quantify their significance by skimming off the +γ-ray emitting diffuse background. The Fermi LAT database currently holds + +Springer Nature 2021 LATEX template +Locating γ-Ray Sources via Modal Clustering +5 +over 1 billion photons in the energy range from about 20 MeV to more than 300 +GeV collected in over a decade of operation. Efficient tools to account for the +computational burden required to analyse huge amounts of data, possibly on +the entire sphere, are also discussed. Our method was calibrated on simulated +data provided by the Fermi LAT Collaboration and will be illustrated on a +real Fermi LAT case-study. +The paper is organized as follows. Section 2 sets the methodological back- +ground of kernel density estimation for directional data. Being able to correctly +specify the right amount of smoothing is crucial for the reliable identifica- +tion of the sources. Optimal bandwidth selection is discussed in Section 3, +while Section 4 presents our proposal of modal clustering on the unit sphere. +In particular, to separate the true signal emitted by a source from the back- +ground, we developed a post-processing procedure that combines the findings +of two parallel quests. One establishes the significance of a candidate mode +using a suitable statistical test as presented in Section 4.2.1. The second skims +off the photons emitted by the γ-ray background using a tree-based classifier +build on previous knowledge provided by the Fermi LAT Collaboration; see +Section 4.2.2. Section 5.1 benchmarks two key aspects of our proposal, namely +the selection of the optimal bandwidth and the classification of the incom- +ing photons. Section 5.2 eventually illustrates the performance of our proposal +when applied to a real sample of high-energy photons accumulated by the +LAT. The paper closes with the concluding remarks of Section 6. +This paper is an extended version of the paper presented at the 51st Sci- +entific Meeting of the Italian Statistical Society on June, 2022 (Montin et al., +2022). +2 Kernel density estimators for directional data +2.1 The von Mises-Fisher distribution +Directions in the 3-dimensional space can be represented using Cartesian +coordinates as unit vectors x, that is, as points on the sphere +Ω2 = {x ∈ R3 : ∥x∥2 = x2 +1 + x2 +2 + x2 +3 = 1} +with unit radius and centre at the origin. These can be retrieved from Galactic +coordinates, that is, from the longitude l ∈ (−180, +180) and the latitude +b ∈ (−90, +90) of a given data point, by +x = [cos l cos b, sin l cos b, sin b]⊤. +A widely used distribution to model γ-ray emission in astrophysics searches +(Banerjee et al., 2006) is the von Mises-Fisher (vMF) distribution +fvMF (x; µ, κ) = C2(κ) exp{κx⊤µ}, + +Springer Nature 2021 LATEX template +6 +Locating γ-Ray Sources via Modal Clustering +which extends the 3-dimensional normal distribution N3(µ, κ−1I3), with I3 +being the 3 × 3 diagonal unit matrix, by restricting its density to the unit +sphere. Here, µ ∈ Ω2 represents the mean direction, while κ ≥ 0 is a con- +centration parameter (Mardia and Jupp, 2000, Section 9.6). As such, the von +Mises-Fisher distribution describes observations which scatter simmetrically +around their mean direction µ. The normalizing constant +C2(κ) = +κ +1 +2 +(2π) +3 +2 I 1 +2 (κ) +includes the modified Bessel function +Iν(z) = +� +z +2 +�ν +π1/2Γ(ν + 1 +2) +� 1 +−1 +(1 − t2)ν− 1 +2 eztdt +of order ν = 1/2. +2.2 Kernel density estimator +Let x1, . . . , xn ∈ Ω2 be a random sample of n observations generated by a +distribution with density f(x) defined on the unit sphere Ω2 such that +� +Ω2 +f(x)ω2(dx) = 1, +where ω2 is the Lebesgue measure on Ω2. We can estimate the density f using +the kernel density estimator proposed by Bai et al. (1988) for directional data, +ˆfh(x) = ch(K) +n +n +� +i=1 +K +�1 − x⊤xi +h2 +� +, +(1) +where K(·) is a suitable kernel function which decreases on [0, ∞), and h > 0 +is the smoothing parameter. The normalizing constant ch(K), is defined by +ch(K)−1 = +� +Ω2 +K +�1 − x⊤xi +h2 +� +ω2(dx) = h2˜ch(K), +where ˜ch(K) = +� 2/h2 +0 +K(u)du. Using the von Mises-Fisher kernel, expression +(1) becomes +ˆfh(x) = 1 +n +n +� +i=1 +fvMF +� +x; xi, 1 +h2 +� += +1 +(2π) +3 +2 I 1 +2 (h−2) +1 +hn +n +� +i=1 +exp +�x⊤xi +h2 +� +. +(2) + +Springer Nature 2021 LATEX template +Locating γ-Ray Sources via Modal Clustering +7 +Fig. 2 von Mises-Fisher kernel density estimate of the high-energy photons tracked by the +Fermi LAT in the validation region (l, b) ∈ [0◦, 60◦]×[10◦, 60◦] for different values of h: 0.01 +(left), 0.001 (center), hi,SE (right). +That is, the kernel density estimator for direction data on the unit sphere is a +mixture of 3-dimensional von Mises-Fisher distributions with κ = h−2. +3 Bandwidth selection +3.1 Data-based methods +A major issue when using a kernel density estimator is the selection of the +smoothing parameter, or bandwidth, h. Being able to correctly specify the right +amount of smoothing is crucial for the reliable identification of the sources. +This is illustrated in Figure 2, which plots the estimated density for the same +sky region using three different values of h, where the latter choice varies +with sky location. If the smoothing parameter is too large (picture on the +left), false peaks may emerge from the background. Conversely, if the kernel +function is too concentrated (middle picture), we may miss some faint sources. +A wealth of data-driven methods were developed over the years for both, fixed +and variable bandwidth kernel density estimation. As far as directional data +goes, the proposals mainly are for circular observations; see e.g. Hall et al. +(1987) and Klemel¨a (2000). Adaptive kernel density estimation, that is, when +the smoothing parameter hi in (2) adapts to the local behaviour of f at xi, +is of special interest to us, as the spatial scattering of the incoming photons +differs among sources, and to an even larger extent if they were emitted from +the background radiation. +Selecting an optimal bandwidth generally entails minimization of a suitable +measure of the error we commit when estimating the target density f by ˆfh. +A common way of measuring this error is the mean integrated squared error +MISE(h) = E +�� +Ω2 +� +ˆfh(x) − f(x) +�2 +ω2(dx) +� +, +where the expectation is taken with respect to the distribution specified by f; +in this case +hMISE = arg min +h>0 MISE(h). + +500 +400 +300 +densita +200 +100 +0 +10 +0 +20 +10 +30 +20 +30 +AO +b +40 +50 +So +6010k +8K +6k +densita +AK +2K +0 +10 +20 +10 +30 +20 +30 +b +AO +90 +50 +So +Og7000 +6000 +5000 +4000 +2000 +1000 +10 +20 +1o +20 +30 +30 +40 +b +O +50 +5o +60Springer Nature 2021 LATEX template +8 +Locating γ-Ray Sources via Modal Clustering +Fig. 3 Left: Photon scattering as a function of their energy content (courtesy of Sottosanti +et al., 2021). Right: Values of hi,SE as a function of energy and event quality, where PSF0 +represents the worst event type. The higher the energy and quality of the event, the smaller +is the smoothing parameter. +The alternative choice hAMISE minimizes the asymptotic approximation of +the mean integrated squared error, that is, when n → ∞. However, both +window widths depend explicitly on the unknown density to be estimated, and +cannot be computed exactly. Simple “plug-in” procedures, where f is replaced +by a suitable pilot estimate ˆf, turned out to be generally unsatisfactory. An +automatic way of determining the optimal bandwidth h is by likelihood cross- +validation, that is, +hLCV = arg max +h>0 CV (h), +where +CV (h) = +n +� +i=1 +log ˆfh,−i(xi). +Here, ˆfh,−i(xi) is the kernel density estimate we obtain after having omitted +observation i, evaluated at xi. A further option is to adapt the most promising +solutions for optimal bandwidth selection on the plane to our problem at hand, +as listed below. The corresponding performance metrics were evaluated on the +simulated sample of high-energy photons emitted by the sources present in the +sky region shown in Figure 2, and will be discussed in Section 5.1.1. +A first possibility is to generalize Garc´ıa-Portugu´es’ (2013) rule of thumb +to spherical data, +hT HUMB = +� +8 sinh2(ˆκ) +ˆκ[(1 + 4ˆκ2) sinh(2ˆκ) − 2ˆκ cosh(2ˆκ)]n +� 1 +6 +, +where the concentration parameter ˆκ is estimated by maximum likelihood. +Conversely, if we want the bandwidth h to depend on the current location xi +of the estimator, a first possibility is to use Abramson’s (1982) rule, which has + +Energy >= 11 GeV +Energy >= 307 GeV +45.0 +0 +QQ +44.5- +8 +44.0 +43.5 - +43.0 - +0 +Q +Energy >= 604 GeV +Energy >= 900 GeV +45.0 - +44.5- +44.0 - +米 +米 +43.5 +43.0- +42.5 +20.5 +21.0 +21.5 +22.0 +22.5 +23.0 +20.5 +21.0 +21.5 +22.0 +22.5 +23.0 +Longitudine0.006 +Quality +0.004 +0123 +i.SE +h +0.002 +0 +250000 +500000 +750000 +1000000 +Energy (MeV)Springer Nature 2021 LATEX template +Locating γ-Ray Sources via Modal Clustering +9 +hi change proportionally with the inverse of the square root of ˆfh(xi), +hA +i,T HUMB = hT HUMB +� +ˆfhT HUMB(xi) +�− 1 +2 +and +hA +i,LCV = hLCV +� +ˆfhLCV (xi) +�− 1 +2 +. +Here, ˆfhT HUMB(xi) and ˆfhLCV (xi) are winsorized (or clipped) versions of a +suitably constructed pilot kernel density estimate with fixed bandwidth h, +which may be hT HUMB or hLCV . A second possibility is to use the modification +proposed by Silverman (1986, Section 5.3), +hS +i,T HUMB = hT HUMB +� 1 +mg +ˆfhT HUMB(xi) +�−β +and +hS +i,LCV = hLCV +� 1 +mg +ˆfhLV C(xi) +�−β +, +where mg is a scale factor defined by the geometric mean of the two pilot +estimates, ˆfhT HUMB(xi) or ˆfhLCV (xi), while β ∈ [0, 1] tunes the sensitivity +of the bandwidth to variations of these. We will set β = 0.5, as this choice +generally entails a better behavior of the kernel density estimator on the tails +of the distribution (Izenman, 1991). +3.2 Using scientific input +A valid alternative for determining the smoothing parameter h is to use scien- +tific input. As mentioned in Section 1.1, the spatial scattering of the photons +around the source direction µ is modelled by the LAT’s point spread function +(PSF). This function depends on the energy of the incoming photon, on its +inclination angle θ (see left panel of Figure 1) and on the quality of the recorded +event (Ackermann et al., 2013). The latter is expressed by the PSF event +type, that is, an event-level quantity which indicates how well the LAT man- +aged to reconstruct the direction of the incoming photon and which assumes +four values, from the lowest quality (PSF0) to the best quality (PSF3). Most +importantly, the PSF depends on the scale factor +S(Ei) ∝ +�� +c0,i +� +Ei +100MeV +�−0.8�2 ++ c2 +1,i, +which describes the uncertainty of the event as a decreasing function of the +energy Ei, expressed in Mega electron Volt (MeV), and of the two parameters +c0,i and c1,i, which are given distinct values for the different event qualities +and can be retrieved from the Fermi LAT web site1. The first constant, ci,0, +represents multiple scattering while the second, c1,i, represents the spatial +resolution of the LAT tracker. How the precision of the measurements depends +on the energy is shown in the left panel of Figure 3 (Sottosanti et al., 2021), +while the right panel of the same figure plots the values we obtain for hi,SE +1https://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/Cicerone LAT IRFs/IRF PSF.html + +Springer Nature 2021 LATEX template +10 +Locating γ-Ray Sources via Modal Clustering +for the four different event types. On this basis, we may specify a variable +bandwidth as +hi,SE = +�� +c0,i +� +Ei +100MeV +�−0.8�2 ++ c2 +1,i, +(3) +which is the one used in the right panel of Figure 2. +4 Modal clustering on the unit sphere +4.1 Mode hunting +Modal clustering associates clusters with the domain of attraction of the modes +of the underlying density f. Two main strands can be identified, depending on +whether the modes are given explicitely or not (Menardi, 2016). A first strand +follows the route of Hartigan (1975) and identifies clusters with high-density +regions of the sample space, defined by the density level sets +Lc(f) = {x ∈ Ω2 : f(x) ≥ c}, +0 ≤ c ≤ max f. +An estimate of the unknown Lc(f) is obtained by replacing f(x) by its non- +parametric estimate ˆf(x). The rationale behind this class of methods is that +any connected component of Lc(f) includes at least one mode of the density +function, and, on the other hand, for each mode of the density function, there +exists λ for which one of the connected components of the associated L(λ) +includes this mode at most. The major drawback is that the identification of +the connected components of a multidimensional set is not straightforward. +As our aim is to discover and identify unknown γ-ray emitting sources, we +want to associate their direction explicitly with the modes of the unknown +density f. Yang et al. (2014) adapted the mean-shift algorithm developed by +Fukunaga and Hostetler (1975) to be used with the directional kernel estimator +(2) and fixed bandwidth h. Starting from a generic point x(0), the algorithm +recursively shifts it to a local weighted mean, until convergence. Denoted by +wi(x(s)) the vector of weights of the components of xi at step s, at the next +step, (s + 1), we have +x(s+1) = +n +� +i=1 +wi(x(s))xi = x(s) + M(x(s)), +where M(x(s)) = �n +i=1 wi(x(s))xi − x(s) denotes the mean shift. Up to a +normalising factor, the weights wi(x) involve the derivative K′(h−2(1−x⊤xi)) +of the kernel function, which leads to the weighted average +ˆx(s+1) = − +�n +i=1 xiK′ +� +1−ˆx(s)⊤xi +h2 +� +��� +��� +�n +i=1 xiK′ +� +1−ˆx(s)⊤xi +h2 +���� +��� +2 +, + +Springer Nature 2021 LATEX template +Locating γ-Ray Sources via Modal Clustering +11 +where || · ||2 is the Euclidean norm. Here, the minus sign is due becasue K(·) +is a decreasing function. If we replace the kernel function K(·) by the von +Mises-Fisher kernel, the above expression becomes +ˆx(s+1) = +�n +i=1 xi exp +� +ˆx(s)⊤xi−1 +h2 +� +��� +��� +�n +i=1 xi exp +� +ˆx(s)⊤xi−1 +h2 +���� +��� +2 +. +Straightforward calculations allowed us to extend the proposal by Yang +et al. (2014) to varying hi, that is, for adaptive kernel density estimation on +the unit sphere. +4.2 Post-processing +As mentioned in Section 1.1, the incoming photons were either emitted from a +high-energy source or are part of the diffuse γ-ray background which spreads +over the entire area observed by the telescope. The directional kernel density +estimator (2) tries and reconstructs the corresponding mixture distribution. +Hence, the small peaks which emerge as modes may identify true sources, but +they may equally well represent a false signal generated by the irregularly +shaped background radiation. To separate the true signal emitted by a source +from the background, we developed a post-processing procedure that combines +the findings of two parallel quests. One establishes the significance of a candi- +date mode using a suitable statistical test. The second skims off the photons +emitted by the γ-ray background using a suitable classifier build on previous +knowledge provided by the Fermi LAT Collaboration. By super-imposing the +findings from these two quests, we identify candidate sources which are both, +statistically significant and qualified as such according to a set of relevant +features. Furthermore, we are now able to distinguish photons emitted by a +candidate source from those pertaining to the background radiation. +4.2.1 Statistical significance +Mathematically, we can verify whether a function reaches a local maximum +by checking whether all eigenvalues of the Hessian matrix evaluated at the +candidate mode are negative. Statistically, developing a suitable test to ver- +ify the existence of a mode and deriving its null distribution using eigenvalues +is tricky, as these are not continuously differentiable functions of the Hes- +sian. This invalidates resampling-based methods such as the bootstrap and +asymptotic expansion by the delta method, which we may use to reconstruct +the finite-sample null distribution of the test statistic. Genovese et al. (2016) +hence suggest to use data splitting to separate the process of finding candidate +modes from the process of hypothesis testing. They furthermore propose to +base inference on confidence intervals, rather than on p-values. The potential +modes are hence estimated on the first half of the data, while the second half + +Springer Nature 2021 LATEX template +12 +Locating γ-Ray Sources via Modal Clustering +is used to construct asymptotically valid bootstrap confidence intervals for the +eigenvalues of the Hessian matrix, which can be used for hypothesis testing. +The extension of this idea to directional data requires some care, as working +on the unit sphere sets some constraints. To calculate the Hessian matrix +H ˆfh(x), we first need the total gradient +∇ ˆfh(x) = C2(h−2) +n +n +� +i=1 +xi +h2 exp +� +x⊤xi − 1 +h2 +� +, +where ∇ represents suitable differentiation. The Hessian matrix hence is +H ˆfh(x) = (I3 − xx⊤) +� +∇∇ ˆfh(x) − ∇ ˆfh(x)⊤xI3 +� +(I3 − xx⊤) += (I3 − xx⊤) +� +C2(h−2) +n +n +� +i=1 +xix⊤ +i +h4 +exp +� +x⊤xi − 1 +h2 +� ++ +− C2(h−2) +n +n +� +i=1 +x⊤xiI3 +h2 +exp +� +x⊤xi − 1 +h2 +�� +(I3 − xx⊤). +Likewise, we may obtain the Hessian matrix associated with an adaptive kernel +density estimator with variable bandwidth hi. The tricky part is that the +eigenvalue of H ˆfh(µ), when ˆfh(x) is evaluated at µ, is always zero, whether µ +corresponds to a true source or not. This entails that inference has to be based +on the remaining two eigenvalues. We hence construct an 1−α level confidence +interval for the largest non null eigenvalue using bootstrap resampling. The +candidate mode is validated if the interval includes only negative values. +A second possibility is to reparametrize the von Mises-Fisher kernel using +polar coordinates +fvMF (θ, φ) = +κ +4πκ exp +� +κ cos θ cos η + k sin θ sin η cos(φ − ζ) +� +sin θ, +where, as in Mardia and Jupp (2000), x = (cos θ, sin θ cos φ, sin θ sin φ)⊤ and +µ = (cos η, sin η cos ζ, sin η sin ζ)⊤. This workaround allows us to directly +apply the results by Genovese et al. (2016). +4.2.2 Feature selection +A further possibility to skim off the photons emitted by extra-galactic sources +from those which originate from the diffuse background is to build a suitable +classification rule which integrates additional information on the photons pro- +vided by the Fermi LAT Collaboration and/or features that can be extracted +at the various steps of the mean-shift algorithm. These include the energy +content of the photons (photon energy) and their incoming direction (longi- +tude, latitude), the number of photons assigned to a mode (n photons), the + +Springer Nature 2021 LATEX template +Locating γ-Ray Sources via Modal Clustering +13 +h +ARI +¯d(s, ˆs) +ns +hi,SE +0.9976 +0.0004 +86 +hT HUMB +0.6841 +0.0079 +10 +hA +i,T HUMB +0.6805 +0.0139 +18 +hS +i,T HUMB +0.8524 +0.0063 +25 +hLCV +0.9777 +0.0092 +142 +hA +i,LCV +0.9777 +0.0092 +142 +hS +i,LCV +0.9777 +0.0092 +142 +Table 1 Performance metrics for +different choices of the bandwidth h of +the von Mises-Fisher kernel density +estimator applied to the sky region +plotted in Figure 2: ARI = adjusted +Rand index; ¯d(s, ˆs) = median angular +distance (in degrees) between the +directions of true sources (s) and +candidate sources (ˆs) identified by the +algorithm; ns = number of identified +sources. The number of true sources is +68. +density estimates for the signal and the background model (density, den- +sity difference) and various types of distances between the photons and their +mode (intra cluster distance, total distance, first step length). We hence sug- +gest to train and test a tree-based classifier on a suitable area of the sky. The +final classifier will then be pruned so as to assign any cluster with a single pho- +ton to the background. Section 5.1.2 reports the performance metrics of our +classification rule when applied to a portion of the Northern sky. +5 Application to Fermi LAT data +5.1 Benchmarking +5.1.1 Optimal bandwidth +Table 1 compares the different proposals for bandwidth selection listed in +Sections 3.1 and 3.2 using three performance metrics, that is, the adjusted +Rand index (ARI), the median angular distance (in degrees) between the direc- +tions of true sources and candidate sources, ¯d(s, ˆs), and the number ns of +identified sources. These metrics were obtained by benchmarking our algorithm +on a simulated sample of 2.335 photons emitted by the 68 sources present in +the validation region (l, b) ∈ [0◦, 60◦] × [10◦, 60◦] shown in Figure 2. The three +proposals based on the rule of thumb oversmooth the true photon density, lead- +ing to rather low ARI values. Likelihood cross validation, on the other hand, +tends to over adapt the true density yielding too many candidate sources: 142 +in place of the 68 present. The best partition of the selected sky region is +obtained when using the variable bandwidth hi,SE, that is, the scale factor of +the LAT’s point spread function. Further support to this choice is provided by +Table 2, which contrasts the selected optimal bandwidths (Columns 3–7) with +the true photon scattering, as measured by its standard deviation (Column +2), for 5 selected sources of varying size, that is, which emit from a minimum +of ns = 7 photons up to a mximum of ns = 151 photons. Again, hi,SE is the +best performing choice. + +Springer Nature 2021 LATEX template +14 +Locating γ-Ray Sources via Modal Clustering +Source +sd +¯hi,SE +¯hA +i,LCV +¯hA +i,T HUMB +¯hS +i,LCV +¯hS +i,T HUMB +ns = 7 +0.0019 +0.0017 +3.2958 · 10−06 +0.1053 +2.8623 · 10−07 +0.0611 +ns = 19 +0.0048 +0.0028 +3.2225 · 10−06 +0.0221 +2.7986 · 10−07 +0.0128 +ns = 31 +0.0042 +0.0027 +2.8184 · 10−06 +0.0501 +2.4477 · 10−07 +0.0290 +ns = 79 +0.0030 +0.0028 +2.1721 · 10−06 +0.0314 +1.8864 · 10−07 +0.0182 +ns = 151 +0.0068 +0.0027 +2.0684 · 10−06 +0.0215 +1.7963 · 10−07 +0.0125 +Table 2 Standard deviation (Column 2) of photon scattering for 5 selected sources of +varying size (Column 1) and average bandwidths computed using the scale factor of the +PSF (Column 3) or selected by Abramson’s or Silverman’s rules (Columns 4–7). +5.1.2 Performance metrics +We bemchmarked our tree-based classifier on a sample of 35,365 simulated +photon emissions in the sky region (l, b) ∈ [100◦, 150◦] × [0◦, 90◦]. This area +covers the entire Northern sky to account for the rater prominent variability +of the diffuse γ-ray background as we move away from the Galactic plane. The +classifier was estimated on the first 2/3 of the sample, for a total of 24,573 +photons, and tested on the remaining 11,062 photons. In both sets, about +85% of the photons were emitted from the background. The final classifier was +pruned so as to assign any cluster with a single photon to the background. +The classifier was hence benchmarked on the sky region shown in Figure 2, +where it selected a total of ns = 86 sources. The average sensibility, computed +on the candidate sources identified by the classifier, was 90,5%, while the +average specificity was 99,5%. The adjusted Rand index (ARI) is 0.9752 and +the median angular distance between the true sources and the identified ones +is 0.0005 degrees. +5.2 Case-study +The yellow region in Figure 1 shows a portion of the Southern sky of size (l, b) ∈ +[95◦, 135◦]×[−40◦, −10◦] for which the LAT accumulated 3,849 photon counts +over a five-year period of observation.2 Of these, about 26% were emitted by +the 44 sources present in the area, while the remaining 74% originated from the +diffuse γ-ray background. The left panel of Figure 4 plots the estimated kernel +density (2) using a von Mises-Fisher kernel. Here, the bandwidth parameter h +was set according to scientific input, as described in Section 3.2. This choice +revealed to be the most performing one in terms of adjusted Rand index (ARI), +median angular distance, ¯d(s, ˆs), between the true source direction and the +reconstructed one and number of identified sources (ns). In all, the mean-shift +algorithm identified 876 modes. To further refine the list of candidate sources +we proceeded in two steps as outlined in Section 4.2. +A tree-based classifier to discriminate between source and background pho- +tons was trained on the 6,814 photon counts highlighted in green in Figure 1. +The importance of the selected predictor variables is shown in the right panel of +Figure 4. The most discriminating features are the number of photons assigned +2https://fermi.gsfc.nasa.gov/ssc/data/access/ + +Springer Nature 2021 LATEX template +Locating γ-Ray Sources via Modal Clustering +15 +Fig. 4 Left: Kernel density estimate using a von Mises-Fisher kernel for the 3,849 γ-ray +photon counts accumulated by the LAT in a 5-year period in the region of the Southern sky +identified by (l, b) ∈ [95◦, 135◦] × [−40◦, −10◦]. About 26% of the photons were emitted by +the 44 sources present in the area. Right: Feature importance plot for the tree-based photon +classifier used to discriminate between source and diffuse γ-ray background emission. +to a cluster (n photons), the difference between the two photon densities for, +respectively, the all sky and background counts only (density differences), and +the density observed for each photon (density). This reduces the original 876 +modes to 39 candidate sources, which are shown as blue circles in the left panel +of Figure 5. The table on the right reports the performance of our classifier in +terms of ARI and median angular distance ¯d(s, ˆs). The true positive rate for +single photon classification is 98.5% rate, while the percentage of false positives +is 22.9%. Indeed, the five missed sources are the less photon emitting ones. +In parallel, we tested all the 555 clusters which contain two or more pho- +tons at a significance level of 5% as outlined in Section 4.2.1 and applying +Bonferroni’s correction. This skimmed off 448 modes, for a total of 107 remain- +ing candidate sources, shown in the left panel of Figure 6 as red crosses. Here, +the true positive rate for single photon classification is 85.0% and the false +positive rate is 11.2%. +By super-imposing these two findings, we obtain in all 27 sources which +are both, statistically significant and qualified as such by the non-parametric +classifier. The global true positive rate for single photon classification is 94.6% +while the false positive rate is 14.1%. +6 Concluding remarks +Astronomical data typically come in the form of big data, whose volumes have +increased over the past years from gigabytes into terabytes and petabytes. +However, the widely used model-based approach to multivariate classifica- +tion, which involves maximizing the likelihood of the mixture model using +e.g. the expectation maximization (EM) algorithm or Markov chain Monte +Carlo (McMC) simulation, is computationally impractical for today’s enor- +mous databases. Suitable machine-learning techniques, that apply to such +volumes of data, have recently made their way into the general knowledge basis +of the astrophysics community. Yet, they miss the flexibility and adaptability + +n_photons +density_difference +density +intra_cluster_distance +total_ distance +first_ step_ length +latitude +longitude +photon_energy +0 +500 +1000 +1500 +Importance2000 +1500 +1000 +density +500 +-40 +95 +-35 +100 +105 +-30 +latitude +110 +-25 +115 +120 +-20 +125 +-15 +130Springer Nature 2021 LATEX template +16 +Locating γ-Ray Sources via Modal Clustering +Results +ARI +0.961 +¯d(s, ˆs) +0.001 +ns +39 +True sources +44 +Fig. 5 Left: Fermi-LAT γ-ray photon count map (in Galactic coordinates) for the anal- +ysed 5-year observation period with superimposed the true (black crosses) and candidate +sources. A red cross pinpoints a candidate source which is statistically significant at the 5% +level, while a blue circle identifies a candidate source on the basis of its features. Right: +Performance measures of the tree-based classifier. +which is required if we want to take account of further pieces of available infor- +mation, such as the energy content and quality of the detected events and/or +temporal aspects. Indeed, this may allow us to more efficiently determine the +physical origin of the signals and to discover rare and/or very faint objects, +leading to major discoveries in astrophysics. +Our proposal represents a fast and scalable computational tool to efficiently +and effectively extract knowledge from such large databases. As our aim is to +analyze whole sky maps in one go, we are currently fine-tuning our algorithm +by including a consensus clustering step. This will allow us to aggregate results +from multiple runs, while guaranteeing more stable and robust results (Monti +et al., 2003; Vega-Pons and Ruiz-Shulcloper, 2011). More precisely, borrowing +from Nordhaug Myhre et al. (2018), we form a clustering ensemble consisting +of separate and bootstrapped runs of the mean-shift algorithm on a given +number of overlapping regions of the sky, as shown in Figure 6. The size +and location of these regions varies on a random basis. The final modes are +identified by selecting the cluster configuration which was observed most of +the times. This way of proceeding guarantees robustness with respect to the +choice of the smoothing parameter h, while at the same time allowing us to +work with tremendous amount of data. +Competing interests and funding +The authors have no relevant financial or non-financial interests to disclose. +References +Abramson, I.S. 1982. Arbitrariness of the pilot estimator in adaptive kernel +methods. 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A survey of clustering ensem- +ble algorithms. International Journal of Pattern Recognition and Artificial +Intelligence 25(3): 337–372 . +Yang, M.S., S.J. Chang-Chien, and H.C. Kuo 2014. On mean shift clustering +for directional data on a hypersphere. In L. Rutkowski, M. Korytkowski, +R. Scherer, R. Tadeusiewicz, L. A. Zadeh, and J. M. Zurada (Eds.), Artificial +Intelligence and Soft Computing, Cham, pp. 809–818. Springer International +Publishing. + diff --git a/k9FIT4oBgHgl3EQfriuF/content/tmp_files/load_file.txt b/k9FIT4oBgHgl3EQfriuF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e1d4aacd7510c697fdf1828c0be844cf6037a2e --- /dev/null +++ b/k9FIT4oBgHgl3EQfriuF/content/tmp_files/load_file.txt @@ -0,0 +1,739 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf,len=738 +page_content='Springer Nature 2021 LATEX template Locating γ-Ray Sources on the Celestial Sphere via Modal Clustering Anna Montin1†, Alessandra R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Brazzale1*† and Giovanna Menardi1† 1*Department of Statistical Sciences, University of Padova, Via C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Battisti 241, Padova, 35121, (PD), Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' E-mail(s): alessandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='brazzale@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Contributing authors: anna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='montin@studenti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' giovanna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='menardi@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Abstract Searching for as yet undetected γ-ray sources is a major target of the Fermi LAT Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' We present an algorithm capable of identify- ing such type of sources by non-parametrically clustering the directions of arrival of the high-energy photons detected by the telescope onboard the Fermi spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' In particular, the sources will be identified using a von Mises-Fisher kernel estimate of the photon count density on the unit sphere via an adjustment of the mean-shift algorithm to account for the directional nature of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This choice entails a number of desir- able benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' It allows us to by-pass the difficulties inherent on the borders of any projection of the photon directions onto a 2-dimensional plane, while guaranteeing high flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The smoothing parameter will be chosen adaptively, by combining scientific input with optimal selec- tion guidelines, as known from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Using statistical tools from hypothesis testing and classification, we furthermore present an automatic way to skim off sound candidate sources from the γ-ray emitting diffuse background and to quantify their significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The algo- rithm was calibrated on simulated data provided by the Fermi LAT Collaboration and will be illustrated on a real Fermi LAT case-study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Keywords: directional data, kernel density estimator, man-shift algorithm, tree-based classification 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='11332v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='IM] 25 Jan 2023 Springer Nature 2021 LATEX template 2 Locating γ-Ray Sources via Modal Clustering 1 Motivation and rationale 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1 High-energy astrophysics The past three decades have been a golden era for Astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Pioneering technology has driven remarkable acceleration in the rate of detection and characterization of celestial objects, and new space missions will have more and better quality data to help find and characterize these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Discoveries in this field are of utmost relevance as they contain a wealth of information about the history of the Universe, and impact on the understanding of our Galaxy and our own Solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' An important example is high-energy astrophysics, which acts at the interface between particle physics and astronomy to study the multitude of extreme phenomena which inhabit the Cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' To date, the observation of γ-ray photons, that is, of quanta of light in the highest energy range, has provided the basis for a large number of astronomical discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' γ- rays are usually generated from accelerated charged particles, such as electrons or protons, boosted by extreme celestial objects such as supermassive black holes, supernova remnants, pulsars and active galactic nuclei, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The study of these γ-ray emitting sources improves our understanding of high- energy astrophysical phenomena, and might even resolve the mystery of the fundamental nature of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The Fermi Gamma-ray Space Telescope is an international and multi- agency space mission launched in June 2008 which studies the Cosmos in the energy range 10 keV – 300 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The primary instrument onboard the Fermi spacecraft is the Large Area Telescope (LAT), a wide field-of-view pair- conversion telescope which was designed to perform an all-sky survey aimed at discovering and locating high-energy emitting sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The standard proce- dure of the Fermi LAT Collaboration for point-like source detection relies on so-called single-source models (Hobson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 2009, par.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='4), which require the sky map to be split into small regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The presence of a possible new source is assessed on a pixel-by-pixel basis: Poisson regression is used to model the num- ber of photons associated with each pixel and likelihood ratio tests assess the significance of the source (Mattox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' See also van Dyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2001) for a Bayesian treatment with appplication to low-count X-ray data collected by the Chandra X-Ray Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Conversely, variable-source-number mod- els address the problem from a more global perspective, as they simultaneously identify and locate all possible sources in a given sky map (Hobson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 2009, par.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Since point-like sources present themselves as spatially concentrated photon emissions, the problem can naturally be recast as a clustering prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Recent examples of variable-source-number modelling of X-ray and γ-ray photon count data using finite and infinite mixtures are Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2015), Costantin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2020), Costantin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2020), Sottosanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2021) and Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The data provided by the Fermi LAT Collaboration typically consist of an event list which gives the direction in the sky of each detected photon together with additional information, the primary one being its energy content and Springer Nature 2021 LATEX template Locating γ-Ray Sources via Modal Clustering 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 1 Left: Polar coordinates as recorded by the LAT (Image credit: Mardia and Jupp, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' φ is the longitude and θ is the co-latitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Center and right: Fermi-LAT γ-ray photon count maps for a 5-year observation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Center: in polar coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Right: in Galactic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Yellow: region of size (l, b) ∈ [95◦, 135◦] × [−40◦, −10◦] analyzed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Green: photon counts used to train the post-processing classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Red: Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' the so-called event type which expresses the quality of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This information is used to determine the number of the emitting extra-galactic sources, measure their intensities, and assign to them the corresponding indi- vidual photon counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A major challenge of trying and detecting high-energy phenomena from astronomical data is to separate the signal of the putative emitting source from noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The Fermi LAT data, in particular, are char- acterized by two types of noise: (i) measurement error associated with the components of the LAT (tracker, calorimeter etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=') and (ii) the diffuse γ-ray background which spreads over the entire area observed by the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The former is expressed through the LAT’s point spread function (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 2013), which is typically included into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Different phenom- ena contribute to the residual γ-ray background (Acero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Broadly speaking, its origins can be brought under two headings: galactic interstel- lar emission (GIE), that is, the interaction of galactic cosmic rays with gas and radiation fields, and a residual all-sky emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The latter is commonly called the isotropic diffuse gamma-ray background (IGRB), and includes the γ-ray emission from faint unresolved sources and any residual galactic emis- sion which is approximately isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Costantin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2020) translate the simulation-based background model developed by Acero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2016) into a workable parametric formulation, while Sottosanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2021) reconstruct it via a flexible Bayesian nonparametric model based on B-splines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A further challenge of analysing the Fermi LAT data refers to the geometry of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' As the distance to the emitting source is not given, the data points are placed on the celestial sphere with Earth at its center and unit radius, as shown in the middle panel of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Directions are expressed in Galactic coordinates, that is longitude l and latitude b, which place the origin of the Cartesian system in the center of our galaxy — the Milky Way — and align the x-axis with the Galactic plane (right panel of Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This is the situation considered by Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2015), Costantin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2020), Sottosanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2021) and Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Instead of projecting data onto a 2- dimensional map, we may rather express directions in 3 dimensions through polar coordinates, that is, co-latitude (θ) and longitude (φ) in geographical terms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' see the left panel of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' These can easily be back-transformed to 44 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='Springer Nature 2021 LATEX template 4 Locating γ-Ray Sources via Modal Clustering Cartesian coordinates x = [cos θ, sin θ cos φ, sin θ sin φ]⊤ on the unit sphere, as done by Costantin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A thorough treatment of directional data can be found in Mardia and Jupp (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2 The statistical state of the art The discovery of celestial objects is an intrinsically interdisciplinary field which combines both, statistical and astrophysical methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Statistical learning, by which we mean the ability of discovering patterns and regularities in the data, plays a central role in knowledge discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This also includes allocating objects to a pre-assigned or unknown number of groups according to a set of observed attributes or features, which is a natural activity of any science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A major distinction is made depending on whether the groups are defined, and known a priori, or need be detected using the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Clustering, or unsuper- vised learning, considers the latter situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A surge of techniques has been proposed over the years, which differ significantly in their definition of what a cluster is and how to identify it (Hennig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A precise statisti- cal notion of what a “group” is, is provided by the density-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Here, the clusters are associated with some specific features of the probability distribution which is assumed to underlie the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This idea has been devel- oped into two distinct directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The model-based or parametric approach represents the probability distribution of the data as a mixture of parametric distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A cluster is associated with each component of the mixture and the observations are allocated to the cluster with maximal density among the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Standard accounts are the seminal works of Fraley and Raftery (1998, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A less widespread density-based clustering formulation is referred to as modal or nonparametric clustering and dates back to Carmichael et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Here, the underlying density is reconstructed from the data using suit- able nonparametric density estimators, and clusters are associated with the domain of attraction of their modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The rather scattered theory is reviewed in Menardi (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Chac´on (2015) provides some new insight into the theoreti- cal foundations of modal clustering making use of Morse theory (Milnor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' In this paper, we advocate the use of nonparametric, or modal clustering for γ-ray source detection using a von Mises-Fisher kernel on the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This choice entails a number of desirable benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' It allows us to by-pass the difficulties inherent on the borders of any 2-dimensional projection of the photon directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' But, it also guarantees high flexibility and adaptability, while posing on a sound theoretical ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The sources will be identified via an adjustment of the mean-shift algorithm to account for the directional nature of the Fermi LAT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The issue of selecting the smoothing parameter is addressed adaptively, by combining scientific input with optimal selection guidelines, as known from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Using known results from hypothesis testing and classification, we furthermore present an automatic way to pinpoint sound candidate sources and to quantify their significance by skimming off the γ-ray emitting diffuse background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The Fermi LAT database currently holds Springer Nature 2021 LATEX template Locating γ-Ray Sources via Modal Clustering 5 over 1 billion photons in the energy range from about 20 MeV to more than 300 GeV collected in over a decade of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Efficient tools to account for the computational burden required to analyse huge amounts of data, possibly on the entire sphere, are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Our method was calibrated on simulated data provided by the Fermi LAT Collaboration and will be illustrated on a real Fermi LAT case-study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Section 2 sets the methodological back- ground of kernel density estimation for directional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Being able to correctly specify the right amount of smoothing is crucial for the reliable identifica- tion of the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Optimal bandwidth selection is discussed in Section 3, while Section 4 presents our proposal of modal clustering on the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' In particular, to separate the true signal emitted by a source from the back- ground, we developed a post-processing procedure that combines the findings of two parallel quests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' One establishes the significance of a candidate mode using a suitable statistical test as presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The second skims off the photons emitted by the γ-ray background using a tree-based classifier build on previous knowledge provided by the Fermi LAT Collaboration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1 benchmarks two key aspects of our proposal, namely the selection of the optimal bandwidth and the classification of the incom- ing photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2 eventually illustrates the performance of our proposal when applied to a real sample of high-energy photons accumulated by the LAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The paper closes with the concluding remarks of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This paper is an extended version of the paper presented at the 51st Sci- entific Meeting of the Italian Statistical Society on June, 2022 (Montin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 2 Kernel density estimators for directional data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1 The von Mises-Fisher distribution Directions in the 3-dimensional space can be represented using Cartesian coordinates as unit vectors x, that is, as points on the sphere Ω2 = {x ∈ R3 : ∥x∥2 = x2 1 + x2 2 + x2 3 = 1} with unit radius and centre at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' These can be retrieved from Galactic coordinates, that is, from the longitude l ∈ (−180, +180) and the latitude b ∈ (−90, +90) of a given data point, by x = [cos l cos b, sin l cos b, sin b]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A widely used distribution to model γ-ray emission in astrophysics searches (Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 2006) is the von Mises-Fisher (vMF) distribution fvMF (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' µ, κ) = C2(κ) exp{κx⊤µ}, Springer Nature 2021 LATEX template 6 Locating γ-Ray Sources via Modal Clustering which extends the 3-dimensional normal distribution N3(µ, κ−1I3), with I3 being the 3 × 3 diagonal unit matrix, by restricting its density to the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Here, µ ∈ Ω2 represents the mean direction, while κ ≥ 0 is a con- centration parameter (Mardia and Jupp, 2000, Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' As such, the von Mises-Fisher distribution describes observations which scatter simmetrically around their mean direction µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The normalizing constant C2(κ) = κ 1 2 (2π) 3 2 I 1 2 (κ) includes the modified Bessel function Iν(z) = � z 2 �ν π1/2Γ(ν + 1 2) � 1 −1 (1 − t2)ν− 1 2 eztdt of order ν = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2 Kernel density estimator Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' , xn ∈ Ω2 be a random sample of n observations generated by a distribution with density f(x) defined on the unit sphere Ω2 such that � Ω2 f(x)ω2(dx) = 1, where ω2 is the Lebesgue measure on Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' We can estimate the density f using the kernel density estimator proposed by Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (1988) for directional data, ˆfh(x) = ch(K) n n � i=1 K �1 − x⊤xi h2 � , (1) where K(·) is a suitable kernel function which decreases on [0, ∞), and h > 0 is the smoothing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The normalizing constant ch(K), is defined by ch(K)−1 = � Ω2 K �1 − x⊤xi h2 � ω2(dx) = h2˜ch(K), where ˜ch(K) = � 2/h2 0 K(u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Using the von Mises-Fisher kernel, expression (1) becomes ˆfh(x) = 1 n n � i=1 fvMF � x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' xi, 1 h2 � = 1 (2π) 3 2 I 1 2 (h−2) 1 hn n � i=1 exp �x⊤xi h2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2) Springer Nature 2021 LATEX template Locating γ-Ray Sources via Modal Clustering 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 2 von Mises-Fisher kernel density estimate of the high-energy photons tracked by the Fermi LAT in the validation region (l, b) ∈ [0◦, 60◦]×[10◦, 60◦] for different values of h: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='01 (left), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='001 (center), hi,SE (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' That is, the kernel density estimator for direction data on the unit sphere is a mixture of 3-dimensional von Mises-Fisher distributions with κ = h−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 3 Bandwidth selection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1 Data-based methods A major issue when using a kernel density estimator is the selection of the smoothing parameter, or bandwidth, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Being able to correctly specify the right amount of smoothing is crucial for the reliable identification of the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This is illustrated in Figure 2, which plots the estimated density for the same sky region using three different values of h, where the latter choice varies with sky location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' If the smoothing parameter is too large (picture on the left), false peaks may emerge from the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Conversely, if the kernel function is too concentrated (middle picture), we may miss some faint sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A wealth of data-driven methods were developed over the years for both, fixed and variable bandwidth kernel density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' As far as directional data goes, the proposals mainly are for circular observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (1987) and Klemel¨a (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Adaptive kernel density estimation, that is, when the smoothing parameter hi in (2) adapts to the local behaviour of f at xi, is of special interest to us, as the spatial scattering of the incoming photons differs among sources, and to an even larger extent if they were emitted from the background radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Selecting an optimal bandwidth generally entails minimization of a suitable measure of the error we commit when estimating the target density f by ˆfh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A common way of measuring this error is the mean integrated squared error MISE(h) = E �� Ω2 � ˆfh(x) − f(x) �2 ω2(dx) � , where the expectation is taken with respect to the distribution specified by f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' in this case hMISE = arg min h>0 MISE(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 500 400 300 densita 200 100 0 10 0 20 10 30 20 30 AO b 40 50 So 6010k 8K 6k densita AK 2K 0 10 20 10 30 20 30 b AO 90 50 So Og7000 6000 5000 4000 2000 1000 10 20 1o 20 30 30 40 b O 50 5o 60Springer Nature 2021 LATEX template 8 Locating γ-Ray Sources via Modal Clustering Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 3 Left: Photon scattering as a function of their energy content (courtesy of Sottosanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Right: Values of hi,SE as a function of energy and event quality, where PSF0 represents the worst event type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The higher the energy and quality of the event, the smaller is the smoothing parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The alternative choice hAMISE minimizes the asymptotic approximation of the mean integrated squared error, that is, when n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' However, both window widths depend explicitly on the unknown density to be estimated, and cannot be computed exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Simple “plug-in” procedures, where f is replaced by a suitable pilot estimate ˆf, turned out to be generally unsatisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' An automatic way of determining the optimal bandwidth h is by likelihood cross- validation, that is, hLCV = arg max h>0 CV (h), where CV (h) = n � i=1 log ˆfh,−i(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Here, ˆfh,−i(xi) is the kernel density estimate we obtain after having omitted observation i, evaluated at xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A further option is to adapt the most promising solutions for optimal bandwidth selection on the plane to our problem at hand, as listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The corresponding performance metrics were evaluated on the simulated sample of high-energy photons emitted by the sources present in the sky region shown in Figure 2, and will be discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A first possibility is to generalize Garc´ıa-Portugu´es’ (2013) rule of thumb to spherical data, hT HUMB = � 8 sinh2(ˆκ) ˆκ[(1 + 4ˆκ2) sinh(2ˆκ) − 2ˆκ cosh(2ˆκ)]n � 1 6 , where the concentration parameter ˆκ is estimated by maximum likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Conversely, if we want the bandwidth h to depend on the current location xi of the estimator, a first possibility is to use Abramson’s (1982) rule, which has Energy >= 11 GeV Energy >= 307 GeV 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0 0 QQ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5- 8 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5 - 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0 - 0 Q Energy >= 604 GeV Energy >= 900 GeV 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0 - 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5- 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0 - 米 米 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0- 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0 Longitudine0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='006 Quality 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='004 0123 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='SE h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='002 0 250000 500000 750000 1000000 Energy (MeV)Springer Nature 2021 LATEX template Locating γ-Ray Sources via Modal Clustering 9 hi change proportionally with the inverse of the square root of ˆfh(xi), hA i,T HUMB = hT HUMB � ˆfhT HUMB(xi) �− 1 2 and hA i,LCV = hLCV � ˆfhLCV (xi) �− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Here, ˆfhT HUMB(xi) and ˆfhLCV (xi) are winsorized (or clipped) versions of a suitably constructed pilot kernel density estimate with fixed bandwidth h, which may be hT HUMB or hLCV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A second possibility is to use the modification proposed by Silverman (1986, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='3), hS i,T HUMB = hT HUMB � 1 mg ˆfhT HUMB(xi) �−β and hS i,LCV = hLCV � 1 mg ˆfhLV C(xi) �−β , where mg is a scale factor defined by the geometric mean of the two pilot estimates, ˆfhT HUMB(xi) or ˆfhLCV (xi), while β ∈ [0, 1] tunes the sensitivity of the bandwidth to variations of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' We will set β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5, as this choice generally entails a better behavior of the kernel density estimator on the tails of the distribution (Izenman, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2 Using scientific input A valid alternative for determining the smoothing parameter h is to use scien- tific input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' As mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1, the spatial scattering of the photons around the source direction µ is modelled by the LAT’s point spread function (PSF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This function depends on the energy of the incoming photon, on its inclination angle θ (see left panel of Figure 1) and on the quality of the recorded event (Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The latter is expressed by the PSF event type, that is, an event-level quantity which indicates how well the LAT man- aged to reconstruct the direction of the incoming photon and which assumes four values, from the lowest quality (PSF0) to the best quality (PSF3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Most importantly, the PSF depends on the scale factor S(Ei) ∝ �� c0,i � Ei 100MeV �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='8�2 + c2 1,i, which describes the uncertainty of the event as a decreasing function of the energy Ei, expressed in Mega electron Volt (MeV), and of the two parameters c0,i and c1,i, which are given distinct values for the different event qualities and can be retrieved from the Fermi LAT web site1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The first constant, ci,0, represents multiple scattering while the second, c1,i, represents the spatial resolution of the LAT tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' How the precision of the measurements depends on the energy is shown in the left panel of Figure 3 (Sottosanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 2021), while the right panel of the same figure plots the values we obtain for hi,SE 1https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='gov/ssc/data/analysis/documentation/Cicerone/Cicerone LAT IRFs/IRF PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='html Springer Nature 2021 LATEX template 10 Locating γ-Ray Sources via Modal Clustering for the four different event types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' On this basis, we may specify a variable bandwidth as hi,SE = �� c0,i � Ei 100MeV �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='8�2 + c2 1,i, (3) which is the one used in the right panel of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 4 Modal clustering on the unit sphere 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1 Mode hunting Modal clustering associates clusters with the domain of attraction of the modes of the underlying density f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Two main strands can be identified, depending on whether the modes are given explicitely or not (Menardi, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A first strand follows the route of Hartigan (1975) and identifies clusters with high-density regions of the sample space, defined by the density level sets Lc(f) = {x ∈ Ω2 : f(x) ≥ c}, 0 ≤ c ≤ max f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' An estimate of the unknown Lc(f) is obtained by replacing f(x) by its non- parametric estimate ˆf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The rationale behind this class of methods is that any connected component of Lc(f) includes at least one mode of the density function, and, on the other hand, for each mode of the density function, there exists λ for which one of the connected components of the associated L(λ) includes this mode at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The major drawback is that the identification of the connected components of a multidimensional set is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' As our aim is to discover and identify unknown γ-ray emitting sources, we want to associate their direction explicitly with the modes of the unknown density f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2014) adapted the mean-shift algorithm developed by Fukunaga and Hostetler (1975) to be used with the directional kernel estimator (2) and fixed bandwidth h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Starting from a generic point x(0), the algorithm recursively shifts it to a local weighted mean, until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Denoted by wi(x(s)) the vector of weights of the components of xi at step s, at the next step, (s + 1), we have x(s+1) = n � i=1 wi(x(s))xi = x(s) + M(x(s)), where M(x(s)) = �n i=1 wi(x(s))xi − x(s) denotes the mean shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Up to a normalising factor, the weights wi(x) involve the derivative K′(h−2(1−x⊤xi)) of the kernel function, which leads to the weighted average ˆx(s+1) = − �n i=1 xiK′ � 1−ˆx(s)⊤xi h2 � ��� ��� �n i=1 xiK′ � 1−ˆx(s)⊤xi h2 ���� ��� 2 , Springer Nature 2021 LATEX template Locating γ-Ray Sources via Modal Clustering 11 where || · ||2 is the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Here, the minus sign is due becasue K(·) is a decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' If we replace the kernel function K(·) by the von Mises-Fisher kernel, the above expression becomes ˆx(s+1) = �n i=1 xi exp � ˆx(s)⊤xi−1 h2 � ��� ��� �n i=1 xi exp � ˆx(s)⊤xi−1 h2 ���� ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Straightforward calculations allowed us to extend the proposal by Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2014) to varying hi, that is, for adaptive kernel density estimation on the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2 Post-processing As mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1, the incoming photons were either emitted from a high-energy source or are part of the diffuse γ-ray background which spreads over the entire area observed by the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The directional kernel density estimator (2) tries and reconstructs the corresponding mixture distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Hence, the small peaks which emerge as modes may identify true sources, but they may equally well represent a false signal generated by the irregularly shaped background radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' To separate the true signal emitted by a source from the background, we developed a post-processing procedure that combines the findings of two parallel quests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' One establishes the significance of a candi- date mode using a suitable statistical test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The second skims off the photons emitted by the γ-ray background using a suitable classifier build on previous knowledge provided by the Fermi LAT Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' By super-imposing the findings from these two quests, we identify candidate sources which are both, statistically significant and qualified as such according to a set of relevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Furthermore, we are now able to distinguish photons emitted by a candidate source from those pertaining to the background radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1 Statistical significance Mathematically, we can verify whether a function reaches a local maximum by checking whether all eigenvalues of the Hessian matrix evaluated at the candidate mode are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Statistically, developing a suitable test to ver- ify the existence of a mode and deriving its null distribution using eigenvalues is tricky, as these are not continuously differentiable functions of the Hes- sian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This invalidates resampling-based methods such as the bootstrap and asymptotic expansion by the delta method, which we may use to reconstruct the finite-sample null distribution of the test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Genovese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2016) hence suggest to use data splitting to separate the process of finding candidate modes from the process of hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' They furthermore propose to base inference on confidence intervals, rather than on p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The potential modes are hence estimated on the first half of the data, while the second half Springer Nature 2021 LATEX template 12 Locating γ-Ray Sources via Modal Clustering is used to construct asymptotically valid bootstrap confidence intervals for the eigenvalues of the Hessian matrix, which can be used for hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The extension of this idea to directional data requires some care, as working on the unit sphere sets some constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' To calculate the Hessian matrix H ˆfh(x), we first need the total gradient ∇ ˆfh(x) = C2(h−2) n n � i=1 xi h2 exp � x⊤xi − 1 h2 � , where ∇ represents suitable differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The Hessian matrix hence is H ˆfh(x) = (I3 − xx⊤) � ∇∇ ˆfh(x) − ∇ ˆfh(x)⊤xI3 � (I3 − xx⊤) = (I3 − xx⊤) � C2(h−2) n n � i=1 xix⊤ i h4 exp � x⊤xi − 1 h2 � + − C2(h−2) n n � i=1 x⊤xiI3 h2 exp � x⊤xi − 1 h2 �� (I3 − xx⊤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Likewise, we may obtain the Hessian matrix associated with an adaptive kernel density estimator with variable bandwidth hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The tricky part is that the eigenvalue of H ˆfh(µ), when ˆfh(x) is evaluated at µ, is always zero, whether µ corresponds to a true source or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This entails that inference has to be based on the remaining two eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' We hence construct an 1−α level confidence interval for the largest non null eigenvalue using bootstrap resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The candidate mode is validated if the interval includes only negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A second possibility is to reparametrize the von Mises-Fisher kernel using polar coordinates fvMF (θ, φ) = κ 4πκ exp � κ cos θ cos η + k sin θ sin η cos(φ − ζ) � sin θ, where, as in Mardia and Jupp (2000), x = (cos θ, sin θ cos φ, sin θ sin φ)⊤ and µ = (cos η, sin η cos ζ, sin η sin ζ)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This workaround allows us to directly apply the results by Genovese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2 Feature selection A further possibility to skim off the photons emitted by extra-galactic sources from those which originate from the diffuse background is to build a suitable classification rule which integrates additional information on the photons pro- vided by the Fermi LAT Collaboration and/or features that can be extracted at the various steps of the mean-shift algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' These include the energy content of the photons (photon energy) and their incoming direction (longi- tude, latitude), the number of photons assigned to a mode (n photons), the Springer Nature 2021 LATEX template Locating γ-Ray Sources via Modal Clustering 13 h ARI ¯d(s, ˆs) ns hi,SE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='9976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0004 86 hT HUMB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='6841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0079 10 hA i,T HUMB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='6805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0139 18 hS i,T HUMB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='8524 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0063 25 hLCV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='9777 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0092 142 hA i,LCV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='9777 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0092 142 hS i,LCV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='9777 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0092 142 Table 1 Performance metrics for different choices of the bandwidth h of the von Mises-Fisher kernel density estimator applied to the sky region plotted in Figure 2: ARI = adjusted Rand index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' ¯d(s, ˆs) = median angular distance (in degrees) between the directions of true sources (s) and candidate sources (ˆs) identified by the algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' ns = number of identified sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The number of true sources is 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' density estimates for the signal and the background model (density, den- sity difference) and various types of distances between the photons and their mode (intra cluster distance, total distance, first step length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' We hence sug- gest to train and test a tree-based classifier on a suitable area of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The final classifier will then be pruned so as to assign any cluster with a single pho- ton to the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2 reports the performance metrics of our classification rule when applied to a portion of the Northern sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 5 Application to Fermi LAT data 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1 Benchmarking 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1 Optimal bandwidth Table 1 compares the different proposals for bandwidth selection listed in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2 using three performance metrics, that is, the adjusted Rand index (ARI), the median angular distance (in degrees) between the direc- tions of true sources and candidate sources, ¯d(s, ˆs), and the number ns of identified sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' These metrics were obtained by benchmarking our algorithm on a simulated sample of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='335 photons emitted by the 68 sources present in the validation region (l, b) ∈ [0◦, 60◦] × [10◦, 60◦] shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The three proposals based on the rule of thumb oversmooth the true photon density, lead- ing to rather low ARI values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Likelihood cross validation, on the other hand, tends to over adapt the true density yielding too many candidate sources: 142 in place of the 68 present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The best partition of the selected sky region is obtained when using the variable bandwidth hi,SE, that is, the scale factor of the LAT’s point spread function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Further support to this choice is provided by Table 2, which contrasts the selected optimal bandwidths (Columns 3–7) with the true photon scattering, as measured by its standard deviation (Column 2), for 5 selected sources of varying size, that is, which emit from a minimum of ns = 7 photons up to a mximum of ns = 151 photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Again, hi,SE is the best performing choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 14 Locating γ-Ray Sources via Modal Clustering Source sd ¯hi,SE ¯hA i,LCV ¯hA i,T HUMB ¯hS i,LCV ¯hS i,T HUMB ns = 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0017 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2958 · 10−06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1053 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='8623 · 10−07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0611 ns = 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0028 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2225 · 10−06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0221 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='7986 · 10−07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0128 ns = 31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0027 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='8184 · 10−06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0501 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='4477 · 10−07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0290 ns = 79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0028 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1721 · 10−06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0314 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='8864 · 10−07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0182 ns = 151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0027 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0684 · 10−06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0215 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='7963 · 10−07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0125 Table 2 Standard deviation (Column 2) of photon scattering for 5 selected sources of varying size (Column 1) and average bandwidths computed using the scale factor of the PSF (Column 3) or selected by Abramson’s or Silverman’s rules (Columns 4–7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2 Performance metrics We bemchmarked our tree-based classifier on a sample of 35,365 simulated photon emissions in the sky region (l, b) ∈ [100◦, 150◦] × [0◦, 90◦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This area covers the entire Northern sky to account for the rater prominent variability of the diffuse γ-ray background as we move away from the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The classifier was estimated on the first 2/3 of the sample, for a total of 24,573 photons, and tested on the remaining 11,062 photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' In both sets, about 85% of the photons were emitted from the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The final classifier was pruned so as to assign any cluster with a single photon to the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The classifier was hence benchmarked on the sky region shown in Figure 2, where it selected a total of ns = 86 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The average sensibility, computed on the candidate sources identified by the classifier, was 90,5%, while the average specificity was 99,5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The adjusted Rand index (ARI) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='9752 and the median angular distance between the true sources and the identified ones is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0005 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2 Case-study The yellow region in Figure 1 shows a portion of the Southern sky of size (l, b) ∈ [95◦, 135◦]×[−40◦, −10◦] for which the LAT accumulated 3,849 photon counts over a five-year period of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2 Of these, about 26% were emitted by the 44 sources present in the area, while the remaining 74% originated from the diffuse γ-ray background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The left panel of Figure 4 plots the estimated kernel density (2) using a von Mises-Fisher kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Here, the bandwidth parameter h was set according to scientific input, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This choice revealed to be the most performing one in terms of adjusted Rand index (ARI), median angular distance, ¯d(s, ˆs), between the true source direction and the reconstructed one and number of identified sources (ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' In all, the mean-shift algorithm identified 876 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' To further refine the list of candidate sources we proceeded in two steps as outlined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A tree-based classifier to discriminate between source and background pho- tons was trained on the 6,814 photon counts highlighted in green in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The importance of the selected predictor variables is shown in the right panel of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The most discriminating features are the number of photons assigned 2https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='gov/ssc/data/access/ Springer Nature 2021 LATEX template Locating γ-Ray Sources via Modal Clustering 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 4 Left: Kernel density estimate using a von Mises-Fisher kernel for the 3,849 γ-ray photon counts accumulated by the LAT in a 5-year period in the region of the Southern sky identified by (l, b) ∈ [95◦, 135◦] × [−40◦, −10◦].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' About 26% of the photons were emitted by the 44 sources present in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Right: Feature importance plot for the tree-based photon classifier used to discriminate between source and diffuse γ-ray background emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' to a cluster (n photons), the difference between the two photon densities for, respectively, the all sky and background counts only (density differences), and the density observed for each photon (density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This reduces the original 876 modes to 39 candidate sources, which are shown as blue circles in the left panel of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The table on the right reports the performance of our classifier in terms of ARI and median angular distance ¯d(s, ˆs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The true positive rate for single photon classification is 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='5% rate, while the percentage of false positives is 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Indeed, the five missed sources are the less photon emitting ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' In parallel, we tested all the 555 clusters which contain two or more pho- tons at a significance level of 5% as outlined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1 and applying Bonferroni’s correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This skimmed off 448 modes, for a total of 107 remain- ing candidate sources, shown in the left panel of Figure 6 as red crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Here, the true positive rate for single photon classification is 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='0% and the false positive rate is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' By super-imposing these two findings, we obtain in all 27 sources which are both, statistically significant and qualified as such by the non-parametric classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The global true positive rate for single photon classification is 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='6% while the false positive rate is 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 6 Concluding remarks Astronomical data typically come in the form of big data, whose volumes have increased over the past years from gigabytes into terabytes and petabytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' However, the widely used model-based approach to multivariate classifica- tion, which involves maximizing the likelihood of the mixture model using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' the expectation maximization (EM) algorithm or Markov chain Monte Carlo (McMC) simulation, is computationally impractical for today’s enor- mous databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Suitable machine-learning techniques, that apply to such volumes of data, have recently made their way into the general knowledge basis of the astrophysics community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Yet, they miss the flexibility and adaptability n_photons density_difference density intra_cluster_distance total_ distance first_ step_ length latitude longitude photon_energy 0 500 1000 1500 Importance2000 1500 1000 density 500 40 95 35 100 105 30 latitude 110 25 115 120 20 125 15 130Springer Nature 2021 LATEX template 16 Locating γ-Ray Sources via Modal Clustering Results ARI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='961 ¯d(s, ˆs) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='001 ns 39 True sources 44 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 5 Left: Fermi-LAT γ-ray photon count map (in Galactic coordinates) for the anal- ysed 5-year observation period with superimposed the true (black crosses) and candidate sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' A red cross pinpoints a candidate source which is statistically significant at the 5% level, while a blue circle identifies a candidate source on the basis of its features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Right: Performance measures of the tree-based classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' which is required if we want to take account of further pieces of available infor- mation, such as the energy content and quality of the detected events and/or temporal aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Indeed, this may allow us to more efficiently determine the physical origin of the signals and to discover rare and/or very faint objects, leading to major discoveries in astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Our proposal represents a fast and scalable computational tool to efficiently and effectively extract knowledge from such large databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' As our aim is to analyze whole sky maps in one go, we are currently fine-tuning our algorithm by including a consensus clustering step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This will allow us to aggregate results from multiple runs, while guaranteeing more stable and robust results (Monti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Vega-Pons and Ruiz-Shulcloper, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' More precisely, borrowing from Nordhaug Myhre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' (2018), we form a clustering ensemble consisting of separate and bootstrapped runs of the mean-shift algorithm on a given number of overlapping regions of the sky, as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The size and location of these regions varies on a random basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' The final modes are identified by selecting the cluster configuration which was observed most of the times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' This way of proceeding guarantees robustness with respect to the choice of the smoothing parameter h, while at the same time allowing us to work with tremendous amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Competing interests and funding The authors have no relevant financial or non-financial interests to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' References Abramson, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Arbitrariness of the pilot estimator in adaptive kernel methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' Journal of Multivariate Analysis 12(4): 562–567 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FIT4oBgHgl3EQfriuF/content/2301.11332v1.pdf'} +page_content=' 10 20 30 40 100 110 120 130 Sources: O discrimination rule (ns) X 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with Offline Data +Shuze Liu +shuzeliu@virginia.edu +Department of Computer Science +University of Virginia +85 Engineer’s Way, Charlottesville, VA, 22903 +Shangtong Zhang +shangtong@virginia.edu +Department of Computer Science +University of Virginia +85 Engineer’s Way, Charlottesville, VA, 22903 +Abstract +Monte Carlo (MC) methods are the most widely used methods to estimate the performance +of a policy. Given an interested policy, MC methods give estimates by repeatedly running +this policy to collect samples and taking the average of the outcomes. Samples collected +during this process are called online samples. To get an accurate estimate, MC methods +consume massive online samples. When online samples are expensive, e.g., online recom- +mendations and inventory management, we want to reduce the number of online samples +while achieving the same estimate accuracy. To this end, we use off-policy MC methods +that evaluate the interested policy by running a different policy called behavior policy. We +design a tailored behavior policy such that the variance of the off-policy MC estimator is +provably smaller than the ordinary MC estimator. Importantly, this tailored behavior pol- +icy can be efficiently learned from existing offline data, i,e., previously logged data, which +are much cheaper than online samples. With reduced variance, our off-policy MC method +requires fewer online samples to evaluate the performance of a policy compared with the +ordinary MC method. Moreover, our off-policy MC estimator is always unbiased. +1. Introduction +Evaluating a policy of interest via a scalar performance metric is a fundamental problem +in reinforcement learning (Sutton and Barto, 2018). Such evaluation makes it possible to +directly compare two policies and lays out the foundations for the more ambitious con- +trol problem, the goal of which is to find the best-performing policy. Monte Carlo methods +(Kakutani, 1945) are the most commonly used methods for such evaluation problems. Given +an interested policy, Monte Carlo methods give estimates by repeatedly executing this pol- +icy to collect trajectories and taking the averaged outcomes. Trajectory samples collected +during this process are called online samples. Online samples are cheap when reliable sim- +ulators exist. For example, board games (Tesauro 1995; Silver et al. 2016) and video games +(Mnih et al. 2015; Vinyals et al. 2019) give a large amount of instant online samples with +little cost. However, in many real-world applications, online samples are usually expensive +and take non-negligible time and financial costs to collect, e.g., online recommendations (Li +et al. 2010; Gauci et al. 2018) and inventory management (Giannoccaro and Pontrandolfo +©2022 Shuze Liu and Shangtong Zhang. +License: CC-BY 4.0, see https://creativecommons.org/licenses/by/4.0/. +arXiv:2301.13734v1 [cs.LG] 31 Jan 2023 + +Liu and Zhang +2002). In those scenarios, reducing the number of the required online samples in Monte +Carlo methods brings in substantial time and economic benefits. +Online sample reduction can be achieved by reducing the variance of Monte Carlo es- +timators (see, e.g., Bernstein (1924)). One well-known variance reduction tool in statistics +is importance sampling (Rubinstein 1981; Benjamin Melamed 1998). In the setting of rein- +forcement learning, it corresponds to evaluating an interested policy by executing a different +policy to collect online samples. The interested policy is called target policy and the exe- +cuted policy is called behavior policy. To get an unbiased estimate, we rely on importance +sampling ratios to reweight online samples from the behavior policy and thus give an un- +biased estimate of the target policy. This idea of estimating a target policy by running a +different behavior policy is called off-policy learning (Geweke 1988; Hesterberg 1995; Precup +et al. 2000; Koller and Friedman 2009). By contrast, estimating a target policy by running +this policy itself is called on-policy learning (Sutton, 1988). When the behavior policy is +properly tailored, the variance of the off-policy Monte Carlo estimator can be significantly +smaller than that of the ordinary on-policy one. In this paper, we propose algorithms to +learn such behavior policies from previously logged, existing offline data, which are much +cheaper than online data. +Notably, the learned behavior policy from our proposed method may suffer from learning +errors, just like any other learning process, due to, e.g., insufficient offline data coverage, +mismatched hypothesis spaces, etc. +Nevertheless, our proposed Monte Carlo-off policy +estimator is always unbiased. We further use a bandit algorithm to strategically switch +between the learned behavior policy and the target policy for online data collection, which +guarantees that the regret is reasonably small even if the learned behavior policy contains +non-negligible learning errors. +2. Background +We consider a finite horizon Markov Decision Process (MDP, Puterman (2014)) with a +finite state space S, a finite action space A, a reward function r : S × A → R, a transition +probability function p : S × S × A → [0, 1], an initial distribution p0 : S → [0, 1], and a +constant horizon length T. Without loss of generality, we consider this undiscounted setting +for simplifying notations. Our results naturally apply to the discounted setting (Puterman, +2014) as long as the horizon is fixed and finite. At time step 0, an initial state S0 is sampled +from p0. At time step t ∈ {0, 1, . . . , T − 1}, an action At is sampled according to πt(·|St) +where πt : A × S → [0, 1] is the policy at time step t. A finite reward Rt+1 .= r(St, At) is +then emitted and a successor state St+1 is sampled from p(·|St, At). We define abbreviations +πi:j .= {πi, πi+1, . . . , πj} and π .= π0:T−1. The return at time step t is defined as +Gt .= +T +� +i=t+1 +Ri, +which allows us to define the state- and action-value functions as +vπ,t(s) .=Eπ [Gt|St = s] , +qπ,t(s, a) .=Eπ [Gt|St = s, At = a] . +2 + +Improving Monte Carlo Evaluation with Offline Data +It is easy to see that +vπ,t(s) = +� +a +πt(a|s)qπ,t(s, a). +(1) +We focus on the total rewards performance metric (Puterman, 2014) to measure the per- +formance of the policy π, which is defined as +J(π) .= +� +s +p0(s)vπ,0(s). +Knowing such a scalar performance metric makes it possible to easily compare two policies +and is also preferred in machine learning applications and research (Ng 2017) because it +offers a clear learning goal. In this paper, we focus on Monte Carlo methods introduced by +Kakutani (1945) to estimate the total rewards J(π). Among many of its variants, the most +straightforward and widely used way is to draw samples of J(π) by executing the policy π +online. As the number of samples increases, the empirical average of the sampled returns +converges to J(π). This idea is called on-policy learning (Sutton 1988) because it estimates +a policy π by executing itself. +From now on, we consider off-policy learning, where we estimate the total rewards J(π) +of an interested policy π called target policy by executing a different policy µ called behavior +policy. Off-policy learning has substantive advantages. First, estimating the value of a target +policy π without actual deployment makes the learning process much safer (Thomas 2015). +Safety and reliability are critical factors in real-world applications. +Second, trajectory +samples collected by one behavior policy can be used to evaluate multiple target policies +(Sutton et al. 2011), making the estimation more efficient. +In off-policy learning, each +trajectory +{S0, A0, R1, S1, A1, R2, . . . , ST−1, AT−1, RT } +is generated by a behavior policy µ with +S0 ∼ p0, At ∼ µt(·|St), t ∈ {0, 1, . . . , T − 1}. +Let τ µt:T −1 +t:T−1 +.= {St, At, Rt+1, . . . , ST−1, AT−1, RT } be a shorthand for a segment of a random +trajectory generated by the behavior policy µ from the time step t to the time step T − 1 +inclusively. In off-policy learning, we use the importance sampling ratio to reweight rewards +collected by µ in order to give an estimate of J(π). The importance sampling ratio at time +step t is defined as +ρt .= πt(At|St) +µt(At|St). +The product of importance sampling ratios from time t to the last step T − 1 is defined as +ρt:T−1 .= +T−1 +� +k=t +πk(Ak|Sk) +µk(Ak|Sk). +3 + +Liu and Zhang +There are various methods to use the importance sampling ratios in off-policy learning. +The most straightforward ordinary importance sampling (IS) estimator is defined as +GIS(τ µt:T −1 +t:T−1 ) .= ρt:T−1Gt. +This ordinary importance sampling estimator is an unbiased estimator when the behavior +policy µ covers the target policy π. That is when µt(a|s) = 0 =⇒ πt(a|s) = 0, we have +E +� +GIS(τ µt:T −1 +t:T−1 )|St = s +� += E [ρt:T−1Gt|St = s] = vπ,t(s) +∀t, s. +However, weighted by the entire product ρ0:T−1, the ordinary importance sampling es- +timator has a high variance. Intensive research has been conducted in finding importance +sampling estimators with reduced variance, e.g. the weighted importance sampling estima- +tor (Geweke 1988; Hesterberg 1995; Koller and Friedman 2009), the per-decision importance +sampling estimator (Precup et al. 2000), the consistent weighted per-decision importance +sampling estimator (Thomas 2015), etc. Our paper is based on the per-decision importance +sampling estimator (PDIS), which is defined as +GPDIS(τ µt:T −1 +t:T−1 ) .= +T−1 +� +k=t +ρ0:kRk+1. +We choose the per-decision importance sampling estimator because it is an unbiased esti- +mator for any behavior policy µ that covers target policy π (Precup et al., 2000). In other +words, when µt(a|s) = 0 =⇒ πt(a|s) = 0, we have +E[GPDIS(τ µt:T −1 +t:T−1 )|St = s] = vπ,t(s) +∀t, s. +We will intensively use the recursive expression of the per-decision importance sampling +estimator +GPDIS(τ µt:T −1 +t:T−1 ) = +� +ρt +� +Rt+1 + GPDIS(τ µt+1:T −1 +t+1:T−1 ) +� +0 ≤ t < T − 1 +ρtRt+1 +t = T − 1. +. +(2) +The per-decision importance sampling estimator has a lower variance than the ordinary +importance sampling estimator since each reward Rk+1 is only weighted by the importance +sampling ratio ρ0:k, instead of the entire product ρ0:T−1. Intuitively, this is still unbiased +because given the current state, the current reward is independent of future actions and +states. +This paper focuses on further reducing the variance of the per-decision importance sam- +pling estimator by using a proper behavior policy. From a statistics perspective, with a +lower variance, fewer trajectory samples are required to achieve an evaluation accuracy of +J(π) with the same confidence (Bernstein 1924; Bertsekas and Tsitsiklis 2002). From a +machine learning perspective, empirical error can be decomposed into bias and variance. +For unbiased methods, a lower variance induces a lower empirical error. In empirical exper- +iments, estimators with a lower variance take fewer steps and data to achieve convergence +in reinforcement learning algorithms (Sutton and Barto 2018). +4 + +Improving Monte Carlo Evaluation with Offline Data +3. Variance Reduction in Statistics +In this section, we provide the mathematical foundation for variance reduction with impor- +tance sampling ratios. The notations here are independent of the rest of this paper – we use +similar notations only for easy interpretation in later sections. Consider a discrete random +variable A taking values from a finite space A according to a probability mass function +π : A → [0, 1] and a function q : A → R mapping a value in A to a real number. We are +interested in estimating +EA∼π[q(A)]. +The ordinary Monte Carlo methods then sample {A1, . . . , AN} from π and use the empirical +average +1 +N +N +� +i=1 +q(Ai) +(3) +as the estimate. +In statistics, importance sampling is introduced as a variance reduc- +tion technique for Monte Carlo methods (Rubinstein 1981). The main idea is to sample +{Ai, . . . , AN} from a different distribution µ and use +1 +N +N +� +i=1 +ρ(Ai)q(Ai) +(4) +as the estimate, where +ρ(A) .= π(A) +µ(A) +is the importance sampling ratio. Assuming µ covers π, i.e., +∀a, µ(a) = 0 =⇒ π(a) = 0 +the estimation (4) is then unbiased because +EA∼π[q(A)] = EA∼µ[ρ(A)q(A)]. +If the sampling distribution µ is carefully designed, the variance of (4) can be smaller than +that of (3). This problem of searching for a variance reducing sampling distribution can be +formulated as an optimization problem: +minµ∈Λ+ +VA∼µ(ρ(A)q(A)). +(5) +Here Λ+ denotes the set of all the policies that give unbiased estimations, i.e., +Λ+ .= {µ ∈ ∆(A) | EA∼µ [ρ(A)q(A)] = EA∼π [q(A)]}, +5 + +Liu and Zhang +where ∆(X) denotes the set of all probability distributions on the set X. Solving (5) is in +general very challenging. To see this, consider a concrete example where A = {a1, a2, a3} +and +� +� +� +� +� +q(a1) = −10 +q(a2) = 2 +q(a3) = 2 +, +� +� +� +� +� +π(a1) = 0.1 +π(a2) = 0.5 +π(a3) = 0.4 +, +� +� +� +� +� +µ(a1) = 0 +µ(a2) = 0 +µ(a3) = 1 +. +(6) +It can be computed that EA∼π [q(A)] = 0.8 and EA∼µ [ρ(A)q(A)] = 0.8. In other words, we +could sample A from µ and use ρ(A)q(A) as an estimator. This estimator is unbiased. But +apparently, this µ does not cover π. Moreover, since this µ is deterministic, the variance +of this estimator is 0, which is the minimum possible variance. In other words, this µ is +an optimal sampling distribution. However, this µ is hand-crafted based on the knowledge +that q(a1)π(a1) + q(a2)π(a2) = 0. Without such knowledge, we argue that there is little +hope to find this µ. This example suggests that searching over the entire Λ+ might be too +ambitious. One natural choice is to restrict the search to +Λ− .= {µ ∈ ∆(A) | ∀a, µ(a) = 0 =⇒ π(a) = 0}. +In other words, we aim to find a variance minimizing sampling distribution among all +distributions that cover π. Because coverage implies unbiasedness, we have Λ− ⊆ Λ+. By +searching over only Λ−, we reduce the search space, with the hope that the problem is more +tractable. It turns out that we can slightly enlarge Λ− to Λ defined as +Λ .= {µ ∈ ∆(A) | ∀a, µ(a) = 0 =⇒ π(a)q(a) = 0}. +(7) +If for some µ and a0, there is π(a0) > 0, q(a0) = 0, µ(a0) = 0. Then this µ would not be in +Λ− but it is still in Λ. Importantly, any distribution in Λ still gives unbiased estimation, +though it may not cover π. The intuition is that the only sample a0 where µ does not cover +π must satisfy q(a0) = 0, i.e., this sample does not contribute to the expectation anyway. +Lemma 1 ∀µ ∈ Λ, EA∼µ [ρ(A)q(A)] = EA∼π [q(A)] +The proof is in Appendix A.1. We now consider the variance minimization problem on Λ, +i.e., +minµ∈Λ +VA∼µ(ρ(A)q(A)). +(8) +The following lemma gives a solution µ∗ to the optimization problem (8). +Lemma 2 Define +µ∗(a) ∝ +� +π(a)|q(a)| +if ∃a0, π0(a0)q(a0) ̸= 0 +1 +|A| +otherwise. +Then µ∗ is an optimal solution to (8). +6 + +Improving Monte Carlo Evaluation with Offline Data +Here by f(a) ∝ g(a), we mean f(a) .= +g(a) +� +b g(b). The proof is detailed in Appendix A.2. +To understand why µ∗ gives the minimum variance, considering an example where ∀a ∈ +A, q(a) > 0, we have ∀a ∈ A +µ∗(a) = π(a)|q(a)| +c +, +where c > 0 is a normalizing constant. By plugging µ∗ to ρ(A)q(A), we get ∀a ∈ A +ρ(a)q(a) = π(a) +µ∗(a)q(a) = +π(a) +π(a)|q(a)| +c +q(a) = c, +i.e., the random variable ρ(·)q(·) is now a constant function. The variance of a constant +random variable is of course zero. The following lemma slightly generalizes this intuition, +whose proof is in Appendix A.3. +Lemma 3 If ∀a ∈ A, q(a) ≥ 0 or ∀a ∈ A, q(a) ≤ 0, then Λ = Λ+, and the µ∗ defined in +Lemma 2 gives a zero variance, i.e., +VA∼µ∗(ρ(A)q(A)) = 0. +A sampling distribution proportional to π(a)|q(a)| dates back to Rubinstein (1981); +Benjamin Melamed (1998) and is also previously used in reinforcement learning (Carpentier +et al., 2015; Mukherjee et al., 2022). This µ∗ is previously deemed as an optimal sampling +distribution. We, however, make two notes here, both of which, to our knowledge, appear +novel. (1) We have to carefully specify the set of distributions under consideration before +claiming the optimality of µ∗. For example, if we compute this µ∗ for the example (6), it can +be easily found that π(A)q(A) +µ∗(A) +has a strictly positive variance because it evaluates negative for +a1 and positive for a2 and a3, while the µ in (6) has a zero variance and is also unbaised. In +other words, µ∗ can actually be suboptimal in the set Λ+. (2) This µ∗ does not necessarily +cover π. It is possible that for some a0, there are π(a0) > 0, µ∗(a0) = 0, q(a0) = 0. Lemma 1, +however, still ensures that µ∗ gives unbaised estimation. +4. Variance Reduction in Reinforcement Learning +We now apply the techniques in Section 3 in the reinforcement learning setting. In par- +ticular, we seek to reduce the variance V +� +GPDIS(τ µ0:T −1 +0:T−1 ) +� +by designing a proper behavior +policy µ. Of course, we need to ensure that the PDIS estimator with this behavior policy +is unbiased. In other words, ideally we should search over +Λ+ .= +� +µ ∈ ∆(A)T | E +� +GPDIS(τ µ0:T −1 +0:T−1 ) +� += J(π) +� +. +As discussed in Section 3, this might be too ambitious without domain-specific knowledge. +Instead, we can search over all policies that cover π, i.e., +Λ− .= +� +µ ∈ ∆(A)T | ∀t, s, a, µt(a|s) = 0 =⇒ πt(a|s) = 0 +� +. +7 + +Liu and Zhang +Set Λ− contains all policies that satisfy the policy coverage constraint in off-policy learn- +ing (Sutton and Barto 2018). We relax the policy coverage constraint while maintaining +unbiasedness. Define +Λ .= +� +µ ∈ ∆(A)T | ∀t, s, a, µt(a|s) = 0 =⇒ πt(a|s)qπ,t(s, a) = 0 +� +. +The following theorem ensures unbiasedness, which is proved in Appendix A.4. +Theorem 4 (Unbiasedness) ∀µ ∈ Λ, t, s, E +� +GPDIS(τ µt:T −1 +t:T−1 )|St = s +� += vπ,t(s). +One immediate consequence of Theorem 4 is that +∀µ ∈ Λ, E +� +GPDIS(τ µ0:T −1 +0:T−1 ) +� += J(π). +It, however, turns out that searching over Λ is still intractable. We instead consider a set +Λ∗ such that Λ− ⊆ Λ∗ ⊆ Λ. This Λ∗ will be defined soon. We now formulate our problem +as +min +µ∈Λ∗ +V +� +GPDIS(τ µ0:T −1 +0:T−1 ) +� +. +(9) +By the law of total variance, given any two random variables X and Y , we have +V[Y ] = E[V(Y |X)] + V(E[Y |X]). +(10) +For any µ ∈ Λ, we decompose the variance of the PDIS estimator as +V +� +GPDIS(τ µ0:T −1 +0:T−1 ) +� +(11) +=ES0 +� +V +� +GPDIS(τ µ0:T −1 +0:T−1 ) | S0 +�� ++ VS0 +� +E +� +GPDIS(τ µ0:T −1 +0:T−1 ) | S0 +�� +=ES0 +� +V +� +GPDIS +0 +(τ µ0:T −1 +0:T−1 ) | S0 +�� ++ VS0 (vπ,0(S0)) . +(Theorem 4) +The second term in (11) is a constant given a target policy π and is unrelated to the choice +of µ. In the first term, the expectation is taken over S0 that is determined by the initial +probability distribution p0. Consequently, solving the problem (9) is equivalent to solving +for each s, +min +µ∈Λ∗ +V +� +GPDIS(τ µ0:T −1 +0:T−1 )|S0 = s +� +. +Denote {0, 1, . . . , n} as [n]. Define the variance of the state value for the next state given +the current state-action pair (s, a) as +νπ,t(s, a) .= +� +VSt+1 (vπ,t+1(St+1) | St = s, At = a) +if t ∈ [T − 2] +0 +if t = T − 1. +(12) +The variance of the PDIS estimator can be expressed in the following recursive form. +8 + +Improving Monte Carlo Evaluation with Offline Data +Lemma 5 For any µ ∈ Λ, we have for t = T − 1, +V +� +GPDIS(τ µt:T −1 +t:T−1 ) | St +� += EAt∼µt +� +ρ2 +t q2 +π,t(St, At) | St +� +− v2 +π,t(St); +For t ∈ [T − 2], +V +� +GPDIS(τ µt:T −1 +t:T−1 ) | St +� +=EAt∼µt +� +ρ2 +t +� +ESt+1 +� +V +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St +� +| St, At +� ++ νπ,t(St, At) + q2 +π,t(St, At) +� +| St +� +− v2 +π,t(St). +Its proof is in Appendix A.5. This recursive form allows us to construct a behavior policy +µ∗ as +µ∗ +t (a|s) ∝ +� +πt(a|s) +� +uπ,t(s, a) +if ∃a′, πt(a′|s)|uπ,t(s, a′)| ̸= 0 +1 +|A| +otherwise +, +(13) +where for t = T − 1, +uπ,t(s, a) .= q2 +π,t(s, a); +for t ∈ [T − 2], +uπ,t(s, a) .= +� +s′ +p(s′|s, a)V +� +GPDIS(τ +µ∗ +t+1:T −1 +t+1:T−1 ) | St+1 = s′� ++ νπ,t(s, a) + q2 +π,t(s, a). (14) +This uπ,t(s, a) is always non-negative because all the summands are non-negative. This µ∗ +is optimal in the following sense. +Theorem 6 (Optimal Behavior Policy) For any t and s, the behavior policy µ∗ +t (a|s) +defined above is an optimal solution to the following problem +min +µt∈Λt,...,µT −1∈ΛT −1 +V +� +GPDIS(τ µt:T −1 +t:T−1 )|St = s +� +, +where +Λt .= {µt ∈ ∆(A) | ∀s, a, µt(a|s) = 0 =⇒ πt(a|s)uπ,t(s, a) = 0}. +Its proof is in Appendix A.6. Theorem 6 indicates that µ∗ achieves optimality in the set +Λ∗ .= Λ0 × · · · × ΛT−1. +Since uπ,t(s, a) = 0 =⇒ qπ,t(s, a) = 0, we have Λ∗ ⊆ Λ. If µt(a|s) = 0 =⇒ πt(a|s) = 0, +it follows immediately that µt(a|s) = 0 =⇒ πt(a|s)uπ,t(s, a) = 0. This indicates Λ− ⊆ Λ∗. +Though the set of policies Λ∗ considered in Theorem 6 is not as broad as Λ, it still includes +all the policies that cover the target policy, which is the setting where most off-policy results +consider (Precup et al., 2000; Maei, 2011; Sutton et al., 2016; Zhang, 2022). It is easy to +see π ∈ Λ∗. So Theorem 6 ensures, by the definition of optimality, that the variance of the +off-policy PDIS estimator with the behavior policy µ∗ is no larger than the variance of the +on-policy PDIS estimator, which reduces to the ordinary on-policy Monte Carlo estimator. +9 + +Liu and Zhang +Corollary 7 For all s, t, +V +� +GPDIS(τ +µ∗ +t:T −1 +t:T−1 )|St = s +� +≤ V +� +GPDIS(τ πt:T −1 +t:T−1 )|St = s +� +∀t, s. +Unfortunately, implementing µ∗ +t requires to know uπ,t (14) that contains the transition +function p. Approximating the transition function is very challenging in MDPs with large +stochasticity and approximation (cf. +model-based reinforcement learning (Sutton, 1990; +Sutton et al., 2008; Deisenroth and Rasmussen, 2011; Chua et al., 2018)). Thus, we seek to +build another policy ˆµ that can be implemented without direct knowledge of the transition +function p (cf. model-free reinforcement learning (Sutton, 1988; Watkins, 1989)). +We achieve this by aiming at local optimality instead of global optimality. In particular, +at a time step t, if we aim for global optimality, we should try to find the best µt assuming +in the future we follow µ∗ +t+1, . . . , µ∗ +T−1. Instead, we aim for the local optimality and try to +find the best µt assuming in the future we follow πt+1, . . . , πT−1. We refer to such a local +optimal behavior policy as ˆµt. Similarly, to define optimality we first need to specify the +set of policies we are concerned about. To this end, we define for t = T − 1, +ˆqπ,t(s, a) .= q2 +π,t(s, a); +for t ∈ [T − 2], +ˆqπ,t(s, a) .= +� +s′ +p(s′|s, a)V +� +GPDIS(τ πt+1:T −1 +t+1:T−1 ) | St+1 = s′� ++ νπ,t(s, a) + q2 +π,t(s, a). +(15) +This ˆqπ,t is always non-negative since all the summands are non-negative. Accordingly, we +define for t ∈ [T − 1], +ˆΛt .= {µt ∈ ∆(A) | ∀s, a, µt(a|s) = 0 =⇒ πt(a|s)ˆqπ,t(s, a) = 0}. +Corollary 7 ensures that ˆqπ,t(s, a) ≥ uπ,t(s, a) ≥ 0 holds ∀s, a, t. As a result, if µt ∈ ˆΛt, we +have +µt(a|s) = 0 =⇒ πt(a|s)ˆqπ,t(a|s) = 0 =⇒ πt(a|s)uπ,t(a|s) = 0, +indicating µt ∈ Λt. In other words, we have ˆΛt ⊆ Λt. To search for ˆµ0:T−1, we work on +ˆΛ .= ˆΛ0 × · · · × ˆΛT−1. +To summarize, we have +Λ− ⊆ ˆΛ ⊆ Λ∗ ⊆ Λ ⊆ Λ+. +Recall that Λ+ is the set of all behavior policies such that the corresponding PDIS estimator +is unbiased. +Λ is a sufficient but not necessary condition to ensure such unbiasedness +(Theorem 4). Λ∗ is a restriction of Λ such that we are able to find an optimal solution. +Here we future restrict Λ∗ to ˆΛ, aiming for a sub-optimal but implementable policy. Our +search space is still larger than Λ−, which contains all behavior policies that cover the target +policy. +10 + +Improving Monte Carlo Evaluation with Offline Data +According to the recursive expression of the variance in Lemma 5 and the aforementioned +goal for local optimality, we let ˆµt be an optimal solution to the following problem +min +µt∈ˆΛt +EAt∼µt +� +ρ2 +t +� +ESt+1 +� +V +� +GPDIS(τ πt+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ νπ,t(St, At) + q2 +π,t(St, At) +� +| St +� +. +(16) +Simple calculation yields +EAt∼µt +� +ρ2 +t +� +ESt+1 +� +V +� +GPDIS(τ πt+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ νπ,t(St, At) + q2 +π,t(St, At) +� +| St +� +=EAt∼µt +� +ρ2 +t ˆqπ,t(St, At)|St +� +=VAt∼µt +� +ρt +� +ˆqπ,t(St, At)|St +� +− E2 +At∼µt +� +ρt +� +ˆqπ,t(St, At)|St +� +=VAt∼µt +� +ρt +� +ˆqπ,t(St, At)|St +� +− E2 +At∼πt +�� +ˆqπ,t(St, At)|St +� +. +(Lemma 1 and µt ∈ ˆΛt) +According to Lemma 2, if we define +ˆµt(a|s) ∝ +� +πt(a|s) +� +ˆqπ,t(s, a) +if +∃a0, πt(a0|s) +� +ˆqπ,t(s, a0) ̸= 0 +1 +|A| +otherwise +, +(17) +then ˆµt is an optimal solution to (16). +To further characterize the property of ˆµ, we need a more explicit treatment of +V +� +GPDIS(τ πt:T −1 +t:T−1 )|St = s +� +, the variance of the on-policy Monte Carlo estimator. To this +end, we make use of the well-known fact that this variance can be expressed recursively +in the form of a Bellman equation (Tamar et al., 2016; O’Donoghue et al., 2018; Sherstan +et al., 2018). Formally speaking, define shorthands +˜rπ,t(s, a) .= νπ,t(s, a) + q2 +π,t(s, a) − v2 +π,t(s) +∀t ∈ [T − 1], +(18) +˜qπ,t(s, a) .= +� +˜rπ,t(s, a) + � +s′,a′ p(s′|s, a)πt+1(a′|s′)˜qπ,t+1(s′, a′) +if t ∈ [T − 2] +˜rπ,t(s, a) +if t = T − 1 . +(19) +Then we have +Lemma 8 +V +� +GPDIS(τ πt:T −1 +t:T−1 )|St = s +� += +� +a +πt(a|s)˜qπ,t(s, a) +∀t, s. +Its proof is in Appendix A.7. Here, this ˜q is exactly the action value function of the target +policy π in the MDP w.r.t. to a new reward function ˜r. Manipulating (15) then yields +ˆqπ,t(s, a) = +� +s′ +p(s′|s, a) +� +a′ +πt+1(a′|s′)˜qπ,t+1(s′, a′) + νt(s, a) + q2 +π,t(s, a) +(20) +=˜qπ,t(s, a) + v2 +π,t(s). +11 + +Liu and Zhang +This behavior policy ˆµ is of course inferior to the optimal behavior policy µ∗. We, however, +argue that ˆµ is provably better than the target policy π. In particular, since we have ˆµ ∈ ˆΛ ⊆ +Λ, Theorem 4 ensures that the PDIS estimator using ˆµ as the behavior policy gives unbiased +estimation, even if ˆµ may not cover π. Moreover, the following theorem confirms that the +PDIS estimator using ˆµ has a provably smaller variance than the PDIS estimator using π +as the behavior policy, which is exactly the ordinary on-policy Monte Carlo estimator. +Theorem 9 (Variance Reduction) For any t and s, +V +� +GPDIS(τ ˆµt:T −1 +t:T−1 ) | St = s +� +≤V +� +GPDIS(τ πt:T −1 +t:T−1 ) | St = s +� +Proof We proceed via induction. For t = T − 1, +V +� +GPDIS(τ ˆµt:T −1 +t:T−1 ) | St +� +=EAt∼ˆµt +� +ρ2 +t q2 +π,t(St, At) | St +� +− v2 +π,t(St) +(Lemma 5) +=EAt∼ˆµt +� +ρ2 +t ˆqπ,t(St, At) | St +� +− v2 +π,t(St) +(Definition of ˆq (15)) +=VAt∼ˆµt +� +ρt +� +ˆqπ,t(St, At)|St +� ++ E2 +At∼ˆµt +� +ρt +� +ˆqπ,t(St, At)|St +� +− v2 +π,t(St) +(Definition of variance and non-negativity of ˆq) +=VAt∼ˆµt +� +ρt +� +ˆqπ,t(St, At)|St +� ++ +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− v2 +π,t(St) +(Lemma 1) += +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− v2 +π,t(St) +(Definition of ˆµ (17) and Lemma 3) +≤ +� +a +πt(a|St)ˆqπ,t(St, a) − v2 +π,t(St) +(Jensen’s inequality) +=V +� +GPDIS(τ πt:T −1 +t:T−1 ) | St +� +. +(By (20) and Lemma 8) +For t ∈ [T − 2], we have +V +� +GPDIS(τ ˆµt:T −1 +t:T−1 ) | St +� +=EAt∼ˆµt +� +ρ2 +t +� +ESt+1 +� +V +� +GPDIS(τ ˆµt+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ νπ,t(St, At) + q2 +π,t(St, At) +� +| St +� +− v2 +π,t(St) +(Lemma 5) +≤EAt∼ˆµt +� +ρ2 +t +� +ESt+1 +� � +a′ +πt+1(a′|St+1)˜qπ,t+1(St+1, a′) | St, At +� ++ νπ,t(St, At) ++ q2 +π,t(St, At) +� +| St +� +− v2 +π,t(St) +(Inductive hypothesis and Lemma 8) +=EAt∼ˆµt +� +ρ2 +t +� +˜qπ,t(St, At) + v2 +π,t(St) +� +| St +� +− v2 +π,t(St) +(Definition of ˜q (19)) +=EAt∼ˆµt +� +ρ2 +t ˆqπ,t(St, At) | St +� +− v2 +π,t(St) +(Definition of ˆq (15)) +12 + +Improving Monte Carlo Evaluation with Offline Data +=VAt∼ˆµt +� +ρt +� +ˆqπ,t(St, At)|St +� ++ E2 +At∼ˆµt +� +ρt +� +ˆqπ,t(St, At)|St +� +− v2 +π,t(St) +(Repeating the arguments for t = T − 1) +=VAt∼ˆµt +� +ρt +� +ˆqπ,t(St, At)|St +� ++ +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− v2 +π,t(St) += +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− v2 +π,t(St) +≤ +� +a +πt(a|St)ˆqπ,t(St, a) − v2 +π,t(St) +=V +� +GPDIS(τ πt:T −1 +t:T−1 ) | St +� +. +This completes the proof. +We also prove a stronger lemma in the following to further compute the exact amount +of the reduced variance. +Theorem 10 For any t and s, +V +� +GPDIS(τ ˆµt:T −1 +t:T−1 ) | St = s +� +≤ V +� +GPDIS(τ πt:T −1 +t:T−1 ) | St = s +� +− ϵt(s). +where +ct(s) .= +� +a +πt(a|s)ˆqπ,t(s, a) − +�� +a +πt(a|s) +� +ˆqπ,t(s, a) +�2 +, +∀t ∈ [T − 1] +ϵt(s) .= +� +ct(s) + mina +� +s′ p(s′|s, a)ϵt+1(s′) +if t ∈ [T − 2] +ct(s) +if t = T − 1. +(21) +The proof is similar to Theorem 9 and is in Appendix A.8. Notably, this ct is always non- +negative thanks to Jensen’s inequality, which ensures that ϵt is also non-negative. If we +regard ct as a cost function, then the reduced variance ϵt is exactly the optimal cost-to-go +function of the stochastic shortest path problem in the MDP induced by the cost function +ct (Bertsekas and Tsitsiklis, 1996). +5. Variance Reduction with Offline Data +Having identified a sub-optimal but provably better policy ˆµ, the next step is to approx- +imate it, preferably with offline data. Since the target policy πt is considered known, to +approximate ˆµt, according to (15), it is sufficient to approximate ˆqπ,t. +This, according +to (20), requires to approximate ˜qπ,t and vπ,t. The state value function vπ,t can be learned +using any existing offline evaluation methods. In particular, we can use fitted Q-learning +(Le et al., 2019) to learn the action value function qπ,t first, then compute vπ,t analytically +using (1). The observant reader may question, if we have learned the action value func- +tion, do we still need to do Monte Carlo evaluation? The answer is affirmative and more +13 + +Liu and Zhang +details are deferred to the discussion regarding model-free offline evaluation in Section 8. +Having learned vπ,t, the remaining is to approximate ˜qπ,t, which is exactly the action value +function w.r.t. a different reward function ˜rπ,t. If ˜rπ,t is known, we could then resort to +fitted Q-learning again. To approximate ˜rπ,t, we, accroding to (18), need to approximate +νπ,t, qπ,t, and vπ,t. We have already learned qπ,t and vπ,t. The remaining is thus to learn +νπ,t, which by its definition in (12) is a variance. Approximating a variance is in general a +supervised learning problem – we can approximate the first and second moments separately. +In particular, we have +VSt+1 (vπ,t+1(St+1) | St = s, At = a) +=ESt+1 +� +v2 +π,t+1(St+1) | St = s, At = a +� +− E2 +St+1 [vπ,t+1(St+1) | St = s, At = a] . +The two expectations can be learned via supervised learning, using our approximation +of vπ,t+1 to generate regression targets. The approximation error in vπ,t+1 will, however, +inevitably compound into the approximation error of this variance term. The approximation +error in this variance term then compounds into the approximation error of ˆqπ,t+1. This +repeated error compounding is the first challenge in learning ˜qπ,t and vπ,t separately. +Moreover, to ensure that ˆµt is well defined, it is necessary to make sure that the approx- +imation of ˜qπ,t(s, a) + v2 +π,t(s) is non-negative. Apparently, v2 +π,t(s) is always non-negative. +There is, however, no guarantee that the learned approximation of ˜qπ,t(s, a) is non-negative. +One way to achieve this is to use a non-negative function class, e.g., (·)2 or ln(1 + exp(·)), +to approximate ˜qπ,t(·, ·), such that our approximation is always non-negative. This unfor- +tunately introduces bias as we use a non-negative function class to approximate a function +whose value can possibly be negative. This bias is the second challenge in learning ˜qπ,t and +vπ,t separately. +We now address those two challenges simultaneously via learning ˆqπ,t(s, a) directly. This +is made possible thanks to the following observation. +Lemma 11 ∀s, a, +ˆqπ,t(s, a) = +� +ˆrπ,t(s, a) + � +s′,a′ p(s′|s, a)πt+1(a′|s′)ˆqπ,t+1(s′, a′) +if t ∈ [T − 2] +ˆrπ,t(s, a) +if t = T − 1 , +where +ˆrπ,t(s, a) .= 2r(s, a)qπ,t(s, a) − r2(s, a). +(22) +Proof For t = T − 1, we have +ˆqπ,t(s, a) = q2 +π,t(s, a) +(Definition of ˆqπ,t (15)) += ˆrπ,t(s, a). +(By qπ,T−1(s, a) = r(s, a) and (22)) +For t ∈ [T − 2], we have +ˆqπ,t(s, a) +=˜qπ,t(s, a) + v2 +π,t(s) +(By (20)) +14 + +Improving Monte Carlo Evaluation with Offline Data +=˜rπ,t(s, a) + v2 +π,t(s) + +� +s′,a′ +p(s′|s, a)πt+1(a′|s′)˜qπ,t+1(s′, a′) +(Definition of ˜q (19)) +=˜rπ,t(s, a) + v2 +π,t(s) + +� +s′,a′ +p(s′|s, a)πt+1(a′|s′)(˜qπ,t+1(s′, a′) + v2 +π,t+1(s′) − v2 +π,t+1(s′)) +=˜rπ,t(s, a) + v2 +π,t(s) + +� +s′,a′ +p(s′|s, a)πt+1(a′|s′)(ˆqπ,t+1(s′, a′) − v2 +π,t+1(s′)) +(By (20)) +=νπ,t(s, a) + q2 +π,t(s, a) − +� +s′ +p(s′|s, a)v2 +π,t+1(s′) + +� +s′,a′ +p(s′|s, a)πt+1(a′|s′)ˆqπ,t+1(s′, a′) +(Definition of ˜r (18)) += − (E[vπ,t+1(St+1) | St = s, At = a])2 + q2 +π,t(s, a) + +� +s′,a′ +p(s′|s, a)πt+1(a′|s′)ˆqπ,t+1(s′, a′) +(Definition of ν (12)) += − (qπ,t(s, a) − r(s, a))2 + q2 +π,t(s, a) + +� +s′,a′ +p(s′|s, a)πt+1(a′|s′)ˆqπ,t+1(s′, a′) +=2r(s, a)qπ,t(s, a) − r2(s, a) + +� +s′,a′ +p(s′|s, a)πt+1(a′|s′)ˆqπ,t+1(s′, a′) +=ˆrπ,t(s, a) + +� +s′,a′ +p(s′|s, a)πt+1(a′|s′)ˆqπ,t+1(s′, a′), +which completes the proof. +In other words, ˆq is exactly the action value function of the policy π w.r.t. the reward +function ˆr. Suppose ˆr is learned, we can then learn ˆq with any offline evaluation methods +for action-value functions, e.g., fitted Q-learning. To learn ˆr, it is sufficient to learn r and +q. Fitted Q-learning can be used to learn q and learning r is a simple regression problem. +Importantly, this regression problem now has accurate targets. By contrast, the regression +of the variance ν use the estimation of vπ,t to compute the target. +As a result, error +compounding is reduced by learning ˆqπ,t directly. +We consider the behavior policy agnostic offline learning setting (Nachum et al., 2019), +where the offline data in the form of +� +(ti, si, ai, ri, s′ +i) +�m +i=1. +consists of m previously logged data tuples. In the i-th data tuple, ti is the time step, si is +the state at time step ti, ai is the action executed on state si, ri is the sampled reward, and +s′ +i is the successor state. Those tuples can be generated by one or more, possibly unknown +behavior policies. And those tuples do not need to form a complete trajectory. +Our proposed algorithm for learning ˆµ is detailed in Algorithm 1. In particular, we first +learn r with supervised learning and qπ,t with fitted Q-learning from the offline data. Then +we compute ˆr analytically and apply fitted Q-learning again to learn ˆqπ,t, which is then +used to compute ˆµ analytically. We split the offline data into training sets and test sets to +tune all the hyperparameters in Algorithm 1, based on the supervised learning loss or the +fitted Q-learning loss on the test set. +15 + +Liu and Zhang +Algorithm 1: Approximating ˆµ with offline data +Input: a differentiable function parameterization rw : S × A × Rd0 → R +a differentiable function parameterization qw : [T] × S × A × Rd1 → R +a differentiable function parameterization ˆqw : [T] × S × A × Rd2 → R +a target policy π +an offline dataset {(ti, si, ai, ri, si)}m +i=1 +Initialize wr ∈ Rd0, wq ∈ Rd1, wˆq ∈ Rd2 arbitrarily. +Algorithm Parameters: learning rates αr, αq, αˆq +Output: a behavior policy ˆµ +Step 1: Augment data with a′ +Sample an action a′ by πt+1(·|s′) for each (t, s, a, r, s′) pair. +Step 2: Approximate rw +Loop for each training step: +Sample a minibatch of (s, a, r) +Perform a mini-batch gradient descent step based on +wr ← wr + αr[r − rw(s, a, wr)]∇rw(s, a, wr) +Step 3: Approximate qw +Loop for each training step: +Sample a minibatch of (t, s, a, r, s′, a′) +Perform a mini-batch gradient descent step based on +wq ← wq + αq[r + qw,t+1(s′, a′, wq) − qw,t(s, a, wq)]∇qw,t(s, a, wq) +Step 4: Approximate ˆqw +Loop for each training step: +Sample a minibatch of (t, s, a, s′, a′) +Perform a mini-batch gradient descent step based on +ˆr ← 2rw(s, a, wr)qw,t(s, a, wq) − r2 +w(s, a, wr) +wˆq ← wˆq + αˆq[ˆr + ˆqw,t+1(s′, a′, wˆq) − ˆqw,t(s, a, wˆq)]∇ˆqw,t(s, a, wˆq) +Step 5: Output ˆµ +ˆµt(a|s) ∝ πt(a|s) +� +ˆqw,t(s, a, wˆq) +16 + +Improving Monte Carlo Evaluation with Offline Data +Algorithm 2: Adaptive Monte Carlo Evaluation +Input: A policy π to be evaluated +A policy ˆµ computed from Algorithm 1 +Algorithm Parameters: UCB parameter c +Initialize: Rewards(π) ← an empty list +Rewards(ˆµ) ← an empty list +n ← 0 +J ← 0 +Output: Total rewards estimation J +Loop for K episodes: +S0 ∼ p0 +b ← argmaxb∈{ˆµ,π}Average +� +Rewards(b) +� ++ c +� +log n +|Rewards(b)| +(UCB) +Generate a trajectory {S0, A0, R1, S1, A1, R2, · · · , ST−1, AT−1, RT } following b +G ← 0 +for t = T − 1, T − 2, · · · , 0: +G ← πt(At|St) +bt(At|St) (Rt+1 + G) +Rewards(b) append −G2 +n ← n + 1 +J ← J + 1 +n(G − J) +6. Variance Reduction in Online Execution +Though the PDIS Monte Carlo estimator with ˆµ is provably better than that with π (The- +orems 4 & 9), we do not have access to ˆµ but only its approximation learned from offline +data. +This learning process suffers from various biases, e.g., insufficient data coverage, +mismatched hypothesis space, incomplete optimization, insufficient hyperparameter tun- +ing, etc, just like any other learning process. As a result, using the learned approximation +of ˆµ is not necessary better than using π directly. With a slight abuse of notation, we in +this section use ˆµ to denote the learned approximation from Algorithm 1. +Thus when we actually collect online data, there are two choices, to use the target policy +π or to use the learned ˆµ. +Though both yield unbiased estimation, their variances are +different. This places us in the exploration and exploitation dilemma (see, e.g., Lattimore +and Szepesv´ari (2020)). On the exploration side, we want to execute both policies more to +know their variance. On the exploitation side, we want to commit to a better policy as soon +as possible. Motivated by the celebrated success in the bandit community in solving the +exploration and exploitation dilemma (Lattimore and Szepesv´ari, 2020), we now formulate +this problem as a multi-armed bandit problem, where the target policy π and the learned +ˆµ are two arms. +To complete the bandit formulation, the next step is to specify a reward function for +each arm. Since we want to identify the policy that yields a lower variance, the natural +choice is then to use the additive inverse of the variance, i.e., −V +� +GPDIS(τ ˆµ0:T −1 +0:T−1 ) +� +and +17 + +Liu and Zhang +−V +� +GPDIS(τ ˆπ0:T −1 +0:T−1 ) +� +, as a reward. +We, however, cannot estimate the variance from a +single trajectory τ0:T−1. Using multiple trajectories either reduces the sample efficiency or +makes the reward function non-stationary. To address this challenge, we use the following +observation, +V +� +GPDIS(τ ˆµ0:T −1 +0:T−1 ) +� += E +� +(GPDIS(τ ˆµ0:T −1 +0:T−1 ))2� +− E2 � +GPDIS(τ ˆµ0:T −1 +0:T−1 ) +� +, +(23) +V +� +GPDIS(τ π0:T −1 +0:T−1 ) +� += E +� +(GPDIS(τ π0:T −1 +0:T−1 ))2� +− E2 � +GPDIS(τ π0:T −1 +0:T−1 ) +� +. +(24) +Since the PDIS estimator is unbiased for both ˆµ and π, the E2 [·] terms above are equal. To +compare the variances, it is sufficient to compare the second moments. We, therefore, use +−(GPDIS(τ ˆµ0:T −1 +0:T−1 ))2 and −(GPDIS(τ π0:T −1 +0:T−1 ))2 as the rewards for the arms ˆµ and π respectively. +Those rewards are immediately available after a trajectory τ0:T−1 is sampled. We use the +Upper Confidence Bound (UCB, Auer (2002)) algorithm to adaptively switch between the +two arms during online executions. The details are documented in Algorithm 2. Notably, +since both ˆµ and π induce unbiased estimation, all trajectories, not just those from the +better policy, contribute to the final estimation. No online data is wasted in this sense. +Denote b(i) ∈ {ˆµ, π} as the chosen behavior policy at the i-th online episode. Define K +as the total number of online episodes and define the regret for the variance as +Regret(K) +.= +K +� +i=1 +V(GPDIS(τ b(i)0:T −1 +0:T−1 +)) − K · min +� +V(GPDIS(τ ˆµ0:T −1 +0:T−1 )), V(GPDIS(τ π0:T −1 +0:T−1 )) +� +. +The following lemma gives a sublinear regret guarantee, regardless of the approximation +error in ˆµ. Its proof is in Appendix A.9. +Lemma 12 E [Regret(K)] = O( +√ +K ln K) +7. Empirical Results +In this section, we present empirical results to answer the following two questions. +1. If the approximation of ˆµ is of high quality, can the PDIS Monte Carlo estimator +outperform the ordinary on-policy Monte Carlo estimator? +2. If the approximation of ˆµ is of poor quality, can the adaptive execution strategy still +ensure a low estimation error? +We use grid worlds with different sizes as our benchmark environments. For a grid world +with size n, its width, height, and time horizon T are all set to n. There are four possible +actions: up, down, left, and right. After taking an action, the agent has 0.9 probability to +move accordingly and 0.1 probability to move uniformly at random. If the agent runs into +a boundary, the agent stays in its current location. The reward function r(s, a) is randomly +generated and fixed after generation. We normalize the rewards across all (s, a) such that +maxs,a r(s, a) = 1. We consider a set of randomly generated target policies. The ground +truth policy performance is estimated using the on-policy Monte Carlo method by running +18 + +Improving Monte Carlo Evaluation with Offline Data +each target policy for 106 episodes. We test three different sizes of the grid world, i.e., +n ∈ {5, 10, 15}. The offline dataset always contains m = 105 randomly generated tuples +regardless of n. +Given an environment and a target policy, we execute Algorithm 1 to approximate +function r, q, and ˆq. We consider a tabular setting where each (t, s, a) pair is represented +by a distinct one-hot vector. As shown in Algorithm 1, we train r using supervised learning +by batch gradient descent. We train q and ˆq using fitted Q-learning. We split the offline +data into a training set and a test set and tune all hyperparameters offline based on the +supervised learning loss and fitted Q-learning loss on the test set. We use the same set +of hyperparameters for all grid worlds and target policies. We end up with learning rates +being 1, training steps being 103, and batch size being 128. +In each grid world environment, we test 30 randomly generated target policies and each +target policy is tested 30 times. For an environment and a target policy, we execute ˆµ +and π for 500 steps to estimate the expected total rewards of the target policy. Each step +is defined as one interaction between the agent and the environment. Thus, the estimate +for the environment with time horizon 15 starts from steps 15. Because we are interested +in estimation accuracy, we define the estimation error at step t as the absolute difference +between the PDIS estimation and the ground truth divided by the ground truth. We use +normalized estimation error which is the estimation error divided by the average estimation +error of the on-policy estimator after the first episode. This ensures that the normalized +estimation error of the on-policy estimator starts from 1. +0 +100 +200 +300 +400 +500 +steps +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +normalized estimation error +n = 5 +on-policy +off-policy +off-policy with UCB +0 +100 +200 +300 +400 +500 +steps +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +n = 10 +on-policy +off-policy +off-policy with UCB +0 +100 +200 +300 +400 +500 +steps +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +n = 15 +on-policy +off-policy +off-policy with UCB +Figure 1: The normalized estimation errors of the off-policy estimator with the learned ˆµ +and the on-policy estimator. Each curve averages the results of 30 random policies and +each policy has 30 independent runs. The shaded regions denote standard errors and are +invisible for some curves because they are too small. +Blue lines in Figure 1 show the experiment results when we always use learned ˆµ as the +actual behavior policy. For the grid world of size n = 5, the off-policy estimator with the +learned ˆµ consumes significantly fewer online steps to achieve the same estimation accuracy, +compared with the on-policy estimator. For example, to achieve 0.2 normalized estimation +error, the off-policy estimator consumes around 40 online steps while the on-policy estimator +consumes around 120 online steps. Substantial improvements also exist in the grid world +environment with size n = 10. However, in the grid world environment with size n = 15, the +off-policy estimator with the learned ˆµ is actually worse than the on-policy estimator. This +19 + +Liu and Zhang +is because the learned ˆµ contains a non-negligible approximation error due to insufficient +data coverage as the environment size increases while the number of offline data remains +unchanged. +To maintain a reasonably low estimation error when this occurs, we use Algorithm 2 +with a standard UCB confidence value c = 1 to identify the better policy and has results +shown in Figure 1. As shown with red lines, after adopting the UCB algorithm to adaptively +switch between the learned policy ˆµ and the target policy π during online executions, the +improvements on the grid world with size 5 or 10 is still significant (e.g., when n = 5, +to achieve 0.2 normalized estimation error, the off-policy estimator takes 60 online steps +while the on-policy estimator takes 120). The jump at the start of the red lines is because +Algorithm 2 initially breaks ties by choosing the learned ˆµ and then chooses to explore the +target policy π. This causes the error to increase because of the large on-policy estimation +error. The performance degeneration when n = 15 has been effectively mitigated shown by +the closer curves in Figure 1. As we examine the variance numerically below, this mitigation +is significant, Define the variance ratio as +V +� +GPDIS(τ b0:T −1 +0:T−1 ) +� +V +� +GPDIS(τ π0:T −1 +0:T−1 ) +� +where b is the actual behavior policy during online executions. By using collected online +samples to estimate the variance, Table 7 shows that when n = 5 or 10, the variance of +the off-policy estimator with the learned ˆµ is much smaller than the variance of the on- +policy estimator. After adopting the UCB algorithm, the significant variance reduction for +n ∈ {5, 10} still exists while the variance when n = 15 has been reduced to almost one-third +of its original size. Thus, Algorithm 2 greatly reduces the regret when the learned ˆµ is of +poor quality. +Gridworld size n +Without UCB +With UCB +5 +0.282 +0.350 +10 +0.352 +0.422 +15 +3.673 +1.390 +Table 1: Variance Ratio for Gridworld Environments. +8. Related Work +Monte Carlo methods. +Reducing the variance of Monte Carlo estimators via learning +a proper behavior policy has been explored before. Hanna et al. (2017) model the problem +of finding a variance-reducing behavior policy as an optimization problem and thus rely +on stochastic gradient descent to update a parameterized behavior policy directly. In par- +ticular, Hanna et al. (2017) consider the ordinary importance sampling. By contrast, we +consider the per-decision importance sampling, which is fundamentally better (Precup et al., +2000). Moreover, Hanna et al. (2017) require new online data to learn this behavior policy. +By contrast, our method works with offline data and does not need any more online data for +20 + +Improving Monte Carlo Evaluation with Offline Data +behavior policy learning. Hanna et al. (2017) also requrie the online data to be complete +trajectories. By contrast, our method copes well with offline tuples. +Mukherjee et al. (2022) also investigate variance-reducing behavior policies for the +per-decision importance sampling estimator. Their results, however, apply to only tree- +structured MDPs, which is rather restrictive because many MDPs of interest are not tree- +structured. For example, in the finite horizon MDP considered in this paper, if two states +at time t can transit to the same successor state at time t + 1, then this MDP is not tree- +structured. Moreover, Mukherjee et al. (2022) require to directly approximate the transition +function in the MDP by counting, making it essentially a model-based approach. Mukherjee +et al. (2022), therefore, suffer from all canonical challenges in model learning (Sutton, 1990; +Sutton et al., 2008; Deisenroth and Rasmussen, 2011; Chua et al., 2018). By contrast, we +work on general MDPs without making any assumption regarding the underlying structures +of the MDPs and do not need to approximate the transition function. +Our approach is +model-free. +Zhong et al. (2022) adjust the behavior policy by encouraging under-sampled data. +Zhong et al. (2022), however, rely on strong assumptions on offline data. Their offline data +has to be complete trajectories generated by known policies. In their experiments, they +also require the policies for generating offline data to be similar to the target policy since +they do not have any importance sampling. By contrast, our method copes well offline data +in the form of incomplete segments from probably unknown behavior policies that can be +arbitrarily different from the target policy. Moreover, there is no theoretical guarantee that +the estimates made by Zhong et al. (2022) are unbiased or consistent. By contrast, our +estimate is always provably unbiased. +Other attempts for variance reduction in Monte Carlo evaluation are mostly by using +control variates based on value function (Zinkevich et al., 2006; White and Bowling, 2009; +Jiang and Li, 2016). Such control variates can be easily integrated into our estimator, which +we, however, save for future work. +Model-based offline evaluation. +One straightforward way to exploit offline data for +policy evaluation is to learn a model of the MDP first, probably with supervised learning +(Jiang and Li, 2016; Paduraru, 2013; Zhang et al., 2021), and then execute Monte Carlo +methods inside the learned model. Learning a high-fidelity model is, however, sometimes +even more challenging than evaluating the policy itself. And the model prediction error can +easily compound over time steps during model rollouts (Wan et al., 2019). Nevertheless, if +a good model can somehow be learned, our work still helps reduce the required rollouts when +Monte Carlo is applied within the learned model. +Model-free offline evaluation. +Offline data can also be exploited for policy evaluation +without explicitly constructing a model. Those model-free offline evaluation methods in- +stead learn some other quantities, including density ratio (a.k.a. marginalized importance +sampling ratio, Liu et al. (2018); Nachum et al. (2019); Zhang et al. (2020); Yang et al. +(2020); Mousavi et al. (2020); Li (2019); Uehara et al. (2020); Xie et al. (2019)) and action +value function (Le et al., 2019; Harutyunyan et al., 2016; Precup et al., 2000; Munos et al., +2016; Farajtabar et al., 2018). But those learning processes bring in bias, just like any +other learning process, either due to the misspecification of the function class or due to the +complexity of optimization. Consequently, the estimation they make is biased and it is hard +21 + +Liu and Zhang +to quantify such bias without restrictive assumptions. To our knowledge, the only practical +way in general settings to certify that their estimation is indeed accurate is to compare those +estimations with estimations made by Monte Carlo methods. We believe this is why Monte +Carlo methods still dominate the evaluation of policies. Even worse, those learning algo- +rithms also have hyperparameters to tune, just like any other learning algorithm. In other +words, we need to evaluate different outputs of those learning algorithms corresponding to +different hyperparameters. This is called model selection. Apparently, we cannot use the +aforementioned density ratio or action value function based model free evaluation methods +for model selection – otherwise we run into a self-loop. In fact, those works (Liu et al., 2018; +Nachum et al., 2019; Zhang et al., 2020; Yang et al., 2020; Mousavi et al., 2020; Li, 2019; +Uehara et al., 2020; Xie et al., 2019) usually use Monte Carlo with online data for evaluating +different candidates. The online data comes from either a simulator or a learned model. As +a result, this work helps reduce the online data used in model selection by those model-free +offline evaluation methods. Efforts have been made to perform model selection with only +offline data without explicitly learning a model as well (Kumar et al., 2021; Paine et al., +2020; Zhang and Jiang, 2021; Xie and Jiang, 2021). Those offline model selection methods, +however, rarely have a correctness guarantee without restrictive assumptions. They can +probably provide a preliminary screen in model selection but Monte Carlo methods make the +final decision when correctness really matters. +To summarize, if obtaining online data is entirely impossible, existing offline evaluation +methods without using any online data might be the only choices. These include model- +based methods and model-free methods augmented by offline model selection. However, +in many real-world problems, it is practical to assume that a small amount of online data +is available. If in addition, evaluation correctness should be honored, then the improved +Monte Carlo method in this work might be a better choice. +Control. +Control algorithms can use online data (Watkins and Dayan, 1992; Sutton et al., +1999; Mnih et al., 2015; Schulman et al., 2017), offline data (Fujimoto et al., 2019; Kumar +et al., 2020; Yu et al., 2020; Kidambi et al., 2020; Schrittwieser et al., 2021), or a mix of online +and offline data (Vecerik et al., 2017; Nair et al., 2020; Lee et al., 2022; Ijspeert et al., 2002; +Kim et al., 2013; Rajeswaran et al., 2017; Ajay et al., 2020). Control algorithms also have +hyperparameters. Consequently, to use control algorithms, model selection is necessary, +where Monte Carlo methods now dominate. As a result, this work makes almost all control +algorithms more efficient, in terms of using online data. Using offline data to help online +model selection in control problems is previously explored by Konyushova et al. (2021). In +particular, it uses offline data to decide which policy, among a given set of policies, should +be given priority to evaluate. When it comes to the actual online evaluation, Konyushova +et al. (2021) still uses the ordinary online Monte Carlo methods. Konyushova et al. (2021), +therefore, again benefit from the improved Monte Carlo method in this paper. +9. Conclusion +Monte Carlo methods are the most dominating approach for evaluating a policy. +The +development and deployment of almost all RL algorithms implicitly or explicitly depend on +Monte Carlo methods more or less. For example, when a reinforcement learning researcher +wants to plot a curve of the agent performance against training steps, Monte Carlo methods +22 + +Improving Monte Carlo Evaluation with Offline Data +are usually the first choice. +This work develops a method to improve the online data +efficiency of Monte Carlo evaluation by learning a tailored behavior policy from offline +data. +The Monte Carlo estimator with this tailored behavior policy is provably better +than the canonical Monte Carlo estimator. The theoretical advantage is also demonstrated +empirically, as a proof-of-concept, in the tested domains. We save the investigation on +large-scale problems for future work. Moreover, this work considers only the total rewards +performance metric on finite horizon MDPs. +One natural next step is to consider the +average reward (Puterman, 2014) or the discounted total rewards (Puterman, 2014) on +infinite horizon MDPs. +Acknowledgments +The authors thank Haifeng Xu for the insightful discussion. SZ is part of the Link Lab at +the University of Virginia. +A. Proofs +A.1 Proof of Lemma 1 +Proof +EA∼µ [ρ(A)q(A)] = +� +a∈{a|µ(a)>0} +µ(a)π(a) +µ(a)q(a) += +� +a∈{a|µ(a)>0} +π(a)q(a) += +� +a∈{a|µ(a)>0} +π(a)q(a) + +� +a∈{a|µ(a)=0} +π(a)q(a) +(µ ∈ Λ) += +� +a +π(a)q(a) +=EA∼π [q(A)] . +A.2 Proof of Lemma 2 +Proof +For a given π and q, define +A+ .= {a | π(a)q(a) ̸= 0}. +For any µ ∈ Λ, we expand the variance as +VA∼µ(ρ(A)q(A)) +=EA∼µ[(ρ(A)q(A))2] − E2 +A∼µ[ρ(A)q(A)] +23 + +Liu and Zhang +=EA∼µ[(ρ(A)q(A))2] − E2 +A∼π[q(A)] +(Lemma 1) += +� +a∈{a|µ(a)>0} +π2(a)q2(a) +µ(a) +− E2 +A∼π[q(A)] += +� +a∈{a|µ(a)>0}∩A+ +π2(a)q2(a) +µ(a) +− E2 +A∼π[q(A)] +(π(a)q(a) = 0, ∀a /∈ A+) += +� +a∈A+ +π2(a)q2(a) +µ(a) +− EA∼π[q(A)]2 +(µ ∈ Λ) +The second term is a constant and is unrelated to µ. Solving the optimization problem (8) +is, therefore, equivalent to solving +minµ∈Λ +� +a∈A+ +π2(a)q2(a) +µ(a) +. +(25) +Case 1: |A+| = 0 +In this case, the variance is always 0 so any µ ∈ Λ is optimal. In particular, µ∗(a) = 1 +A is +optimal. +Case 2: |A+| > 0 +The definition of Λ in (7) can be equivalently expressed, using contraposition, as +Λ = {µ ∈ ∆(A) | ∀a, a ∈ A+ =⇒ µ(a) > 0}. +The optimization problem (25) can then be equivalently written as +minµ∈∆(A) +� +a∈A+ +π2(a)q2(a) +µ(a) +(26) +s.t. +µ(a) > 0 +∀a ∈ A+. +If for some µ we have � +a∈A+ µ(a) < 1, then there must exist some a0 /∈ A+ such that +µ(a0) > 0. Since a0 does not contribute to the summation in the objective function of (26), +we can move the probability mass on a0 to some other a1 ∈ A+ to increase µ(a1) to further +decrease the objective. In other words, any optimal solution µ to (26) must put all its mass +on A+. This motivates the following problem +minz∈∆(A+) +� +a∈A+ +π2(a)q2(a) +z(a) +(27) +s.t. +z(a) > 0 +∀a ∈ A+. +In particular, if z∗ is an optimal solution to (27), then an optimal solution to (26) can be +constructed as +µ∗(a) = +� +z∗(a) +a ∈ A+ +0 +otherwise. +(28) +24 + +Improving Monte Carlo Evaluation with Offline Data +Let R++ .= (0, +∞). +According to the Cauchy-Schwarz inequality, for any z ∈ R|A+| +++ , we have +� +� � +a∈A+ +π2(a)q2(a) +z(a) +� +� +� +� � +a∈A+ +z(a) +� +� ≥ +� +� � +a∈A+ +π(a)|q(a)| +� +z(a) +� +z(a) +� +� +2 += +� +� � +a∈A+ +π(a)|q(a)| +� +� +2 +. +It can be easily verified that the equality holds for +z∗(a) .= +π(a)|q(a)| +� +b π(b)|q(b)| > 0. +Since � +a∈A+ z∗(a) = 1, we conclude that z∗ is an optimal solution to (27). An optimal +solution µ∗ to (8) can then be constructed according to (28). Making use of the fact that +π(a)|q(a)| = 0 for a /∈ A+, this µ∗ can be equivalently expressed as +µ∗(a) = +π(a)|q(a)| +� +b∈A π(b)q(b), +which completes the proof. +A.3 Proof of Lemma 3 +Proof We start with showing Λ = Λ+. Lemma 1 ensures that µ ∈ Λ =⇒ µ ∈ Λ+. We +now show that µ ∈ Λ+ =⇒ µ ∈ Λ. For any µ ∈ Λ+, we have +� +a∈{a|µ(a)>0} +µ(a)π(a) +µ(a)q(a) = +� +a +π(a)q(a). +This indicates that +� +a∈{a|µ(a)=0} +π(a)q(a) = 0. +Since π(a) ≥ 0 and all q(a) has the same sign, we must have +π(a)q(a) = 0, ∀a ∈ {a|µ(a) = 0}. +This is exactly µ(a) = 0 =⇒ π(a)q(a) = 0, yielding µ ∈ Λ. This completes the proof of +Λ+ = Λ. +We now show the zero variance. When ∀a ∈ A, q(a) ≥ 0, if ∃a0, π0(a0)q(a0) ̸= 0, we +have ∀a ∈ A +µ∗(a) = π(a)|q(a)| +c +and c > 0 is a normalizing constant. Plugging µ∗ to ρ(A)q(A), we get ∀a ∈ A +ρ(a)q(a) = π(a) +µ∗(a)q(a) = +π(a) +π(a)|q(a)| +c +q(a) = c. +25 + +Liu and Zhang +This means in this setting, with the optimal distribution µ∗, the random variable ρ(·)q(·) +is a constant function. Thus, +VA∼µ∗(ρ(A)q(A)) = 0. +When ∀a ∈ A, q(a) ≥ 0, if ∀a0, π0(a0)q(a0) = 0, we have ∀a ∈ A +µ∗(a) = +1 +|A|. +Plugging µ∗ to ρ(A)q(A), we get ∀a ∈ A +ρ(a)q(a) = π(a) +µ∗(a)q(a) = π(a)q(a) +1 +|A| += 0. +This shows ρ(A)q(A) is a also constant. Thus, +VA∼µ∗(ρ(A)q(A)) = 0. +The proof is similar for ∀a ∈ A, q(a) ≤ 0 and is thus omitted. +A.4 Proof of Theorem 4 +Proof We proceed via induction. For t = T − 1, we have +E +� +GPDIS(τ µt:T −1 +t:T−1 ) | St +� +=E [ρtRt+1 | St] = E [ρtqπ,t(St, At) | St] +=EAt∼πt(·|St) [qπ,t(St, At)|St] +(Lemma 1) +=vπ,t(St). +For t ∈ [T − 2], we have +E +� +GPDIS(τ µt:T −1 +t:T−1 ) | St +� +=E +� +ρtRt+1 + ρtGPDIS(τ µt+1:T −1 +t+1:T−1 ) | St +� +=E [ρtRt+1 | St] + E +� +ρtGPDIS(τ µt+1:T −1 +t+1:T−1 ) | St +� +=E [ρtRt+1 | St] + EAt∼µt(·|St),St+1∼p(·|St,At) +� +E +� +ρtGPDIS(τ µt+1:T −1 +t+1:T−1 ) | St, At, St+1 +�� +(Law of total expectation) +=E [ρtRt+1 | St] + EAt∼µt(·|St),St+1∼p(·|St,At) +� +ρtE +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St+1 +� +|St +� +(Conditional independence and Markov property) +=E [ρtRt+1 | St] + EAt∼µt(·|St),St+1∼p(·|St,At) [ρtvπ,t+1(St+1)|St] +(Inductive hypothesis) +=EAt∼µt(·|St) [ρtqπ,t(St, At)|St] +(Definition of qπ,t) +=EAt∼πt(·|St) [qπ,t(St, At)|St] +(Lemma 1) +=vπ,t(St), +which completes the proof. +26 + +Improving Monte Carlo Evaluation with Offline Data +A.5 Proof of Lemma 5 +Proof When t ∈ [T − 2], we have +V +� +GPDIS(τ µt:T −1 +t:T−1 ) | St +� +(29) +=EAt +� +V +� +GPDIS(τ µt:T −1 +t:T−1 ) | St, At +� +| St +� ++ VAt +� +E +� +GPDIS(τ µt:T −1 +t:T−1 ) | St, At +� +| St +� +(Law of total variance (10)) +=EAt +� +ρ2 +t V +� +r(St, At) + GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St, At +� +| St +� ++ VAt +� +ρtE +� +r(St, At) + GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St, At +� +| St +� +(Using (2)) +=EAt +� +ρ2 +t V +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St, At +� +| St +� ++ VAt (ρtqπ,t(St, At) | St) . +(Deterministic reward r) +Further decomposing the first term, we have +V +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St, At +� +(30) +=ESt+1 +� +V +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St, At, St+1 +� +| St, At +� ++ VSt+1 +� +E +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St, At, St+1 +� +| St, At +� +(Law of total variance (10)) +=ESt+1 +� +V +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ VSt+1 +� +E +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� +(Markov property) +=ESt+1 +� +V +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ VSt+1 (vπ,t+1(St+1) | St, At) . (Theorem 4) +With νπ,t defined in (12), plugging (30) back to (29) yields +V +� +GPDIS(τ µt:T −1 +t:T−1 ) | St +� +=EAt +� +ρ2 +t +� +ESt+1 +� +V +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ νt(St, At) +� +| St +� ++ VAt (ρtqπ,t(St, At) | St) +=EAt +� +ρ2 +t +� +ESt+1 +� +V +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ νt(St, At) +� +| St +� ++ EAt +� +ρ2 +t q2 +π,t(St, At) | St +� +− (EAt [ρtqπ,t(St, At) | St])2 +=EAt +� +ρ2 +t +� +ESt+1 +� +V +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ νt(St, At) +� +| St +� ++ EAt +� +ρ2 +t q2 +π,t(St, At) | St +� +− v2 +π,t(St). +(Lemma 1) +When t = T − 1, we have +V +� +GPDIS(τ µt:T −1 +t:T−1 ) | St +� +=V (ρtr(St, At) | St) +=V (ρtqπ,t(St, At) | St) +=EAt +� +ρ2 +t q2 +π,t(St, At) | St +� +− v2 +π,t(St), +which completes the proof. +27 + +Liu and Zhang +A.6 Proof of Theorem 6 +Proof We proceed via induction. When t = T − 1, we have +V +� +GPDIS(τ µT −1:T −1 +T−1:T−1 ) | ST−1 = s +� +=VAT −1 (ρT−1r(s, AT−1) | ST−1 = s) +=VAT −1 (ρT−1qπ,T−1(s, AT−1) | ST−1 = s) +The definition of µ∗ +T−1 in (13) and Lemma 2 ensure that µ∗ +T−1 is an optimal solution to +min +µT −1∈ΛT −1 +V +� +GPDIS � +τ µT −1 +T−1 +� +|ST−1 = s +� +. +Now, suppose for some t ∈ [T − 2], µ∗ +t+1:T−1 is an optimal solution to +min +µt+1∈Λt+1,...,µT −1∈ΛT −1 +V +� +GPDIS � +τ µt+1:T −1 +t+1:T−1 +� +|St+1 = s +� +. +To complete induction, we proceed to proving that µ∗ +t:T−1 is an optimal solution to +min +µt∈Λt,...,µT −1∈ΛT −1 +V +� +GPDIS � +τ µt:T −1 +t:T−1 +� +|St = s +� +. +(31) +In the rest of this proof, we omit the domain Λt, . . . , ΛT−1 for simplying notations. For any +µt:T−1, we have +V +� +GPDIS(τ µt:T −1 +t:T−1 ) | St +� +=EAt +� +ρ2 +t +� +ESt+1 +� +V +� +GPDIS(τ µt+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ νt(St, At) + q2 +π,t(St, At) +� +| St +� +− v2 +π,t(St) +(By Lemma 5) +(a) +≥EAt +� +ρ2 +t +� +ESt+1 +� +min +µ′ +t+1:T −1 +V +� +GPDIS(τ +µ′ +t+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ νt(St, At) + q2 +π,t(St, At) +� +| St +� +− v2 +π,t(St) +(Monotonically non-increasing in V(·)) +=EAt +� +ρ2 +t +� +ESt+1 +� +V +� +GPDIS(τ +µ∗ +t+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ νt(St, At) + q2 +π,t(St, At) +� +| St +� +− v2 +π,t(St) +(Inductive hypothesis) +=EAt +� +ρ2 +t uπ,t(St, At) | St +� +− v2 +π,t(St) +(By (14)) +=VAt +� +ρt +� +uπ,t(St, At) | St +� ++ EAt +� +ρt +� +uπ,t(St, At)|St +�2 +− v2 +π,t(St) +(Definition of variance) +=VAt +� +ρt +� +uπ,t(St, At) | St +� ++ EAt∼πt(·|St) +�� +uπ,t(St, At)|St +�2 +− v2 +π,t(St) +(Lemma 1 and µt ∈ Λt) +(b) +≥EAt∼πt(·|St) +�� +uπ,t(St, At)|St +�2 +− v2 +π,t(St). +(Non-negativity of variance) +28 + +Improving Monte Carlo Evaluation with Offline Data +According to the inductive hypothesis, the equality in (a) can be achieved when µt+1:T−1 = +µ∗ +t+1:T−1. According to the construnction of µ∗ +t in (13) and Lemma 3, the equality in (b) +can be achieved when µt = µ∗ +t . This suggests that µ∗ +t:T−1 achieves the lower bound and +is thus an optimal solution to (31), which completes the induction and thus completes the +proof. +A.7 Proof of Lemma 8 +Proof We proceed via induction. When t = T − 1, we have +V +� +GPDIS(τ πt:T −1 +t:T−1 ) | St +� +=VAt (ρtr(St, At) | St) +=VAt (r(St, At) | St) +(By on-policy) +=VAt (qπ,t(St, At) | St) +=EAt +� +q2 +π,t(St, At) | St +� +− v2 +π,t(St) += +� +a +πt(a|St)˜qπ,t(St, a). +(By (19) and νπ,T−1(s, a) = 0) +For t ∈ [T − 2], we have +V +� +GPDIS(τ πt:T −1 +t:T−1 ) | St +� +=EAt +� +ESt+1 +� +V +� +GPDIS(τ πt+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ q2 +π,t(St, At) + νπ,t(St, At) | St +� +− v2 +π,t(St) +(Lemma 5 and on-policy) += +� +a +πt(a|St) +�� +s′ +p(s′|St, a)V +� +GPDIS(τ πt+1:T −1 +t+1:T−1 ) | St+1 = s′� ++ ˜r(St, a) +� += +� +a +πt(a|St) +�� +s′ +p(s′|St, a) +� +a′ +πt+1(a′|s′)˜qπ,t+1(s′, a′) + ˜r(St, a) +� +(Inductive hypothesis) += +� +a +πt(a|St)˜qπ,t(St, a), +(By (19)) +which completes the proof. +A.8 Proof of Theorem 10 +Proof We proceed via induction. For t = T − 1, we have +V +� +GPDIS(τ ˆµt:T −1 +t:T−1 ) | St +� +=EAt∼ˆµt +� +ρ2 +t q2 +π,t(St, At) | St +� +− v2 +π,t(St) +(Lemma 5) +=EAt∼ˆµt +� +ρ2 +t ˆqπ,t(St, At) | St +� +− v2 +π,t(St) +(Definition of ˆq (15)) +29 + +Liu and Zhang +=VAt∼ˆµt +� +ρt +� +ˆqπ,t(St, At)|St +� ++ E2 +At∼ˆµt +� +ρt +� +ˆqπ,t(St, At)|St +� +− v2 +π,t(St) +(Definition of variance and non-negativity of ˆq) +=VAt∼ˆµt +� +ρt +� +ˆqπ,t(St, At)|St +� ++ +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− v2 +π,t(St) +(Lemma 1) += +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− v2 +π,t(St) +(Definition of ˆµ (17) and Lemma 3) += +� +a +πt(a|St)ˆqπ,t(St, a) + +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− +� +a +πt(a|St)ˆqπ,t(St, a) − v2 +π,t(St) +=V +� +GPDIS(τ πt:T −1 +t:T−1 ) | St +� ++ +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− +� +a +πt(a|St)ˆqπ,t(St, a) +(By (20) and Lemma 8) +=V +� +GPDIS(τ πt:T −1 +t:T−1 ) | St +� +− ϵt(St) +(Definition of ϵ (21)) +For t ∈ [T − 2], we have +V +� +GPDIS(τ ˆµt:T −1 +t:T−1 ) | St +� +=EAt∼ˆµt +� +ρ2 +t +� +ESt+1 +� +V +� +GPDIS(τ ˆµt+1:T −1 +t+1:T−1 ) | St+1 +� +| St, At +� ++ νπ,t(St, At) + q2 +π,t(St, At) +� +| St +� +− v2 +π,t(St) +(Lemma 5) +≤EAt∼ˆµt +� +ρ2 +t +� +ESt+1 +� � +a′ +πt+1(a′|St+1)˜qπ,t+1(St+1, a′) | St, At +� ++ νπ,t(St, At) ++ q2 +π,t(St, At) +� +| St +� +− v2 +π,t(St) − EAt∼ˆµt +� +ρ2 +t ESt+1 [ϵt+1(St+1)|St, At] +� +(Inductive hypothesis and Lemma 8) +=EAt∼ˆµt +� +ρ2 +t +� +˜qπ,t(St, At) + v2 +π,t(St) +� +| St +� +− v2 +π,t(St) − EAt∼ˆµt +� +ρ2 +t ESt+1 [ϵt+1(St+1)|St, At] +� +(Definition of ˜q (19)) +=EAt∼ˆµt +� +ρ2 +t ˆqπ,t(St, At) | St +� +− v2 +π,t(St) − EAt∼ˆµt +� +ρ2 +t ESt+1 [ϵt+1(St+1)|St, At] +� +(Definition of ˆq (15)) +=VAt∼ˆµt +� +ρt +� +ˆqπ,t(St, At)|St +� ++ E2 +At∼ˆµt +� +ρt +� +ˆqπ,t(St, At)|St +� +− v2 +π,t(St) +− EAt∼ˆµt +� +ρ2 +t ESt+1 [ϵt+1(St+1)|St, At] +� +(Definition of variance and non-negativity of ˆq) +=VAt∼ˆµt +� +ρt +� +ˆqπ,t(St, At)|St +� ++ +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− v2 +π,t(St) +− EAt∼ˆµt +� +ρ2 +t ESt+1 [ϵt+1(St+1)|St, At] +� +(Lemma 1) +30 + +Improving Monte Carlo Evaluation with Offline Data += +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− v2 +π,t(St) − EAt∼ˆµt +� +ρ2 +t ESt+1 [ϵt+1(St+1)|St, At] +� +(Definition of ˆµ (17) and Lemma 3) += +� +a +πt(a|St)ˆqπ,t(St, a) − v2 +π,t(St) + +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− +� +a +πt(a|St)ˆqπ,t(St, a) +− EAt∼ˆµt +� +ρ2 +t ESt+1 [ϵt+1(St+1)|St, At] +� +=V +� +GPDIS(τ πt:T −1 +t:T−1 ) | St +� ++ +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− +� +a +πt(a|St)ˆqπ,t(St, a) +− EAt∼ˆµt +� +ρ2 +t ESt+1 [ϵt+1(St+1)|St, At] +� +(By (20) and Lemma 8) +=V +� +GPDIS(τ πt:T −1 +t:T−1 ) | St +� ++ +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− +� +a +πt(a|St)ˆqπ,t(St, a) +− VAt∼ˆµt +� +ρt +� +ESt+1 [ϵt+1(St+1)|St, At] +� +− +�� +a +π(At|St) +� +ESt+1 [ϵt+1(St+1)|St, At] +�2 +(Definition of variance and Lemma 1) +≤V +� +GPDIS(τ πt:T −1 +t:T−1 ) | St +� ++ +�� +a +πt(a|St) +� +ˆqπ,t(St, a) +�2 +− +� +a +πt(a|St)ˆqπ,t(St, a) +− min +a ESt+1 [ϵt+1(St+1)|St, At] +(Non-negativity of variance, property of min) +=V +� +GPDIS(τ πt:T −1 +t:T−1 ) | St +� +− ϵt(St). +(Definition of ϵ (21)) +A.9 Proof of Lemma 12 +Proof Based on the definition of the variance regret: +Regret(K) += +K +� +i=1 +V(GPDIS(τ b(i)0:T −1 +0:T−1 +)) − K · min +� +V(GPDIS(τ ˆµ0:T −1 +0:T−1 )), V(GPDIS(τ π0:T −1 +0:T−1 )) +� += +K +� +i=1 +E +� +(GPDIS(τ b(i)0:T −1 +0:T−1 +))2� +− K · min +� +E +� +(GPDIS(τ ˆµ0:T −1 +0:T−1 ))2� +, E +� +(GPDIS(τ π0:T −1 +0:T−1 ))2�� +(By (23), (24), and unbiasedness) += max +� +E +� +−(GPDIS(τ ˆµ0:T −1 +0:T−1 ))2� +, E +� +−(GPDIS(τ π0:T −1 +0:T−1 ))2�� +− +K +� +i=1 +E +� +−(GPDIS(τ b(i)0:T −1 +0:T−1 +))2� +. +The last equation gives a standard regret equation with −(GPDIS(τ b(i)0:T −1 +0:T−1 +))2 as collected +arm rewards. These rewards are arm rewards we used in Algorithm 2. The UCB regret +bound theorem (see, e.g., Agrawal (2018)) can be restated as +31 + +Liu and Zhang +Theorem 13 When rewards are finite, with N arms, the expected total regret achieved by +the UCB algorithm in round K is bounded by O( +√ +NK ln K). +Thus, to bound the regret, it suffices to show that our arm rewards are finite. Because +the reward emitted by the MDP is finite, we have Rt ∈ [−rb, rb] with rb > 0. Let A+ +t (s) .= +{a|ˆµt(a|s) > 0}. Define +ϱ .= +max +t,s,a∈A+ +t (s) +πt(a|s) +µt(a|s) +as the largest importance sampling ratio. Define +Cub .= ( +T +� +t=1 +rbϱt)2. +We have Cub as the upper bound of GPDIS(τ ˆµt:T −1 +t:T−1 )2 because ∀τ ˆµt:T −1 +t:T−1 , +Cub = ( +T +� +t=1 +rbϱt)2 ≥ ( +T +� +t=1 +Rtϱt)2 ≥ GPDIS(τ ˆµt:T −1 +t:T−1 )2. +Because ϱ ≥ 1, we also have ∀τ πt:T −1 +t:T−1 , Cub ≥ (�T +t=1 rb)2 = GPDIS(τ πt:T −1 +t:T−1 )2. 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In Proceedings of the AAAI +Conference on Artificial Intelligence, 2006. +38 + diff --git a/l9FST4oBgHgl3EQfJzgq/content/tmp_files/load_file.txt b/l9FST4oBgHgl3EQfJzgq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bdc31eab45a26358c27af525ac27904f90b7a518 --- /dev/null +++ b/l9FST4oBgHgl3EQfJzgq/content/tmp_files/load_file.txt @@ -0,0 +1,1505 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf,len=1504 +page_content='Improving Monte Carlo Evaluation with Offline Data Shuze Liu shuzeliu@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='edu Department of Computer Science University of Virginia 85 Engineer’s Way, Charlottesville, VA, 22903 Shangtong Zhang shangtong@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='edu Department of Computer Science University of Virginia 85 Engineer’s Way, Charlottesville, VA, 22903 Abstract Monte Carlo (MC) methods are the most widely used methods to estimate the performance of a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Given an interested policy, MC methods give estimates by repeatedly running this policy to collect samples and taking the average of the outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Samples collected during this process are called online samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To get an accurate estimate, MC methods consume massive online samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' When online samples are expensive, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', online recom- mendations and inventory management, we want to reduce the number of online samples while achieving the same estimate accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To this end, we use off-policy MC methods that evaluate the interested policy by running a different policy called behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We design a tailored behavior policy such that the variance of the off-policy MC estimator is provably smaller than the ordinary MC estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Importantly, this tailored behavior pol- icy can be efficiently learned from existing offline data, i,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', previously logged data, which are much cheaper than online samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' With reduced variance, our off-policy MC method requires fewer online samples to evaluate the performance of a policy compared with the ordinary MC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Moreover, our off-policy MC estimator is always unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Introduction Evaluating a policy of interest via a scalar performance metric is a fundamental problem in reinforcement learning (Sutton and Barto, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Such evaluation makes it possible to directly compare two policies and lays out the foundations for the more ambitious con- trol problem, the goal of which is to find the best-performing policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Monte Carlo methods (Kakutani, 1945) are the most commonly used methods for such evaluation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Given an interested policy, Monte Carlo methods give estimates by repeatedly executing this pol- icy to collect trajectories and taking the averaged outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Trajectory samples collected during this process are called online samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Online samples are cheap when reliable sim- ulators exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For example, board games (Tesauro 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 2016) and video games (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 2019) give a large amount of instant online samples with little cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' However, in many real-world applications, online samples are usually expensive and take non-negligible time and financial costs to collect, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', online recommendations (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Gauci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 2018) and inventory management (Giannoccaro and Pontrandolfo ©2022 Shuze Liu and Shangtong Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' License: CC-BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='0, see https://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='0/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='13734v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='LG] 31 Jan 2023 Liu and Zhang 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In those scenarios, reducing the number of the required online samples in Monte Carlo methods brings in substantial time and economic benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Online sample reduction can be achieved by reducing the variance of Monte Carlo es- timators (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', Bernstein (1924)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' One well-known variance reduction tool in statistics is importance sampling (Rubinstein 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Benjamin Melamed 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In the setting of rein- forcement learning, it corresponds to evaluating an interested policy by executing a different policy to collect online samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The interested policy is called target policy and the exe- cuted policy is called behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To get an unbiased estimate, we rely on importance sampling ratios to reweight online samples from the behavior policy and thus give an un- biased estimate of the target policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This idea of estimating a target policy by running a different behavior policy is called off-policy learning (Geweke 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Hesterberg 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Precup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Koller and Friedman 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' By contrast, estimating a target policy by running this policy itself is called on-policy learning (Sutton, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' When the behavior policy is properly tailored, the variance of the off-policy Monte Carlo estimator can be significantly smaller than that of the ordinary on-policy one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In this paper, we propose algorithms to learn such behavior policies from previously logged, existing offline data, which are much cheaper than online data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Notably, the learned behavior policy from our proposed method may suffer from learning errors, just like any other learning process, due to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', insufficient offline data coverage, mismatched hypothesis spaces, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Nevertheless, our proposed Monte Carlo-off policy estimator is always unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We further use a bandit algorithm to strategically switch between the learned behavior policy and the target policy for online data collection, which guarantees that the regret is reasonably small even if the learned behavior policy contains non-negligible learning errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Background We consider a finite horizon Markov Decision Process (MDP, Puterman (2014)) with a finite state space S, a finite action space A, a reward function r : S × A → R, a transition probability function p : S × S × A → [0, 1], an initial distribution p0 : S → [0, 1], and a constant horizon length T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Without loss of generality, we consider this undiscounted setting for simplifying notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Our results naturally apply to the discounted setting (Puterman, 2014) as long as the horizon is fixed and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At time step 0, an initial state S0 is sampled from p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At time step t ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' , T − 1}, an action At is sampled according to πt(·|St) where πt : A × S → [0, 1] is the policy at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' A finite reward Rt+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= r(St, At) is then emitted and a successor state St+1 is sampled from p(·|St, At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We define abbreviations πi:j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= {πi, πi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' , πj} and π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= π0:T−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The return at time step t is defined as Gt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= T � i=t+1 Ri, which allows us to define the state- and action-value functions as vπ,t(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='=Eπ [Gt|St = s] , qπ,t(s, a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='=Eπ [Gt|St = s, At = a] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 2 Improving Monte Carlo Evaluation with Offline Data It is easy to see that vπ,t(s) = � a πt(a|s)qπ,t(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (1) We focus on the total rewards performance metric (Puterman, 2014) to measure the per- formance of the policy π, which is defined as J(π) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= � s p0(s)vπ,0(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Knowing such a scalar performance metric makes it possible to easily compare two policies and is also preferred in machine learning applications and research (Ng 2017) because it offers a clear learning goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In this paper, we focus on Monte Carlo methods introduced by Kakutani (1945) to estimate the total rewards J(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Among many of its variants, the most straightforward and widely used way is to draw samples of J(π) by executing the policy π online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' As the number of samples increases, the empirical average of the sampled returns converges to J(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This idea is called on-policy learning (Sutton 1988) because it estimates a policy π by executing itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' From now on, we consider off-policy learning, where we estimate the total rewards J(π) of an interested policy π called target policy by executing a different policy µ called behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Off-policy learning has substantive advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' First, estimating the value of a target policy π without actual deployment makes the learning process much safer (Thomas 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Safety and reliability are critical factors in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Second, trajectory samples collected by one behavior policy can be used to evaluate multiple target policies (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 2011), making the estimation more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In off-policy learning, each trajectory {S0, A0, R1, S1, A1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' , ST−1, AT−1, RT } is generated by a behavior policy µ with S0 ∼ p0, At ∼ µt(·|St), t ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' , T − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Let τ µt:T −1 t:T−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= {St, At, Rt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' , ST−1, AT−1, RT } be a shorthand for a segment of a random trajectory generated by the behavior policy µ from the time step t to the time step T − 1 inclusively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In off-policy learning, we use the importance sampling ratio to reweight rewards collected by µ in order to give an estimate of J(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The importance sampling ratio at time step t is defined as ρt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= πt(At|St) µt(At|St).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The product of importance sampling ratios from time t to the last step T − 1 is defined as ρt:T−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= T−1 � k=t πk(Ak|Sk) µk(Ak|Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 3 Liu and Zhang There are various methods to use the importance sampling ratios in off-policy learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The most straightforward ordinary importance sampling (IS) estimator is defined as GIS(τ µt:T −1 t:T−1 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= ρt:T−1Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This ordinary importance sampling estimator is an unbiased estimator when the behavior policy µ covers the target policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' That is when µt(a|s) = 0 =⇒ πt(a|s) = 0, we have E � GIS(τ µt:T −1 t:T−1 )|St = s � = E [ρt:T−1Gt|St = s] = vπ,t(s) ∀t, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' However, weighted by the entire product ρ0:T−1, the ordinary importance sampling es- timator has a high variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Intensive research has been conducted in finding importance sampling estimators with reduced variance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' the weighted importance sampling estima- tor (Geweke 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Hesterberg 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Koller and Friedman 2009), the per-decision importance sampling estimator (Precup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 2000), the consistent weighted per-decision importance sampling estimator (Thomas 2015), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Our paper is based on the per-decision importance sampling estimator (PDIS), which is defined as GPDIS(τ µt:T −1 t:T−1 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= T−1 � k=t ρ0:kRk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We choose the per-decision importance sampling estimator because it is an unbiased esti- mator for any behavior policy µ that covers target policy π (Precup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In other words, when µt(a|s) = 0 =⇒ πt(a|s) = 0, we have E[GPDIS(τ µt:T −1 t:T−1 )|St = s] = vπ,t(s) ∀t, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We will intensively use the recursive expression of the per-decision importance sampling estimator GPDIS(τ µt:T −1 t:T−1 ) = � ρt � Rt+1 + GPDIS(τ µt+1:T −1 t+1:T−1 ) � 0 ≤ t < T − 1 ρtRt+1 t = T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2) The per-decision importance sampling estimator has a lower variance than the ordinary importance sampling estimator since each reward Rk+1 is only weighted by the importance sampling ratio ρ0:k, instead of the entire product ρ0:T−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Intuitively, this is still unbiased because given the current state, the current reward is independent of future actions and states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This paper focuses on further reducing the variance of the per-decision importance sam- pling estimator by using a proper behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' From a statistics perspective, with a lower variance, fewer trajectory samples are required to achieve an evaluation accuracy of J(π) with the same confidence (Bernstein 1924;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Bertsekas and Tsitsiklis 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' From a machine learning perspective, empirical error can be decomposed into bias and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For unbiased methods, a lower variance induces a lower empirical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In empirical exper- iments, estimators with a lower variance take fewer steps and data to achieve convergence in reinforcement learning algorithms (Sutton and Barto 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 4 Improving Monte Carlo Evaluation with Offline Data 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Variance Reduction in Statistics In this section, we provide the mathematical foundation for variance reduction with impor- tance sampling ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The notations here are independent of the rest of this paper – we use similar notations only for easy interpretation in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Consider a discrete random variable A taking values from a finite space A according to a probability mass function π : A → [0, 1] and a function q : A → R mapping a value in A to a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We are interested in estimating EA∼π[q(A)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The ordinary Monte Carlo methods then sample {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' , AN} from π and use the empirical average 1 N N � i=1 q(Ai) (3) as the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In statistics, importance sampling is introduced as a variance reduc- tion technique for Monte Carlo methods (Rubinstein 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The main idea is to sample {Ai, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' , AN} from a different distribution µ and use 1 N N � i=1 ρ(Ai)q(Ai) (4) as the estimate, where ρ(A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= π(A) µ(A) is the importance sampling ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Assuming µ covers π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', ∀a, µ(a) = 0 =⇒ π(a) = 0 the estimation (4) is then unbiased because EA∼π[q(A)] = EA∼µ[ρ(A)q(A)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' If the sampling distribution µ is carefully designed, the variance of (4) can be smaller than that of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This problem of searching for a variance reducing sampling distribution can be formulated as an optimization problem: minµ∈Λ+ VA∼µ(ρ(A)q(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (5) Here Λ+ denotes the set of all the policies that give unbiased estimations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', Λ+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= {µ ∈ ∆(A) | EA∼µ [ρ(A)q(A)] = EA∼π [q(A)]}, 5 Liu and Zhang where ∆(X) denotes the set of all probability distributions on the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Solving (5) is in general very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To see this, consider a concrete example where A = {a1, a2, a3} and � � � � � q(a1) = −10 q(a2) = 2 q(a3) = 2 , � � � � � π(a1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='1 π(a2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='5 π(a3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='4 , � � � � � µ(a1) = 0 µ(a2) = 0 µ(a3) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (6) It can be computed that EA∼π [q(A)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='8 and EA∼µ [ρ(A)q(A)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In other words, we could sample A from µ and use ρ(A)q(A) as an estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This estimator is unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' But apparently, this µ does not cover π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Moreover, since this µ is deterministic, the variance of this estimator is 0, which is the minimum possible variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In other words, this µ is an optimal sampling distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' However, this µ is hand-crafted based on the knowledge that q(a1)π(a1) + q(a2)π(a2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Without such knowledge, we argue that there is little hope to find this µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This example suggests that searching over the entire Λ+ might be too ambitious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' One natural choice is to restrict the search to Λ− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= {µ ∈ ∆(A) | ∀a, µ(a) = 0 =⇒ π(a) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In other words, we aim to find a variance minimizing sampling distribution among all distributions that cover π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Because coverage implies unbiasedness, we have Λ− ⊆ Λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' By searching over only Λ−, we reduce the search space, with the hope that the problem is more tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' It turns out that we can slightly enlarge Λ− to Λ defined as Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= {µ ∈ ∆(A) | ∀a, µ(a) = 0 =⇒ π(a)q(a) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (7) If for some µ and a0, there is π(a0) > 0, q(a0) = 0, µ(a0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Then this µ would not be in Λ− but it is still in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Importantly, any distribution in Λ still gives unbiased estimation, though it may not cover π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The intuition is that the only sample a0 where µ does not cover π must satisfy q(a0) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', this sample does not contribute to the expectation anyway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Lemma 1 ∀µ ∈ Λ, EA∼µ [ρ(A)q(A)] = EA∼π [q(A)] The proof is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We now consider the variance minimization problem on Λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', minµ∈Λ VA∼µ(ρ(A)q(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (8) The following lemma gives a solution µ∗ to the optimization problem (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Lemma 2 Define µ∗(a) ∝ � π(a)|q(a)| if ∃a0, π0(a0)q(a0) ̸= 0 1 |A| otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Then µ∗ is an optimal solution to (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 6 Improving Monte Carlo Evaluation with Offline Data Here by f(a) ∝ g(a), we mean f(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= g(a) � b g(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The proof is detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To understand why µ∗ gives the minimum variance, considering an example where ∀a ∈ A, q(a) > 0, we have ∀a ∈ A µ∗(a) = π(a)|q(a)| c , where c > 0 is a normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' By plugging µ∗ to ρ(A)q(A), we get ∀a ∈ A ρ(a)q(a) = π(a) µ∗(a)q(a) = π(a) π(a)|q(a)| c q(a) = c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', the random variable ρ(·)q(·) is now a constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The variance of a constant random variable is of course zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The following lemma slightly generalizes this intuition, whose proof is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Lemma 3 If ∀a ∈ A, q(a) ≥ 0 or ∀a ∈ A, q(a) ≤ 0, then Λ = Λ+, and the µ∗ defined in Lemma 2 gives a zero variance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', VA∼µ∗(ρ(A)q(A)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' A sampling distribution proportional to π(a)|q(a)| dates back to Rubinstein (1981);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Benjamin Melamed (1998) and is also previously used in reinforcement learning (Carpentier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This µ∗ is previously deemed as an optimal sampling distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We, however, make two notes here, both of which, to our knowledge, appear novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (1) We have to carefully specify the set of distributions under consideration before claiming the optimality of µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For example, if we compute this µ∗ for the example (6), it can be easily found that π(A)q(A) µ∗(A) has a strictly positive variance because it evaluates negative for a1 and positive for a2 and a3, while the µ in (6) has a zero variance and is also unbaised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In other words, µ∗ can actually be suboptimal in the set Λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2) This µ∗ does not necessarily cover π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' It is possible that for some a0, there are π(a0) > 0, µ∗(a0) = 0, q(a0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Lemma 1, however, still ensures that µ∗ gives unbaised estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Variance Reduction in Reinforcement Learning We now apply the techniques in Section 3 in the reinforcement learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In par- ticular, we seek to reduce the variance V � GPDIS(τ µ0:T −1 0:T−1 ) � by designing a proper behavior policy µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Of course, we need to ensure that the PDIS estimator with this behavior policy is unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In other words, ideally we should search over Λ+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= � µ ∈ ∆(A)T | E � GPDIS(τ µ0:T −1 0:T−1 ) � = J(π) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' As discussed in Section 3, this might be too ambitious without domain-specific knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Instead, we can search over all policies that cover π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', Λ− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= � µ ∈ ∆(A)T | ∀t, s, a, µt(a|s) = 0 =⇒ πt(a|s) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 7 Liu and Zhang Set Λ− contains all policies that satisfy the policy coverage constraint in off-policy learn- ing (Sutton and Barto 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We relax the policy coverage constraint while maintaining unbiasedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Define Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= � µ ∈ ∆(A)T | ∀t, s, a, µt(a|s) = 0 =⇒ πt(a|s)qπ,t(s, a) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The following theorem ensures unbiasedness, which is proved in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Theorem 4 (Unbiasedness) ∀µ ∈ Λ, t, s, E � GPDIS(τ µt:T −1 t:T−1 )|St = s � = vπ,t(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' One immediate consequence of Theorem 4 is that ∀µ ∈ Λ, E � GPDIS(τ µ0:T −1 0:T−1 ) � = J(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' It, however, turns out that searching over Λ is still intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We instead consider a set Λ∗ such that Λ− ⊆ Λ∗ ⊆ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This Λ∗ will be defined soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We now formulate our problem as min µ∈Λ∗ V � GPDIS(τ µ0:T −1 0:T−1 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (9) By the law of total variance, given any two random variables X and Y , we have V[Y ] = E[V(Y |X)] + V(E[Y |X]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (10) For any µ ∈ Λ, we decompose the variance of the PDIS estimator as V � GPDIS(τ µ0:T −1 0:T−1 ) � (11) =ES0 � V � GPDIS(τ µ0:T −1 0:T−1 ) | S0 �� + VS0 � E � GPDIS(τ µ0:T −1 0:T−1 ) | S0 �� =ES0 � V � GPDIS 0 (τ µ0:T −1 0:T−1 ) | S0 �� + VS0 (vπ,0(S0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (Theorem 4) The second term in (11) is a constant given a target policy π and is unrelated to the choice of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In the first term, the expectation is taken over S0 that is determined by the initial probability distribution p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Consequently, solving the problem (9) is equivalent to solving for each s, min µ∈Λ∗ V � GPDIS(τ µ0:T −1 0:T−1 )|S0 = s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Denote {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' , n} as [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Define the variance of the state value for the next state given the current state-action pair (s, a) as νπ,t(s, a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= � VSt+1 (vπ,t+1(St+1) | St = s, At = a) if t ∈ [T − 2] 0 if t = T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (12) The variance of the PDIS estimator can be expressed in the following recursive form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 8 Improving Monte Carlo Evaluation with Offline Data Lemma 5 For any µ ∈ Λ, we have for t = T − 1, V � GPDIS(τ µt:T −1 t:T−1 ) | St � = EAt∼µt � ρ2 t q2 π,t(St, At) | St � − v2 π,t(St);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For t ∈ [T − 2], V � GPDIS(τ µt:T −1 t:T−1 ) | St � =EAt∼µt � ρ2 t � ESt+1 � V � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St � | St, At � + νπ,t(St, At) + q2 π,t(St, At) � | St � − v2 π,t(St).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Its proof is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This recursive form allows us to construct a behavior policy µ∗ as µ∗ t (a|s) ∝ � πt(a|s) � uπ,t(s, a) if ∃a′, πt(a′|s)|uπ,t(s, a′)| ̸= 0 1 |A| otherwise , (13) where for t = T − 1, uπ,t(s, a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= q2 π,t(s, a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' for t ∈ [T − 2], uπ,t(s, a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= � s′ p(s′|s, a)V � GPDIS(τ µ∗ t+1:T −1 t+1:T−1 ) | St+1 = s′� + νπ,t(s, a) + q2 π,t(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (14) This uπ,t(s, a) is always non-negative because all the summands are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This µ∗ is optimal in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Theorem 6 (Optimal Behavior Policy) For any t and s, the behavior policy µ∗ t (a|s) defined above is an optimal solution to the following problem min µt∈Λt,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=',µT −1∈ΛT −1 V � GPDIS(τ µt:T −1 t:T−1 )|St = s � , where Λt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= {µt ∈ ∆(A) | ∀s, a, µt(a|s) = 0 =⇒ πt(a|s)uπ,t(s, a) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Its proof is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Theorem 6 indicates that µ∗ achieves optimality in the set Λ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= Λ0 × · · · × ΛT−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Since uπ,t(s, a) = 0 =⇒ qπ,t(s, a) = 0, we have Λ∗ ⊆ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' If µt(a|s) = 0 =⇒ πt(a|s) = 0, it follows immediately that µt(a|s) = 0 =⇒ πt(a|s)uπ,t(s, a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This indicates Λ− ⊆ Λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Though the set of policies Λ∗ considered in Theorem 6 is not as broad as Λ, it still includes all the policies that cover the target policy, which is the setting where most off-policy results consider (Precup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Maei, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Zhang, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' It is easy to see π ∈ Λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' So Theorem 6 ensures, by the definition of optimality, that the variance of the off-policy PDIS estimator with the behavior policy µ∗ is no larger than the variance of the on-policy PDIS estimator, which reduces to the ordinary on-policy Monte Carlo estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 9 Liu and Zhang Corollary 7 For all s, t, V � GPDIS(τ µ∗ t:T −1 t:T−1 )|St = s � ≤ V � GPDIS(τ πt:T −1 t:T−1 )|St = s � ∀t, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Unfortunately, implementing µ∗ t requires to know uπ,t (14) that contains the transition function p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Approximating the transition function is very challenging in MDPs with large stochasticity and approximation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' model-based reinforcement learning (Sutton, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Deisenroth and Rasmussen, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Chua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Thus, we seek to build another policy ˆµ that can be implemented without direct knowledge of the transition function p (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' model-free reinforcement learning (Sutton, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Watkins, 1989)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We achieve this by aiming at local optimality instead of global optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In particular, at a time step t, if we aim for global optimality, we should try to find the best µt assuming in the future we follow µ∗ t+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' , µ∗ T−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Instead, we aim for the local optimality and try to find the best µt assuming in the future we follow πt+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' , πT−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We refer to such a local optimal behavior policy as ˆµt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Similarly, to define optimality we first need to specify the set of policies we are concerned about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To this end, we define for t = T − 1, ˆqπ,t(s, a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= q2 π,t(s, a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' for t ∈ [T − 2], ˆqπ,t(s, a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= � s′ p(s′|s, a)V � GPDIS(τ πt+1:T −1 t+1:T−1 ) | St+1 = s′� + νπ,t(s, a) + q2 π,t(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (15) This ˆqπ,t is always non-negative since all the summands are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Accordingly, we define for t ∈ [T − 1], ˆΛt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= {µt ∈ ∆(A) | ∀s, a, µt(a|s) = 0 =⇒ πt(a|s)ˆqπ,t(s, a) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Corollary 7 ensures that ˆqπ,t(s, a) ≥ uπ,t(s, a) ≥ 0 holds ∀s, a, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' As a result, if µt ∈ ˆΛt, we have µt(a|s) = 0 =⇒ πt(a|s)ˆqπ,t(a|s) = 0 =⇒ πt(a|s)uπ,t(a|s) = 0, indicating µt ∈ Λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In other words, we have ˆΛt ⊆ Λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To search for ˆµ0:T−1, we work on ˆΛ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= ˆΛ0 × · · · × ˆΛT−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To summarize, we have Λ− ⊆ ˆΛ ⊆ Λ∗ ⊆ Λ ⊆ Λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Recall that Λ+ is the set of all behavior policies such that the corresponding PDIS estimator is unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Λ is a sufficient but not necessary condition to ensure such unbiasedness (Theorem 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Λ∗ is a restriction of Λ such that we are able to find an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Here we future restrict Λ∗ to ˆΛ, aiming for a sub-optimal but implementable policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Our search space is still larger than Λ−, which contains all behavior policies that cover the target policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 10 Improving Monte Carlo Evaluation with Offline Data According to the recursive expression of the variance in Lemma 5 and the aforementioned goal for local optimality, we let ˆµt be an optimal solution to the following problem min µt∈ˆΛt EAt∼µt � ρ2 t � ESt+1 � V � GPDIS(τ πt+1:T −1 t+1:T−1 ) | St+1 � | St, At � + νπ,t(St, At) + q2 π,t(St, At) � | St � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (16) Simple calculation yields EAt∼µt � ρ2 t � ESt+1 � V � GPDIS(τ πt+1:T −1 t+1:T−1 ) | St+1 � | St, At � + νπ,t(St, At) + q2 π,t(St, At) � | St � =EAt∼µt � ρ2 t ˆqπ,t(St, At)|St � =VAt∼µt � ρt � ˆqπ,t(St, At)|St � − E2 At∼µt � ρt � ˆqπ,t(St, At)|St � =VAt∼µt � ρt � ˆqπ,t(St, At)|St � − E2 At∼πt �� ˆqπ,t(St, At)|St � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (Lemma 1 and µt ∈ ˆΛt) According to Lemma 2, if we define ˆµt(a|s) ∝ � πt(a|s) � ˆqπ,t(s, a) if ∃a0, πt(a0|s) � ˆqπ,t(s, a0) ̸= 0 1 |A| otherwise , (17) then ˆµt is an optimal solution to (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To further characterize the property of ˆµ, we need a more explicit treatment of V � GPDIS(τ πt:T −1 t:T−1 )|St = s � , the variance of the on-policy Monte Carlo estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To this end, we make use of the well-known fact that this variance can be expressed recursively in the form of a Bellman equation (Tamar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' O’Donoghue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Sherstan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Formally speaking, define shorthands ˜rπ,t(s, a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= νπ,t(s, a) + q2 π,t(s, a) − v2 π,t(s) ∀t ∈ [T − 1], (18) ˜qπ,t(s, a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= � ˜rπ,t(s, a) + � s′,a′ p(s′|s, a)πt+1(a′|s′)˜qπ,t+1(s′, a′) if t ∈ [T − 2] ˜rπ,t(s, a) if t = T − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (19) Then we have Lemma 8 V � GPDIS(τ πt:T −1 t:T−1 )|St = s � = � a πt(a|s)˜qπ,t(s, a) ∀t, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Its proof is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Here, this ˜q is exactly the action value function of the target policy π in the MDP w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' to a new reward function ˜r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Manipulating (15) then yields ˆqπ,t(s, a) = � s′ p(s′|s, a) � a′ πt+1(a′|s′)˜qπ,t+1(s′, a′) + νt(s, a) + q2 π,t(s, a) (20) =˜qπ,t(s, a) + v2 π,t(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 11 Liu and Zhang This behavior policy ˆµ is of course inferior to the optimal behavior policy µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We, however, argue that ˆµ is provably better than the target policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In particular, since we have ˆµ ∈ ˆΛ ⊆ Λ, Theorem 4 ensures that the PDIS estimator using ˆµ as the behavior policy gives unbiased estimation, even if ˆµ may not cover π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Moreover, the following theorem confirms that the PDIS estimator using ˆµ has a provably smaller variance than the PDIS estimator using π as the behavior policy, which is exactly the ordinary on-policy Monte Carlo estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Theorem 9 (Variance Reduction) For any t and s, V � GPDIS(τ ˆµt:T −1 t:T−1 ) | St = s � ≤V � GPDIS(τ πt:T −1 t:T−1 ) | St = s � Proof We proceed via induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For t = T − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' V � GPDIS(τ ˆµt:T −1 t:T−1 ) | St � =EAt∼ˆµt � ρ2 t q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Lemma 5) =EAt∼ˆµt � ρ2 t ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Definition of ˆq (15)) =VAt∼ˆµt � ρt � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St � + E2 At∼ˆµt � ρt � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Definition of variance and non-negativity of ˆq) =VAt∼ˆµt � ρt � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St � + �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Lemma 1) = �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Definition of ˆµ (17) and Lemma 3) ≤ � a πt(a|St)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Jensen’s inequality) =V � GPDIS(τ πt:T −1 t:T−1 ) | St � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (By (20) and Lemma 8) For t ∈ [T − 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' we have V � GPDIS(τ ˆµt:T −1 t:T−1 ) | St � =EAt∼ˆµt � ρ2 t � ESt+1 � V � GPDIS(τ ˆµt+1:T −1 t+1:T−1 ) | St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + νπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) + q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) � | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Lemma 5) ≤EAt∼ˆµt � ρ2 t � ESt+1 � � a′ πt+1(a′|St+1)˜qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(St+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′) | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + νπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) + q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) � | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Inductive hypothesis and Lemma 8) =EAt∼ˆµt � ρ2 t � ˜qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) + v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) � | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Definition of ˜q (19)) =EAt∼ˆµt � ρ2 t ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Definition of ˆq (15)) 12 Improving Monte Carlo Evaluation with Offline Data =VAt∼ˆµt � ρt � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St � + E2 At∼ˆµt � ρt � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Repeating the arguments for t = T − 1) =VAt∼ˆµt � ρt � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St � + �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) = �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) ≤ � a πt(a|St)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) =V � GPDIS(τ πt:T −1 t:T−1 ) | St � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We also prove a stronger lemma in the following to further compute the exact amount of the reduced variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Theorem 10 For any t and s, V � GPDIS(τ ˆµt:T −1 t:T−1 ) | St = s � ≤ V � GPDIS(τ πt:T −1 t:T−1 ) | St = s � − ϵt(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' where ct(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= � a πt(a|s)ˆqπ,t(s, a) − �� a πt(a|s) � ˆqπ,t(s, a) �2 , ∀t ∈ [T − 1] ϵt(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= � ct(s) + mina � s′ p(s′|s, a)ϵt+1(s′) if t ∈ [T − 2] ct(s) if t = T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (21) The proof is similar to Theorem 9 and is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Notably, this ct is always non- negative thanks to Jensen’s inequality, which ensures that ϵt is also non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' If we regard ct as a cost function, then the reduced variance ϵt is exactly the optimal cost-to-go function of the stochastic shortest path problem in the MDP induced by the cost function ct (Bertsekas and Tsitsiklis, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Variance Reduction with Offline Data Having identified a sub-optimal but provably better policy ˆµ, the next step is to approx- imate it, preferably with offline data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Since the target policy πt is considered known, to approximate ˆµt, according to (15), it is sufficient to approximate ˆqπ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This, according to (20), requires to approximate ˜qπ,t and vπ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The state value function vπ,t can be learned using any existing offline evaluation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In particular, we can use fitted Q-learning (Le et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2019) to learn the action value function qπ,t first, then compute vπ,t analytically using (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The observant reader may question, if we have learned the action value func- tion, do we still need to do Monte Carlo evaluation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The answer is affirmative and more 13 Liu and Zhang details are deferred to the discussion regarding model-free offline evaluation in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Having learned vπ,t, the remaining is to approximate ˜qπ,t, which is exactly the action value function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a different reward function ˜rπ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' If ˜rπ,t is known, we could then resort to fitted Q-learning again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To approximate ˜rπ,t, we, accroding to (18), need to approximate νπ,t, qπ,t, and vπ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We have already learned qπ,t and vπ,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The remaining is thus to learn νπ,t, which by its definition in (12) is a variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Approximating a variance is in general a supervised learning problem – we can approximate the first and second moments separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In particular, we have VSt+1 (vπ,t+1(St+1) | St = s, At = a) =ESt+1 � v2 π,t+1(St+1) | St = s, At = a � − E2 St+1 [vπ,t+1(St+1) | St = s, At = a] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The two expectations can be learned via supervised learning, using our approximation of vπ,t+1 to generate regression targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The approximation error in vπ,t+1 will, however, inevitably compound into the approximation error of this variance term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The approximation error in this variance term then compounds into the approximation error of ˆqπ,t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This repeated error compounding is the first challenge in learning ˜qπ,t and vπ,t separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Moreover, to ensure that ˆµt is well defined, it is necessary to make sure that the approx- imation of ˜qπ,t(s, a) + v2 π,t(s) is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Apparently, v2 π,t(s) is always non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' There is, however, no guarantee that the learned approximation of ˜qπ,t(s, a) is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' One way to achieve this is to use a non-negative function class, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', (·)2 or ln(1 + exp(·)), to approximate ˜qπ,t(·, ·), such that our approximation is always non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This unfor- tunately introduces bias as we use a non-negative function class to approximate a function whose value can possibly be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This bias is the second challenge in learning ˜qπ,t and vπ,t separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We now address those two challenges simultaneously via learning ˆqπ,t(s, a) directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This is made possible thanks to the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Lemma 11 ∀s, a, ˆqπ,t(s, a) = � ˆrπ,t(s, a) + � s′,a′ p(s′|s, a)πt+1(a′|s′)ˆqπ,t+1(s′, a′) if t ∈ [T − 2] ˆrπ,t(s, a) if t = T − 1 , where ˆrπ,t(s, a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= 2r(s, a)qπ,t(s, a) − r2(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (22) Proof For t = T − 1, we have ˆqπ,t(s, a) = q2 π,t(s, a) (Definition of ˆqπ,t (15)) = ˆrπ,t(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (By qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='T−1(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) = r(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) and (22)) For t ∈ [T − 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' we have ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) =˜qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) + v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s) (By (20)) 14 Improving Monte Carlo Evaluation with Offline Data =˜rπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) + v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s) + � s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='a′ p(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a)πt+1(a′|s′)˜qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′) (Definition of ˜q (19)) =˜rπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) + v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s) + � s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='a′ p(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a)πt+1(a′|s′)(˜qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′) + v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′) − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′)) =˜rπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) + v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s) + � s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='a′ p(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a)πt+1(a′|s′)(ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′) − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′)) (By (20)) =νπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) + q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) − � s′ p(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a)v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′) + � s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='a′ p(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a)πt+1(a′|s′)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′) (Definition of ˜r (18)) = − (E[vπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(St+1) | St = s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At = a])2 + q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) + � s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='a′ p(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a)πt+1(a′|s′)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′) (Definition of ν (12)) = − (qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) − r(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a))2 + q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) + � s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='a′ p(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a)πt+1(a′|s′)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′) =2r(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a)qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) − r2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) + � s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='a′ p(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a)πt+1(a′|s′)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′) =ˆrπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) + � s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='a′ p(s′|s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a)πt+1(a′|s′)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In other words, ˆq is exactly the action value function of the policy π w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' the reward function ˆr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Suppose ˆr is learned, we can then learn ˆq with any offline evaluation methods for action-value functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', fitted Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To learn ˆr, it is sufficient to learn r and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Fitted Q-learning can be used to learn q and learning r is a simple regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Importantly, this regression problem now has accurate targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' By contrast, the regression of the variance ν use the estimation of vπ,t to compute the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' As a result, error compounding is reduced by learning ˆqπ,t directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We consider the behavior policy agnostic offline learning setting (Nachum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2019), where the offline data in the form of � (ti, si, ai, ri, s′ i) �m i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' consists of m previously logged data tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In the i-th data tuple, ti is the time step, si is the state at time step ti, ai is the action executed on state si, ri is the sampled reward, and s′ i is the successor state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Those tuples can be generated by one or more, possibly unknown behavior policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' And those tuples do not need to form a complete trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Our proposed algorithm for learning ˆµ is detailed in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In particular, we first learn r with supervised learning and qπ,t with fitted Q-learning from the offline data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Then we compute ˆr analytically and apply fitted Q-learning again to learn ˆqπ,t, which is then used to compute ˆµ analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We split the offline data into training sets and test sets to tune all the hyperparameters in Algorithm 1, based on the supervised learning loss or the fitted Q-learning loss on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 15 Liu and Zhang Algorithm 1: Approximating ˆµ with offline data Input: a differentiable function parameterization rw : S × A × Rd0 → R a differentiable function parameterization qw : [T] × S × A × Rd1 → R a differentiable function parameterization ˆqw : [T] × S × A × Rd2 → R a target policy π an offline dataset {(ti, si, ai, ri, si)}m i=1 Initialize wr ∈ Rd0, wq ∈ Rd1, wˆq ∈ Rd2 arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Algorithm Parameters: learning rates αr, αq, αˆq Output: a behavior policy ˆµ Step 1: Augment data with a′ Sample an action a′ by πt+1(·|s′) for each (t, s, a, r, s′) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Step 2: Approximate rw Loop for each training step: Sample a minibatch of (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' r) Perform a mini-batch gradient descent step based on wr ← wr + αr[r − rw(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' wr)]∇rw(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' wr) Step 3: Approximate qw Loop for each training step: Sample a minibatch of (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′) Perform a mini-batch gradient descent step based on wq ← wq + αq[r + qw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' wq) − qw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' wq)]∇qw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' wq) Step 4: Approximate ˆqw Loop for each training step: Sample a minibatch of (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′) Perform a mini-batch gradient descent step based on ˆr ← 2rw(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' wr)qw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' wq) − r2 w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' wr) wˆq ← wˆq + αˆq[ˆr + ˆqw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' wˆq) − ˆqw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' wˆq)]∇ˆqw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' wˆq) Step 5: Output ˆµ ˆµt(a|s) ∝ πt(a|s) � ˆqw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' wˆq) 16 Improving Monte Carlo Evaluation with Offline Data Algorithm 2: Adaptive Monte Carlo Evaluation Input: A policy π to be evaluated A policy ˆµ computed from Algorithm 1 Algorithm Parameters: UCB parameter c Initialize: Rewards(π) ← an empty list Rewards(ˆµ) ← an empty list n ← 0 J ← 0 Output: Total rewards estimation J Loop for K episodes: S0 ∼ p0 b ← argmaxb∈{ˆµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='π}Average � Rewards(b) � + c � log n |Rewards(b)| (UCB) Generate a trajectory {S0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' R1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' ST−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' AT−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' RT } following b G ← 0 for t = T − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' T − 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 0: G ← πt(At|St) bt(At|St) (Rt+1 + G) Rewards(b) append −G2 n ← n + 1 J ← J + 1 n(G − J) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Variance Reduction in Online Execution Though the PDIS Monte Carlo estimator with ˆµ is provably better than that with π (The- orems 4 & 9), we do not have access to ˆµ but only its approximation learned from offline data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This learning process suffers from various biases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', insufficient data coverage, mismatched hypothesis space, incomplete optimization, insufficient hyperparameter tun- ing, etc, just like any other learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' As a result, using the learned approximation of ˆµ is not necessary better than using π directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' With a slight abuse of notation, we in this section use ˆµ to denote the learned approximation from Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Thus when we actually collect online data, there are two choices, to use the target policy π or to use the learned ˆµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Though both yield unbiased estimation, their variances are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This places us in the exploration and exploitation dilemma (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', Lattimore and Szepesv´ari (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' On the exploration side, we want to execute both policies more to know their variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' On the exploitation side, we want to commit to a better policy as soon as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Motivated by the celebrated success in the bandit community in solving the exploration and exploitation dilemma (Lattimore and Szepesv´ari, 2020), we now formulate this problem as a multi-armed bandit problem, where the target policy π and the learned ˆµ are two arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To complete the bandit formulation, the next step is to specify a reward function for each arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Since we want to identify the policy that yields a lower variance, the natural choice is then to use the additive inverse of the variance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', −V � GPDIS(τ ˆµ0:T −1 0:T−1 ) � and 17 Liu and Zhang −V � GPDIS(τ ˆπ0:T −1 0:T−1 ) � , as a reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We, however, cannot estimate the variance from a single trajectory τ0:T−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Using multiple trajectories either reduces the sample efficiency or makes the reward function non-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To address this challenge, we use the following observation, V � GPDIS(τ ˆµ0:T −1 0:T−1 ) � = E � (GPDIS(τ ˆµ0:T −1 0:T−1 ))2� − E2 � GPDIS(τ ˆµ0:T −1 0:T−1 ) � , (23) V � GPDIS(τ π0:T −1 0:T−1 ) � = E � (GPDIS(τ π0:T −1 0:T−1 ))2� − E2 � GPDIS(τ π0:T −1 0:T−1 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (24) Since the PDIS estimator is unbiased for both ˆµ and π, the E2 [·] terms above are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To compare the variances, it is sufficient to compare the second moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We, therefore, use −(GPDIS(τ ˆµ0:T −1 0:T−1 ))2 and −(GPDIS(τ π0:T −1 0:T−1 ))2 as the rewards for the arms ˆµ and π respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Those rewards are immediately available after a trajectory τ0:T−1 is sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We use the Upper Confidence Bound (UCB, Auer (2002)) algorithm to adaptively switch between the two arms during online executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The details are documented in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Notably, since both ˆµ and π induce unbiased estimation, all trajectories, not just those from the better policy, contribute to the final estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' No online data is wasted in this sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Denote b(i) ∈ {ˆµ, π} as the chosen behavior policy at the i-th online episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Define K as the total number of online episodes and define the regret for the variance as Regret(K) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= K � i=1 V(GPDIS(τ b(i)0:T −1 0:T−1 )) − K · min � V(GPDIS(τ ˆµ0:T −1 0:T−1 )), V(GPDIS(τ π0:T −1 0:T−1 )) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The following lemma gives a sublinear regret guarantee, regardless of the approximation error in ˆµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Its proof is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Lemma 12 E [Regret(K)] = O( √ K ln K) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Empirical Results In this section, we present empirical results to answer the following two questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' If the approximation of ˆµ is of high quality, can the PDIS Monte Carlo estimator outperform the ordinary on-policy Monte Carlo estimator?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' If the approximation of ˆµ is of poor quality, can the adaptive execution strategy still ensure a low estimation error?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We use grid worlds with different sizes as our benchmark environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For a grid world with size n, its width, height, and time horizon T are all set to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' There are four possible actions: up, down, left, and right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' After taking an action, the agent has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='9 probability to move accordingly and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='1 probability to move uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' If the agent runs into a boundary, the agent stays in its current location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The reward function r(s, a) is randomly generated and fixed after generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We normalize the rewards across all (s, a) such that maxs,a r(s, a) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We consider a set of randomly generated target policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The ground truth policy performance is estimated using the on-policy Monte Carlo method by running 18 Improving Monte Carlo Evaluation with Offline Data each target policy for 106 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We test three different sizes of the grid world, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', n ∈ {5, 10, 15}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The offline dataset always contains m = 105 randomly generated tuples regardless of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Given an environment and a target policy, we execute Algorithm 1 to approximate function r, q, and ˆq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We consider a tabular setting where each (t, s, a) pair is represented by a distinct one-hot vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' As shown in Algorithm 1, we train r using supervised learning by batch gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We train q and ˆq using fitted Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We split the offline data into a training set and a test set and tune all hyperparameters offline based on the supervised learning loss and fitted Q-learning loss on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We use the same set of hyperparameters for all grid worlds and target policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We end up with learning rates being 1, training steps being 103, and batch size being 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In each grid world environment, we test 30 randomly generated target policies and each target policy is tested 30 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For an environment and a target policy, we execute ˆµ and π for 500 steps to estimate the expected total rewards of the target policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Each step is defined as one interaction between the agent and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Thus, the estimate for the environment with time horizon 15 starts from steps 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Because we are interested in estimation accuracy, we define the estimation error at step t as the absolute difference between the PDIS estimation and the ground truth divided by the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We use normalized estimation error which is the estimation error divided by the average estimation error of the on-policy estimator after the first episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This ensures that the normalized estimation error of the on-policy estimator starts from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 0 100 200 300 400 500 steps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='0 normalized estimation error n = 5 on-policy off-policy off-policy with UCB 0 100 200 300 400 500 steps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='0 n = 10 on-policy off-policy off-policy with UCB 0 100 200 300 400 500 steps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='2 n = 15 on-policy off-policy off-policy with UCB Figure 1: The normalized estimation errors of the off-policy estimator with the learned ˆµ and the on-policy estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Each curve averages the results of 30 random policies and each policy has 30 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The shaded regions denote standard errors and are invisible for some curves because they are too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Blue lines in Figure 1 show the experiment results when we always use learned ˆµ as the actual behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For the grid world of size n = 5, the off-policy estimator with the learned ˆµ consumes significantly fewer online steps to achieve the same estimation accuracy, compared with the on-policy estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For example, to achieve 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='2 normalized estimation error, the off-policy estimator consumes around 40 online steps while the on-policy estimator consumes around 120 online steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Substantial improvements also exist in the grid world environment with size n = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' However, in the grid world environment with size n = 15, the off-policy estimator with the learned ˆµ is actually worse than the on-policy estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This 19 Liu and Zhang is because the learned ˆµ contains a non-negligible approximation error due to insufficient data coverage as the environment size increases while the number of offline data remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To maintain a reasonably low estimation error when this occurs, we use Algorithm 2 with a standard UCB confidence value c = 1 to identify the better policy and has results shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' As shown with red lines, after adopting the UCB algorithm to adaptively switch between the learned policy ˆµ and the target policy π during online executions, the improvements on the grid world with size 5 or 10 is still significant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', when n = 5, to achieve 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='2 normalized estimation error, the off-policy estimator takes 60 online steps while the on-policy estimator takes 120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The jump at the start of the red lines is because Algorithm 2 initially breaks ties by choosing the learned ˆµ and then chooses to explore the target policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This causes the error to increase because of the large on-policy estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The performance degeneration when n = 15 has been effectively mitigated shown by the closer curves in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' As we examine the variance numerically below, this mitigation is significant, Define the variance ratio as V � GPDIS(τ b0:T −1 0:T−1 ) � V � GPDIS(τ π0:T −1 0:T−1 ) � where b is the actual behavior policy during online executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' By using collected online samples to estimate the variance, Table 7 shows that when n = 5 or 10, the variance of the off-policy estimator with the learned ˆµ is much smaller than the variance of the on- policy estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' After adopting the UCB algorithm, the significant variance reduction for n ∈ {5, 10} still exists while the variance when n = 15 has been reduced to almost one-third of its original size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Thus, Algorithm 2 greatly reduces the regret when the learned ˆµ is of poor quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Gridworld size n Without UCB With UCB 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='282 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='350 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='422 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='673 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='390 Table 1: Variance Ratio for Gridworld Environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Related Work Monte Carlo methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Reducing the variance of Monte Carlo estimators via learning a proper behavior policy has been explored before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Hanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2017) model the problem of finding a variance-reducing behavior policy as an optimization problem and thus rely on stochastic gradient descent to update a parameterized behavior policy directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In par- ticular, Hanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2017) consider the ordinary importance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' By contrast, we consider the per-decision importance sampling, which is fundamentally better (Precup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Moreover, Hanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2017) require new online data to learn this behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' By contrast, our method works with offline data and does not need any more online data for 20 Improving Monte Carlo Evaluation with Offline Data behavior policy learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Hanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2017) also requrie the online data to be complete trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' By contrast, our method copes well with offline tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2022) also investigate variance-reducing behavior policies for the per-decision importance sampling estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Their results, however, apply to only tree- structured MDPs, which is rather restrictive because many MDPs of interest are not tree- structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For example, in the finite horizon MDP considered in this paper, if two states at time t can transit to the same successor state at time t + 1, then this MDP is not tree- structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Moreover, Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2022) require to directly approximate the transition function in the MDP by counting, making it essentially a model-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2022), therefore, suffer from all canonical challenges in model learning (Sutton, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Deisenroth and Rasmussen, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Chua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' By contrast, we work on general MDPs without making any assumption regarding the underlying structures of the MDPs and do not need to approximate the transition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Our approach is model-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2022) adjust the behavior policy by encouraging under-sampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2022), however, rely on strong assumptions on offline data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Their offline data has to be complete trajectories generated by known policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In their experiments, they also require the policies for generating offline data to be similar to the target policy since they do not have any importance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' By contrast, our method copes well offline data in the form of incomplete segments from probably unknown behavior policies that can be arbitrarily different from the target policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Moreover, there is no theoretical guarantee that the estimates made by Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2022) are unbiased or consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' By contrast, our estimate is always provably unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Other attempts for variance reduction in Monte Carlo evaluation are mostly by using control variates based on value function (Zinkevich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' White and Bowling, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Jiang and Li, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Such control variates can be easily integrated into our estimator, which we, however, save for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Model-based offline evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' One straightforward way to exploit offline data for policy evaluation is to learn a model of the MDP first, probably with supervised learning (Jiang and Li, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Paduraru, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2021), and then execute Monte Carlo methods inside the learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Learning a high-fidelity model is, however, sometimes even more challenging than evaluating the policy itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' And the model prediction error can easily compound over time steps during model rollouts (Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Nevertheless, if a good model can somehow be learned, our work still helps reduce the required rollouts when Monte Carlo is applied within the learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Model-free offline evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Offline data can also be exploited for policy evaluation without explicitly constructing a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Those model-free offline evaluation methods in- stead learn some other quantities, including density ratio (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' marginalized importance sampling ratio, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Nachum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Mousavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Li (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Uehara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2019)) and action value function (Le et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Harutyunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Precup et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Farajtabar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' But those learning processes bring in bias, just like any other learning process, either due to the misspecification of the function class or due to the complexity of optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Consequently, the estimation they make is biased and it is hard 21 Liu and Zhang to quantify such bias without restrictive assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To our knowledge, the only practical way in general settings to certify that their estimation is indeed accurate is to compare those estimations with estimations made by Monte Carlo methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We believe this is why Monte Carlo methods still dominate the evaluation of policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Even worse, those learning algo- rithms also have hyperparameters to tune, just like any other learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In other words, we need to evaluate different outputs of those learning algorithms corresponding to different hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This is called model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Apparently, we cannot use the aforementioned density ratio or action value function based model free evaluation methods for model selection – otherwise we run into a self-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In fact, those works (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Nachum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Mousavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Li, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Uehara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2019) usually use Monte Carlo with online data for evaluating different candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The online data comes from either a simulator or a learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' As a result, this work helps reduce the online data used in model selection by those model-free offline evaluation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Efforts have been made to perform model selection with only offline data without explicitly learning a model as well (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Paine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Zhang and Jiang, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Xie and Jiang, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Those offline model selection methods, however, rarely have a correctness guarantee without restrictive assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' They can probably provide a preliminary screen in model selection but Monte Carlo methods make the final decision when correctness really matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To summarize, if obtaining online data is entirely impossible, existing offline evaluation methods without using any online data might be the only choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' These include model- based methods and model-free methods augmented by offline model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' However, in many real-world problems, it is practical to assume that a small amount of online data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' If in addition, evaluation correctness should be honored, then the improved Monte Carlo method in this work might be a better choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Control algorithms can use online data (Watkins and Dayan, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2017), offline data (Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Kidambi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Schrittwieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2021), or a mix of online and offline data (Vecerik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Nair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Ijspeert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Rajeswaran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Ajay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Control algorithms also have hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Consequently, to use control algorithms, model selection is necessary, where Monte Carlo methods now dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' As a result, this work makes almost all control algorithms more efficient, in terms of using online data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Using offline data to help online model selection in control problems is previously explored by Konyushova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In particular, it uses offline data to decide which policy, among a given set of policies, should be given priority to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' When it comes to the actual online evaluation, Konyushova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2021) still uses the ordinary online Monte Carlo methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Konyushova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (2021), therefore, again benefit from the improved Monte Carlo method in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Conclusion Monte Carlo methods are the most dominating approach for evaluating a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The development and deployment of almost all RL algorithms implicitly or explicitly depend on Monte Carlo methods more or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For example, when a reinforcement learning researcher wants to plot a curve of the agent performance against training steps, Monte Carlo methods 22 Improving Monte Carlo Evaluation with Offline Data are usually the first choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This work develops a method to improve the online data efficiency of Monte Carlo evaluation by learning a tailored behavior policy from offline data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The Monte Carlo estimator with this tailored behavior policy is provably better than the canonical Monte Carlo estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The theoretical advantage is also demonstrated empirically, as a proof-of-concept, in the tested domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We save the investigation on large-scale problems for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Moreover, this work considers only the total rewards performance metric on finite horizon MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' One natural next step is to consider the average reward (Puterman, 2014) or the discounted total rewards (Puterman, 2014) on infinite horizon MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Acknowledgments The authors thank Haifeng Xu for the insightful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' SZ is part of the Link Lab at the University of Virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Proofs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='1 Proof of Lemma 1 Proof EA∼µ [ρ(A)q(A)] = � a∈{a|µ(a)>0} µ(a)π(a) µ(a)q(a) = � a∈{a|µ(a)>0} π(a)q(a) = � a∈{a|µ(a)>0} π(a)q(a) + � a∈{a|µ(a)=0} π(a)q(a) (µ ∈ Λ) = � a π(a)q(a) =EA∼π [q(A)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='2 Proof of Lemma 2 Proof For a given π and q, define A+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= {a | π(a)q(a) ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For any µ ∈ Λ, we expand the variance as VA∼µ(ρ(A)q(A)) =EA∼µ[(ρ(A)q(A))2] − E2 A∼µ[ρ(A)q(A)] 23 Liu and Zhang =EA∼µ[(ρ(A)q(A))2] − E2 A∼π[q(A)] (Lemma 1) = � a∈{a|µ(a)>0} π2(a)q2(a) µ(a) − E2 A∼π[q(A)] = � a∈{a|µ(a)>0}∩A+ π2(a)q2(a) µ(a) − E2 A∼π[q(A)] (π(a)q(a) = 0, ∀a /∈ A+) = � a∈A+ π2(a)q2(a) µ(a) − EA∼π[q(A)]2 (µ ∈ Λ) The second term is a constant and is unrelated to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Solving the optimization problem (8) is, therefore, equivalent to solving minµ∈Λ � a∈A+ π2(a)q2(a) µ(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (25) Case 1: |A+| = 0 In this case, the variance is always 0 so any µ ∈ Λ is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In particular, µ∗(a) = 1 A is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Case 2: |A+| > 0 The definition of Λ in (7) can be equivalently expressed, using contraposition, as Λ = {µ ∈ ∆(A) | ∀a, a ∈ A+ =⇒ µ(a) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The optimization problem (25) can then be equivalently written as minµ∈∆(A) � a∈A+ π2(a)q2(a) µ(a) (26) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' µ(a) > 0 ∀a ∈ A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' If for some µ we have � a∈A+ µ(a) < 1, then there must exist some a0 /∈ A+ such that µ(a0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Since a0 does not contribute to the summation in the objective function of (26), we can move the probability mass on a0 to some other a1 ∈ A+ to increase µ(a1) to further decrease the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In other words, any optimal solution µ to (26) must put all its mass on A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This motivates the following problem minz∈∆(A+) � a∈A+ π2(a)q2(a) z(a) (27) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' z(a) > 0 ∀a ∈ A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In particular, if z∗ is an optimal solution to (27), then an optimal solution to (26) can be constructed as µ∗(a) = � z∗(a) a ∈ A+ 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (28) 24 Improving Monte Carlo Evaluation with Offline Data Let R++ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' According to the Cauchy-Schwarz inequality, for any z ∈ R|A+| ++ , we have � � � a∈A+ π2(a)q2(a) z(a) � � � � � a∈A+ z(a) � � ≥ � � � a∈A+ π(a)|q(a)| � z(a) � z(a) � � 2 = � � � a∈A+ π(a)|q(a)| � � 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' It can be easily verified that the equality holds for z∗(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= π(a)|q(a)| � b π(b)|q(b)| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Since � a∈A+ z∗(a) = 1, we conclude that z∗ is an optimal solution to (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' An optimal solution µ∗ to (8) can then be constructed according to (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Making use of the fact that π(a)|q(a)| = 0 for a /∈ A+, this µ∗ can be equivalently expressed as µ∗(a) = π(a)|q(a)| � b∈A π(b)q(b), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='3 Proof of Lemma 3 Proof We start with showing Λ = Λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Lemma 1 ensures that µ ∈ Λ =⇒ µ ∈ Λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We now show that µ ∈ Λ+ =⇒ µ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For any µ ∈ Λ+, we have � a∈{a|µ(a)>0} µ(a)π(a) µ(a)q(a) = � a π(a)q(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This indicates that � a∈{a|µ(a)=0} π(a)q(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Since π(a) ≥ 0 and all q(a) has the same sign, we must have π(a)q(a) = 0, ∀a ∈ {a|µ(a) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This is exactly µ(a) = 0 =⇒ π(a)q(a) = 0, yielding µ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This completes the proof of Λ+ = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We now show the zero variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' When ∀a ∈ A, q(a) ≥ 0, if ∃a0, π0(a0)q(a0) ̸= 0, we have ∀a ∈ A µ∗(a) = π(a)|q(a)| c and c > 0 is a normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Plugging µ∗ to ρ(A)q(A), we get ∀a ∈ A ρ(a)q(a) = π(a) µ∗(a)q(a) = π(a) π(a)|q(a)| c q(a) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 25 Liu and Zhang This means in this setting, with the optimal distribution µ∗, the random variable ρ(·)q(·) is a constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Thus, VA∼µ∗(ρ(A)q(A)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' When ∀a ∈ A, q(a) ≥ 0, if ∀a0, π0(a0)q(a0) = 0, we have ∀a ∈ A µ∗(a) = 1 |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Plugging µ∗ to ρ(A)q(A), we get ∀a ∈ A ρ(a)q(a) = π(a) µ∗(a)q(a) = π(a)q(a) 1 |A| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This shows ρ(A)q(A) is a also constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Thus, VA∼µ∗(ρ(A)q(A)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The proof is similar for ∀a ∈ A, q(a) ≤ 0 and is thus omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='4 Proof of Theorem 4 Proof We proceed via induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For t = T − 1, we have E � GPDIS(τ µt:T −1 t:T−1 ) | St � =E [ρtRt+1 | St] = E [ρtqπ,t(St, At) | St] =EAt∼πt(·|St) [qπ,t(St, At)|St] (Lemma 1) =vπ,t(St).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For t ∈ [T − 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' we have E � GPDIS(τ µt:T −1 t:T−1 ) | St � =E � ρtRt+1 + ρtGPDIS(τ µt+1:T −1 t+1:T−1 ) | St � =E [ρtRt+1 | St] + E � ρtGPDIS(τ µt+1:T −1 t+1:T−1 ) | St � =E [ρtRt+1 | St] + EAt∼µt(·|St),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='St+1∼p(·|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='At) � E � ρtGPDIS(τ µt+1:T −1 t+1:T−1 ) | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' St+1 �� (Law of total expectation) =E [ρtRt+1 | St] + EAt∼µt(·|St),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='St+1∼p(·|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='At) � ρtE � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St+1 � |St � (Conditional independence and Markov property) =E [ρtRt+1 | St] + EAt∼µt(·|St),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='St+1∼p(·|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='At) [ρtvπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(St+1)|St] (Inductive hypothesis) =EAt∼µt(·|St) [ρtqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St] (Definition of qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t) =EAt∼πt(·|St) [qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St] (Lemma 1) =vπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 26 Improving Monte Carlo Evaluation with Offline Data A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='5 Proof of Lemma 5 Proof When t ∈ [T − 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' we have V � GPDIS(τ µt:T −1 t:T−1 ) | St � (29) =EAt � V � GPDIS(τ µt:T −1 t:T−1 ) | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � | St � + VAt � E � GPDIS(τ µt:T −1 t:T−1 ) | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � | St � (Law of total variance (10)) =EAt � ρ2 t V � r(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) + GPDIS(τ µt+1:T −1 t+1:T−1 ) | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � | St � + VAt � ρtE � r(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) + GPDIS(τ µt+1:T −1 t+1:T−1 ) | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � | St � (Using (2)) =EAt � ρ2 t V � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � | St � + VAt (ρtqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (Deterministic reward r) Further decomposing the first term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' we have V � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � (30) =ESt+1 � V � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + VSt+1 � E � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � (Law of total variance (10)) =ESt+1 � V � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + VSt+1 � E � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � (Markov property) =ESt+1 � V � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + VSt+1 (vπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(St+1) | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (Theorem 4) With νπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t defined in (12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' plugging (30) back to (29) yields V � GPDIS(τ µt:T −1 t:T−1 ) | St � =EAt � ρ2 t � ESt+1 � V � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + νt(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) � | St � + VAt (ρtqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St) =EAt � ρ2 t � ESt+1 � V � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + νt(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) � | St � + EAt � ρ2 t q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St � − (EAt [ρtqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St])2 =EAt � ρ2 t � ESt+1 � V � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + νt(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) � | St � + EAt � ρ2 t q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (Lemma 1) When t = T − 1, we have V � GPDIS(τ µt:T −1 t:T−1 ) | St � =V (ρtr(St, At) | St) =V (ρtqπ,t(St, At) | St) =EAt � ρ2 t q2 π,t(St, At) | St � − v2 π,t(St), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 27 Liu and Zhang A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='6 Proof of Theorem 6 Proof We proceed via induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' When t = T − 1, we have V � GPDIS(τ µT −1:T −1 T−1:T−1 ) | ST−1 = s � =VAT −1 (ρT−1r(s, AT−1) | ST−1 = s) =VAT −1 (ρT−1qπ,T−1(s, AT−1) | ST−1 = s) The definition of µ∗ T−1 in (13) and Lemma 2 ensure that µ∗ T−1 is an optimal solution to min µT −1∈ΛT −1 V � GPDIS � τ µT −1 T−1 � |ST−1 = s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Now, suppose for some t ∈ [T − 2], µ∗ t+1:T−1 is an optimal solution to min µt+1∈Λt+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=',µT −1∈ΛT −1 V � GPDIS � τ µt+1:T −1 t+1:T−1 � |St+1 = s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' To complete induction, we proceed to proving that µ∗ t:T−1 is an optimal solution to min µt∈Λt,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=',µT −1∈ΛT −1 V � GPDIS � τ µt:T −1 t:T−1 � |St = s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (31) In the rest of this proof, we omit the domain Λt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' , ΛT−1 for simplying notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For any µt:T−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' we have V � GPDIS(τ µt:T −1 t:T−1 ) | St � =EAt � ρ2 t � ESt+1 � V � GPDIS(τ µt+1:T −1 t+1:T−1 ) | St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + νt(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) + q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) � | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (By Lemma 5) (a) ≥EAt � ρ2 t � ESt+1 � min µ′ t+1:T −1 V � GPDIS(τ µ′ t+1:T −1 t+1:T−1 ) | St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + νt(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) + q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) � | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Monotonically non-increasing in V(·)) =EAt � ρ2 t � ESt+1 � V � GPDIS(τ µ∗ t+1:T −1 t+1:T−1 ) | St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + νt(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) + q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) � | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Inductive hypothesis) =EAt � ρ2 t uπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (By (14)) =VAt � ρt � uπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St � + EAt � ρt � uπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St �2 − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Definition of variance) =VAt � ρt � uπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St � + EAt∼πt(·|St) �� uπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St �2 − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Lemma 1 and µt ∈ Λt) (b) ≥EAt∼πt(·|St) �� uπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St �2 − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (Non-negativity of variance) 28 Improving Monte Carlo Evaluation with Offline Data According to the inductive hypothesis, the equality in (a) can be achieved when µt+1:T−1 = µ∗ t+1:T−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' According to the construnction of µ∗ t in (13) and Lemma 3, the equality in (b) can be achieved when µt = µ∗ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' This suggests that µ∗ t:T−1 achieves the lower bound and is thus an optimal solution to (31), which completes the induction and thus completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='7 Proof of Lemma 8 Proof We proceed via induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' When t = T − 1, we have V � GPDIS(τ πt:T −1 t:T−1 ) | St � =VAt (ρtr(St, At) | St) =VAt (r(St, At) | St) (By on-policy) =VAt (qπ,t(St, At) | St) =EAt � q2 π,t(St, At) | St � − v2 π,t(St) = � a πt(a|St)˜qπ,t(St, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (By (19) and νπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='T−1(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) = 0) For t ∈ [T − 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' we have V � GPDIS(τ πt:T −1 t:T−1 ) | St � =EAt � ESt+1 � V � GPDIS(τ πt+1:T −1 t+1:T−1 ) | St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) + νπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Lemma 5 and on-policy) = � a πt(a|St) �� s′ p(s′|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a)V � GPDIS(τ πt+1:T −1 t+1:T−1 ) | St+1 = s′� + ˜r(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) � = � a πt(a|St) �� s′ p(s′|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) � a′ πt+1(a′|s′)˜qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′) + ˜r(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) � (Inductive hypothesis) = � a πt(a|St)˜qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (By (19)) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='8 Proof of Theorem 10 Proof We proceed via induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' For t = T − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' we have V � GPDIS(τ ˆµt:T −1 t:T−1 ) | St � =EAt∼ˆµt � ρ2 t q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Lemma 5) =EAt∼ˆµt � ρ2 t ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Definition of ˆq (15)) 29 Liu and Zhang =VAt∼ˆµt � ρt � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St � + E2 At∼ˆµt � ρt � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Definition of variance and non-negativity of ˆq) =VAt∼ˆµt � ρt � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St � + �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Lemma 1) = �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Definition of ˆµ (17) and Lemma 3) = � a πt(a|St)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) + �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − � a πt(a|St)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) =V � GPDIS(τ πt:T −1 t:T−1 ) | St � + �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − � a πt(a|St)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) (By (20) and Lemma 8) =V � GPDIS(τ πt:T −1 t:T−1 ) | St � − ϵt(St) (Definition of ϵ (21)) For t ∈ [T − 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' we have V � GPDIS(τ ˆµt:T −1 t:T−1 ) | St � =EAt∼ˆµt � ρ2 t � ESt+1 � V � GPDIS(τ ˆµt+1:T −1 t+1:T−1 ) | St+1 � | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + νπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) + q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) � | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) (Lemma 5) ≤EAt∼ˆµt � ρ2 t � ESt+1 � � a′ πt+1(a′|St+1)˜qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t+1(St+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a′) | St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At � + νπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) + q2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) � | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) − EAt∼ˆµt � ρ2 t ESt+1 [ϵt+1(St+1)|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At] � (Inductive hypothesis and Lemma 8) =EAt∼ˆµt � ρ2 t � ˜qπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) + v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) � | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) − EAt∼ˆµt � ρ2 t ESt+1 [ϵt+1(St+1)|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At] � (Definition of ˜q (19)) =EAt∼ˆµt � ρ2 t ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At) | St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) − EAt∼ˆµt � ρ2 t ESt+1 [ϵt+1(St+1)|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At] � (Definition of ˆq (15)) =VAt∼ˆµt � ρt � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St � + E2 At∼ˆµt � ρt � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St � − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) − EAt∼ˆµt � ρ2 t ESt+1 [ϵt+1(St+1)|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At] � (Definition of variance and non-negativity of ˆq) =VAt∼ˆµt � ρt � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At)|St � + �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) − EAt∼ˆµt � ρ2 t ESt+1 [ϵt+1(St+1)|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At] � (Lemma 1) 30 Improving Monte Carlo Evaluation with Offline Data = �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) − EAt∼ˆµt � ρ2 t ESt+1 [ϵt+1(St+1)|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At] � (Definition of ˆµ (17) and Lemma 3) = � a πt(a|St)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) − v2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St) + �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − � a πt(a|St)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) − EAt∼ˆµt � ρ2 t ESt+1 [ϵt+1(St+1)|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At] � =V � GPDIS(τ πt:T −1 t:T−1 ) | St � + �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − � a πt(a|St)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) − EAt∼ˆµt � ρ2 t ESt+1 [ϵt+1(St+1)|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At] � (By (20) and Lemma 8) =V � GPDIS(τ πt:T −1 t:T−1 ) | St � + �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − � a πt(a|St)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) − VAt∼ˆµt � ρt � ESt+1 [ϵt+1(St+1)|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At] � − �� a π(At|St) � ESt+1 [ϵt+1(St+1)|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At] �2 (Definition of variance and Lemma 1) ≤V � GPDIS(τ πt:T −1 t:T−1 ) | St � + �� a πt(a|St) � ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) �2 − � a πt(a|St)ˆqπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='t(St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' a) − min a ESt+1 [ϵt+1(St+1)|St,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' At] (Non-negativity of variance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' property of min) =V � GPDIS(τ πt:T −1 t:T−1 ) | St � − ϵt(St).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (Definition of ϵ (21)) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='9 Proof of Lemma 12 Proof Based on the definition of the variance regret: Regret(K) = K � i=1 V(GPDIS(τ b(i)0:T −1 0:T−1 )) − K · min � V(GPDIS(τ ˆµ0:T −1 0:T−1 )),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' V(GPDIS(τ π0:T −1 0:T−1 )) � = K � i=1 E � (GPDIS(τ b(i)0:T −1 0:T−1 ))2� − K · min � E � (GPDIS(τ ˆµ0:T −1 0:T−1 ))2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' E � (GPDIS(τ π0:T −1 0:T−1 ))2�� (By (23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' and unbiasedness) = max � E � −(GPDIS(τ ˆµ0:T −1 0:T−1 ))2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' E � −(GPDIS(τ π0:T −1 0:T−1 ))2�� − K � i=1 E � −(GPDIS(τ b(i)0:T −1 0:T−1 ))2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The last equation gives a standard regret equation with −(GPDIS(τ b(i)0:T −1 0:T−1 ))2 as collected arm rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' These rewards are arm rewards we used in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' The UCB regret bound theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=', Agrawal (2018)) can be restated as 31 Liu and Zhang Theorem 13 When rewards are finite, with N arms, the expected total regret achieved by the UCB algorithm in round K is bounded by O( √ NK ln K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Thus, to bound the regret, it suffices to show that our arm rewards are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Because the reward emitted by the MDP is finite, we have Rt ∈ [−rb, rb] with rb > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Let A+ t (s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= {a|ˆµt(a|s) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Define ϱ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= max t,s,a∈A+ t (s) πt(a|s) µt(a|s) as the largest importance sampling ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Define Cub .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content='= ( T � t=1 rbϱt)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' We have Cub as the upper bound of GPDIS(τ ˆµt:T −1 t:T−1 )2 because ∀τ ˆµt:T −1 t:T−1 , Cub = ( T � t=1 rbϱt)2 ≥ ( T � t=1 Rtϱt)2 ≥ GPDIS(τ ˆµt:T −1 t:T−1 )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Because ϱ ≥ 1, we also have ∀τ πt:T −1 t:T−1 , Cub ≥ (�T t=1 rb)2 = GPDIS(τ πt:T −1 t:T−1 )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Since Cub is a constant, ∀τ ˆµt:T −1 t:T−1 , GPDIS(τ ˆµt:T −1 t:T−1 )2 is finite and ∀τ πt:T −1 t:T−1 , GPDIS(τ πt:T −1 t:T−1 )2 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Since their additive inverses are also finite, we proved the arm rewards are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Because we only have two arms, by Theorem 13, the expected total regret of Algorithm 2 is bounded by O( √ K ln K) References Shipra Agrawal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' A course on multi-armed bandits and reinforcement learning, 2018.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Martin Zinkevich, Michael Bowling, Nolan Bard, Morgan Kan, and Darse Billings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' Opti- mal unbiased estimators for evaluating agent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} +page_content=' 38' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FST4oBgHgl3EQfJzgq/content/2301.13734v1.pdf'} diff --git a/m9AyT4oBgHgl3EQflPjP/content/tmp_files/2301.00450v1.pdf.txt b/m9AyT4oBgHgl3EQflPjP/content/tmp_files/2301.00450v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..86e23b7de2daa00c5fbe65252be604e3e560d6af --- /dev/null +++ b/m9AyT4oBgHgl3EQflPjP/content/tmp_files/2301.00450v1.pdf.txt @@ -0,0 +1,2835 @@ +MNRAS 000, 1–19 (2022) +Preprint 3 January 2023 +Compiled using MNRAS LATEX style file v3.0 +A comparison of numerical methods for computing the reionization of +intergalacitc hydrogen and helium by a central radiating source +Ka-Hou Leong1⋆, Avery Meiksin1, Althea Lai1, K. H. To2 +1SUPA†, 1Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, UK +2Department of Physics, University of Tokyo, 7 Chome-3-1 Hongo, Bunkyo City, Tokyo 113-8654, Japan +Accepted . Received ; in original form +ABSTRACT +We compare numerical methods for solving the radiative transfer equation in the context of the photoionization of +intergalactic gaseous hydrogen and helium by a central radiating source. Direct integration of the radiative transfer +equation and solutions using photon packets are examined, both for solutions to the time-dependent radiative transfer +equation and in the infinite-speed-of-light approximation. The photon packet schemes are found to be more generally +computationally efficient than a direct integration scheme. Whilst all codes accurately describe the growth rate of +hydrogen and helium ionization zones, it is shown that a fully time-dependent method is required to capture the gas +temperature and ionization structure in the near zone of a source when an ionization front expands at a speed close +to the speed of light. Applied to Quasi-Stellar Objects in the Epoch of Reionization (EoR), temperature differences as +high as 5×104 K result in the near-zone for solutions of the time-dependent radiative transfer equation compared with +solutions in the infinite-speed-of-light approximation. Smaller temperature differences are found following the nearly +full photoionization of helium in gas in which the hydrogen was already ionized and the helium was singly ionized. +Variations found in the temperature and ionization structure far from the source, where the gas is predominantly +neutral, may affect some predictions for 21-cm EoR experiments. +Key words: radiative transfer – quasars: absorption lines – quasars: general – dark ages, reionization, first stars – +cosmology: large-scale structure of Universe +1 INTRODUCTION +Numerical simulations are now a staple method for providing +detailed descriptions of the complex processes in virtually all +areas of astrophysics. The range in physical processes treated +has increased from gravity and gas dynamics to include mag- +netic fields, a host of atomic and molecular reaction networks +and radiative transport. +Including radiative transport is in particular numerically +challenging when the gas is not everywhere optically thick. +In this case, the transport of radiation is not diffusive so that +devising efficient methods to solve the full set of radiative +transfer (RT) equations must be addressed, particularly when +the mean free path of the radiation exceeds other important +length scales in an application that must be spatially resolved. +As radiative transport is a long-range effect in these applica- +tions, its introduction seriously hampers computations both +because of the required increase in memory to represent the +radiation field and, potentially, because of a severely curtailed +time step. +Applications to galactic and cosmological structure forma- +tion for which radiative transport is essential include star +⋆ E-mail: KH.Leong@ed.ac.uk (KHL) +† Scottish Universities Physics Alliance +formation and the impact of stars and Quasi-Stellar Objects +(QSOs) on intergalactic gas, including the reionization of the +intergalactic medium (IGM). Various approximation meth- +ods have been introduced into numerical simulations to solve +the radiative transfer equations. Applications of the meth- +ods in numerical simulations are typically done in a post- +processing stage, ie, on top of previously computed solutions +to the gas dynamical equations. A comparison of many of +these schemes to applications to static gas problems is pre- +sented in Iliev et al. (2006). These tests are confined to the +photoionization of hydrogen. Whilst the results from the dif- +ferent schemes are generally in good agreement on the place- +ment of the ionization front, with discrepancies limited to 5– +10%, a much broader range of differences were obtained for +other quantities. The ionization fractions showed differences +up to a factor of a few to several, and temperatures differed +by up to a few tens of percent. Some of these differences do +not necessarily reflect differences in the algorithms, but may +be attributed in part to the different frequency ranges cov- +ered for the sources. Some of the differences, however, appear +intrinsic to the schemes. +Fully coupled radiative hydrodynamical codes have also +been formulated, but often for restricted applications, either +through a suppression of one or more spatial dimensions or +through approximations made to the radiative transfer equa- +© 2022 The Authors +arXiv:2301.00450v1 [astro-ph.IM] 1 Jan 2023 + +2 +tions. A comparison of some of these methods is presented in +Iliev et al. (2009). +Whilst the reionization of intergalactic hydrogen has been +addressed in a wide variety of simulations (see Gnedin & +Madau 2022, for a recent review), the reionization of he- +lium has received much less attention. Yet its solution is vital +for understanding the temperature evolution and small scale +structure of the IGM, which has been used to place con- +straints on various dark matter candidates (eg Baur et al. +2016; Garzilli et al. 2017; Irˇsiˇc et al. 2017; Leong et al. +2019) and for placing limits on the neutrino mass (Viel +et al. 2010). The temperature evolution of the IGM in turn +places constraints on the nature, abundance and evolution +of the sources that reionized the hydrogen and helium in +the IGM (eg Theuns et al. 2002; Tittley & Meiksin 2007; +Bolton et al. 2012; Upton Sanderbeck et al. 2016; Walther +et al. 2019; Keating et al. 2020). Unlike for hydrogen reioniza- +tion, the two ionization states of helium result in a broadened +singly-ionized (He II) zone. As intergalactic helium is detected +through the He II Lyα absorption signature, it is important +to obtain an accurate solution for this zone. Additional appli- +cations of helium reionization include the detailed structure +of the proximity zones around QSOs for constraints on the +reionization of both hydrogen and helium, which depend on +the lifetime of the sources, the size of the ionized regions +they produce and on the temperature of the ionized gas in +the zone (eg Zheng et al. 2015; Davies et al. 2020; Worseck +et al. 2021). The possibility of the detection of a 21-cm signal +during the Epoch of Reionization (EoR) also requires under- +standing the heating of the still neutral gas by high energy +photons, as only a small amount of heating is able to affect +the absorption signal against the Cosmic Microwave Back- +ground (CMB), and even convert it into an emission signal +(Tozzi et al. 2000; Ross et al. 2019; Ma et al. 2020). +In the context of IGM reionization simulations, two broadly +different algorithmic approaches have been used to compute +the photoionization driven by isolated radiation sources, one +based on representing the radiation field as photon packets +and the other based on a direct integration of the radiative +transfer equation. Photon packet schemes have generally been +used rather than direct integration for numerical simulations +because of their greater numerical efficiency (Bolton et al. +2004). Several three-dimensional numerical RT packages in- +corporating multi-frequency radiation have been used to com- +pute the photoionization of both hydrogen and helium in the +IGM, including LICORICE (Baek et al. 2010), RADAMESH (Can- +talupo & Porciani 2011), TRAPHIC (Pawlik & Schaye 2011), +C2-RAY (Friedrich et al. 2012), CRASH (Graziani et al. 2013) +and RADHYDRO (La Plante et al. 2017). +With the exception of TRAPHIC, these codes provide only +quasi-time-dependent solutions in the sense that they allow +for evolution in the gas properties and the sources, but they +solve only the static RT equation by taking the speed of +light to be infinite. The infinite-speed-of-light approximation +(ISLA) has two shortcomings: the ionization fronts may prop- +agate to acausally large distances (greater than the distance +light could travel), and the approximation is not valid when +the ionization front is propagating near the speed of light. The +propagation of superluminal ionization fronts is particularly a +concern for the ionization zones produced by QSOs: because +of their high luminosities, the ionization fronts may move +at superluminal velocities over the lifetime of the sources, +producing unphysically large ionization regions. This may be +partly overcome by enforcing a light-horizon, eg, by remov- +ing photon packets when they exceed their causal horizon. +Such a solution, however, does not conserve the radiative en- +ergy carried by the photons and may alter the post-ionization +temperature, and therefore ionization fractions, of the gas. +The computation of ionization fronts propagating at near +the speed of light requires an algorithm that solves the time- +dependent radiative transfer equation to obtain an accurate +post-ionization temperature, as will be demonstrated in this +paper. One goal of this paper is to provide a means for imple- +menting an approximate practical photon packet RT scheme +for cosmological simulations that handles near-luminal ion- +ization front expansion without solving the time-dependent +RT equation, which is prohibitively computationally expen- +sive. +Photon packet schemes have been applied to the reioniza- +tion of intergalactic hydrogen and helium both as a post- +processing step (Sokasian et al. 2002; Bolton et al. 2004; +McQuinn et al. 2009; Ciardi et al. 2012; Compostella et al. +2013; Kakiichi et al. 2017; Eide et al. 2018, 2020) and in fully +coupled radiative hydrodynamics (RHD) schemes (Meiksin & +Tittley 2012; La Plante et al. 2017). Moments based schemes +for solving the radiative transfer equations to study the reion- +ization of hydrogen and helium in the IGM have been em- +ployed both in a multi-step scheme (Puchwein et al. 2022) +and in RHD simulations (Wu et al. 2019; Kannan et al. 2022). +Codes that compute the reionization of hydrogen and he- +lium by a QSO in a static (or comoving) gas and assuming +spherical symmetry have been used to estimate the gas prop- +erties around a QSO using both direct integration (Madau +et al. 1997; Tozzi et al. 2000; Friedrich et al. 2012) and pho- +ton packet schemes (Zheng et al. 2008; Davies et al. 2016; +Khrykin et al. 2016; Graziani et al. 2018; Eilers et al. 2021; +Morey et al. 2021). +Rather than comparing results from a broad range of nu- +merical codes, the focus of this paper is on the requirements +for achieving convergent results using different numerical RT +methods, with a particular view to application to the reioniza- +tion of helium in the IGM. We compare two photon packet al- +gorithms and a direct integration algorithm applied to sources +with spectra similar to stars and QSOs. Whilst previous con- +vergence tests focused on the extent of ionization zones and +their properties using solutions of the time-independent RT +equation (ISLA methods), special attention is given here to +differences arising between solutions to the time-dependent +(for which the speed of light is finite) and time-independent +RT equations. In addition to eliminating spurious fast-than- +light growth of the ionization zones, we show there are sig- +nificant differences in the post-ionization temperature in the +near-zone of QSO sources when solving the time-dependent +RT equation compared with ISLA solutions. With a view to +applications to the 21-cm EoR signal, we also examine con- +vergence in the far-zone, where the gas is still largely neutral. +We focus on idealised test problems, as these provide the best +means for isolating differences in the behaviour of the algo- +rithms. To demonstrate the consequences of these differences +in a cosmological setting, we also provide results for the ion- +ization by a source in a cosmological simulation. In the next +section, we describe the numerical methods. In Sec. 3, test +problems are described and results presented. These are dis- +cussed in Sec. 4, where also results for a cosmological simula- +MNRAS 000, 1–19 (2022) + +Numerical methods for IGM reionization +3 +tion are presented, and conclusions are summarised in Sec. 5. +Comparisons between the time-dependent solutions to the RT +equation and published ISLA solutions for test problems are +presented in Appendix A. One of the photon packet schemes +is a module of the gravity-hydrodynamics code ENZO. Con- +vergence test results for ENZO are provided in Appendix B. +Alterations to the code required to carry out the tests are +described in Appendex C. +2 NUMERICAL SOLUTIONS OF THE REIONIZATION +EQUATIONS +2.1 The Reionization Equations +The ionization equations for hydrogen and helium are +dxH I +dt += +−xHI [ΓH I + neγH I(T)] + xH IIneαH II(T), +dxH II +dt += +−dxH I +dt , +dxHe I +dt += +−xHe I [ΓHe I + neγHe I(T)] + xHe IIneαHe II, +dxHe II +dt += +−dxHe I +dt +− dxHe III +dt +, +dxHe III +dt += +xHe II [ΓHe II + neγHe II(T)] − xHe IIIneαHe III,(1) +where αH II, αHe II and αHe III are the total radiative recom- +bination rates to all levels of H I, He I and He II, respectively, +ΓH I, ΓHe I and ΓHe II are the corresponding photoionization +rates, and γH I(T), γHe I(T) and γHe II(T) are the correspond- +ing collisional ionization coefficients. For hard spectra, sec- +ondary ionizations produced by ejected electrons may help +to partially ionize the gas. We do not include this effect to +focus comparisons between the codes on the numerical solu- +tions of the radiative transfer equations; including secondary +electron ionizations could in principle complicate the inter- +pretation of any differences. Their correct implementation is +moreover hampered by the long path lengths of the secondary +electrons compared with the length scales of interest. A com- +parison between different treatments is provided by Davies +et al. (2016). To illustrate the magnitude of the effects of sec- +ondary electron ionizations, we present some results for test +problems without and with secondary electron ionizations in +Appendix A. Generally the effects are only moderate, but +warrant inclusion for more precise solutions; the effects of +secondary electron ionizations are particularly large in the +partially ionized regions outside the main ionized zones. The +respective H I, H II, He I, He II and He III fractions are denoted +by xHI, xHII, xHeI, xHeII and xHeIII. The total electron density +is ne = nHII + nHeII + 2nHeIII. The time-derivatives are la- +grangian, so that Eqs. (1) are valid in the presence of velocity +flows. +The photoionization rate per atom (or ion) of a species i +(H I, He I or He II) is +Γi = c +� ∞ +νT,i +dν uν +hν ai,ν, +(2) +where ai,ν is the photoelectric cross section of species i, νT,i +is the threshold frequency required to ionize species i, uν is +the specific energy density of the ambient radiation field, and +h is Planck’s constant. The specific energy density is related +to the specific intensity of the radiation field Iν(r, t, ˆn) by +uν = 4πJν/c, where Jν(r, t) = (1/4π) +� +dΩIν(r, t, ˆn) is the +angle-averaged specific intensity at position r at time t in the +direction ˆn. +The equation of radiative transfer for Iν(r, t, ˆn) in a static +medium with absorption coefficient αν(r, t, ˆn) and emission +coefficient jν(r, t, ˆn) is +1 +c +∂Iν(r, t, ˆn) +∂t ++ ˆn · ∇Iν(r, t, ˆn) = +−αν(r, t, ˆn)Iν(r, t, ˆn) + jν(r, t, ˆn). +(3) +In the context of reionization, the emission coefficient repre- +sents a source like a star, galaxy or QSO, although in princi- +ple it may also account for photoionizing radiation following +radiative recombination within the gas. Since only contin- +uum radiation is considered, as distinct from resonance lines, +the static medium approximation is adequate on scales small +compared with the cosmic horizon. +The value of Iν at any given time t and position s along +the direction ˆn will be given by any incoming intensity Iinc +ν +at position s0 at time tret = t − (s − s0)/c, absorbed by +intervening material at positions s′ at the retarded times +t′ +ret = t − (s − s′)/c, along with contributions from sources +at positions s′′ that emitted at the retarded times t′′ +ret = +t − (s − s′′)/c, followed by absorption. Accordingly, the for- +mal solution to Eq. (3) is +Iν(s, t) = +� +Iinc +ν +� +s0,tret +exp +� +− +� s +s0 +ds′ (αν)s′,t′ +ret +� ++ +� s +s0 +ds′′ (jν)s′′,t′′ +ret exp +� +− +� s +s′′ ds′ (αν)s′,t′ +ret +� +. +(4) +We shall refer to such solutions as solutions to the time- +dependent RT equation. By contrast, most of the literature +makes the infinite-speed-of-light approximation, which corre- +sponds to solving the time-independent (or static) RT equa- +tion, for which the term involving the time-derivative of Iν +in Eq. (3) is absent, and only the instantaneous properties +of the gas appear in Eq. (4) rather than the time-retarded +properties. The solutions may none the less be quasi-time- +dependent when using the ISLA if the properties of the gas +and the source vary with time, but only on timescales long +compared with the light propagation time from the source. +The photoionization will also heat the gas, while radiative +recombination and collisional effects will cool the gas. We +follow the approach outlined in Meiksin (2009). Specifically, +we solve the energy equation in the form +dSE +dt += (γ − 1)ρ−γ (G − L) , +(5) +where SE = p/ργ for gas pressure p, mass density ρ and +ratio of specific heats γ. Here, G is the heating rate per unit +volume of the gas and L is the radiative energy loss rate per +unit volume. The photoionization heating rate per volume for +a species i of number density ni is +Gi = nic +� ∞ +νT,i +dν uν +hν ai,νh (ν − νT,i) . +(6) +The total heating rate from all species is G = GH I + GHe I + +GHe II. The radiative energy loss term L includes energy +losses from radiative recombination, collisional excitation and +MNRAS 000, 1–19 (2022) + +4 +inverse Compton cooling off the Cosmic Microwave Back- +ground, using the rates referenced in Meiksin (2009). For ap- +plications in Appendix A, energy losses from secondary elec- +tron ionizations and adiabatic cooling from cosmic expansion +(allowing for an evolving mass density) are included for some +of the test problems. The gas temperature is computed from +T = ¯m +k SEργ−1, +(7) +where ¯m is the mean mass per particle and k is Boltzmann’s +constant. The ionization scenarios treated are idealized in +that they do not allow for additional energy losses from dust +or metals, as may occur in the ionized regions of high red- +shift QSO sources. Modelling such effects is well beyond the +intent of this paper and would only complicate the interpre- +tation of differences in the results arising from the different +photoionization algorithms considered. +2.2 Methods of Numerical Solution +We confine the discussion to algorithms that solve the radia- +tive transfer equation along individual rays, as distinct from +moment methods. Full 3D RT is achieved through the means +of constructing the rays, a topic we shall not discuss, simply +adopting the existing framework for the 3D code used (ENZO). +We consider two types of numerical methods for solving the +1D RT equation, one based on a direct integration of the +time-independent RT equation (an ISLA method) and the +other for which the radiation is represented by photon pack- +ets. Two versions of the latter are tested, one corresponding +to integrating the time-independent RT equation (an ISLA +method) and the other corresponding to integrating the time- +dependent RT equation (retaining the differential time oper- +ator). Each method is outlined in turn. +2.2.1 Instantaneous direct integration +At each time step, the 1D radiative transfer equation is inte- +grated along the line of sight. Retaining the past absorption +coefficient for all positions along the line of sight at previ- +ous times is normally prohibitively expensive in a simulation, +so instead new rays are cast for each time step and the ab- +sorption coefficient for that time step is used. The solution is +taken as +Iν(s, t) = +� +Iinc +ν +� +s0,t exp +� +− +� s +s0 +ds′ (αν)s′,t +� ++ +� s +s0 +ds′′ (jν)s′′,t exp +� +− +� s +s′′ ds′ (αν)s′,t +� +. +(8) +This in effect treats the speed of light as infinite, although +a cut-off in the distance the radiation reaches is imposed to +ensure the extent of the region affected by a source preserves +causality. This instantaneous solution may be a good approxi- +mation when the ionization front moves much faster than the +gas flows. In the implementation used here, a spatial grid +is used along the line of sight, with the width of each zone +chosen to ensure the optical depth of still neutral hydrogen +or singly ionized helium is approximately 1. This ensures an +accurate integration of the radiative transfer equation. The +time step is chosen to be a fraction of the shortest ionization +or cooling time, sufficient to provide convergence at the few +percent level. The energy equation is solved with an explicit +second order time integrator, and an implicit scheme is used +to solve the time-dependent ionization equations (Meiksin +1994). +2.2.2 Rays and photon packets +An alternative approach traces packets of photons of distinct +energy groups along rays. The algorithm consists of two parts: +casting rays through a simulation volume from each source, +and propagating photon packets along the rays. For more +than a single spatial dimension, rays bunch together near a +source and the separations between the rays increase with +distance. Adequate cell coverage of the regions affected by +each source is assured by splitting rays as required, as in +Abel & Wandelt (2002). Photon packets are then propagated +along the rays on each time step, until the packets are either +completely absorbed or escape the simulation volume. +The gas component in the simulations is computed on a +spatial grid. The fraction of the photon packets absorbed on +crossing a grid zone depends on the optical depth through +the zone. An advantage of this method over the direct inte- +gration scheme above is that the optical depth may be chosen +to be above 1 while maintaining good accuracy; up to 10 is +typical, but even larger values are able accurately to recover +the expansion of an ionization front (Abel et al. 1999). This +considerably reduces the computational time. A consequence, +however, is an ambiguity in how to share the photons when +more than a single species may be ionized in a zone, as they +will compete for the same photons. How the gas is ionized +in a zone, were it completely resolved, can change the proba- +bility for photons of different energies to be absorbed. In the +simplest implementation, such as in ENZO, the fraction of pho- +tons of energy i absorbed by species j is 1−exp[−τj(νi)], but +in looping over j, different fractions may result depending on +how the species are ordered in the loop. We adopt instead the +more balanced probabilistic approach of Bolton et al. (2004), +and have introduced this into the version of ENZO we use. The +absorption probabilities of a photon in a packet of frequency +ν by H I, He I and He II are respectively +P HI +abs = pHIqHeIqHeII[1 − exp(−τ total +ν +)]/D, +(9) +P HeI +abs = qHIpHeIqHeII[1 − exp(−τ total +ν +)]/D, +(10) +P HeII +abs += qHIqHeIpHeII[1 − exp(−τ total +ν +)]/D. +(11) +Here pi = 1 − exp(−τ i +ν) and qi = exp(−τ i +ν) are the auxil- +iary absorption and transmission probabilities for the species +i, D = pHIqHeIqHeII + qHIpHeIqHeII + qHIqHeIpHeII is the nor- +malisation factor, and τ total +ν += τ HI +ν ++ τ HeI +ν ++ τ HeII +ν +is the total +optical depth, where τ i +ν is the optical depth of species i.1 +1 In practice, for the test problems presented here, the different +ionization zones are sufficiently distinct that the results are nearly +the same using a simpler formulation treating the absorption by +each species independently, with results agreeing typically to better +than a percent. Differences up to 20 percent, however, may arise for +low ionization fractions in some regions. We retain the formulation +described here for generality. +MNRAS 000, 1–19 (2022) + +Numerical methods for IGM reionization +5 +We test the implementation of the photon packet scheme +used in the numerical-hydrodynamics code ENZO v2.6 (Wise +& Abel 2011; Bryan et al. 2014), modified as described +in Appendix C, including the photon absorption probabil- +ities above, with photoionization cross-sections from Anni- +nos et al. (1997)2 , and the chemistry and cooling solver +GRACKLE 3(Smith et al. 2017). Because the code runs in 3D, +the memory cost of retaining photon packets from previous +time steps rapidly becomes prohibitive in reionization prob- +lems. For this reason, instantaneous radiative transfer is as- +sumed in the code (ISLA). New packets are generated on each +time step. This corresponds to neglecting the time derivative +in Eq. (3), or equivalently adopting an infinite speed of light. +To preserve causality, we also permanently delete any surviv- +ing photon packets that travel outside the light cone of the +source (see Appendix C3.) This necessarily results in a loss +of radiant energy and an artificially reduced photoionization +heating rate of the gas. +For a more physical contrast compared with the other two +methods, we also present results using a 1D spherically sym- +metric code, PhRay, that propagates the photon packets at a +finite velocity. The photon packets are kept between time +steps. They then have a memory of the gas they passed +through on all previous time steps. This coresponds to re- +taining the time derivative in Eq. (3). Adopting the speed of +light for the packet velocity requires very short time steps, +as the code is written to ensure photons cannot move more +than a single grid zone in one time step. (A more efficient +code could be written to allow movement through multiple +grid zones, but this would entail considerable additional com- +putational overheads.) The grid zones are adjusted to a preset +maximum optical depth per zone. In outline, the basic steps +of the code are: +(i) Choose the length of the ray. +(ii) Compute the minimum cell width to ensure the H I, +He I and He II optical depths do not exceed preset maximum +values for a cell. Set the time step to the time it takes a +photon to cross a single cell. +(iii) On the first time step, add photon packages to the first +cell nearest the source according to the source luminosity and +time step. +(iv) On subsequent time steps, move photon packages in +each cell to the next cell away from the source. Solve the +ionization equations Eqs. (1) and energy equation Eq.(5) in +each cell. Remove photon packages according to the optical +depth in the new cell, using Eqs. (9) - (11). Add new photons +to the first cell nearest the source. +(v) Repeat step (iv) until the final integration time. +As long as the advance of the fastest ionization front is +highly sub-luminal, nearly identical results are produced al- +lowing for a packet velocity smaller than the speed of light, +provided the packets still move quickly compared with the +ionization fronts. This has the advantage of allowing the code +to take longer time steps. In our tests, we use the physical +speed of light. +For both photon packet codes, the energy and ionization +2 The He I photoionization cross-section is updated to that of +Verner et al. (1996). +3 https://grackle.readthedocs.io/ +equations are solved using a first order Eulerian time integra- +tor. +2.3 Frequency integration +For the photon packet schemes, the integration of the radi- +ation field over the photoionization cross-section to compute +the ionization and heating rates is accomplished using Gaus- +sian Legendre quadrature. This is more efficient than a uni- +form grid in frequencies, and yields extremely accurate inte- +grations when the integrand is well-approximated by a poly- +nomial. We experimented with several implementations, and +use one we find ensures accuracy in the frequency integrations +typically to within a percent for the applications we present. +Specifically, the frequency range is divided into intervals be- +tween the photoelectric edges for H I (13.6 eV), He I (24.6 eV) +and He II (54.4 eV), and extending to an upper limiting value +dependent on the RT method used and the application. For +the spherically symmetric photon packet method (PhRay), 8 +energy bins were used in each frequency interval. The inte- +gration variable for the final integration to infinite frequency +is changed to 1/ν, so that the integration ranges from 0 to +1/νL,He II, where νL,He II is the ionization threshold for He II. +For the ENZO simulations, instead an upper frequency is im- +posed, provided in the test problem descriptions below. The +number of frequency bins between threshold energies is set at +5, with 10 between the He II photoelectric threshold and the +maximum energy. The energy binnings used for the photon +packet codes ensure accuracy in the frequency integrations to +within a percent. +For the direct integration method, Gaussian quadrature +offers some advantage over a mixed linear-logarithmic fre- +quency grid in allowing a reduction by a factor of three in +the number of frequencies used for an accurate solution within +the ionized region, however the ionization front moves some- +what too quickly unless a comparable number of frequen- +cies is used. We consequently use a mixed linear-logarithmic +frequency grid for the direct integration scheme, with the +number of frequencies typically 200, half placed uniformly +between the H I and He II photoelectric thresholds and the +remainder placed logarithmically to a maximum energy of +2 keV. This choice ensures temperatures and placements of +ionization fronts are converged to better than a percent at a +given spatial grid resolution. +3 TEST RESULTS +3.1 Test problems +The algorithms are tested on four problems, reionization by a +black body spectrum with a temperature of 105 K or 106 K, +and reionization under intergalactic medium conditions by +a power-law spectrum before and after hydrogen is ionized. +The 105 K black body roughly represents a high mass Pop II +or Pop III star (Bond et al. 1984), whilst the 106 K black +body (hotter than expected for stellar atmospheres), is pre- +sented as a contrasting source with a greater proportion of +ionizing photons able to fully ionize helium. The power-law +spectra represent QSOs, although they also approximate the +photon emission rates for galaxies dominated by Pop II or +Pop III stars (Meiksin 2005). The mass-fraction abundances +MNRAS 000, 1–19 (2022) + +6 +of hydrogen and helium are 0.76 and 0.24, respectively. In all +cases, the surrounding gas is static and its hydrodynamical +response is not included, since the focus is on the ionization +structure. The problems and results are discussed in greater +detail below. +3.2 Black body spectra +The luminosity function of the black body spectrum is mod- +elled as +LBB,ν = L0 × +(hν)3 +exp hν +kT − 1 eVs−1Hz−1. +(12) +The +surrounding +hydrogen +number +density +is +0.76 × +10−3cm−3. Hydrogen and helium are initially neutral in all +simulations. The physical boxsize in all the ENZO 3D instan- +taneous simulations with 2563 cells is 6.6kpc.4 The optical +depths of all species per zone are below 1. The total number +of cells in the 1D simulations is adjusted in different situa- +tions, ensuring the optical depth of neutral hydrogen in each +cell at the start of the computation is approximately equal +to 1. This ensures the positions of the ionization fronts are +converged to within a few percent. +We show that in the two black body radiation problems, +all algorithms show consistent temperature patterns and the +differences in the gaseous ionisation levels are negligible for +practical applications. +3.2.1 TBB = 105 K +The initial gas temperature is T = 100 K. The coefficient +L0 = 1.9 × 1031 eV−2 corresponds to a photon emission +rate above the hydrogen ionization threshold of +˙NH,γ += +5×1048 s−1. For the ENZO computations, the maximum energy +bin used is 200 eV to ensure convergence on the temperature. +The temperature and ionization structure at times t = +10 Myr and 30 Myr after the source turns on are shown in +Fig. 1. The temperature profiles from all the codes are in +substantial agreement, although the direct integration code +slightly anticipates the position of the knee in the tempera- +ture profile, where it starts its decline to T < 104 K, just past +the He II-front. +The knee in the temperature profile is reflected in the ion- +ization profiles, for which the H II and He II fronts from the +direct integration scheme slightly lead the results from the +photon packet codes. At distances beyond the He III-front, at +0.9 kpc (1.2 kpc) at t = 10 Myr (30 Myr), the He III fraction +from the direct integration scheme first declines gently, then +decreases precipitously near 2 kpc (3 kpc), near the H II-front +at 2.2 kpc (3.2 kpc) at t = 10 Myr (30 Myr). By contrast, +both photon packet codes (PhRay and ENZO), allow leakage of +He II-ionizing photons to larger distances sufficient to main- +tain partial He III ionization. The direct integration scheme +none the less produces He II fractions similar to the photon +packet codes at all radii. At distances beyond 5 kpc, the He II +and He III fractions from ENZO decline faster than the corre- +sponding fractions from PhRay. This is found to be a spatial +4 The parameters are adopted from the PhotonTest test problem +in ENZO v2.6. +103 +104 +105 +Temperature (K) +10 Myr +PhRay +DI +ENZO +103 +104 +105 +Temperature (K) +30 Myr +2 +4 +6 +8 +radius (kpc) +10(4 +10(3 +10(2 +10(1 +100 +fractio +10 Myr +HI +HeI +HeII +HeIII +2 +4 +6 +8 +radius (kpc) +10(4 +10(3 +10(2 +10(1 +100 +fractio +30 Myr +Figure 1. Reionization by a black body source with temperature +TBB = 105 K. Shown are results for the time-dependent pho- +ton packet code (PhRay, black lines), the direct integration scheme +(DI, cyan lines) and ENZO (blue lines). Upper panels: Temperature +profiles. The dotted (magenta) vertical line shows the H II-front +according to the PhRay calculation; the dot-dashed (green) verti- +cal line shows the He II-front. Lower panels: ionization profiles for +H I (dotted lines), He I (dashed lines), He II (dot-dashed lines) and +He III(dot-dot-dashed lines). +103 +104 +105 +Temperature (K) +10 Myr +PhRay +DI +ENZO +103 +104 +105 +Temperature (K) +30 Myr +2 +4 +6 +8 +radius (kpc) +10(4 +10(3 +10(2 +10(1 +100 +fractio +10 Myr +HI +HeI +HeII +HeIII +2 +4 +6 +8 +radius (kpc) +10(4 +10(3 +10(2 +10(1 +100 +fractio +30 Myr +Figure 2. As in Fig. 1, but for a black body source with temperature +TBB = 106 K. +resolution effect in the simulation volume: increasing the res- +olution in ENZO increases the range of agreement with the +results from PhRay. +3.2.2 T = 106 K +The initial gas conditions are identical to those used in the +105 K black body problem. The coefficient L0 = 1.9 × +1027 eV−2 corresponds to ˙NH,γ = 7×1047 s−1. To allow for the +higher frequency peak in the Planck distribution, the upper +energy bin in the ENZO computation is increased to 1000 eV +to ensure convergence on the temperature. +The temperature and ionization structure at times t = +10 Myr and 30 Myr after the source turns on are shown in +MNRAS 000, 1–19 (2022) + +Numerical methods for IGM reionization +7 +Fig. 2. Whilst the fraction of emitted photons able to fully +ionize helium is higher compared with the TBB = 105 K black +body spectrum, the results are qualitatively very similar to +those for the 105 K source. The exception is the position of +the temperature knee, where the gas temperature starts to de- +cline below 104 K. For the TBB = 106 K source, the plateau +in temperature is maintained at T ≃ (1.3 − 1.5) × 104 K +somewhat beyond the He II-front, with the front (shown by +the dot-dashed green vertical line) positioned about half way +through the temperature plateau. The temperature falls be- +low 104 K only once the He II fraction declines to below about +10 percent. +3.3 Power-law spectra +3.3.1 QSO reionization +We consider two reionization problems: (1) the reionization +of the IGM at z = 6 and (2) the reionization of the He II +component of the IGM at z = 4, both by a QSO spec- +trum modelled as a power law in frequency for values above +the frequency νL of the hydrogen photoelectric threshold, +Lν = LL(ν/νL)−αQ. Typical initial gas densities and ion- +ization states are adopted at these redshifts, as explained +below. The reionization is followed for times short compared +with the Hubble time, so that cosmological expansion is not +included: the gas is static. Inverse Compton cooling off the +CMB is also neglected, as the characteristic cooling time is +0.5 Gyr at z = 6 and 2 Gyr at z = 4. The QSO spectra are +modelled as LS,ν = 0.56 × 1031 ergs−1Hz−1(ν/νL)−0.5 and +LS,ν = 2.0 × 1031 ergs−1Hz−1(ν/νL)−1.73. Only the photon +packet codes are run for this problem since convergent re- +sults become computationally prohibitively expensive for the +direct integration code. +The cut-off energies for the ENZO simulations for the LS,ν ∼ +ν−0.5 and LS,ν ∼ ν−1.73 spectra are both set to 1000eV (see +Appendix B for details). The physical box size used in all the +simulations is 25 Mpc, with 2563 cubic cells. For the PhRay +spherically symmetric simulations, the grid cell size is chosen +to assure the maximum initial optical depths in a single cell +for H I, He I and He II do not exceed unity. +3.3.2 Reionization at z = 6 +The surrounding hydrogen density is 6.5 × 10−5 cm−3, cor- +responding to the mean IGM density at redshift z = 6, the +typical redshift when QSOs begin photoionizing the IGM. +The hydrogen and helium are assumed neutral5, with initial +gas temperature set to T = 100 K. +Outside the central 3 Mpc, the gas temperature profiles for +PhRay and ENZO substantially agree for αQ = 0.5, as shown in +Fig. 3. The temperature takes a sharp step down, by about +∆T ≃ 5000 K, at the H II-front (shown by the dotted magenta +lines). Both codes capture the temperature step as well as the +temperature decline beyond the He II ionized zone. Within +the inner 3 Mpc, however, the temperature from PhRay is +5 At z = 6, ∼ 20 percent of the volume of the IGM is expected to +be neutral (Gnedin & Madau 2022). +104 +105 +Temperature (K) +20 Myr +PhRay +ENZO +104 +105 +Temperature (K) +30 Myr +5 +10 +15 +20 +radius (Mpc) +104 +105 +Temperature (K) +50 Myr +5 +10 +15 +20 +radius (Mpc) +104 +105 +Temperature (K) +100 Myr +104 +105 +Temperature (K) +20 Myr +PhRay +ENZO +104 +105 +Temperature (K) +30 Myr +1.5 +3.0 +4.5 +6.0 +7.5 +radius (Mpc) +104 +105 +Temperature (K) +50 Myr +1.5 +3.0 +4.5 +6.0 +7.5 +radius (Mpc) +104 +105 +Temperature (K) +100 Myr +Figure 3. Temperature profiles for reionization at z = 6 by source +LS,ν ∼ ν−0.5. The bottom set of panels shows the detailed temper- +ature structure within the ionization fronts. Shown are the results +for PhRay (black solid lines) and ENZO (dashed blue lines). The dot- +ted magenta line in each panel shows the position of the H II-front. +The dot-dashed green line shows the leading edge of the He II zone. +boosted compared with that from ENZO, reaching values ex- +ceeding 105 K;6 the H II region is expanding nearly at the +speed of light out to this distance. This region is discussed in +more detail in Sec. 4 below. +The ionization fractions from the two codes similarly track +each other closely, as shown in Fig. 4, although the ionized hy- +drogen and helium regions tend to lead slightly in the ENZO +computation. In spite of the difference in gas temperature +within the central 3 Mpc, the rise in He II fractions agree +well in this region. The leading edge of the He II-ionized re- +gion (shown in Fig. 3 by the dot-dashed green line) extends +slightly beyond the H II region (shown by the dotted magenta +line). Comparison with the TBB = 106 K black body spec- +trum case in Fig. 2, for which the H II end He II fronts are +more clearly separated, shows that the ledge in high temper- +ature actually extends beyond the He II-front, into the region +6 We confirmed that this result is largely unaffected by inverse +Compton cooling off the CMB. Including inverse Compton cooling +lowers the peak temperature by only 10 percent by t = 100 Myr. +MNRAS 000, 1–19 (2022) + +8 +10−4 +10−3 +10−2 +10−1 +100 +fraction +20 M r +PhRa +ENZO +10−4 +10−3 +10−2 +10−1 +100 +fraction +30 M r +HI +HeI +HeII +HeIII +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +fraction +50 M r +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +fraction +100 M r +Figure 4. Ionization profiles for reionization at z = 6 by source +LS,ν ∼ ν−0.5. Shown are the results for PhRay (solid lines) and +ENZO (dotted lines), for the H I (black lines), He I (blue lines), He II +(green lines) and He III (yellow lines) ionization fractions. +104 +105 +Temperature (K) +20 Myr +PhRay +ENZO +104 +105 +Temperature (K) +30 Myr +5 +10 +15 +20 +radius (Mpc) +104 +105 +Temperature (K) +50 Myr +5 +10 +15 +20 +radius (Mpc) +104 +105 +Temperature (K) +100 Myr +Figure 5. Temperature profiles for reionization at z = 6 by source +LS,ν ∼ ν−1.73. Shown are the results for PhRay (black solid lines) +and ENZO (dashed blue lines). The dotted magenta line shows the +position of the H II front. +of partial He II ionization. For the power-law spectrum case +here, the temperature falls below 2 × 104 K only for a He II +fraction below 5–7 percent. At t = 100 Myr, a low level of +He II ionization persists to the edge of the simulation volume, +with ionization fraction xHeII > 5 × 10−4 and T > 350 K, +large compared with the initial temperature of 100 K. +For the αQ = 1.73 spectrum, as shown in Fig. 5, the boost +in temperature in the central 3 Mpcs for PhRay compared +with ENZO is smaller than for the αQ = 0.5 source. The gas +temperature declines abruptly at the H II front, shown by the +dotted magenta lines in Fig. 5. The leading edge of the He II +region almost exactly tracks the H II front (with positions +agreeing to better than 1 percent) at all times, with no ledge +in high temperature extending beyond as in the αQ = 0.5 +spectrum case. +The ionized regions from ENZO again slightly lead those +10−4 +10−3 +10−2 +10−1 +100 +fraction +20 M r +PhRa +ENZO +10−4 +10−3 +10−2 +10−1 +100 +fraction +30 M r +HI +HeI +HeII +HeIII +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +fraction +50 M r +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +fraction +100 M r +Figure 6. Ionization profiles for reionization at z = 6 by source +LS,ν ∼ ν−1.73, as in Fig. 4. +from PhRay, as shown in Fig. 6. The region of low He II ion- +ization extends further for the PhRay calculation than for +ENZO; the extent is limited by the higher energy photon cut- +off in ENZO. At t = 100 Myr, xHeII > 1.8 × 10−5 out to the +edge of the PhRay simulation volume of 26 Mpc radius, with +T > 108 K. Compared with the initial temperature of 100 K, +the amount of heating is small at these radii. +3.3.3 Reionization of He II at z = 4 +The surrounding hydrogen density is 2.4 × 10−5 cm−3, corre- +sponding to the mean IGM density at redshift z = 4, the typi- +cal redshift when QSOs begin photoionizing He II in the IGM. +The initial hydrogen neutral fraction is set at xHI = 2 × 10−5 +and the He I and He III helium fractions xHeI = 9 × 10−6 and +xHeIII = 0. These correspond approximately to the ionization +levels for the ultra-violet (UV) metagalactic background at +z = 4 (Haardt & Madau 2012) in a region for which He II +has not yet been ionized. The initial gas temperature is set +to T = 104 K. +For the αQ = 0.5 spectrum, as shown in Fig. 7, the gas +temperature is elevated behind the He III-front relative to the +temperature of the ambient gas. Whilst the PhRay and ENZO +ionization levels agree well within the He III region, as shown +in Fig. 8, the PhRay temperature somewhat exceeds that of +ENZO by about 4000 K. As discussed below, this is a conse- +quence of near luminal expansion of the He III-front once the +QSO turns on. Ahead of the He III-front, the temperatures are +in good agreement, although the ENZO temperature is slightly +higher than the PhRay temperature. +As for the z = 6 simulations, the ENZO ionization regions +slightly lead those from PhRay (Fig. 8), with the more ionized +H II and He II regions expanding somewhat more rapidly for +ENZO. Otherwise the ionization fractions are in good agree- +ment outside the He III region. At distances from the source +beyond the light front, the ionization fractions remain con- +stant with distance, reflecting the initial conditions. Because +there is no ambient UV photoionizing background field, the +ionization level at these distances is evolving as hydrogen and +helium gradually recombine. +The temperature is again elevated out to the He III-front +MNRAS 000, 1–19 (2022) + +Numerical methods for IGM reionization +9 +104 +105 +Temperature (K) +20 Myr +PhRay +ENZO +104 +105 +Temperature (K) +30 Myr +5 +10 +15 +20 +radius (Mpc) +104 +105 +Temperature (K) +50 Myr +5 +10 +15 +20 +radius (Mpc) +104 +105 +Temperature (K) +100 Myr +Figure 7. Temperature profiles following He II reionization at z = 4 +by a source LS,ν ∼ ν−0.5. Shown are the results for PhRay (black +solid lines) and ENZO (dashed blue lines). The dot-dashed green +line in each panel shows the position of the He III-front. +10−4 +10−3 +10−2 +10−1 +100 +fraction +20 M r +PhRa +ENZO +10−4 +10−3 +10−2 +10−1 +100 +fraction +30 M r +HI +HeI +HeII +HeIII +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +fraction +50 M r +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +fraction +100 M r +Figure 8. Ionization profiles following He II reionization at z = 4 by +a source LS,ν ∼ ν−0.5, as in Fig. 4. +for the αQ = 1.73 spectrum relative to the ambient gas tem- +perature, as shown in Fig. 9, but not by as much as for the +αQ = 0.5 spectrum. The temperatures from PhRay and ENZO +agree well, although the ENZO temperature slightly exceeds +that of PhRay beyond the He III-front. This is consistent with +a slightly faster expansion of the He III-front from ENZO com- +pared with PhRay, as shown in Fig. 10. +4 DISCUSSION +Both the direct integration and photon packet codes recover +the principal ionized zones of hydrogen and helium produced +by the black-body and power-law spectral sources. Several +discrepancies, however, are found. We discuss the differences +that are particularly pertinent to measurements of the IGM. +We focus on differences in the near zones, where hydrogen +and helium are nearly fully ionized, and the far zones, where +the hydrogen and helium are nearly neutral. +104 +105 +Temperature (K) +20 Myr +PhRay +ENZO +104 +105 +Temperature (K) +30 Myr +5 +10 +15 +20 +radius (Mpc) +104 +105 +Temperature (K) +50 Myr +5 +10 +15 +20 +radius (Mpc) +104 +105 +Temperature (K) +100 Myr +Figure 9. Temperature profiles following He II reionization at z = 4 +by a source LS,ν ∼ ν−1.73. Shown are the results for PhRay (black +solid lines) and ENZO (dashed blue lines). The dot-dashed green +line shows the position of the He III-front. +10−4 +10−3 +10−2 +10−1 +100 +fraction +20 M r +PhRa +ENZO +10−4 +10−3 +10−2 +10−1 +100 +fraction +30 M r +HI +HeI +HeII +HeIII +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +fraction +50 M r +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +fraction +100 M r +Figure 10. Ionization profiles following He II reionization at z = 4 +by a source LS,ν ∼ ν−1.73, as in Fig. 4. +4.1 Near zone +4.1.1 Black-body spectra +For the TBB = 105 K black-body spectrum, the hydrogen- +ionizing photon emission rate ˙NH,γ = 5×1048 s−1 corresponds +to an expansion rate of the H II-front, before radiative recom- +binations become important, given by balancing the emission +rate to the rate at which hydrogen atoms are ionized: +rHII = +� 3 +4π +˙NH,γt +nH +�1/3 +≃ 1.2t1/3 +Myr kpc, +(13) +where tMyr is the time since the source turned on in units +of 106 yr and a hydrogen density nH = 0.76 × 10−3 cm−3 +has been assumed7. For the TBB = 106 K black-body spec- +7 Eq. (13) is an approximation assuming all ionizing photons are +absorbed at the ionization front. In practice, sufficiently high en- +MNRAS 000, 1–19 (2022) + +10 +10−4 +10−3 +10−2 +10−1 +100 +fraction +5 Myr +10−4 +10−3 +10−2 +10−1 +100 +fraction +10 Myr +PhRay:HI +PhRay:HeI +PhRay:HeII +PhRay:HeIII +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +radius/ct +10−4 +10−3 +10−2 +10−1 +100 +fraction +15 Myr +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +radius/ct +10−4 +10−3 +10−2 +10−1 +100 +fraction +20 Myr +Figure 11. Ionization profiles shown vs the light front distance for +reionization at z = 6 by a source LS,ν ∼ ν−0.5, from solving the +full time-dependent RT equation. +trum, with the lower hydrogen-ionizing photon emission rate +˙NH,γ = 7 × 1047 s−1, the expansion rate is about half as fast, +rHII ≃ 0.6t1/3 +Myr kpc. +The growth of the H II region for the TBB = 105 K source +agrees well with the theoretical expectation, with the H II- +front (defined at the position where xHII = 0.5), occurring +within 15 percent of the prediction of Eq. (13), although +falling systematically slightly short. The agreement is poorer +for the harder TBB = 106 K spectrum, with the H II-front +lagging far behind the prediction. The discrepancies may be +attributed to the presence of helium. For TBB = 105 K, about +half the hydrogen-ionizing photons may ionize helium. After +subtracting these, the predicted position of the H II-front de- +creases by about 20 percent. For TBB = 106 K, 99 percent +of the hydrogen-ionizing photons may also ionize helium. Re- +moving these decreases the predicted radius of the H II-front +by about a factor of 5, in good agreement with the compu- +tations. The inclusion of helium thus requires accounting for +the sharing of photons that may ionize more than a single +species, which will depend on the relative abundances of the +species in general, as well as on their relative cross sections. +The ionization structures for the black-body spectra also +agree well between the codes. Nearly perfect agreement is +found for the photon packet codes PhRay and ENZO. The ion- +ization fronts from the direct integration scheme, however, +slightly lead the positions from the photon packet codes. +4.1.2 Power-law spectra +Outside the inner 3 Mpc, but still within the highly ionized +regions, the temperatures found by PhRay and ENZO for the +test problems for IGM conditions at z = 6 agree well. This is +a significant achievement of the probabilistic formulation of +the radiative transfer problem, as the initial optical depth per +cell in the ENZO computation is 124, compared with an optical +ergy photons continue un-absorbed because of their long mean free +paths, but they make up only a small fraction of all the photons. +ν/νL +10−2 +10−1 +100 +Transmission +time-dependent +ν/νL +10−2 +10−1 +100 +HI transmission +instantaneous +5Myr +8Myr +10Myr +15Myr +20Myr +100 +101 +102 +ν/νL +100 +101 +Lν +time-dependent +100 +101 +102 +ν/νL +100 +101 +Lν +instantaneous +Figure 12. Comparison of the transmitted radiation for reionization +at z = 6 by a source LS,ν ∼ ν−0.5, for the solution to the full +time-dependent RT equation (left panels) and in the instantaneous +(ISLA) limit (right panels). Upper panels: Transmission profiles +e−τν at H II-front. Lower panels: Transmitted luminosity Lν = +LS,νe−τν at H II-front. +depth of unity in PhRay. The sharpness of the H II-front per- +mits a generous optical depth criterion, making the problem +practical. By contrast, the convergence requirements that the +optical depth per zone not exceed unity, along with a higher +number of frequency bins, renders a direct integration of the +radiative transfer equation computationally impractical. +For the test problem with IGM conditions at z = 4, the +temperatures between PhRay and ENZO agree well, although +the temperature from PhRay somewhat exceeds that of ENZO +within the He III region. The ionization fronts from ENZO also +slightly lead those from PhRay. This could over-estimate the +size of the expected He III zone predicted for a given QSO +spectrum and age. Agreement improves on increasing the spa- +tial resolution for ENZO from 1283 to 2563 zones, correspond- +ing to decreasing the He II optical depth at the photoelectric +threshold from 1.9 to 0.9. +The higher temperature found by PhRay compared with +ENZO within the inner 3 Mpc for IGM conditions at z = 6, and +for IGM conditions at z = 4 out to the He III-front, especially +for the αQ = 0.5 spectrum, is a consequence of the rapid +expansion of the ionized region around the source. In the +test problem for z = 6, the H II region expands nearly at the +velocity of light, whilst in the z = 4 problem, the He III region +has near luminal expansion. This may be seen by equating +the output rate of ionizing photons to the rate at which gas +is photoionized, when recombinations may be neglected. The +criterion that the expansion of the zone becomes subluminal +is then +1 +4πr2 +i +˙Nγ +n < c, +(14) +where +˙Nγ is the production rate of ionizing photons and n +is the density of the species being ionized.8 The criterion in +8 White et al. (2003) give for the evolution of an I-front radius +RI, allowing for the finite travel time of light, +˙Nγ(t − RI/c) = +MNRAS 000, 1–19 (2022) + +Numerical methods for IGM reionization +11 +Eq. (14) then corresponds to an I-front velocity of 0.5c. The +production rate of all photons above the H I threshold for +αQ = 0.5 and 1.73 is 1.7 × 1057 s−1. For a hydrogen density +6.5 × 10−5 cm−3, the H II region expansion will become sub- +luminal only at r >∼ 2.7 Mpc, or somewhat smaller allowing +for some photons to ionize helium. This is consistent with +the agreement in the temperatures between PhRay and ENZO +at r > 2.5 Mpc, where they both give T ≃ 5.5 × 104 K for +αQ = 0.5 and T ≃ 4 × 104 K for αQ = 1.73, as shown in +Figs. 3 and 5. +Similarly, as shown in Figs. 7 and 9, enhancements in tem- +perature are found at z = 4 where the hydrogen has already +been ionised (and the helium singly ionised). (The enhance- +ment is small for the softer αQ = 1.73 spectrum.) In this case, +the boost in temperature results from the rapidly expanding +He III-fronts. From Eq. (14), taking n = nHe and considering +He II-ionizing photons, the expansion of the He III region be- +comes sub-luminal only for r > 11 Mpc for αQ = 0.5 and +r > 5 Mpc for αQ = 1.73. The temperature enhancement +is confined to the region with substantial He III ionization +(xHeIII > 0.5), shown by the magenta lines in Figs. 7 and 9. +The boost persists until the region of xHeIII > 0.5 reaches the +luminal expansion limiting radius. +The reason near-luminal expansion of an ionization region +gives rise to a boost in temperature is illustrated in Figs. 11 +and 12 for the αQ = 0.5 spectrum at z = 6. Fig. 11 shows +the ionization fractions as a function of position in units of +the light front (r = ct) from the time-dependent code PhRay, +which tracks all photon packets since they were emitted until +they are absorbed. The time for the ionization fronts to reach +∼ 2.5 Mpc and become sub-luminal is 8 Myr. The ionization +fronts then begin to slip increasingly behind the light front. +As long as the ionization fronts keep up with the light front, +the gas encountered by the photons is largely neutral. As a +consequence, the lower energy photons are rapidly absorbed +by the gas. Most of the photoionization is carried out by +the surviving most energetic photons. Once the ionization +front becomes sub-luminal, the photon packets that arrive at +the front include proportionately more lower energy photons +from the source, and the amount of energy deposited in the +gas per ionization decreases. This is shown in the left panels +of Fig. 12. The median energy at which photons are trans- +mitted is higher at times t < 8 Myr, with the peak in the +transmitted luminosity Lν shifted towards higher energies. +By t > 8 Myr, the photon luminosity profile at the H II-front +reflects the transmission through the intervening ionization +structure. +By contrast, rather than tracking photon packets since they +were emitted, ENZO recasts new rays at each time step and +computes the instantaneous radiative transfer along the rays +with a new set of photon packets launched from the source. +At t = 5 Myr, the intervening gas between the source and the +H II-front removes fewer low energy photons (above the ion- +ization threshold energy) than would have been removed from +photon packets that were moving only very slightly ahead of +the ionization front, as in the PhRay computation. This is +shown in the right panels of Fig. 12. The transmitted lumi- +(4π/3)R3 +In, where n is the density of the species being pho- +toionized. This corresponds to an expansion velocity dRI/dt = +c ˙Nγ/( ˙Nγ + 4πR2 +Inc). +nosity Lν at the H II-front peaks at a lower energy compared +with the time-dependent computation in the left panel. By +t > 15 Myr, the transmission factor and Lν have relaxed to +those for the time-dependent RT equation solution once the +H II-front has become sub-luminal. Thereafter, the temper- +atures from the time-dependent (PhRay) and instantaneous +(ENZO) computations agree. The gas heated earlier, during +the luminal expansion phase, in the time-dependent RT equa- +tion solution from PhRay at r < 2.5 Mpc, however, remains +hotter compared with the temperature computed in the in- +stantaneous (ISLA) limit by ENZO because of the long cooling +time. +The difference in temperature in the near-zone has possi- +ble implications for metagalactic UV background (UVBG) or +QSO lifetime estimates from proximity zone measurements, +as these depend on the H I or He II fraction in the vicinity of +QSOs. Models based on instantaneous photo-ionization may +underestimate the gas temperature, and so overestimate the +recombination rate and H I or He II fraction. This may result +in an under-estimate in the size of the proximity zone around +a QSO for a given UVBG level or QSO age, and so to an +under-estimate of the UVBG level or over-estimate of the +QSO age needed to agree with the proximity zone measured. +The additional near zone heating may also boost the Lyα +photon emission rate through collisional excitation of H I in +the ionization front during the luminal expansion phase. The +increase in temperature may also affect the Lyα forest power +spectrum at wavenumbers k +>∼ 0.003 s km−1, corresponding +to the sizes of the luminal expansion regions. +The discrepancy in the predictions for the near zone be- +tween the time-dependent and ISLA solutions to the radia- +tive transfer equation when ionization fronts expand near the +speed of light poses a dilemma for photon packet radiative +transfer codes. Solving the time dependent radiative trans- +fer equation requires assigning a finite velocity to the photon +packets and retaining all photon packets emitted during any +previous time step until they exit the grid. This imposes an +impractical memory demand on the computations. The cor- +rect size of the ionization regions may instead be computed +using an ISLA method by removing surviving photon pack- +ets able to reach their causal horizons, but this results in +an artificial loss of radiative energy from the source and too +low a temperature in the main ionized region. We suggest a +compromise solution in Sec. 4.3 below. +4.2 Far zone +4.2.1 Black-body spectra +The temperature profiles beyond the ionization fronts agree +closely between all three codes for both the TBB = 105 K +and 106 K black-body spectra, including the position of the +temperature knee, where the temperature begins its decline +to the ambient IGM value. The code results for the tempera- +ture begin to depart from each other well beyond the ionized +gas region once the temperature declines below ∼ 5000 K. +No direct observational consequences are expected. +The ionization structures for hydrogen and helium agree +closely well beyond the ionization fronts, with the exception +of the direct integration code result for He III, for which the +ionization fraction plummets abruptly beyond the He III-front +for both black-body spectra. Virtually all of the He II-ionizing +MNRAS 000, 1–19 (2022) + +12 +radiation is absorbed just beyond the He III-front. This ap- +pears to be a failing of the scheme. Since He III is not directly +measured, it has no direct observational consequences. +4.2.2 Power-law spectra +The temperature and ionization structure beyond the ion- +ization fronts agree well between PhRay and ENZO for the +αQ = 0.5 spectrum for IGM densities at both z = 4 and +z = 6, although the ENZO temperatures begin to decline some- +what more rapidly at large distances. +For the softer αQ = 1.73 spectrum, the He II fraction from +ENZO for the z = 6 IGM density, while first tracking the PhRay +result, suddenly declines at t >∼ 50 Myr. The He II fraction +adheres to the PhRay result to greater distances as the spa- +tial resolution is increased for ENZO: going from 1283 to 2563 +cells corresponds to decreasing the He II optical depth at the +photoelectric edge from 5.2 to 2.6. As the gas temperature +from PhRay remains well above 100 K to distances exceeding +12 Mpc at t = 50 Myr and 15 Mpc at t = 100 Myr, ENZO +would under-estimate the range around a QSO to which the +IGM was heated above the CMB temperature, and so under- +estimate the range to which the 21-cm signal would be seen in +emission against the CMB around the QSO. In practice, the +signal would be complicated by heating from galactic sources, +which may well have already warmed the IGM to tempera- +tures above the CMB (Madau & Fragos 2017; Meiksin et al. +2017). +4.3 Hybrid RT scheme +We develop a hybrid ISLA method applied to sources reion- +izing their local environment for our revised version of ENZO, +to alleviate the discrepancies in temperature and ionisation +structures between the time-dependent RT equation solution +and the ISLA solution when removing photon packets that +exceed their causal horizon. In the hybrid solution, the prop- +agation speed of the photon packets remains infinite, corre- +sponding to the instantaneous solution of the RT equation, +but the travelling distance restriction imposed by causality is +enabled only for photons in the sub-luminal region, as given +by Eq. (14).9his switch is applied only when the hydrogen +around a source is still predominantly neutral, or the helium +predominantly neutral or singly ionized. It is also only ap- +plied to the photon packets that would effect the reionization. +In other situations, the gas will have already been heated +by photoionization, with little additional heating from the +source, so that no special measures need be taken to ensure +an accurate temperature solution. In these cases, the trav- +elling distance restriction is applied to the relevant photon +packets to ensure the changing ionization fractions around +the source remain causal. +This hybrid approach captures the accumulated attenua- +tion of the radiation field in the near-luminal expansion re- +gion, as the attenuation is mainly at the ionization front. This +ensures the gas is heated to approximately the same temper- +ature it would have using time-dependent RT (ie, a finite +9 T +104 +105 +Temperature (K) +20 Myr +PhRay +ENZO cfin +ENZO cinf +ENZO hybrid +104 +105 +Temperature (K) +30 Myr +R=3Mpc +light horizon +5 +10 +15 +20 +radius (Mpc) +104 +105 +Temperature (K) +50 Myr +5 +10 +15 +20 +radius (Mpc) +104 +105 +Temperature (K) +100 Myr +Figure 13. Temperature profiles for reionization at z = 6 by source +LS,ν ∼ ν−0.5. Shown are the results for PhRay (black solid lines), +ENZO cfin, for which photon packets are continuously removed +when they reach their causal horizon (dot-dashed blue lines), ENZO +cinf, for which photon packets are removed only when absorbed +or exit the grid (dot-dot-dashed yellow lines) and ENZO hybrid +(dashed green lines), for which photons are removed when they +reach their causal horizon only if they are located in the sub- +luminal region (R > 3 Mpc, shown by the vertical dashed magenta +line). The dashed red line in each panel shows the position of the +light horizon. +photon velocity). Once the ionization front slows down to be- +coming sub-luminal, RT proceeds in a time-independent man- +ner (or quasi-time-dependent allowing for slow changes in the +gas or source properties), so that the ISLA method becomes +increasingly accurate (Sec. 3.3). At the earliest times after +the source turns on, however, before the light front reaches +the radius at which the ionization front should become sub- +luminal, the scheme may produce an ionization front that is +acausally large, extending beyond the light front. +Fig.13 and Fig.14 illustrate the temperature and ionisation +profiles for reionization at z = 6 by a source LS,ν ∼ ν−0.5 +computed using three methods: (1) the ISLA method (case +‘cinf’), (2) removing photons everywhere when they exceed +their light horizon (case ‘cfin’), and (3) the hybrid scheme. +The ionization zone is too large in the ISLA method. Remov- +ing photons everywhere when they exceed their causal radius +results in too great an energy loss in the near luminal expan- +sion region, with the resulting temperature too low in the re- +gion. For the hybrid method (ENZO hybrid), the agreement of +the temperature and ionisation structures with those of the +time-dependent RT solution from PhRay is much improved +not only in the far zone but in the near zone as well. For +the softer αQ = 1.73 spectrum, we find improved agreement +by defining the sub-luminal region according to the radius at +which the expansion speed of the ionization front declines to +0.2c instead of 0.5c. Interpolation may be used for intermedi- +ate values of αQ. These choices may be applied for each source +individually in a multiple-source simulation with a range of +source spectra, although fixing the radius according to an +ionization front speed of 0.5c may be adequate, as the tem- +perature differences between the static and time-dependent +RT solutions are smaller for softer spectra. +MNRAS 000, 1–19 (2022) + +Numerical methods for IGM reionization +13 +10−4 +10−3 +10−2 +10−1 +100 +fraction +20 My +PhRay +ENZO hyb id +R=3Mpc +light ho izon +10−4 +10−3 +10−2 +10−1 +100 +f action +30 My +HI +HeI +HeII +HeIII +5 +10 +15 +20 + adius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +f action +50 My +5 +10 +15 +20 + adius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +f action +100 My +Figure 14. Ionisation profiles for reionization at z = 6 by source +LS,ν ∼ ν−0.5. Shown are the results for PhRay (solid lines) and +ENZO hybrid (dotted lines), for which photon packets reaching +their causal horizon are removed only in the sub-luminal region +(R > 3 Mpc, shown by the vertical dashed magenta line), for the +H I (black lines), He I (blue lines), He II (green lines) and He III (yel- +low lines) ionization fractions. The dashed red line in each panel +shows the position of the light horizon. +4.4 Cosmological simulation application +We apply the three different ISLA methods (ENZO cinf: the +original ISLA, ENZO cfin: ISLA, but adopting the causal +travel distance restriction throughout the entire simulation +volume and ENZO hybrid: ISLA, but applying the causal +travel distance restriction only in the sub-luminal ionization +front expansion region) to a cosmological hydrodynamic sim- +ulation using our revised version of ENZO to study the tem- +perature and ionization structure around a QSO. Assuming +for simplicity that no metagalactic ultraviolet background +(UVB) is present, we turn on a beamed QSO-like radiation +source with an αQ = 0.5 power-law spectrum at the centre +of the simulation box at z = 7. The total hydrogen-ionizing +photon emission rate is +˙Nγ = 1.5 × 1057 s−1 and the open- +ing angle of the source is 10◦. The cosmological parameters +assumed are Ωm = 0.27, Ωb = 0.046, ΩΛ = 0.73, h = 0.70, +σ8 = 0.811 and ns = 0.961, with a primordial helium mass +fraction Y = 0.24, consistent with PLANCK measurements +(Planck Collaboration et al. 2018). The code is run in uni- +grid mode with a comoving box size of 120h−1 Mpc and 2563 +cubic cells. (The spatial resolution in proper units at z = 7 is +comparable to the spatial resolution in the test problems in +Sec.3.3.) The Cold Dark Matter initial conditions at z = 50 +are generated by the MUSIC code (Hahn & Abel 2011); the +code also sets the baryon properties, with a low tempera- +ture given by adiabatic expansion following the recombina- +tion epoch. The chemical and cooling processes are computed +by GRACKLE 10 (Smith et al. 2017). The RT equations are +solved in a sub-cycle process in ENZO, so that the cosmological +simulations are fully coupled radiation hydrodynamics sim- +ulations, rather than being performed as a post-processing +step, like in most cosmological RT simulations (eg Sokasian +10 https://grackle.readthedocs.io/ +−10.0 −7.5 +−5.0 +−2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +z (Mpc) +−10.0 +−7.5 +−5.0 +−2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +x (Mpc) +100 +101 +102 +103 +104 +Temperature +(K) +Figure 15. Temperature slice plot at z ∼ 6.93 for a beamed QSO- +like radiation source (LS,ν ∼ ν−0.5) after 30 Myr. The ENZO hy- +brid RT method is used, for which the causal travel distance re- +striction for photon packets is applied only in the sub-luminal ion- +ization front expansion region (see text). +et al. 2002; Bolton et al. 2004; McQuinn et al. 2009; Ciardi +et al. 2012; Compostella et al. 2013; Kakiichi et al. 2017; Eide +et al. 2018, 2020). +The temperature around the source is shown in Fig. 15, +with the temperature declining with distance from the source, +and modulated by the large-scale structure of the gas. The +sharp ends to the temperature cone correspond to the light +fronts. In Fig. 16, the temperature along a line of sight +through the beam centre is shown for the three methods. +The ISLA method (ENZO cinf) produces high excess tem- +peratures away from the source. The ENZO cfin and ENZO +hybrid methods agree in temperature on large scales, but the +temperature from the ENZO cfin method is too low in the lu- +minal ionization expansion region, within the inner 2 Mpc +from the source, by up to 5×104 K. The different predictions +for the ionization fractions are shown in Fig. 17. The ISLA +method again gives excess ionization on large scales. The ENZO +cfin and ENZO hybrid methods agree, except within the in- +ner 2 Mpc. The near zone Lyα forest is shown in Fig. 18. +The lower temperatures using ENZO cfin result in a larger +radiative recombination rate and so a greater amount of ab- +sorption near the source. Whilst the result from ENZO hybrid +more faithfully recovers the expected gas temperature within +the luminal region (Sec. 4.3), the actual amount of absorp- +tion would be still somewhat smaller in this region because +of the temperature boost allowing for the time-dependent RT +in the luminal zone. For precision work, a time-dependent so- +lution to the RT equation would be required for such a hard +spectrum. The discrepancy is smaller for a softer spectrum +(Sec. 3.3.2). +5 CONCLUSIONS +We compare three radiative transfer codes applied to pho- +toionization problems for sources with spectra typical of stars +(black body) and QSOs (power law). One code integrates the +MNRAS 000, 1–19 (2022) + +14 +103 +104 +105 +Temperature (K) +20 Myr +ENZO cinf +103 +104 +105 +Tempera ure (K) +30 Myr +ENZO hybrid +25 +50 +75 +100 +comoving radius (h−1Mpc) +103 +104 +105 +Tempera ure (K) +50 Myr +ENZO cfin +25 +50 +75 +100 +comoving radius (h−1Mpc) +103 +104 +105 +Tempera ure (K) +100 Myr +Figure 16. Temperature profiles for a QSO-like radiation source +(LS,ν ∼ ν−0.5) at the indicated times after the source turns on at +z = 7: 20 Myr, 30 Myr, 50 Myr and 100 Myr. The radiation source +is located at the centre of the line of sight. Results are shown for +the ENZO cinf method (solid black lines), the ENZO cfin method +(dot-dashed red lines) and for ENZO hybrid (dashed blue lines). +Temperature profiles in the central luminal ionization front ex- +pansion region are underestimated by ENZO cfin. The tempera- +ture discrepancy in the region surrounding the radiation source is +as high as 5 × 104 K. +10−4 +10−3 +10−2 +10−1 +100 +fraction +20 Myr +cinf +cfin +hybrid +10 4 +10 3 +10 2 +10 1 +100 +fraction +30 Myr +HI +HeI +HeII +HeIII +25 +50 +75 +100 +comoving radius (h 1Mpc) +10 4 +10 3 +10 2 +10 1 +100 +fraction +50 Myr +25 +50 +75 +100 +comoving radius (h 1Mpc) +10 4 +10 3 +10 2 +10 1 +100 +fraction +100 Myr +Figure 17. Ionisation profiles for the QSO-like radiation source +(LS,ν ∼ ν−0.5) at times 20 Myr, 30 Myr, 50 Myr and 100 Myr. The +radiation source is located at the centre of the line of sight. Re- +sults are for the ENZO cinf method (solid lines), ENZO cfin method +(dashed lines) and for ENZO hybrid (dotted lines). +time-independent radiative transfer equation directly, and is +applied only to the black-body spectra problems. The other +two use photon packets to solve for the radiative transfer, +one assuming instantaneous photoionization (with the dis- +tance photon packets travel limited by the speed of light) +and the other retaining fully the time-dependent term in the +radiative transfer equation. Our main findings are: +1. Photon packet codes are far more efficient at solving +the radiative transfer problem for photoionization compared +with direct integration. Fewer photon frequencies and coarser +0 +1000 +2000 +3000 +4000 +velocity (s−1km) +0.0 +0.2 +0.4 +0.6 +0.8 +Normalised flu +50 Myr +hybrid +cfin +Figure 18. Normalised Lyα flux spectra emitted by a QSO-like radi- +ation source 50 Myr after the source turns on at z = 7, allowing for +intervening absorption along the line of sight. Results are shown +for ENZO hybrid (solid black line) and for ENZO cfin (dashed blue +line). (The Lyα absorption spectra are computed following Leong +et al. (2019).) +spatial gridding are tolerated by the photon packet codes, +with optical depths at the threshold energy able to exceed +unity with accurate solutions. Another shortcoming of the +direct integration code is that it may fail to propagate low +levels of doubly ionized helium beyond the He III-front as far +as do the photon packet codes. +2. All methods agree well on the growth of the nearly fully +ionized regions, although the ionization fronts from the di- +rect integration scheme tend slightly to lead those from the +photon packet codes. The successful solution of the ionized re- +gions is a significant achievement of the photon packet codes +particularly for hydrogen ionization, as the spatial grid used +for the instantaneous photoionization version corresponds to +a hydrogen optical depth per grid zone exceeding 100 at the +photoelectric threshold. We recommend, however, that for +ionizing singly ionized helium, the optical depth at the singly +ionized helium threshold should be close to unity or smaller. +3. Including the time-dependent differential operator in the +radiative transfer equation is essential when ionization fronts +expand near the speed of light. Solutions to the radiative +transfer equation in the infinite-speed-of-light approximation +(corresponding to solving the time-independent RT equation) +may substantially under-estimate the temperature in these +regions. The under-estimate increases with the hardness of +the spectrum, with the temperature discrepancy exceeding +5 × 104 K for gas that was initially neutral, as may arise for +reionization at high redshifts by QSOs with hard spectra. A +scheme that solves the time-dependent RT equation is thus +required to obtain an accurate solution in the near zones of +QSOs that photoionize the IGM. +4. The boost in temperature due to time-dependent RT is +larger when both hydrogen and helium are initially predomi- +nantly neutral compared with the case when the hydrogen is +predominantly ionized and the helium singly ionized, as may +arise when the gas is initially ionized by a metagalactic UV +background radiation field dominated by galactic sources. +5. Outside the luminal expansion region, the gas temper- +MNRAS 000, 1–19 (2022) + +Numerical methods for IGM reionization +15 +ature and ionization structure agree well between the time- +dependent and infinite-speed-of-light photon packet codes, al- +though some differences arise at large distances where the gas +is predominantly neutral. These differences appear to result +from differences in spatial resolution, rather than from the +assumption of an infinite speed of light. +6. A photon packet code recovers the correct solutions to +the time-dependent RT equation for an ionization front to +good approximation using a hybrid scheme. In this scheme, +the RT equation is solved in the infinite-speed-of-light ap- +proximation only out to the radius at which the velocity of +the ionization front declines to approximately half the speed +of light. Photon packets outside this radius are removed if +they travel to distances beyond the light front of the source. +7. Photon energies well above the photoionization thresh- +olds must be included to capture the warming of the largely +neutral gas well outside the ionization regions for power-law +spectra. The required maximum photon energy increases for +softer spectra. +ACKNOWLEDGMENTS +The authors thank B. Smith and J. Wise for helpful con- +versations, and the referee for numerous suggestions to im- +prove the manuscript, including the suggestion to make di- +rect comparisons between our results and the published liter- +ature. KHL acknowledges financial support from the School +of Physics and Astronomy, University of Edinburgh. KHL +thanks the Computational Astrophysics Lab at National Tai- +wan University for support. KHT thanks the Robert Cormack +Bequest fund for a Summer Vacation Research Scholarship. +Computations described in this work were performed using +the ENZO code developed by the Laboratory for Computa- +tional Astrophysics at the University of California in San +Diego (http://lca.ucsd.edu). +DATA AVAILABILITY +No new observational data were generated or analysed in sup- +port of this research. +REFERENCES +Abel T., Haehnelt M. G., 1999, ApJ, 520, L13 +Abel T., Wandelt B. D., 2002, MNRAS, 330, L53 +Abel T., Norman M. L., Madau P., 1999, ApJ, 523, 66 +Anninos P., Zhang Y., Abel T., Norman M. L., 1997, New Astron., +2, 209 +Baek S., Semelin B., Di Matteo P., Revaz Y., Combes F., 2010, +A&A, 523, A4 +Baur J., Palanque-Delabrouille N., Y`eche C., Magneville C., Viel +M., 2016, J. Cosmology Astropart. Phys., 2016, 012 +Bolton J., Meiksin A., White M., 2004, MNRAS, 348, L43 +Bolton J. S., Becker G. D., Raskutti S., Wyithe J. S. B., Haehnelt +M. G., Sargent W. L. 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A., Schneider D. P., +York D. G., Anderson S. F., 2015, ApJ, 806, 142 +APPENDIX A: COMPARISONS WITH TEST PROBLEMS +IN THE LITERATURE +Solutions of the time-dependent radiative transfer equation +for power-law spectra using PhRay are compared with pub- +lished test problems of the photoionization of hydrogen and +helium using ISLA methods, as provided by Abel & Haehnelt +(1999), Davies et al. (2016), Graziani et al. (2018) and Chen +& Gnedin (2021). The thermodynamics are governed by pho- +toelectric heating and radiative cooling, including inverse +Compton cooling off the Cosmic Microwave Background at +the indicated redshift. Secondary electron ionizations and as- +sociated energy losses are included as indicated, using the +fits from Ricotti et al. (2002) to the Monte Carlo computa- +tions of Shull & van Steenberg (1985). Parameters for the +test problems are provided in Table A1, showing the net hy- +drogen ionizing photon production rate +˙Nγ, the power-law +exponent αQ for the QSO spectrum, the upper photon en- +ergy cutoff hνmax for the spectrum, the hydrogen density nH +of the surrounding gas and the redshift, which controls the +inverse Compton cooling rate. +The top left hand panel of Fig. A1 compares the solutions +from PhRay without and with secondary ionizations to those +provided by Davies et al. (2016) using a spherically symmet- +ric 1D ISLA method for a QSO spectrum with αQ = 1.5. +The medium surrounding the source is assumed static in this +problem. The ISLA solutions from Davies et al. (2016) (blue +dashed and cyan dot-dashed lines) extend some distance be- +yond the solution of the time-dependent RT equation using +the correct speed of light (thick and thin black solid lines) +given by PhRay, and even beyond the light front (shown as +the vertical magenta dotted line). The central, main ionized +2 +4 +6 +8 +10 +radius (Mpc) +103 +104 +105 +Temperature (K) +10 Myr +˙Nγ = 1057 s−1 +α = 1. 5 +PhRay: no sec +D16: no sec +PhRay: sec +D16: sec +2 +4 +6 +8 +10 +radius (Mpc) +103 +104 +105 +Temperature (K) +100 Myr +˙Nγ = 1057 s−1 +α = 1. 5 +PhRay: no sec +CG21: no sec +2 +4 +6 +8 +10 +radius (Mpc) +103 +104 +105 +Temperature (K) +10 Myr +˙Nγ = 3 × 1056 s−1 +α = 1. 8 +PhRay: no sec +AH99: no sec +2 +4 +6 +8 +10 +radius (Mpc) +103 +104 +105 +Temperature (K) +60 Myr +˙Nγ = 1. 4 × 1056 s−1 +α = 1. 5 +PhRay: no sec +G18: no sec +PhRay: sec +G18: sec +Figure A1. Comparison between solutions to test problems pro- +vided by PhRay using a finite speed of light and published results +using ISLA methods. The vertical green dashed lines indicate the +theoretical position of the hydrogen ionisation front when its ad- +vance slows to speed c/2. The vertical black dotted lines show the +maximum possible radius of the hydrogen ionization front. The +vertical magenta dotted lines show the position of the light front. +In the top left and bottom right panels, solutions both without (‘no +sec’) and with (‘sec’) secondary electron ionizations are shown; for +the other panels, the published results did not include secondary +electron ionizations. +region from Davies et al. (2016) reaches the maximum possi- +ble radius of the H II-front, given by Eq. (13)11 (vertical black +dotted line), which is nearly coincident with the light front +(vertical magenta dotted line). Keeping up with the maxi- +mum radius is expected for an ISLA scheme, which allows +photons to travel until absorbed, but for the ionization front +to have reached this distance, it had to travel superluminally +at earlier times. By contrast, at 10 Myr the dominant ionized +region from PhRay, where T > 104 K, extends just beyond +the distance to which the H II-front travels at a speed exceed- +ing c/2, as given by Eq. (14) (shown by the vertical green +dotted line). In the central, main ionized region, the tem- +peratures agree well between the two calculations, although +the temperatures from PhRay are somewhat higher by about +10%. Significant cooling is provided by secondary electron +ionization losses both in the central ionized region and in the +extended region where the gas temperature is below 104 K. +The top right hand panel compares the solutions from +PhRay and Chen & Gnedin (2021), who use an algorithm +very similar to that of Davies et al. (2016) for solving the +static 1D RT problem, although modified to allow for a vari- +able timestep within regions of very different ionization lev- +els. No secondary electron ionizations are allowed for, and +the surrounding medium is assumed static in this problem. +The ISLA solution of Chen & Gnedin (2021) has an H II-front +that extends nearly to its maximum possible radius (vertical +black dotted line), and is somewhat beyond that obtained +by PhRay. The difference reflects an earlier superluminal ex- +11 Here and for the other test problems, the full value for +˙Nγ is +used. Since helium also absorbs photons above the helium ioniza- +tion thresholds, the maximum radius will be somewhat smaller. +MNRAS 000, 1–19 (2022) + +Numerical methods for IGM reionization +17 +pansion phase of the H II region in the ISLA computation. In +the inner main ionized region, the temperature from PhRay +mildly exceeds that obtained by Chen & Gnedin (2021) by +about 15%. +In the lower left panel, the result for a steeper spectrum +(αQ = 1.8) from PhRay is compared with the ISLA solution of +Abel & Haehnelt (1999). Adiabatic cooling is included in this +problem, although it negligibly affects the temperature over +the brief interval of 10 Myr of the computation. Secondary +electron ionizations are not accounted for in the problem. The +temperatures agree well in the central region, with a slightly +higher temperature obtained by Abel & Haehnelt (1999). The +ISLA solution also has a slightly advanced H II-front com- +pared with the time-dependent RT solution from PhRay, more +nearly reaching its maximum possible radius (vertical black +dotted line). The reason the central temperature from the +ISLA solution slightly exceeds that of the time-dependent +RT solution is unclear; the grid and frequency resolution are +not provided by Abel & Haehnelt (1999) and there is no de- +scription of convergence tests on either. +Lastly, in the lower right panel we compare the solution +of PhRay with the ISLA solution of Graziani et al. (2018). +Computations both without and with secondary electron ion- +izations were performed. The solutions of Graziani et al. +(2018) are anomalous in that the ionization fronts advance +too slowly. From Eq. (13), the H II-front should be located +at rHII ≃ 2.8 Mpc (vertical black dotted line), in good agree- +ment with the result from PhRay, whilst Graziani et al. (2018) +find the front to be located at ∼ 2.0 Mpc. Even allowing for +all photons above the He I threshold to be absorbed by he- +lium atoms, a production rate of purely hydrogen-ionizing +photons of 8.0 × 1055 s−1 would remain, giving an H II-front +position of 2.3 Mpc. As the radiative recombination time is +longer than 109 yr, this position should have been reached. +Another anomaly is their temperature allowing for secondary +electron ionization energy losses, which unexpectedly exceeds +the temperature without secondary electron ionization losses, +contrary to the result from PhRay. The discrepancies may be +consequences of the large optical depths per grid zone in the +computation of Graziani et al. (2018). At the photoelectric +edges, the optical depths before the gas is photoionized are +∼ 70 for H I and ∼ 1.4 for He II. The high H I optical depth +may result in too little penetration of ionizing photon packets +into the still neutral gas. By comparison, for another simula- +tion in Graziani et al. (2018) of a QSO embedded in a halo +with higher spatial resolution, the H I and He II optical depths +at the average IGM gas density are ∼ 16 and 0.3, respectively, +and the size of the H II region found is in good agreement with +the analytic estimate. +We also ran PhRay on a test problem with a 105 K black +body spectrum source emitting at a hydrogen-ionizing photon +rate 1051 s−1 into a static medium with a cosmic abundance +of hydrogen and helium, hydrogen density nH = 0.1 cm−3 and +initial temperature 100 K, to compare with Test 1, without +metals but with secondary electron ionizations, of Graziani +et al. (2013). After 105 yr, the temperature 50 pc from the +source is 4.3 × 104 K, declining gradually to 4.0 × 104 K at +100 pc, 3.2 × 104 K at 200 pc and 2.3 × 104 K at 500 pc. +The temperatures are comparable to, but slightly in excess +by about 0.1 dex of, the temperatures from the CLOUDY ioniza- +tion code reported by Graziani et al. (2013). From Eq. (14), +the ionization front for this problem will expand at a speed +104 +105 +Temperature (K) +100 Myr +10−4 +10−3 +10−2 +10−1 +100 +HII fraction +128 +256 +512 +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +HeI fraction +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +HeIII fraction +Figure B1. Profiles at 100 Myr for reionization at z = 6 by a source +LS,ν ∼ ν−0.5. The upper left panel shows that the relative dif- +ference in temperature between (25 Mpc, 2563) (blue dashed line) +and (25 Mpc, 5123) (black dotted line) simulations is within 10% +in the fully ionized region. +exceeding c/2 until it reaches 5 kpc, so the slightly higher +temperatures are expected since CLOUDY is not designed to +track the relaxation of temperatures following the heating by +near luminal H II-front expansion to their steady-state value. +APPENDIX B: CONVERGENCE TESTS +We show convergence tests for QSO reionization simulations +which are performed by ENZO v2.6. For all the convergence +tests, we adopt identical parameters relating to the ray- +tracing method of ENZO. In particular, the minimum ray an- +gular resolution parameter is Φc = 15.1 and the HEALPix +Level is 6 (see the definitions in Wise & Abel 2011). The +spacial resolution is the only code parameter varied for the +convergence tests. We also use an identical set of energy inter- +vals and energy bins for simulations with various power-law +indices. For all the power-law spectra, the energy interval +ranges from 13.6−1000.0 eV and the selected energy bins are +[14.12, 16.13, 19.10, 22.06, 24.08, 26.00, 31.48, 39.50, 47.52, +53.00, 66.74, 118.20, 205.98, 322.29, 456.81, 597.59, 732.11, +848.42, 936.20, 987.66] eV. +The convergence test results are shown in Figs. B1 - B4, +for the power-law test problems for IGM mean densities at +z = 6 and 4, and for spectral indices αQ = 0.5 and 1.73. +Convergence is generally reached in the inner ionised regions +for the 1283 simulations, but, particularly for the softer spec- +trum, convergence in temperature and the ionization struc- +ture is improved on going to 2563. The latter corresponds to +an initial optical depth per cell at the hydrogen photoelectric +threshold of 124 at z = 6 and an initial optical depth per cell +at the singly ionised helium photoelectric threshold of 0.9 at +z = 4. +APPENDIX C: REVISIONS TO ENZO +We describe revisions to ENZO v2.6 to implement photoion- +ization by a central source. +MNRAS 000, 1–19 (2022) + +18 +104 +105 +Temperature (K) +100 Myr +10−4 +10−3 +10−2 +10−1 +100 +HII fraction +128 +256 +512 +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +HeI fraction +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +HeIII fraction +Figure B2. Profiles at 100 Myr for reionization at z = 6 by a source +LS,ν ∼ ν−1.73. The upper left panel shows that the relative differ- +ence of temperature between (25 Mpc, 2563) and (25 Mpc, 5123) +simulations is within 10% in the fully ionized region. All the pan- +els show that increasing the resolution of the simulations slightly +advances the positions of the ionisation fronts. +104 +105 +Temperature (K) +50 Myr +10−4 +10−3 +10−2 +10−1 +100 +HII fraction +128 +256 +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +HeI fraction +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +HeIII fraction +Figure B3. Profiles at 50 Myr for reionization at z = 4 by a source +LS,ν ∼ ν−0.5. All the quantities are converged in these (25 Mpc, +1283) and (25 Mpc, 2563) simulations. +We check the consistency of the source codes, especially the +consistency of the codes relevant to the ray-tracing module. +We find bugs in the implementation in ENZO v2.6 significantly +affect the accuracy of the results. These are demonstrated in +Appendix C1 by simulating a classical ray-tracing problem, +the formation of a Str¨omgren sphere (Iliev et al. 2006). +We impose new methods and restrictions on the ray-tracing +module to make ENZO suitable for both static and cosmo- +logical hydrodynamical simulations with high-luminosity ra- +diation sources. The modifications are: a) Probabilistic Ab- +sorption Method (Eqs. [9]–[11]); b) He III Ionisation Adaptive +Time Step Scheme (Appendix C2); c) Restriction on Photon +Package Travel Distance (Appendix C3). +104 +105 +Temperature (K) +50 Myr +10−4 +10−3 +10−2 +10−1 +100 +HII fraction +128 +256 +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +HeI fraction +5 +10 +15 +20 +radius (Mpc) +10−4 +10−3 +10−2 +10−1 +100 +HeIII fraction +Figure B4. Profiles at 50 Myr for reionization at z = 4 by a source +LS,ν ∼ ν−1.73. All the quantities are converged in these (25 Mpc, +1283) and (25 Mpc, 2563) simulations. +C1 Test problem: Str¨omgren sphere +A Str¨omgren sphere is the final stage of an isotropically ex- +panding ionization region with a central source in a uniform +medium once ionizations are balanced by recombinations. As +a test problem, the Str¨omgren sphere simulation has a few +key benefits: a) the solution is analytical, hence it is easy to +check the accuracy of the results; b) the solution is isotropic; +as a result, any artificial inhomogeneity caused by the algo- +rithm is visible. The analytic solution for the radius of the +ionisation front is: +R(t) = RS +� +1 − exp +� +− t +trec +�� 1 +3 +, +(C1) +where RS = (3 ˙Nγ/4πnHneα)1/3 is the final radius of the +ionised region (the Str¨omgren radius), ˙Nγ is the photon emis- +sion rate, nH is the hydrogen number density, ne is the elec- +tron number density, α is the radiative recombination rate +within the ionised region and trec = 1/neα is the recombina- +tion time (assumed constant). +We adopt a similar parameter set to that used in Iliev +et al. (2006) to compare with the results therein. Specifi- +cally, the monochromatic source emits photons at the rate +˙Nγ = 5 × 1048 s−1, the simulation box size is 6.6 kpc and +the total number of cells is 1283, the minimum ray angu- +lar resolution parameter is Φc = 5.1 (see the definition in +Wise & Abel 2011), the number density of hydrogen atoms +is nH = 10−3 cm−3 and the recombination rate is α = +2.59 × 10−13 cm3 s−1 at T = 104 K, leading to RS = 5.4 kpc +and trec = 122 Myr. We note that in Iliev et al. (2006), +the temperature is fixed at T = 104 K; however, ENZO does +not support this setting due to its formulation of the inter- +nal chemistry and energy solvers. To make the simulation +close to the analytical problem, we adjust the energy of the +monochromatic photon to 23.26 eV and set the adiabatic in- +dex to γ = 1.667 to ensure that the gas temperature in the +ionized region is close to T = 104 K. This approximation and +these parameters keep the maximum temperature deviation +to within 20% in all Str¨omgren sphere simulations. This en- +MNRAS 000, 1–19 (2022) + +Numerical methods for IGM reionization +19 +2 +3 +4 +5 +rIF (kpc) +fixed version +original Enzo +analytical +0 +50 +100 +150 +200 +250 +Time (Myr) +0.92 +0.94 +0.96 +0.98 +rIF/ranyl +fixed version +original Enzo +Figure C1. In the upper panel, the black solid line shows the an- +alytic solution for the time development of the ionization radius. +The red circles show the ionization radii computed by ENZO with +our revisions, and the blue squares are the results from ENZO v2.6. +The lower panel shows the ratio of the computed and analytic so- +lutions for the ionization radius. The modifications significantly +improve agreement with the analytic solution. +sures the Str¨omgren radius will be matched to better than +4%. +Figure C1 shows the evolution of the ionization radius com- +puted by our revised version of ENZO and by ENZO v2.6. The +result from our version very closely matches the analytical +result, validating our revisions to ENZO v2.6. +C2 He III Ionisation Adaptive Time Step Scheme +We include the maximum changing rate of the He III frac- +tion as an additional condition on the ray-tracing adaptive +time step scheme in ENZO. The original ray-tracing adaptive +time step in ENZO is based on the maximum changing rate +of the H II fraction (see Wise & Abel 2011, for more details). +However, during the helium reionization epoch, the hydrogen +atoms in the IGM have already been fully ionised. As a con- +sequence, hydrogen is optically thin to the ionising photons +and the H II fraction changes very slowly. By contrast, the +ionisation fraction of He II is rapidly changing in this period, +as the gas is initially optically thick to He II ionising photons. +Therefore, an additional restriction based on the maximum +changing rate of He III is implemented in our simulations. +We also extend the applicability of the ray-tracing adap- +tive time step scheme: the original algorithm of the time step +calculator does not consider the influence caused by the ex- +pansion of the Universe. This feature leads to overestimating +the ray-tracing time steps in cosmological simulations, bring- +ing artificial effects into cosmological simulations. +C3 Restriction on Photon Package Travel Distance +The comoving light travel distance from the radiation sources +is used to avoid photon packages from moving beyond the +particle horizon of the photons in our implementation. Sim- +ilarly to other ray-tracing algorithms in cosmological simu- +lation codes (Abel & Wandelt 2002; Iliev et al. 2006), the +ray-tracing algorithm in ENZO assumes the propagation speed +of the photon package is infinite at every radiation transfer +(RT) time step. This approximation is used for the following +reasons: +a) It is computationally cheaper. In a simulation with a +radiation source, where the propagation speed is finite, the +total number of photon packages, which are stored in sys- +tem memory (such as RAM) between two RT time steps, +is proportional to the volume of the ionized region. Also, +in such an algorithm, various information about a photon +package, such as its position, direction, luminosity and birth +time, are required to trace the photon package. Hence, assum- +ing a finite propagation speed significantly increases the cost +to system RAMs and makes the simulation computationally +prohibitive. On the other hand, in a simulation with an infi- +nite speed of light, all the photon packages are generated and +deleted at every RT time step. Thus, the photon packages do +not occupy any system RAM in between two RT time steps. +As a consequence, the infinite propagation speed approxima- +tion is more practical from a technical point of view. This +instantaneous RT approximation amounts to neglecting the +time differential operator on the left hand side of Eq. (3). +b) The expansion speed of the ionisation bubbles is gener- +ally much slower than the speed of light. Also, most of the +ionising photons associated with a ray are absorbed in the +ionisation front. Therefore, when the particle horizon is dis- +tant from the ionisation front, the influence caused by the +infinite propagation speed approximation is marginal. +Although assuming the infinite propagation speed is a safe +approximation in most simulated situations, the particle hori- +zon needs to be imposed as the maximum travel distance +of rays in many situations. These include: the radius of the +ionised bubble could be comparable to the particle horizon +when the source just begins to shine; in a large-scale simula- +tion, the box size of the simulation could be larger than the +particle horizon during the simulation period. In these situ- +ations, forbidding photon packages from transferring across +their corresponding particle horizons also reduces the com- +putation time. In our revised version of ENZO, we delete a +photon packet if it reaches its particle horizon. +MNRAS 000, 1–19 (2022) + diff --git a/m9AyT4oBgHgl3EQflPjP/content/tmp_files/load_file.txt b/m9AyT4oBgHgl3EQflPjP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f788d060c35a4bcc2122f16f81b1ae41922c20a --- /dev/null +++ b/m9AyT4oBgHgl3EQflPjP/content/tmp_files/load_file.txt @@ -0,0 +1,1407 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf,len=1406 +page_content='MNRAS 000, 1–19 (2022) Preprint 3 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 A comparison of numerical methods for computing the reionization of intergalacitc hydrogen and helium by a central radiating source Ka-Hou Leong1⋆, Avery Meiksin1, Althea Lai1, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' To2 1SUPA†, 1Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, UK 2Department of Physics, University of Tokyo, 7 Chome-3-1 Hongo, Bunkyo City, Tokyo 113-8654, Japan Accepted .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Received ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' in original form ABSTRACT We compare numerical methods for solving the radiative transfer equation in the context of the photoionization of intergalactic gaseous hydrogen and helium by a central radiating source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Direct integration of the radiative transfer equation and solutions using photon packets are examined, both for solutions to the time-dependent radiative transfer equation and in the infinite-speed-of-light approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The photon packet schemes are found to be more generally computationally efficient than a direct integration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Whilst all codes accurately describe the growth rate of hydrogen and helium ionization zones, it is shown that a fully time-dependent method is required to capture the gas temperature and ionization structure in the near zone of a source when an ionization front expands at a speed close to the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Applied to Quasi-Stellar Objects in the Epoch of Reionization (EoR), temperature differences as high as 5×104 K result in the near-zone for solutions of the time-dependent radiative transfer equation compared with solutions in the infinite-speed-of-light approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Smaller temperature differences are found following the nearly full photoionization of helium in gas in which the hydrogen was already ionized and the helium was singly ionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Variations found in the temperature and ionization structure far from the source, where the gas is predominantly neutral, may affect some predictions for 21-cm EoR experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Key words: radiative transfer – quasars: absorption lines – quasars: general – dark ages, reionization, first stars – cosmology: large-scale structure of Universe 1 INTRODUCTION Numerical simulations are now a staple method for providing detailed descriptions of the complex processes in virtually all areas of astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The range in physical processes treated has increased from gravity and gas dynamics to include mag- netic fields, a host of atomic and molecular reaction networks and radiative transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Including radiative transport is in particular numerically challenging when the gas is not everywhere optically thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In this case, the transport of radiation is not diffusive so that devising efficient methods to solve the full set of radiative transfer (RT) equations must be addressed, particularly when the mean free path of the radiation exceeds other important length scales in an application that must be spatially resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As radiative transport is a long-range effect in these applica- tions, its introduction seriously hampers computations both because of the required increase in memory to represent the radiation field and, potentially, because of a severely curtailed time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Applications to galactic and cosmological structure forma- tion for which radiative transport is essential include star ⋆ E-mail: KH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='Leong@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='uk (KHL) † Scottish Universities Physics Alliance formation and the impact of stars and Quasi-Stellar Objects (QSOs) on intergalactic gas, including the reionization of the intergalactic medium (IGM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Various approximation meth- ods have been introduced into numerical simulations to solve the radiative transfer equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Applications of the meth- ods in numerical simulations are typically done in a post- processing stage, ie, on top of previously computed solutions to the gas dynamical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' A comparison of many of these schemes to applications to static gas problems is pre- sented in Iliev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' These tests are confined to the photoionization of hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Whilst the results from the dif- ferent schemes are generally in good agreement on the place- ment of the ionization front, with discrepancies limited to 5– 10%, a much broader range of differences were obtained for other quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ionization fractions showed differences up to a factor of a few to several, and temperatures differed by up to a few tens of percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Some of these differences do not necessarily reflect differences in the algorithms, but may be attributed in part to the different frequency ranges cov- ered for the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Some of the differences, however, appear intrinsic to the schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Fully coupled radiative hydrodynamical codes have also been formulated, but often for restricted applications, either through a suppression of one or more spatial dimensions or through approximations made to the radiative transfer equa- © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='00450v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='IM] 1 Jan 2023 2 tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' A comparison of some of these methods is presented in Iliev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Whilst the reionization of intergalactic hydrogen has been addressed in a wide variety of simulations (see Gnedin & Madau 2022, for a recent review), the reionization of he- lium has received much less attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Yet its solution is vital for understanding the temperature evolution and small scale structure of the IGM, which has been used to place con- straints on various dark matter candidates (eg Baur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Garzilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Irˇsiˇc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Leong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2019) and for placing limits on the neutrino mass (Viel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The temperature evolution of the IGM in turn places constraints on the nature, abundance and evolution of the sources that reionized the hydrogen and helium in the IGM (eg Theuns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Tittley & Meiksin 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Upton Sanderbeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Walther et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Keating et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Unlike for hydrogen reioniza- tion, the two ionization states of helium result in a broadened singly-ionized (He II) zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As intergalactic helium is detected through the He II Lyα absorption signature, it is important to obtain an accurate solution for this zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Additional appli- cations of helium reionization include the detailed structure of the proximity zones around QSOs for constraints on the reionization of both hydrogen and helium, which depend on the lifetime of the sources, the size of the ionized regions they produce and on the temperature of the ionized gas in the zone (eg Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Worseck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The possibility of the detection of a 21-cm signal during the Epoch of Reionization (EoR) also requires under- standing the heating of the still neutral gas by high energy photons, as only a small amount of heating is able to affect the absorption signal against the Cosmic Microwave Back- ground (CMB), and even convert it into an emission signal (Tozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In the context of IGM reionization simulations, two broadly different algorithmic approaches have been used to compute the photoionization driven by isolated radiation sources, one based on representing the radiation field as photon packets and the other based on a direct integration of the radiative transfer equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Photon packet schemes have generally been used rather than direct integration for numerical simulations because of their greater numerical efficiency (Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Several three-dimensional numerical RT packages in- corporating multi-frequency radiation have been used to com- pute the photoionization of both hydrogen and helium in the IGM, including LICORICE (Baek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2010), RADAMESH (Can- talupo & Porciani 2011), TRAPHIC (Pawlik & Schaye 2011), C2-RAY (Friedrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2012), CRASH (Graziani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2013) and RADHYDRO (La Plante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' With the exception of TRAPHIC, these codes provide only quasi-time-dependent solutions in the sense that they allow for evolution in the gas properties and the sources, but they solve only the static RT equation by taking the speed of light to be infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The infinite-speed-of-light approximation (ISLA) has two shortcomings: the ionization fronts may prop- agate to acausally large distances (greater than the distance light could travel), and the approximation is not valid when the ionization front is propagating near the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The propagation of superluminal ionization fronts is particularly a concern for the ionization zones produced by QSOs: because of their high luminosities, the ionization fronts may move at superluminal velocities over the lifetime of the sources, producing unphysically large ionization regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This may be partly overcome by enforcing a light-horizon, eg, by remov- ing photon packets when they exceed their causal horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Such a solution, however, does not conserve the radiative en- ergy carried by the photons and may alter the post-ionization temperature, and therefore ionization fractions, of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The computation of ionization fronts propagating at near the speed of light requires an algorithm that solves the time- dependent radiative transfer equation to obtain an accurate post-ionization temperature, as will be demonstrated in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' One goal of this paper is to provide a means for imple- menting an approximate practical photon packet RT scheme for cosmological simulations that handles near-luminal ion- ization front expansion without solving the time-dependent RT equation, which is prohibitively computationally expen- sive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Photon packet schemes have been applied to the reioniza- tion of intergalactic hydrogen and helium both as a post- processing step (Sokasian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Ciardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Compostella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Kakiichi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Eide et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2018, 2020) and in fully coupled radiative hydrodynamics (RHD) schemes (Meiksin & Tittley 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' La Plante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Moments based schemes for solving the radiative transfer equations to study the reion- ization of hydrogen and helium in the IGM have been em- ployed both in a multi-step scheme (Puchwein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2022) and in RHD simulations (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Kannan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Codes that compute the reionization of hydrogen and he- lium by a QSO in a static (or comoving) gas and assuming spherical symmetry have been used to estimate the gas prop- erties around a QSO using both direct integration (Madau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Tozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Friedrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2012) and pho- ton packet schemes (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Khrykin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Graziani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Eilers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Morey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Rather than comparing results from a broad range of nu- merical codes, the focus of this paper is on the requirements for achieving convergent results using different numerical RT methods, with a particular view to application to the reioniza- tion of helium in the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We compare two photon packet al- gorithms and a direct integration algorithm applied to sources with spectra similar to stars and QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Whilst previous con- vergence tests focused on the extent of ionization zones and their properties using solutions of the time-independent RT equation (ISLA methods), special attention is given here to differences arising between solutions to the time-dependent (for which the speed of light is finite) and time-independent RT equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In addition to eliminating spurious fast-than- light growth of the ionization zones, we show there are sig- nificant differences in the post-ionization temperature in the near-zone of QSO sources when solving the time-dependent RT equation compared with ISLA solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' With a view to applications to the 21-cm EoR signal, we also examine con- vergence in the far-zone, where the gas is still largely neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We focus on idealised test problems, as these provide the best means for isolating differences in the behaviour of the algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' To demonstrate the consequences of these differences in a cosmological setting, we also provide results for the ion- ization by a source in a cosmological simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In the next section, we describe the numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3, test problems are described and results presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' These are dis- cussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4, where also results for a cosmological simula- MNRAS 000, 1–19 (2022) Numerical methods for IGM reionization 3 tion are presented, and conclusions are summarised in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Comparisons between the time-dependent solutions to the RT equation and published ISLA solutions for test problems are presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' One of the photon packet schemes is a module of the gravity-hydrodynamics code ENZO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Con- vergence test results for ENZO are provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Alterations to the code required to carry out the tests are described in Appendex C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2 NUMERICAL SOLUTIONS OF THE REIONIZATION EQUATIONS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 The Reionization Equations The ionization equations for hydrogen and helium are dxH I dt = −xHI [ΓH I + neγH I(T)] + xH IIneαH II(T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' dxH II dt = −dxH I dt ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' dxHe I dt = −xHe I [ΓHe I + neγHe I(T)] + xHe IIneαHe II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' dxHe II dt = −dxHe I dt − dxHe III dt ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' dxHe III dt = xHe II [ΓHe II + neγHe II(T)] − xHe IIIneαHe III,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='(1) where αH II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' αHe II and αHe III are the total radiative recom- bination rates to all levels of H I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' He I and He II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' ΓH I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' ΓHe I and ΓHe II are the corresponding photoionization rates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' and γH I(T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' γHe I(T) and γHe II(T) are the correspond- ing collisional ionization coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For hard spectra, sec- ondary ionizations produced by ejected electrons may help to partially ionize the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We do not include this effect to focus comparisons between the codes on the numerical solu- tions of the radiative transfer equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' including secondary electron ionizations could in principle complicate the inter- pretation of any differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Their correct implementation is moreover hampered by the long path lengths of the secondary electrons compared with the length scales of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' A com- parison between different treatments is provided by Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' To illustrate the magnitude of the effects of sec- ondary electron ionizations, we present some results for test problems without and with secondary electron ionizations in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Generally the effects are only moderate, but warrant inclusion for more precise solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' the effects of secondary electron ionizations are particularly large in the partially ionized regions outside the main ionized zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The respective H I, H II, He I, He II and He III fractions are denoted by xHI, xHII, xHeI, xHeII and xHeIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The total electron density is ne = nHII + nHeII + 2nHeIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The time-derivatives are la- grangian, so that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (1) are valid in the presence of velocity flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The photoionization rate per atom (or ion) of a species i (H I, He I or He II) is Γi = c � ∞ νT,i dν uν hν ai,ν, (2) where ai,ν is the photoelectric cross section of species i, νT,i is the threshold frequency required to ionize species i, uν is the specific energy density of the ambient radiation field, and h is Planck’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The specific energy density is related to the specific intensity of the radiation field Iν(r, t, ˆn) by uν = 4πJν/c, where Jν(r, t) = (1/4π) � dΩIν(r, t, ˆn) is the angle-averaged specific intensity at position r at time t in the direction ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The equation of radiative transfer for Iν(r, t, ˆn) in a static medium with absorption coefficient αν(r, t, ˆn) and emission coefficient jν(r, t, ˆn) is 1 c ∂Iν(r, t, ˆn) ∂t + ˆn · ∇Iν(r, t, ˆn) = −αν(r, t, ˆn)Iν(r, t, ˆn) + jν(r, t, ˆn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (3) In the context of reionization, the emission coefficient repre- sents a source like a star, galaxy or QSO, although in princi- ple it may also account for photoionizing radiation following radiative recombination within the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Since only contin- uum radiation is considered, as distinct from resonance lines, the static medium approximation is adequate on scales small compared with the cosmic horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The value of Iν at any given time t and position s along the direction ˆn will be given by any incoming intensity Iinc ν at position s0 at time tret = t − (s − s0)/c, absorbed by intervening material at positions s′ at the retarded times t′ ret = t − (s − s′)/c, along with contributions from sources at positions s′′ that emitted at the retarded times t′′ ret = t − (s − s′′)/c, followed by absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Accordingly, the for- mal solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (3) is Iν(s, t) = � Iinc ν � s0,tret exp � − � s s0 ds′ (αν)s′,t′ ret � + � s s0 ds′′ (jν)s′′,t′′ ret exp � − � s s′′ ds′ (αν)s′,t′ ret � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (4) We shall refer to such solutions as solutions to the time- dependent RT equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' By contrast, most of the literature makes the infinite-speed-of-light approximation, which corre- sponds to solving the time-independent (or static) RT equa- tion, for which the term involving the time-derivative of Iν in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (3) is absent, and only the instantaneous properties of the gas appear in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (4) rather than the time-retarded properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The solutions may none the less be quasi-time- dependent when using the ISLA if the properties of the gas and the source vary with time, but only on timescales long compared with the light propagation time from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The photoionization will also heat the gas, while radiative recombination and collisional effects will cool the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We follow the approach outlined in Meiksin (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Specifically, we solve the energy equation in the form dSE dt = (γ − 1)ρ−γ (G − L) , (5) where SE = p/ργ for gas pressure p, mass density ρ and ratio of specific heats γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Here, G is the heating rate per unit volume of the gas and L is the radiative energy loss rate per unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The photoionization heating rate per volume for a species i of number density ni is Gi = nic � ∞ νT,i dν uν hν ai,νh (ν − νT,i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (6) The total heating rate from all species is G = GH I + GHe I + GHe II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The radiative energy loss term L includes energy losses from radiative recombination, collisional excitation and MNRAS 000, 1–19 (2022) 4 inverse Compton cooling off the Cosmic Microwave Back- ground, using the rates referenced in Meiksin (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For ap- plications in Appendix A, energy losses from secondary elec- tron ionizations and adiabatic cooling from cosmic expansion (allowing for an evolving mass density) are included for some of the test problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The gas temperature is computed from T = ¯m k SEργ−1, (7) where ¯m is the mean mass per particle and k is Boltzmann’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ionization scenarios treated are idealized in that they do not allow for additional energy losses from dust or metals, as may occur in the ionized regions of high red- shift QSO sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Modelling such effects is well beyond the intent of this paper and would only complicate the interpre- tation of differences in the results arising from the different photoionization algorithms considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 Methods of Numerical Solution We confine the discussion to algorithms that solve the radia- tive transfer equation along individual rays, as distinct from moment methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Full 3D RT is achieved through the means of constructing the rays, a topic we shall not discuss, simply adopting the existing framework for the 3D code used (ENZO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We consider two types of numerical methods for solving the 1D RT equation, one based on a direct integration of the time-independent RT equation (an ISLA method) and the other for which the radiation is represented by photon pack- ets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Two versions of the latter are tested, one corresponding to integrating the time-independent RT equation (an ISLA method) and the other corresponding to integrating the time- dependent RT equation (retaining the differential time oper- ator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Each method is outlined in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 Instantaneous direct integration At each time step, the 1D radiative transfer equation is inte- grated along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Retaining the past absorption coefficient for all positions along the line of sight at previ- ous times is normally prohibitively expensive in a simulation, so instead new rays are cast for each time step and the ab- sorption coefficient for that time step is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The solution is taken as Iν(s, t) = � Iinc ν � s0,t exp � − � s s0 ds′ (αν)s′,t � + � s s0 ds′′ (jν)s′′,t exp � − � s s′′ ds′ (αν)s′,t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (8) This in effect treats the speed of light as infinite, although a cut-off in the distance the radiation reaches is imposed to ensure the extent of the region affected by a source preserves causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This instantaneous solution may be a good approxi- mation when the ionization front moves much faster than the gas flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In the implementation used here, a spatial grid is used along the line of sight, with the width of each zone chosen to ensure the optical depth of still neutral hydrogen or singly ionized helium is approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This ensures an accurate integration of the radiative transfer equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The time step is chosen to be a fraction of the shortest ionization or cooling time, sufficient to provide convergence at the few percent level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The energy equation is solved with an explicit second order time integrator, and an implicit scheme is used to solve the time-dependent ionization equations (Meiksin 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 Rays and photon packets An alternative approach traces packets of photons of distinct energy groups along rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The algorithm consists of two parts: casting rays through a simulation volume from each source, and propagating photon packets along the rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For more than a single spatial dimension, rays bunch together near a source and the separations between the rays increase with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Adequate cell coverage of the regions affected by each source is assured by splitting rays as required, as in Abel & Wandelt (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Photon packets are then propagated along the rays on each time step, until the packets are either completely absorbed or escape the simulation volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The gas component in the simulations is computed on a spatial grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The fraction of the photon packets absorbed on crossing a grid zone depends on the optical depth through the zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' An advantage of this method over the direct inte- gration scheme above is that the optical depth may be chosen to be above 1 while maintaining good accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' up to 10 is typical, but even larger values are able accurately to recover the expansion of an ionization front (Abel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This considerably reduces the computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' A consequence, however, is an ambiguity in how to share the photons when more than a single species may be ionized in a zone, as they will compete for the same photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' How the gas is ionized in a zone, were it completely resolved, can change the proba- bility for photons of different energies to be absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In the simplest implementation, such as in ENZO, the fraction of pho- tons of energy i absorbed by species j is 1−exp[−τj(νi)], but in looping over j, different fractions may result depending on how the species are ordered in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We adopt instead the more balanced probabilistic approach of Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2004), and have introduced this into the version of ENZO we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The absorption probabilities of a photon in a packet of frequency ν by H I, He I and He II are respectively P HI abs = pHIqHeIqHeII[1 − exp(−τ total ν )]/D, (9) P HeI abs = qHIpHeIqHeII[1 − exp(−τ total ν )]/D, (10) P HeII abs = qHIqHeIpHeII[1 − exp(−τ total ν )]/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (11) Here pi = 1 − exp(−τ i ν) and qi = exp(−τ i ν) are the auxil- iary absorption and transmission probabilities for the species i, D = pHIqHeIqHeII + qHIpHeIqHeII + qHIqHeIpHeII is the nor- malisation factor, and τ total ν = τ HI ν + τ HeI ν + τ HeII ν is the total optical depth, where τ i ν is the optical depth of species i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 1 In practice, for the test problems presented here, the different ionization zones are sufficiently distinct that the results are nearly the same using a simpler formulation treating the absorption by each species independently, with results agreeing typically to better than a percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Differences up to 20 percent, however, may arise for low ionization fractions in some regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We retain the formulation described here for generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' MNRAS 000, 1–19 (2022) Numerical methods for IGM reionization 5 We test the implementation of the photon packet scheme used in the numerical-hydrodynamics code ENZO v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6 (Wise & Abel 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Bryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2014), modified as described in Appendix C, including the photon absorption probabil- ities above, with photoionization cross-sections from Anni- nos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (1997)2 , and the chemistry and cooling solver GRACKLE 3(Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Because the code runs in 3D, the memory cost of retaining photon packets from previous time steps rapidly becomes prohibitive in reionization prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For this reason, instantaneous radiative transfer is as- sumed in the code (ISLA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' New packets are generated on each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This corresponds to neglecting the time derivative in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (3), or equivalently adopting an infinite speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' To preserve causality, we also permanently delete any surviv- ing photon packets that travel outside the light cone of the source (see Appendix C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=') This necessarily results in a loss of radiant energy and an artificially reduced photoionization heating rate of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For a more physical contrast compared with the other two methods, we also present results using a 1D spherically sym- metric code, PhRay, that propagates the photon packets at a finite velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The photon packets are kept between time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' They then have a memory of the gas they passed through on all previous time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This coresponds to re- taining the time derivative in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Adopting the speed of light for the packet velocity requires very short time steps, as the code is written to ensure photons cannot move more than a single grid zone in one time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (A more efficient code could be written to allow movement through multiple grid zones, but this would entail considerable additional com- putational overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=') The grid zones are adjusted to a preset maximum optical depth per zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In outline, the basic steps of the code are: (i) Choose the length of the ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (ii) Compute the minimum cell width to ensure the H I, He I and He II optical depths do not exceed preset maximum values for a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Set the time step to the time it takes a photon to cross a single cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (iii) On the first time step, add photon packages to the first cell nearest the source according to the source luminosity and time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (iv) On subsequent time steps, move photon packages in each cell to the next cell away from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Solve the ionization equations Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (1) and energy equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (5) in each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Remove photon packages according to the optical depth in the new cell, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (9) - (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Add new photons to the first cell nearest the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (v) Repeat step (iv) until the final integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As long as the advance of the fastest ionization front is highly sub-luminal, nearly identical results are produced al- lowing for a packet velocity smaller than the speed of light, provided the packets still move quickly compared with the ionization fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This has the advantage of allowing the code to take longer time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In our tests, we use the physical speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For both photon packet codes, the energy and ionization 2 The He I photoionization cross-section is updated to that of Verner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3 https://grackle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='io/ equations are solved using a first order Eulerian time integra- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3 Frequency integration For the photon packet schemes, the integration of the radi- ation field over the photoionization cross-section to compute the ionization and heating rates is accomplished using Gaus- sian Legendre quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This is more efficient than a uni- form grid in frequencies, and yields extremely accurate inte- grations when the integrand is well-approximated by a poly- nomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We experimented with several implementations, and use one we find ensures accuracy in the frequency integrations typically to within a percent for the applications we present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Specifically, the frequency range is divided into intervals be- tween the photoelectric edges for H I (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6 eV), He I (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6 eV) and He II (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='4 eV), and extending to an upper limiting value dependent on the RT method used and the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the spherically symmetric photon packet method (PhRay), 8 energy bins were used in each frequency interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The inte- gration variable for the final integration to infinite frequency is changed to 1/ν, so that the integration ranges from 0 to 1/νL,He II, where νL,He II is the ionization threshold for He II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the ENZO simulations, instead an upper frequency is im- posed, provided in the test problem descriptions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The number of frequency bins between threshold energies is set at 5, with 10 between the He II photoelectric threshold and the maximum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The energy binnings used for the photon packet codes ensure accuracy in the frequency integrations to within a percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the direct integration method, Gaussian quadrature offers some advantage over a mixed linear-logarithmic fre- quency grid in allowing a reduction by a factor of three in the number of frequencies used for an accurate solution within the ionized region, however the ionization front moves some- what too quickly unless a comparable number of frequen- cies is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We consequently use a mixed linear-logarithmic frequency grid for the direct integration scheme, with the number of frequencies typically 200, half placed uniformly between the H I and He II photoelectric thresholds and the remainder placed logarithmically to a maximum energy of 2 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This choice ensures temperatures and placements of ionization fronts are converged to better than a percent at a given spatial grid resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3 TEST RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 Test problems The algorithms are tested on four problems, reionization by a black body spectrum with a temperature of 105 K or 106 K, and reionization under intergalactic medium conditions by a power-law spectrum before and after hydrogen is ionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The 105 K black body roughly represents a high mass Pop II or Pop III star (Bond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 1984), whilst the 106 K black body (hotter than expected for stellar atmospheres), is pre- sented as a contrasting source with a greater proportion of ionizing photons able to fully ionize helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The power-law spectra represent QSOs, although they also approximate the photon emission rates for galaxies dominated by Pop II or Pop III stars (Meiksin 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The mass-fraction abundances MNRAS 000, 1–19 (2022) 6 of hydrogen and helium are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='76 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='24, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In all cases, the surrounding gas is static and its hydrodynamical response is not included, since the focus is on the ionization structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The problems and results are discussed in greater detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 Black body spectra The luminosity function of the black body spectrum is mod- elled as LBB,ν = L0 × (hν)3 exp hν kT − 1 eVs−1Hz−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (12) The surrounding hydrogen number density is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='76 × 10−3cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Hydrogen and helium are initially neutral in all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The physical boxsize in all the ENZO 3D instan- taneous simulations with 2563 cells is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='4 The optical depths of all species per zone are below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The total number of cells in the 1D simulations is adjusted in different situa- tions, ensuring the optical depth of neutral hydrogen in each cell at the start of the computation is approximately equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This ensures the positions of the ionization fronts are converged to within a few percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We show that in the two black body radiation problems, all algorithms show consistent temperature patterns and the differences in the gaseous ionisation levels are negligible for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 TBB = 105 K The initial gas temperature is T = 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The coefficient L0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='9 × 1031 eV−2 corresponds to a photon emission rate above the hydrogen ionization threshold of ˙NH,γ = 5×1048 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the ENZO computations, the maximum energy bin used is 200 eV to ensure convergence on the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The temperature and ionization structure at times t = 10 Myr and 30 Myr after the source turns on are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The temperature profiles from all the codes are in substantial agreement, although the direct integration code slightly anticipates the position of the knee in the tempera- ture profile, where it starts its decline to T < 104 K, just past the He II-front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The knee in the temperature profile is reflected in the ion- ization profiles, for which the H II and He II fronts from the direct integration scheme slightly lead the results from the photon packet codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' At distances beyond the He III-front, at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='9 kpc (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 kpc) at t = 10 Myr (30 Myr), the He III fraction from the direct integration scheme first declines gently, then decreases precipitously near 2 kpc (3 kpc), near the H II-front at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 kpc (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 kpc) at t = 10 Myr (30 Myr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' By contrast, both photon packet codes (PhRay and ENZO), allow leakage of He II-ionizing photons to larger distances sufficient to main- tain partial He III ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The direct integration scheme none the less produces He II fractions similar to the photon packet codes at all radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' At distances beyond 5 kpc, the He II and He III fractions from ENZO decline faster than the corre- sponding fractions from PhRay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This is found to be a spatial 4 The parameters are adopted from the PhotonTest test problem in ENZO v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 103 104 105 Temperature (K) 10 Myr PhRay DI ENZO 103 104 105 Temperature (K) 30 Myr 2 4 6 8 radius (kpc) 10(4 10(3 10(2 10(1 100 fractio 10 Myr HI HeI HeII HeIII 2 4 6 8 radius (kpc) 10(4 10(3 10(2 10(1 100 fractio 30 Myr Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Reionization by a black body source with temperature TBB = 105 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Shown are results for the time-dependent pho- ton packet code (PhRay, black lines), the direct integration scheme (DI, cyan lines) and ENZO (blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Upper panels: Temperature profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The dotted (magenta) vertical line shows the H II-front according to the PhRay calculation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' the dot-dashed (green) verti- cal line shows the He II-front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Lower panels: ionization profiles for H I (dotted lines), He I (dashed lines), He II (dot-dashed lines) and He III(dot-dot-dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 103 104 105 Temperature (K) 10 Myr PhRay DI ENZO 103 104 105 Temperature (K) 30 Myr 2 4 6 8 radius (kpc) 10(4 10(3 10(2 10(1 100 fractio 10 Myr HI HeI HeII HeIII 2 4 6 8 radius (kpc) 10(4 10(3 10(2 10(1 100 fractio 30 Myr Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 1, but for a black body source with temperature TBB = 106 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' resolution effect in the simulation volume: increasing the res- olution in ENZO increases the range of agreement with the results from PhRay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 T = 106 K The initial gas conditions are identical to those used in the 105 K black body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The coefficient L0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='9 × 1027 eV−2 corresponds to ˙NH,γ = 7×1047 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' To allow for the higher frequency peak in the Planck distribution, the upper energy bin in the ENZO computation is increased to 1000 eV to ensure convergence on the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The temperature and ionization structure at times t = 10 Myr and 30 Myr after the source turns on are shown in MNRAS 000, 1–19 (2022) Numerical methods for IGM reionization 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Whilst the fraction of emitted photons able to fully ionize helium is higher compared with the TBB = 105 K black body spectrum, the results are qualitatively very similar to those for the 105 K source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The exception is the position of the temperature knee, where the gas temperature starts to de- cline below 104 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the TBB = 106 K source, the plateau in temperature is maintained at T ≃ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5) × 104 K somewhat beyond the He II-front, with the front (shown by the dot-dashed green vertical line) positioned about half way through the temperature plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The temperature falls be- low 104 K only once the He II fraction declines to below about 10 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3 Power-law spectra 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 QSO reionization We consider two reionization problems: (1) the reionization of the IGM at z = 6 and (2) the reionization of the He II component of the IGM at z = 4, both by a QSO spec- trum modelled as a power law in frequency for values above the frequency νL of the hydrogen photoelectric threshold, Lν = LL(ν/νL)−αQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Typical initial gas densities and ion- ization states are adopted at these redshifts, as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The reionization is followed for times short compared with the Hubble time, so that cosmological expansion is not included: the gas is static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Inverse Compton cooling off the CMB is also neglected, as the characteristic cooling time is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 Gyr at z = 6 and 2 Gyr at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The QSO spectra are modelled as LS,ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='56 × 1031 ergs−1Hz−1(ν/νL)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 and LS,ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 × 1031 ergs−1Hz−1(ν/νL)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Only the photon packet codes are run for this problem since convergent re- sults become computationally prohibitively expensive for the direct integration code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The cut-off energies for the ENZO simulations for the LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 and LS,ν ∼ ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73 spectra are both set to 1000eV (see Appendix B for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The physical box size used in all the simulations is 25 Mpc, with 2563 cubic cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the PhRay spherically symmetric simulations, the grid cell size is chosen to assure the maximum initial optical depths in a single cell for H I, He I and He II do not exceed unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 Reionization at z = 6 The surrounding hydrogen density is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 × 10−5 cm−3, cor- responding to the mean IGM density at redshift z = 6, the typical redshift when QSOs begin photoionizing the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The hydrogen and helium are assumed neutral5, with initial gas temperature set to T = 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Outside the central 3 Mpc, the gas temperature profiles for PhRay and ENZO substantially agree for αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The temperature takes a sharp step down, by about ∆T ≃ 5000 K, at the H II-front (shown by the dotted magenta lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Both codes capture the temperature step as well as the temperature decline beyond the He II ionized zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Within the inner 3 Mpc, however, the temperature from PhRay is 5 At z = 6, ∼ 20 percent of the volume of the IGM is expected to be neutral (Gnedin & Madau 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 104 105 Temperature (K) 20 Myr PhRay ENZO 104 105 Temperature (K) 30 Myr 5 10 15 20 radius (Mpc) 104 105 Temperature (K) 50 Myr 5 10 15 20 radius (Mpc) 104 105 Temperature (K) 100 Myr 104 105 Temperature (K) 20 Myr PhRay ENZO 104 105 Temperature (K) 30 Myr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 radius (Mpc) 104 105 Temperature (K) 50 Myr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 radius (Mpc) 104 105 Temperature (K) 100 Myr Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Temperature profiles for reionization at z = 6 by source LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The bottom set of panels shows the detailed temper- ature structure within the ionization fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Shown are the results for PhRay (black solid lines) and ENZO (dashed blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The dot- ted magenta line in each panel shows the position of the H II-front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The dot-dashed green line shows the leading edge of the He II zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' boosted compared with that from ENZO, reaching values ex- ceeding 105 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6 the H II region is expanding nearly at the speed of light out to this distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This region is discussed in more detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ionization fractions from the two codes similarly track each other closely, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4, although the ionized hy- drogen and helium regions tend to lead slightly in the ENZO computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In spite of the difference in gas temperature within the central 3 Mpc, the rise in He II fractions agree well in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The leading edge of the He II-ionized re- gion (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3 by the dot-dashed green line) extends slightly beyond the H II region (shown by the dotted magenta line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Comparison with the TBB = 106 K black body spec- trum case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2, for which the H II end He II fronts are more clearly separated, shows that the ledge in high temper- ature actually extends beyond the He II-front, into the region 6 We confirmed that this result is largely unaffected by inverse Compton cooling off the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Including inverse Compton cooling lowers the peak temperature by only 10 percent by t = 100 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' MNRAS 000, 1–19 (2022) 8 10−4 10−3 10−2 10−1 100 fraction 20 M r PhRa ENZO 10−4 10−3 10−2 10−1 100 fraction 30 M r HI HeI HeII HeIII 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 fraction 50 M r 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 fraction 100 M r Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Ionization profiles for reionization at z = 6 by source LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Shown are the results for PhRay (solid lines) and ENZO (dotted lines), for the H I (black lines), He I (blue lines), He II (green lines) and He III (yellow lines) ionization fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 104 105 Temperature (K) 20 Myr PhRay ENZO 104 105 Temperature (K) 30 Myr 5 10 15 20 radius (Mpc) 104 105 Temperature (K) 50 Myr 5 10 15 20 radius (Mpc) 104 105 Temperature (K) 100 Myr Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Temperature profiles for reionization at z = 6 by source LS,ν ∼ ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Shown are the results for PhRay (black solid lines) and ENZO (dashed blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The dotted magenta line shows the position of the H II front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' of partial He II ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the power-law spectrum case here, the temperature falls below 2 × 104 K only for a He II fraction below 5–7 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' At t = 100 Myr, a low level of He II ionization persists to the edge of the simulation volume, with ionization fraction xHeII > 5 × 10−4 and T > 350 K, large compared with the initial temperature of 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the αQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73 spectrum, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 5, the boost in temperature in the central 3 Mpcs for PhRay compared with ENZO is smaller than for the αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The gas temperature declines abruptly at the H II front, shown by the dotted magenta lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The leading edge of the He II region almost exactly tracks the H II front (with positions agreeing to better than 1 percent) at all times, with no ledge in high temperature extending beyond as in the αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 spectrum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ionized regions from ENZO again slightly lead those 10−4 10−3 10−2 10−1 100 fraction 20 M r PhRa ENZO 10−4 10−3 10−2 10−1 100 fraction 30 M r HI HeI HeII HeIII 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 fraction 50 M r 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 fraction 100 M r Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Ionization profiles for reionization at z = 6 by source LS,ν ∼ ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' from PhRay, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The region of low He II ion- ization extends further for the PhRay calculation than for ENZO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' the extent is limited by the higher energy photon cut- off in ENZO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' At t = 100 Myr, xHeII > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='8 × 10−5 out to the edge of the PhRay simulation volume of 26 Mpc radius, with T > 108 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Compared with the initial temperature of 100 K, the amount of heating is small at these radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3 Reionization of He II at z = 4 The surrounding hydrogen density is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='4 × 10−5 cm−3, corre- sponding to the mean IGM density at redshift z = 4, the typi- cal redshift when QSOs begin photoionizing He II in the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The initial hydrogen neutral fraction is set at xHI = 2 × 10−5 and the He I and He III helium fractions xHeI = 9 × 10−6 and xHeIII = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' These correspond approximately to the ionization levels for the ultra-violet (UV) metagalactic background at z = 4 (Haardt & Madau 2012) in a region for which He II has not yet been ionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The initial gas temperature is set to T = 104 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 spectrum, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 7, the gas temperature is elevated behind the He III-front relative to the temperature of the ambient gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Whilst the PhRay and ENZO ionization levels agree well within the He III region, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 8, the PhRay temperature somewhat exceeds that of ENZO by about 4000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As discussed below, this is a conse- quence of near luminal expansion of the He III-front once the QSO turns on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Ahead of the He III-front, the temperatures are in good agreement, although the ENZO temperature is slightly higher than the PhRay temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As for the z = 6 simulations, the ENZO ionization regions slightly lead those from PhRay (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 8), with the more ionized H II and He II regions expanding somewhat more rapidly for ENZO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Otherwise the ionization fractions are in good agree- ment outside the He III region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' At distances from the source beyond the light front, the ionization fractions remain con- stant with distance, reflecting the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Because there is no ambient UV photoionizing background field, the ionization level at these distances is evolving as hydrogen and helium gradually recombine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The temperature is again elevated out to the He III-front MNRAS 000, 1–19 (2022) Numerical methods for IGM reionization 9 104 105 Temperature (K) 20 Myr PhRay ENZO 104 105 Temperature (K) 30 Myr 5 10 15 20 radius (Mpc) 104 105 Temperature (K) 50 Myr 5 10 15 20 radius (Mpc) 104 105 Temperature (K) 100 Myr Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Temperature profiles following He II reionization at z = 4 by a source LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Shown are the results for PhRay (black solid lines) and ENZO (dashed blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The dot-dashed green line in each panel shows the position of the He III-front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 10−4 10−3 10−2 10−1 100 fraction 20 M r PhRa ENZO 10−4 10−3 10−2 10−1 100 fraction 30 M r HI HeI HeII HeIII 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 fraction 50 M r 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 fraction 100 M r Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Ionization profiles following He II reionization at z = 4 by a source LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' for the αQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73 spectrum relative to the ambient gas tem- perature, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 9, but not by as much as for the αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The temperatures from PhRay and ENZO agree well, although the ENZO temperature slightly exceeds that of PhRay beyond the He III-front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This is consistent with a slightly faster expansion of the He III-front from ENZO com- pared with PhRay, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4 DISCUSSION Both the direct integration and photon packet codes recover the principal ionized zones of hydrogen and helium produced by the black-body and power-law spectral sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Several discrepancies, however, are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We discuss the differences that are particularly pertinent to measurements of the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We focus on differences in the near zones, where hydrogen and helium are nearly fully ionized, and the far zones, where the hydrogen and helium are nearly neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 104 105 Temperature (K) 20 Myr PhRay ENZO 104 105 Temperature (K) 30 Myr 5 10 15 20 radius (Mpc) 104 105 Temperature (K) 50 Myr 5 10 15 20 radius (Mpc) 104 105 Temperature (K) 100 Myr Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Temperature profiles following He II reionization at z = 4 by a source LS,ν ∼ ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Shown are the results for PhRay (black solid lines) and ENZO (dashed blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The dot-dashed green line shows the position of the He III-front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 10−4 10−3 10−2 10−1 100 fraction 20 M r PhRa ENZO 10−4 10−3 10−2 10−1 100 fraction 30 M r HI HeI HeII HeIII 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 fraction 50 M r 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 fraction 100 M r Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Ionization profiles following He II reionization at z = 4 by a source LS,ν ∼ ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 Near zone 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 Black-body spectra For the TBB = 105 K black-body spectrum, the hydrogen- ionizing photon emission rate ˙NH,γ = 5×1048 s−1 corresponds to an expansion rate of the H II-front, before radiative recom- binations become important, given by balancing the emission rate to the rate at which hydrogen atoms are ionized: rHII = � 3 4π ˙NH,γt nH �1/3 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2t1/3 Myr kpc, (13) where tMyr is the time since the source turned on in units of 106 yr and a hydrogen density nH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='76 × 10−3 cm−3 has been assumed7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the TBB = 106 K black-body spec- 7 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (13) is an approximation assuming all ionizing photons are absorbed at the ionization front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In practice, sufficiently high en- MNRAS 000, 1–19 (2022) 10 10−4 10−3 10−2 10−1 100 fraction 5 Myr 10−4 10−3 10−2 10−1 100 fraction 10 Myr PhRay:HI PhRay:HeI PhRay:HeII PhRay:HeIII 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 radius/ct 10−4 10−3 10−2 10−1 100 fraction 15 Myr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 radius/ct 10−4 10−3 10−2 10−1 100 fraction 20 Myr Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Ionization profiles shown vs the light front distance for reionization at z = 6 by a source LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5, from solving the full time-dependent RT equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' trum, with the lower hydrogen-ionizing photon emission rate ˙NH,γ = 7 × 1047 s−1, the expansion rate is about half as fast, rHII ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6t1/3 Myr kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The growth of the H II region for the TBB = 105 K source agrees well with the theoretical expectation, with the H II- front (defined at the position where xHII = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5), occurring within 15 percent of the prediction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (13), although falling systematically slightly short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The agreement is poorer for the harder TBB = 106 K spectrum, with the H II-front lagging far behind the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The discrepancies may be attributed to the presence of helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For TBB = 105 K, about half the hydrogen-ionizing photons may ionize helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' After subtracting these, the predicted position of the H II-front de- creases by about 20 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For TBB = 106 K, 99 percent of the hydrogen-ionizing photons may also ionize helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Re- moving these decreases the predicted radius of the H II-front by about a factor of 5, in good agreement with the compu- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The inclusion of helium thus requires accounting for the sharing of photons that may ionize more than a single species, which will depend on the relative abundances of the species in general, as well as on their relative cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ionization structures for the black-body spectra also agree well between the codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Nearly perfect agreement is found for the photon packet codes PhRay and ENZO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ion- ization fronts from the direct integration scheme, however, slightly lead the positions from the photon packet codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 Power-law spectra Outside the inner 3 Mpc, but still within the highly ionized regions, the temperatures found by PhRay and ENZO for the test problems for IGM conditions at z = 6 agree well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This is a significant achievement of the probabilistic formulation of the radiative transfer problem, as the initial optical depth per cell in the ENZO computation is 124, compared with an optical ergy photons continue un-absorbed because of their long mean free paths, but they make up only a small fraction of all the photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' ν/νL 10−2 10−1 100 Transmission time-dependent ν/νL 10−2 10−1 100 HI transmission instantaneous 5Myr 8Myr 10Myr 15Myr 20Myr 100 101 102 ν/νL 100 101 Lν time-dependent 100 101 102 ν/νL 100 101 Lν instantaneous Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Comparison of the transmitted radiation for reionization at z = 6 by a source LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5, for the solution to the full time-dependent RT equation (left panels) and in the instantaneous (ISLA) limit (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Upper panels: Transmission profiles e−τν at H II-front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Lower panels: Transmitted luminosity Lν = LS,νe−τν at H II-front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' depth of unity in PhRay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The sharpness of the H II-front per- mits a generous optical depth criterion, making the problem practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' By contrast, the convergence requirements that the optical depth per zone not exceed unity, along with a higher number of frequency bins, renders a direct integration of the radiative transfer equation computationally impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the test problem with IGM conditions at z = 4, the temperatures between PhRay and ENZO agree well, although the temperature from PhRay somewhat exceeds that of ENZO within the He III region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ionization fronts from ENZO also slightly lead those from PhRay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This could over-estimate the size of the expected He III zone predicted for a given QSO spectrum and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Agreement improves on increasing the spa- tial resolution for ENZO from 1283 to 2563 zones, correspond- ing to decreasing the He II optical depth at the photoelectric threshold from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='9 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The higher temperature found by PhRay compared with ENZO within the inner 3 Mpc for IGM conditions at z = 6, and for IGM conditions at z = 4 out to the He III-front, especially for the αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 spectrum, is a consequence of the rapid expansion of the ionized region around the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In the test problem for z = 6, the H II region expands nearly at the velocity of light, whilst in the z = 4 problem, the He III region has near luminal expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This may be seen by equating the output rate of ionizing photons to the rate at which gas is photoionized, when recombinations may be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The criterion that the expansion of the zone becomes subluminal is then 1 4πr2 i ˙Nγ n < c, (14) where ˙Nγ is the production rate of ionizing photons and n is the density of the species being ionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='8 The criterion in 8 White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2003) give for the evolution of an I-front radius RI, allowing for the finite travel time of light, ˙Nγ(t − RI/c) = MNRAS 000, 1–19 (2022) Numerical methods for IGM reionization 11 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (14) then corresponds to an I-front velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The production rate of all photons above the H I threshold for αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='7 × 1057 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For a hydrogen density 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 × 10−5 cm−3, the H II region expansion will become sub- luminal only at r >∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='7 Mpc, or somewhat smaller allowing for some photons to ionize helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This is consistent with the agreement in the temperatures between PhRay and ENZO at r > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 Mpc, where they both give T ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 × 104 K for αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 and T ≃ 4 × 104 K for αQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Similarly, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 7 and 9, enhancements in tem- perature are found at z = 4 where the hydrogen has already been ionised (and the helium singly ionised).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (The enhance- ment is small for the softer αQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73 spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=') In this case, the boost in temperature results from the rapidly expanding He III-fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (14), taking n = nHe and considering He II-ionizing photons, the expansion of the He III region be- comes sub-luminal only for r > 11 Mpc for αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 and r > 5 Mpc for αQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The temperature enhancement is confined to the region with substantial He III ionization (xHeIII > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5), shown by the magenta lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 7 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The boost persists until the region of xHeIII > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 reaches the luminal expansion limiting radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The reason near-luminal expansion of an ionization region gives rise to a boost in temperature is illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 11 and 12 for the αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 spectrum at z = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 11 shows the ionization fractions as a function of position in units of the light front (r = ct) from the time-dependent code PhRay, which tracks all photon packets since they were emitted until they are absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The time for the ionization fronts to reach ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 Mpc and become sub-luminal is 8 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ionization fronts then begin to slip increasingly behind the light front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As long as the ionization fronts keep up with the light front, the gas encountered by the photons is largely neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As a consequence, the lower energy photons are rapidly absorbed by the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Most of the photoionization is carried out by the surviving most energetic photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Once the ionization front becomes sub-luminal, the photon packets that arrive at the front include proportionately more lower energy photons from the source, and the amount of energy deposited in the gas per ionization decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This is shown in the left panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The median energy at which photons are trans- mitted is higher at times t < 8 Myr, with the peak in the transmitted luminosity Lν shifted towards higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' By t > 8 Myr, the photon luminosity profile at the H II-front reflects the transmission through the intervening ionization structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' By contrast, rather than tracking photon packets since they were emitted, ENZO recasts new rays at each time step and computes the instantaneous radiative transfer along the rays with a new set of photon packets launched from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' At t = 5 Myr, the intervening gas between the source and the H II-front removes fewer low energy photons (above the ion- ization threshold energy) than would have been removed from photon packets that were moving only very slightly ahead of the ionization front, as in the PhRay computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This is shown in the right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The transmitted lumi- (4π/3)R3 In, where n is the density of the species being pho- toionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This corresponds to an expansion velocity dRI/dt = c ˙Nγ/( ˙Nγ + 4πR2 Inc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' nosity Lν at the H II-front peaks at a lower energy compared with the time-dependent computation in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' By t > 15 Myr, the transmission factor and Lν have relaxed to those for the time-dependent RT equation solution once the H II-front has become sub-luminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Thereafter, the temper- atures from the time-dependent (PhRay) and instantaneous (ENZO) computations agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The gas heated earlier, during the luminal expansion phase, in the time-dependent RT equa- tion solution from PhRay at r < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 Mpc, however, remains hotter compared with the temperature computed in the in- stantaneous (ISLA) limit by ENZO because of the long cooling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The difference in temperature in the near-zone has possi- ble implications for metagalactic UV background (UVBG) or QSO lifetime estimates from proximity zone measurements, as these depend on the H I or He II fraction in the vicinity of QSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Models based on instantaneous photo-ionization may underestimate the gas temperature, and so overestimate the recombination rate and H I or He II fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This may result in an under-estimate in the size of the proximity zone around a QSO for a given UVBG level or QSO age, and so to an under-estimate of the UVBG level or over-estimate of the QSO age needed to agree with the proximity zone measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The additional near zone heating may also boost the Lyα photon emission rate through collisional excitation of H I in the ionization front during the luminal expansion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The increase in temperature may also affect the Lyα forest power spectrum at wavenumbers k >∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='003 s km−1, corresponding to the sizes of the luminal expansion regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The discrepancy in the predictions for the near zone be- tween the time-dependent and ISLA solutions to the radia- tive transfer equation when ionization fronts expand near the speed of light poses a dilemma for photon packet radiative transfer codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Solving the time dependent radiative trans- fer equation requires assigning a finite velocity to the photon packets and retaining all photon packets emitted during any previous time step until they exit the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This imposes an impractical memory demand on the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The cor- rect size of the ionization regions may instead be computed using an ISLA method by removing surviving photon pack- ets able to reach their causal horizons, but this results in an artificial loss of radiative energy from the source and too low a temperature in the main ionized region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We suggest a compromise solution in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 Far zone 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 Black-body spectra The temperature profiles beyond the ionization fronts agree closely between all three codes for both the TBB = 105 K and 106 K black-body spectra, including the position of the temperature knee, where the temperature begins its decline to the ambient IGM value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The code results for the tempera- ture begin to depart from each other well beyond the ionized gas region once the temperature declines below ∼ 5000 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' No direct observational consequences are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ionization structures for hydrogen and helium agree closely well beyond the ionization fronts, with the exception of the direct integration code result for He III, for which the ionization fraction plummets abruptly beyond the He III-front for both black-body spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Virtually all of the He II-ionizing MNRAS 000, 1–19 (2022) 12 radiation is absorbed just beyond the He III-front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This ap- pears to be a failing of the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Since He III is not directly measured, it has no direct observational consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 Power-law spectra The temperature and ionization structure beyond the ion- ization fronts agree well between PhRay and ENZO for the αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 spectrum for IGM densities at both z = 4 and z = 6, although the ENZO temperatures begin to decline some- what more rapidly at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the softer αQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73 spectrum, the He II fraction from ENZO for the z = 6 IGM density, while first tracking the PhRay result, suddenly declines at t >∼ 50 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The He II fraction adheres to the PhRay result to greater distances as the spa- tial resolution is increased for ENZO: going from 1283 to 2563 cells corresponds to decreasing the He II optical depth at the photoelectric edge from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As the gas temperature from PhRay remains well above 100 K to distances exceeding 12 Mpc at t = 50 Myr and 15 Mpc at t = 100 Myr, ENZO would under-estimate the range around a QSO to which the IGM was heated above the CMB temperature, and so under- estimate the range to which the 21-cm signal would be seen in emission against the CMB around the QSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In practice, the signal would be complicated by heating from galactic sources, which may well have already warmed the IGM to tempera- tures above the CMB (Madau & Fragos 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Meiksin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3 Hybrid RT scheme We develop a hybrid ISLA method applied to sources reion- izing their local environment for our revised version of ENZO, to alleviate the discrepancies in temperature and ionisation structures between the time-dependent RT equation solution and the ISLA solution when removing photon packets that exceed their causal horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In the hybrid solution, the prop- agation speed of the photon packets remains infinite, corre- sponding to the instantaneous solution of the RT equation, but the travelling distance restriction imposed by causality is enabled only for photons in the sub-luminal region, as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='9his switch is applied only when the hydrogen around a source is still predominantly neutral, or the helium predominantly neutral or singly ionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' It is also only ap- plied to the photon packets that would effect the reionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In other situations, the gas will have already been heated by photoionization, with little additional heating from the source, so that no special measures need be taken to ensure an accurate temperature solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In these cases, the trav- elling distance restriction is applied to the relevant photon packets to ensure the changing ionization fractions around the source remain causal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This hybrid approach captures the accumulated attenua- tion of the radiation field in the near-luminal expansion re- gion, as the attenuation is mainly at the ionization front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This ensures the gas is heated to approximately the same temper- ature it would have using time-dependent RT (ie, a finite 9 T 104 105 Temperature (K) 20 Myr PhRay ENZO cfin ENZO cinf ENZO hybrid 104 105 Temperature (K) 30 Myr R=3Mpc light horizon 5 10 15 20 radius (Mpc) 104 105 Temperature (K) 50 Myr 5 10 15 20 radius (Mpc) 104 105 Temperature (K) 100 Myr Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Temperature profiles for reionization at z = 6 by source LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Shown are the results for PhRay (black solid lines),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' ENZO cfin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' for which photon packets are continuously removed when they reach their causal horizon (dot-dashed blue lines),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' ENZO cinf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' for which photon packets are removed only when absorbed or exit the grid (dot-dot-dashed yellow lines) and ENZO hybrid (dashed green lines),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' for which photons are removed when they reach their causal horizon only if they are located in the sub- luminal region (R > 3 Mpc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' shown by the vertical dashed magenta line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The dashed red line in each panel shows the position of the light horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' photon velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Once the ionization front slows down to be- coming sub-luminal, RT proceeds in a time-independent man- ner (or quasi-time-dependent allowing for slow changes in the gas or source properties), so that the ISLA method becomes increasingly accurate (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' At the earliest times after the source turns on, however, before the light front reaches the radius at which the ionization front should become sub- luminal, the scheme may produce an ionization front that is acausally large, extending beyond the light front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='13 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='14 illustrate the temperature and ionisation profiles for reionization at z = 6 by a source LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 computed using three methods: (1) the ISLA method (case ‘cinf’), (2) removing photons everywhere when they exceed their light horizon (case ‘cfin’), and (3) the hybrid scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ionization zone is too large in the ISLA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Remov- ing photons everywhere when they exceed their causal radius results in too great an energy loss in the near luminal expan- sion region, with the resulting temperature too low in the re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the hybrid method (ENZO hybrid), the agreement of the temperature and ionisation structures with those of the time-dependent RT solution from PhRay is much improved not only in the far zone but in the near zone as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For the softer αQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73 spectrum, we find improved agreement by defining the sub-luminal region according to the radius at which the expansion speed of the ionization front declines to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2c instead of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Interpolation may be used for intermedi- ate values of αQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' These choices may be applied for each source individually in a multiple-source simulation with a range of source spectra, although fixing the radius according to an ionization front speed of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5c may be adequate, as the tem- perature differences between the static and time-dependent RT solutions are smaller for softer spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' MNRAS 000, 1–19 (2022) Numerical methods for IGM reionization 13 10−4 10−3 10−2 10−1 100 fraction 20 My PhRay ENZO hyb id R=3Mpc light ho izon 10−4 10−3 10−2 10−1 100 f action 30 My HI HeI HeII HeIII 5 10 15 20 adius (Mpc) 10−4 10−3 10−2 10−1 100 f action 50 My 5 10 15 20 adius (Mpc) 10−4 10−3 10−2 10−1 100 f action 100 My Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Ionisation profiles for reionization at z = 6 by source LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Shown are the results for PhRay (solid lines) and ENZO hybrid (dotted lines), for which photon packets reaching their causal horizon are removed only in the sub-luminal region (R > 3 Mpc, shown by the vertical dashed magenta line), for the H I (black lines), He I (blue lines), He II (green lines) and He III (yel- low lines) ionization fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The dashed red line in each panel shows the position of the light horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='4 Cosmological simulation application We apply the three different ISLA methods (ENZO cinf: the original ISLA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' ENZO cfin: ISLA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' but adopting the causal travel distance restriction throughout the entire simulation volume and ENZO hybrid: ISLA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' but applying the causal travel distance restriction only in the sub-luminal ionization front expansion region) to a cosmological hydrodynamic sim- ulation using our revised version of ENZO to study the tem- perature and ionization structure around a QSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Assuming for simplicity that no metagalactic ultraviolet background (UVB) is present, we turn on a beamed QSO-like radiation source with an αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 power-law spectrum at the centre of the simulation box at z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The total hydrogen-ionizing photon emission rate is ˙Nγ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 × 1057 s−1 and the open- ing angle of the source is 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The cosmological parameters assumed are Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='27, Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='046, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='70, σ8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='811 and ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='961, with a primordial helium mass fraction Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='24, consistent with PLANCK measurements (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The code is run in uni- grid mode with a comoving box size of 120h−1 Mpc and 2563 cubic cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (The spatial resolution in proper units at z = 7 is comparable to the spatial resolution in the test problems in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=') The Cold Dark Matter initial conditions at z = 50 are generated by the MUSIC code (Hahn & Abel 2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' the code also sets the baryon properties, with a low tempera- ture given by adiabatic expansion following the recombina- tion epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The chemical and cooling processes are computed by GRACKLE 10 (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The RT equations are solved in a sub-cycle process in ENZO, so that the cosmological simulations are fully coupled radiation hydrodynamics sim- ulations, rather than being performed as a post-processing step, like in most cosmological RT simulations (eg Sokasian 10 https://grackle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='io/ −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 z (Mpc) −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 x (Mpc) 100 101 102 103 104 Temperature (K) Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Temperature slice plot at z ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='93 for a beamed QSO- like radiation source (LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5) after 30 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ENZO hy- brid RT method is used, for which the causal travel distance re- striction for photon packets is applied only in the sub-luminal ion- ization front expansion region (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Ciardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Compostella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Kakiichi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Eide et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2018, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The temperature around the source is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 15, with the temperature declining with distance from the source, and modulated by the large-scale structure of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The sharp ends to the temperature cone correspond to the light fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 16, the temperature along a line of sight through the beam centre is shown for the three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ISLA method (ENZO cinf) produces high excess tem- peratures away from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ENZO cfin and ENZO hybrid methods agree in temperature on large scales, but the temperature from the ENZO cfin method is too low in the lu- minal ionization expansion region, within the inner 2 Mpc from the source, by up to 5×104 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The different predictions for the ionization fractions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ISLA method again gives excess ionization on large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ENZO cfin and ENZO hybrid methods agree, except within the in- ner 2 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The near zone Lyα forest is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The lower temperatures using ENZO cfin result in a larger radiative recombination rate and so a greater amount of ab- sorption near the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Whilst the result from ENZO hybrid more faithfully recovers the expected gas temperature within the luminal region (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3), the actual amount of absorp- tion would be still somewhat smaller in this region because of the temperature boost allowing for the time-dependent RT in the luminal zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For precision work, a time-dependent so- lution to the RT equation would be required for such a hard spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The discrepancy is smaller for a softer spectrum (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 5 CONCLUSIONS We compare three radiative transfer codes applied to pho- toionization problems for sources with spectra typical of stars (black body) and QSOs (power law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' One code integrates the MNRAS 000, 1–19 (2022) 14 103 104 105 Temperature (K) 20 Myr ENZO cinf 103 104 105 Tempera ure (K) 30 Myr ENZO hybrid 25 50 75 100 comoving radius (h−1Mpc) 103 104 105 Tempera ure (K) 50 Myr ENZO cfin 25 50 75 100 comoving radius (h−1Mpc) 103 104 105 Tempera ure (K) 100 Myr Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Temperature profiles for a QSO-like radiation source (LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5) at the indicated times after the source turns on at z = 7: 20 Myr, 30 Myr, 50 Myr and 100 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The radiation source is located at the centre of the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Results are shown for the ENZO cinf method (solid black lines), the ENZO cfin method (dot-dashed red lines) and for ENZO hybrid (dashed blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Temperature profiles in the central luminal ionization front ex- pansion region are underestimated by ENZO cfin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The tempera- ture discrepancy in the region surrounding the radiation source is as high as 5 × 104 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 10−4 10−3 10−2 10−1 100 fraction 20 Myr cinf cfin hybrid 10 4 10 3 10 2 10 1 100 fraction 30 Myr HI HeI HeII HeIII 25 50 75 100 comoving radius (h 1Mpc) 10 4 10 3 10 2 10 1 100 fraction 50 Myr 25 50 75 100 comoving radius (h 1Mpc) 10 4 10 3 10 2 10 1 100 fraction 100 Myr Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Ionisation profiles for the QSO-like radiation source (LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5) at times 20 Myr, 30 Myr, 50 Myr and 100 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The radiation source is located at the centre of the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Re- sults are for the ENZO cinf method (solid lines), ENZO cfin method (dashed lines) and for ENZO hybrid (dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' time-independent radiative transfer equation directly, and is applied only to the black-body spectra problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The other two use photon packets to solve for the radiative transfer, one assuming instantaneous photoionization (with the dis- tance photon packets travel limited by the speed of light) and the other retaining fully the time-dependent term in the radiative transfer equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Our main findings are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Photon packet codes are far more efficient at solving the radiative transfer problem for photoionization compared with direct integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Fewer photon frequencies and coarser 0 1000 2000 3000 4000 velocity (s−1km) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='8 Normalised flu 50 Myr hybrid cfin Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Normalised Lyα flux spectra emitted by a QSO-like radi- ation source 50 Myr after the source turns on at z = 7, allowing for intervening absorption along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Results are shown for ENZO hybrid (solid black line) and for ENZO cfin (dashed blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (The Lyα absorption spectra are computed following Leong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=') spatial gridding are tolerated by the photon packet codes, with optical depths at the threshold energy able to exceed unity with accurate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Another shortcoming of the direct integration code is that it may fail to propagate low levels of doubly ionized helium beyond the He III-front as far as do the photon packet codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' All methods agree well on the growth of the nearly fully ionized regions, although the ionization fronts from the di- rect integration scheme tend slightly to lead those from the photon packet codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The successful solution of the ionized re- gions is a significant achievement of the photon packet codes particularly for hydrogen ionization, as the spatial grid used for the instantaneous photoionization version corresponds to a hydrogen optical depth per grid zone exceeding 100 at the photoelectric threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We recommend, however, that for ionizing singly ionized helium, the optical depth at the singly ionized helium threshold should be close to unity or smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Including the time-dependent differential operator in the radiative transfer equation is essential when ionization fronts expand near the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Solutions to the radiative transfer equation in the infinite-speed-of-light approximation (corresponding to solving the time-independent RT equation) may substantially under-estimate the temperature in these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The under-estimate increases with the hardness of the spectrum, with the temperature discrepancy exceeding 5 × 104 K for gas that was initially neutral, as may arise for reionization at high redshifts by QSOs with hard spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' A scheme that solves the time-dependent RT equation is thus required to obtain an accurate solution in the near zones of QSOs that photoionize the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The boost in temperature due to time-dependent RT is larger when both hydrogen and helium are initially predomi- nantly neutral compared with the case when the hydrogen is predominantly ionized and the helium singly ionized, as may arise when the gas is initially ionized by a metagalactic UV background radiation field dominated by galactic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Outside the luminal expansion region, the gas temper- MNRAS 000, 1–19 (2022) Numerical methods for IGM reionization 15 ature and ionization structure agree well between the time- dependent and infinite-speed-of-light photon packet codes, al- though some differences arise at large distances where the gas is predominantly neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' These differences appear to result from differences in spatial resolution, rather than from the assumption of an infinite speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' A photon packet code recovers the correct solutions to the time-dependent RT equation for an ionization front to good approximation using a hybrid scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In this scheme, the RT equation is solved in the infinite-speed-of-light ap- proximation only out to the radius at which the velocity of the ionization front declines to approximately half the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Photon packets outside this radius are removed if they travel to distances beyond the light front of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Photon energies well above the photoionization thresh- olds must be included to capture the warming of the largely neutral gas well outside the ionization regions for power-law spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The required maximum photon energy increases for softer spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Smith and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Wise for helpful con- versations, and the referee for numerous suggestions to im- prove the manuscript, including the suggestion to make di- rect comparisons between our results and the published liter- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' KHL acknowledges financial support from the School of Physics and Astronomy, University of Edinburgh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' KHL thanks the Computational Astrophysics Lab at National Tai- wan University for support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' KHT thanks the Robert Cormack Bequest fund for a Summer Vacation Research Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='7 × 10−5 7 G18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='36 × 1056 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6 × 10−5 7 CG21 1057 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='7 × 10−5 7 Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=', Anderson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=', 2015, ApJ, 806, 142 APPENDIX A: COMPARISONS WITH TEST PROBLEMS IN THE LITERATURE Solutions of the time-dependent radiative transfer equation for power-law spectra using PhRay are compared with pub- lished test problems of the photoionization of hydrogen and helium using ISLA methods, as provided by Abel & Haehnelt (1999), Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2016), Graziani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2018) and Chen & Gnedin (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The thermodynamics are governed by pho- toelectric heating and radiative cooling, including inverse Compton cooling off the Cosmic Microwave Background at the indicated redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Secondary electron ionizations and as- sociated energy losses are included as indicated, using the fits from Ricotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2002) to the Monte Carlo computa- tions of Shull & van Steenberg (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Parameters for the test problems are provided in Table A1, showing the net hy- drogen ionizing photon production rate ˙Nγ, the power-law exponent αQ for the QSO spectrum, the upper photon en- ergy cutoff hνmax for the spectrum, the hydrogen density nH of the surrounding gas and the redshift, which controls the inverse Compton cooling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The top left hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' A1 compares the solutions from PhRay without and with secondary ionizations to those provided by Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2016) using a spherically symmet- ric 1D ISLA method for a QSO spectrum with αQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The medium surrounding the source is assumed static in this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ISLA solutions from Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2016) (blue dashed and cyan dot-dashed lines) extend some distance be- yond the solution of the time-dependent RT equation using the correct speed of light (thick and thin black solid lines) given by PhRay, and even beyond the light front (shown as the vertical magenta dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The central, main ionized 2 4 6 8 10 radius (Mpc) 103 104 105 Temperature (K) 10 Myr ˙Nγ = 1057 s−1 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 5 PhRay: no sec D16: no sec PhRay: sec D16: sec 2 4 6 8 10 radius (Mpc) 103 104 105 Temperature (K) 100 Myr ˙Nγ = 1057 s−1 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 5 PhRay: no sec CG21: no sec 2 4 6 8 10 radius (Mpc) 103 104 105 Temperature (K) 10 Myr ˙Nγ = 3 × 1056 s−1 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 8 PhRay: no sec AH99: no sec 2 4 6 8 10 radius (Mpc) 103 104 105 Temperature (K) 60 Myr ˙Nγ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 4 × 1056 s−1 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 5 PhRay: no sec G18: no sec PhRay: sec G18: sec Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Comparison between solutions to test problems pro- vided by PhRay using a finite speed of light and published results using ISLA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The vertical green dashed lines indicate the theoretical position of the hydrogen ionisation front when its ad- vance slows to speed c/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The vertical black dotted lines show the maximum possible radius of the hydrogen ionization front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The vertical magenta dotted lines show the position of the light front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In the top left and bottom right panels, solutions both without (‘no sec’) and with (‘sec’) secondary electron ionizations are shown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' for the other panels, the published results did not include secondary electron ionizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' region from Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2016) reaches the maximum possi- ble radius of the H II-front, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (13)11 (vertical black dotted line), which is nearly coincident with the light front (vertical magenta dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Keeping up with the maxi- mum radius is expected for an ISLA scheme, which allows photons to travel until absorbed, but for the ionization front to have reached this distance, it had to travel superluminally at earlier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' By contrast, at 10 Myr the dominant ionized region from PhRay, where T > 104 K, extends just beyond the distance to which the H II-front travels at a speed exceed- ing c/2, as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (14) (shown by the vertical green dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In the central, main ionized region, the tem- peratures agree well between the two calculations, although the temperatures from PhRay are somewhat higher by about 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Significant cooling is provided by secondary electron ionization losses both in the central ionized region and in the extended region where the gas temperature is below 104 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The top right hand panel compares the solutions from PhRay and Chen & Gnedin (2021), who use an algorithm very similar to that of Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2016) for solving the static 1D RT problem, although modified to allow for a vari- able timestep within regions of very different ionization lev- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' No secondary electron ionizations are allowed for, and the surrounding medium is assumed static in this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ISLA solution of Chen & Gnedin (2021) has an H II-front that extends nearly to its maximum possible radius (vertical black dotted line), and is somewhat beyond that obtained by PhRay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The difference reflects an earlier superluminal ex- 11 Here and for the other test problems, the full value for ˙Nγ is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Since helium also absorbs photons above the helium ioniza- tion thresholds, the maximum radius will be somewhat smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' MNRAS 000, 1–19 (2022) Numerical methods for IGM reionization 17 pansion phase of the H II region in the ISLA computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In the inner main ionized region, the temperature from PhRay mildly exceeds that obtained by Chen & Gnedin (2021) by about 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In the lower left panel, the result for a steeper spectrum (αQ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='8) from PhRay is compared with the ISLA solution of Abel & Haehnelt (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Adiabatic cooling is included in this problem, although it negligibly affects the temperature over the brief interval of 10 Myr of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Secondary electron ionizations are not accounted for in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The temperatures agree well in the central region, with a slightly higher temperature obtained by Abel & Haehnelt (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The ISLA solution also has a slightly advanced H II-front com- pared with the time-dependent RT solution from PhRay, more nearly reaching its maximum possible radius (vertical black dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The reason the central temperature from the ISLA solution slightly exceeds that of the time-dependent RT solution is unclear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' the grid and frequency resolution are not provided by Abel & Haehnelt (1999) and there is no de- scription of convergence tests on either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Lastly, in the lower right panel we compare the solution of PhRay with the ISLA solution of Graziani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Computations both without and with secondary electron ion- izations were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The solutions of Graziani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2018) are anomalous in that the ionization fronts advance too slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (13), the H II-front should be located at rHII ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='8 Mpc (vertical black dotted line), in good agree- ment with the result from PhRay, whilst Graziani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2018) find the front to be located at ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Even allowing for all photons above the He I threshold to be absorbed by he- lium atoms, a production rate of purely hydrogen-ionizing photons of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 × 1055 s−1 would remain, giving an H II-front position of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As the radiative recombination time is longer than 109 yr, this position should have been reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Another anomaly is their temperature allowing for secondary electron ionization energy losses, which unexpectedly exceeds the temperature without secondary electron ionization losses, contrary to the result from PhRay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The discrepancies may be consequences of the large optical depths per grid zone in the computation of Graziani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' At the photoelectric edges, the optical depths before the gas is photoionized are ∼ 70 for H I and ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='4 for He II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The high H I optical depth may result in too little penetration of ionizing photon packets into the still neutral gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' By comparison, for another simula- tion in Graziani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2018) of a QSO embedded in a halo with higher spatial resolution, the H I and He II optical depths at the average IGM gas density are ∼ 16 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3, respectively, and the size of the H II region found is in good agreement with the analytic estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We also ran PhRay on a test problem with a 105 K black body spectrum source emitting at a hydrogen-ionizing photon rate 1051 s−1 into a static medium with a cosmic abundance of hydrogen and helium, hydrogen density nH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 cm−3 and initial temperature 100 K, to compare with Test 1, without metals but with secondary electron ionizations, of Graziani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' After 105 yr, the temperature 50 pc from the source is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3 × 104 K, declining gradually to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 × 104 K at 100 pc, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='2 × 104 K at 200 pc and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='3 × 104 K at 500 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The temperatures are comparable to, but slightly in excess by about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 dex of, the temperatures from the CLOUDY ioniza- tion code reported by Graziani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (14), the ionization front for this problem will expand at a speed 104 105 Temperature (K) 100 Myr 10−4 10−3 10−2 10−1 100 HII fraction 128 256 512 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 HeI fraction 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 HeIII fraction Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Profiles at 100 Myr for reionization at z = 6 by a source LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The upper left panel shows that the relative dif- ference in temperature between (25 Mpc, 2563) (blue dashed line) and (25 Mpc, 5123) (black dotted line) simulations is within 10% in the fully ionized region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' exceeding c/2 until it reaches 5 kpc, so the slightly higher temperatures are expected since CLOUDY is not designed to track the relaxation of temperatures following the heating by near luminal H II-front expansion to their steady-state value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' APPENDIX B: CONVERGENCE TESTS We show convergence tests for QSO reionization simulations which are performed by ENZO v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For all the convergence tests, we adopt identical parameters relating to the ray- tracing method of ENZO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In particular, the minimum ray an- gular resolution parameter is Φc = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 and the HEALPix Level is 6 (see the definitions in Wise & Abel 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The spacial resolution is the only code parameter varied for the convergence tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We also use an identical set of energy inter- vals and energy bins for simulations with various power-law indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' For all the power-law spectra, the energy interval ranges from 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6−1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='0 eV and the selected energy bins are [14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='12, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='13, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='10, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='06, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='08, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='00, 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='48, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='50, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='52, 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='00, 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='74, 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='20, 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='98, 322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='29, 456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='81, 597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='59, 732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='11, 848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='42, 936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='20, 987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='66] eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The convergence test results are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' B1 - B4, for the power-law test problems for IGM mean densities at z = 6 and 4, and for spectral indices αQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Convergence is generally reached in the inner ionised regions for the 1283 simulations, but, particularly for the softer spec- trum, convergence in temperature and the ionization struc- ture is improved on going to 2563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The latter corresponds to an initial optical depth per cell at the hydrogen photoelectric threshold of 124 at z = 6 and an initial optical depth per cell at the singly ionised helium photoelectric threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='9 at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' APPENDIX C: REVISIONS TO ENZO We describe revisions to ENZO v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6 to implement photoion- ization by a central source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' MNRAS 000, 1–19 (2022) 18 104 105 Temperature (K) 100 Myr 10−4 10−3 10−2 10−1 100 HII fraction 128 256 512 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 HeI fraction 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 HeIII fraction Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Profiles at 100 Myr for reionization at z = 6 by a source LS,ν ∼ ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The upper left panel shows that the relative differ- ence of temperature between (25 Mpc, 2563) and (25 Mpc, 5123) simulations is within 10% in the fully ionized region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' All the pan- els show that increasing the resolution of the simulations slightly advances the positions of the ionisation fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 104 105 Temperature (K) 50 Myr 10−4 10−3 10−2 10−1 100 HII fraction 128 256 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 HeI fraction 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 HeIII fraction Figure B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Profiles at 50 Myr for reionization at z = 4 by a source LS,ν ∼ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' All the quantities are converged in these (25 Mpc, 1283) and (25 Mpc, 2563) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We check the consistency of the source codes, especially the consistency of the codes relevant to the ray-tracing module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We find bugs in the implementation in ENZO v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6 significantly affect the accuracy of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' These are demonstrated in Appendix C1 by simulating a classical ray-tracing problem, the formation of a Str¨omgren sphere (Iliev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We impose new methods and restrictions on the ray-tracing module to make ENZO suitable for both static and cosmo- logical hydrodynamical simulations with high-luminosity ra- diation sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The modifications are: a) Probabilistic Ab- sorption Method (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' [9]–[11]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' b) He III Ionisation Adaptive Time Step Scheme (Appendix C2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' c) Restriction on Photon Package Travel Distance (Appendix C3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 104 105 Temperature (K) 50 Myr 10−4 10−3 10−2 10−1 100 HII fraction 128 256 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 HeI fraction 5 10 15 20 radius (Mpc) 10−4 10−3 10−2 10−1 100 HeIII fraction Figure B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Profiles at 50 Myr for reionization at z = 4 by a source LS,ν ∼ ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' All the quantities are converged in these (25 Mpc, 1283) and (25 Mpc, 2563) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' C1 Test problem: Str¨omgren sphere A Str¨omgren sphere is the final stage of an isotropically ex- panding ionization region with a central source in a uniform medium once ionizations are balanced by recombinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As a test problem, the Str¨omgren sphere simulation has a few key benefits: a) the solution is analytical, hence it is easy to check the accuracy of the results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' b) the solution is isotropic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' as a result, any artificial inhomogeneity caused by the algo- rithm is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The analytic solution for the radius of the ionisation front is: R(t) = RS � 1 − exp � − t trec �� 1 3 , (C1) where RS = (3 ˙Nγ/4πnHneα)1/3 is the final radius of the ionised region (the Str¨omgren radius), ˙Nγ is the photon emis- sion rate, nH is the hydrogen number density, ne is the elec- tron number density, α is the radiative recombination rate within the ionised region and trec = 1/neα is the recombina- tion time (assumed constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We adopt a similar parameter set to that used in Iliev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2006) to compare with the results therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Specifi- cally, the monochromatic source emits photons at the rate ˙Nγ = 5 × 1048 s−1, the simulation box size is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6 kpc and the total number of cells is 1283, the minimum ray angu- lar resolution parameter is Φc = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='1 (see the definition in Wise & Abel 2011), the number density of hydrogen atoms is nH = 10−3 cm−3 and the recombination rate is α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='59 × 10−13 cm3 s−1 at T = 104 K, leading to RS = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='4 kpc and trec = 122 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We note that in Iliev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (2006), the temperature is fixed at T = 104 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' however, ENZO does not support this setting due to its formulation of the inter- nal chemistry and energy solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' To make the simulation close to the analytical problem, we adjust the energy of the monochromatic photon to 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='26 eV and set the adiabatic in- dex to γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='667 to ensure that the gas temperature in the ionized region is close to T = 104 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This approximation and these parameters keep the maximum temperature deviation to within 20% in all Str¨omgren sphere simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This en- MNRAS 000, 1–19 (2022) Numerical methods for IGM reionization 19 2 3 4 5 rIF (kpc) fixed version original Enzo analytical 0 50 100 150 200 250 Time (Myr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='98 rIF/ranyl fixed version original Enzo Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In the upper panel, the black solid line shows the an- alytic solution for the time development of the ionization radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The red circles show the ionization radii computed by ENZO with our revisions, and the blue squares are the results from ENZO v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The lower panel shows the ratio of the computed and analytic so- lutions for the ionization radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The modifications significantly improve agreement with the analytic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' sures the Str¨omgren radius will be matched to better than 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Figure C1 shows the evolution of the ionization radius com- puted by our revised version of ENZO and by ENZO v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The result from our version very closely matches the analytical result, validating our revisions to ENZO v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' C2 He III Ionisation Adaptive Time Step Scheme We include the maximum changing rate of the He III frac- tion as an additional condition on the ray-tracing adaptive time step scheme in ENZO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' The original ray-tracing adaptive time step in ENZO is based on the maximum changing rate of the H II fraction (see Wise & Abel 2011, for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' However, during the helium reionization epoch, the hydrogen atoms in the IGM have already been fully ionised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As a con- sequence, hydrogen is optically thin to the ionising photons and the H II fraction changes very slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' By contrast, the ionisation fraction of He II is rapidly changing in this period, as the gas is initially optically thick to He II ionising photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Therefore, an additional restriction based on the maximum changing rate of He III is implemented in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' We also extend the applicability of the ray-tracing adap- tive time step scheme: the original algorithm of the time step calculator does not consider the influence caused by the ex- pansion of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This feature leads to overestimating the ray-tracing time steps in cosmological simulations, bring- ing artificial effects into cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' C3 Restriction on Photon Package Travel Distance The comoving light travel distance from the radiation sources is used to avoid photon packages from moving beyond the particle horizon of the photons in our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Sim- ilarly to other ray-tracing algorithms in cosmological simu- lation codes (Abel & Wandelt 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Iliev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' 2006), the ray-tracing algorithm in ENZO assumes the propagation speed of the photon package is infinite at every radiation transfer (RT) time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This approximation is used for the following reasons: a) It is computationally cheaper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In a simulation with a radiation source, where the propagation speed is finite, the total number of photon packages, which are stored in sys- tem memory (such as RAM) between two RT time steps, is proportional to the volume of the ionized region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Also, in such an algorithm, various information about a photon package, such as its position, direction, luminosity and birth time, are required to trace the photon package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Hence, assum- ing a finite propagation speed significantly increases the cost to system RAMs and makes the simulation computationally prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' On the other hand, in a simulation with an infi- nite speed of light, all the photon packages are generated and deleted at every RT time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Thus, the photon packages do not occupy any system RAM in between two RT time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' As a consequence, the infinite propagation speed approxima- tion is more practical from a technical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' This instantaneous RT approximation amounts to neglecting the time differential operator on the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' b) The expansion speed of the ionisation bubbles is gener- ally much slower than the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Also, most of the ionising photons associated with a ray are absorbed in the ionisation front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Therefore, when the particle horizon is dis- tant from the ionisation front, the influence caused by the infinite propagation speed approximation is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' Although assuming the infinite propagation speed is a safe approximation in most simulated situations, the particle hori- zon needs to be imposed as the maximum travel distance of rays in many situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' These include: the radius of the ionised bubble could be comparable to the particle horizon when the source just begins to shine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' in a large-scale simula- tion, the box size of the simulation could be larger than the particle horizon during the simulation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In these situ- ations, forbidding photon packages from transferring across their corresponding particle horizons also reduces the com- putation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' In our revised version of ENZO, we delete a photon packet if it reaches its particle horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} +page_content=' MNRAS 000, 1–19 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQflPjP/content/2301.00450v1.pdf'} diff --git a/mtE1T4oBgHgl3EQfhQRC/content/tmp_files/2301.03238v1.pdf.txt b/mtE1T4oBgHgl3EQfhQRC/content/tmp_files/2301.03238v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f313366886f3fceb85dc0da3a88ad9dc7b42bd3 --- /dev/null +++ b/mtE1T4oBgHgl3EQfhQRC/content/tmp_files/2301.03238v1.pdf.txt @@ -0,0 +1,1007 @@ +arXiv:2301.03238v1 [cs.CL] 9 Jan 2023 +MAQA: A Multimodal QA Benchmark for Negation +Judith Yue Li +Google Research +judithyueli@google.com +Aren Jansen +Google Research +arenjansen@google.com +Qingqing Huang +Google Research +qqhuang@google.com +Joonseok Lee +Google Research +& Seoul National University +joonseok@google.com +Ravi Ganti +Google Research +gmravi@google.com +Dima Kuzmin +Google Research +gmravi@google.com +Abstract +Multimodal learning can benefit from the representation power of pretrained Large +Language Models (LLMs). However, state-of-the-art transformer based LLMs of- +ten ignore negations in natural language and there is no existing benchmark to +quantitatively evaluate whether multimodal transformers inherit this weakness. In +this study, we present a new multimodal question answering (QA) benchmark +adapted from labeled music videos in AudioSet (Gemmeke et al., 2017) with the +goal of systematically evaluating if multimodal transformers can perform complex +reasoning to recognize new concepts as negation of previously learned concepts. +We show that with standard fine-tuning approach multimodal transformers are still +incapable of correctly interpreting negation irrespective of model size. However, +our experiments demonstrate that augmenting the original training task distribu- +tions with negated QA examples allow the model to reliably reason with negation. +To do this, we describe a novel data generation procedure that prompts the 540B- +parameter PaLM model to automatically generate negated QA examples as com- +positions of easily accessible video tags. The generated examples contain more +natural linguistic patterns and the gains compared to template-based task augmen- +tation approach are significant. +1 +Introduction +Large language models (LLMs) have difficulty understanding negation in natural language. Pre- +trained LLMs often ignore negation in cloze questions and give same prediction for negated ("Birds +cannot [MASK]") and non-negated ("Birds can [MASK]") queries (Kassner and Schütze, 2019; +Hosseini et al., 2021). Hossain et al. (2022) analyzed the training corpora of state-of-the-art LLMs +and found that negation is rarely present, leading to the poor handling of negation at inference time. +State-of-the-art multimodal learning leverages pretrained LLMs for fusing different modali- +ties (Jia et al., 2021; Radford et al., 2021; Oncescu et al., 2021; Kilgour et al., 2022; Nagrani et al., +2022). Will the fine-tuned LLMs intended for multimodal applications inherit the negation problem? +Huang et al. (2022) showed that the zero-shot performance on the text query based audio retrieval +task degrades when the text query includes negation cues, e.g., "no vocals". Yu et al. (2022) showed +that the text-to-image model generates items that are mentioned in the text prompt, even when the +NeurIPS 2022 Workshop on Synthetic Data for Empowering ML Research. + +prompt suggests the absence of the item. However, there is no benchmark for quantitatively evalua- +tion of how well negation patterns in the text are handled in such multimodal settings. +In this study, we created MAQA, a binary music audio question answering benchmark, to evalu- +ate how well the multimodal transformers understand negation in music related questions. This +benchmark is created from labeled videos in the music-related portion of AudioSet (Gemmeke et al., +2017). While the original benchmarks features 5000 hours of audios labeled with 527 audio event +classes and only contains a handful of labels including negation, the proposed benchmark MAQA +features a significant portion of negated questions that are generated programmatically from the orig- +inal audio labels. Our goal is to evaluate if multimodal transformer can be fine-tuned to understand +new concepts, e.g., “no vocals” as negation of the previously learned concept, e.g., “vocals” through +compositional generalization. +The main contributions of the paper are: (1) A compositional generalization experiment that demon- +strates standard fine-tuning prevents our baseline model, a multimodal transformer modified from +the multilingual T5 (MT5) (Raffel et al., 2019; Xue et al., 2021) from generalizing to new concepts +that are negation of learned concepts. (2) A PaLM-based data generation approach that automat- +ically generate negated QA examples from easily access video tags. (3) Two task augmentation +strategies that lead to a significant boost of the model performance on portion of MAQA with text +negation. +The rest of this paper is organized as follows. Section 2 provides relevant background and related +work on negation, compositional generalization and multimodal learning. Section 3 provides an +overview of the MAQA dataset and its statistics. Section 4 details how we create the benchmark +through data generation. The models and experiment results are presented in Section 5 and 6. The +paper closes with a discussion on the limitations, implications of our results and future work. +2 +Related Works +Negation. Despite improvements of LLMs in many NLP tasks such as natural language under- +standing, reading comprehension, zero-shot text generation, negation remains a challenge for pre- +trained LLMs (Kassner and Schütze, 2019; Hosseini et al., 2021). +Data augmentation has been +used to tackle negation in the NLP literature. For example, modification of the natural language +understanding corpora by adding negation to the verb or adjective and reversing the labels was +proposed in (Hossain et al., 2020), and an unlikelihood loss for the span corruption pre-training +tasks was proposed in (Hosseini et al., 2021). Negation is also addressed in the meta-learning litera- +ture (Murty et al., 2021), where it is treated as one of the reasoning categories that requires additional +few-shot classification tasks to augment the original task distribution. +Compositional Generalization. +Compositional Generalization refers to the ability to under- +stand novel concept as compositions of previously learned concept or atoms. Negation can be +thought as a form of composition. +In the field of semantic parsing, several benchmarks have +been proposed to evaluate compositional generalization (Lake and Baroni, 2018; Keysers et al., +2020; Kim and Linzen, 2020), which have encouraged development of techniques and architec- +tures to make LLMs better at solving compositional tasks (Furrer et al., 2020; Ontanon et al., +2022; Csordás et al., 2021; Qiu et al., 2022). +Several multimodal benchmarks have shown vi- +sually grounded LLMs often struggle with compositional generalization in visual reasoning +tasks (Johnson et al., 2017), visual grounded command following tasks (Ruis et al., 2020), text-to- +image matching (Zhang et al., 2021), etc. Our study focus on evaluating audio grounded LLMs on +compositional tasks involves negation. +Multimodal QA. Multimodal question answering benchmarks are used to probe the multimodal +models to evaluate their perception and reasoning capability on different modalities. +Visual +Question Answering benchmarks (Zhang et al., 2015; Agrawal et al., 2015) commonly consist of +triplets of (image, a natural language question about the image, answer), and the task is to an- +swer the question based on the visual cue in the image. +In the field of audio perception, au- +dio QA benchmarks (Fayek and Johnson, 2020) are less common than audio classification bench- +marks (Gemmeke et al., 2017). In the music domain, most benchmarks are music information re- +trieval tasks (Law and Von Ahn, 2009), where the text labels are usually in the format of short form +music tags. +2 + +Table 1: Examples of generated Binary Audio QA Pairs in MAQA. The original AudioSet example +is a music audio clip associated with the following tags: Bass guitar, Guitar, Acoustic guitar and +Strum. Questions and their negated counterpart are generated from the sampled attributes with the +PaLM based approach. Negative attributes steel guitar, slide guitar are sampled from the sibling +nodes in the AudioSet ontology. +Sampled Attributes +Question +Answer +Negated +Bass guitar (+) +Q1)The musical instrument played +TRUE +No +Bass guitar (+) +in the song is Bass guitar +TRUE +No +Bass guitar (+) +Q2)Bass guitar is not played in the song +FALSE +Yes +steel guitar, slide guitar (−) +Q3)The song has steel guitar or slide guitar +FALSE +No +steel guitar, slide guitar (−) +Q4)The song does not have slide guitar or steel guitar +TRUE +Yes +Table 2: Statistics on Music Audio QA (MAQA) Benchmark. +Statistics on Music Audio QA +(MAQA) Benchmark. Both evaluation sets ASBaseEval and ASNegationEval are generated by +PaLM. Each training set has a template-generated and a PaLM-generated version. All the datasets +have balanced binary label distributions. ASNegationEval contains ASBaseEval and its negated +counterparts. The music attributes have a simialr distribution in training and evaluation split. +Label Stats +# of QA Pairs +# of mentions +Data Version +True Negated +Non- +negated +Negated +Genre +Mood +Instrument +Role +ASBaseEval +50% 0% +17,028 +0 +5574 +730 +9740 +984 +ASNegationEval +50% 50% +17,028 +17,028 +(32.7%) (4.3%) +(57.2%) +(5.8%) +ASBaseTrain +50% 0% +1,263,004 +0 +439,904 28,634 +740,126 +54,340 +ASNegationTrain +50% 50% +1,263,004 +1,263,004 (34.8%) (2.3%) +(58.6%) +(4.3%) +Multimodal Transformers. A series of Transformer-based multimodal models (Sun et al., 2019; +Tan and Bansal, 2019; Lu et al., 2019), referred to as “Multimodal Transformers” in this study, ex- +plored using Transformer encoder as a join encoder for multimodal fusion achieve state-of-the-art +results on a range of multimodal QA tasks. Changpinyo et al. (2022) proposed a multimodal version +of T5 (Raffel et al., 2019). Given the image and the question in a VQA example, the multimodal +T5 takes the global and regional image features generated by a pre-trained visual encoder and text +tokens of the question as inputs, and solves a classification problem with pre-defined classes of an- +swers for the VQA task. The parameters of the visual encoder are frozen during T5 fine-tuning. We +follow the same approach but use pre-trained audio encoders (Gemmeke et al., 2017; Huang et al., +2022) to extract global representation of the music audio. A detailed survey of audio representation +can be found in Huang et al. (2022). +3 +Music Audio Question Answering (MAQA) +To evaluate the ability of multimodal models to reason with negation, we create a music au- +dio QA benchmark (MAQA) which emphasizes on correct understanding of text negation. The +music audio QA pairs are generated programmatically from the music related portion of Au- +dioSet (Gemmeke et al., 2017), which contains music audio clips annotated with music attributes +and an ontology describing their relationship. There are 388, 262 and 4, 497 unique music audio +clips in the train and evaluation split, respectively. Each clip is labeled with one or more music tags +out of the 141 unique music attributes covering music genres, music roles, music instruments and +music moods. +Table 1 presents an example in MAQA, which consists of four QA pairs generated from an example +of music attribute labeled audio clip in AudioSet. Q1 and Q2 are questions generated from the same +seed attribute, and essentially probe about the same musical skill, i.e., listen to a music audio and +try to identify if a bass guitar is played. Q2 is a negated form of Q1. If a model answer Q1 correctly +and fail on its negated counterpart Q2, it suggests that the model does not understand the negation +logic in the question and unable to perform compositional generalization. +MAQA contains two evaluation sets ASBaseEval and ASNegationEval and two training sets AS- +BaseTrain and ASNegationTrain as shown in Table 2 with balanced binary label distribution, featur- +ing QA pairs about music moods, genre, instrument and roles. ASBaseTrain / ASBaseEval contains +3 + +non-negated QA pairs about music audio recordings. ASNegationTrain / ASNegationEval is a su- +perset of ASBaseTrain / ASBaseEval, and it also includes their negated counterparts of the QA pairs. +A multimodal model with strong music audio understanding capabilities should score high on AS- +BaseEval. Moreover, to demonstrate its ability of reasoning about negation logic, it has to also score +high on ASNegationEval. +4 +Data Generation +Since music descriptive text that involves negation rarely occur in the standard text corpora +Hossain et al. (2022), we propose the following 3-step approach to programmatically generate bi- +nary audio QA pairs that involve text negation: 1. For each music audio-attribute pair in the original +dataset, we sample a negative attribute that is not associated with the audio clip. 2. Convert the +positive and the negative audio-attribute pair into a binary AQA example in the format of a triplet +(audio clip, question on the attribute, True / False label). 3. Perform a text negation on the question +and flip the binary label simultaneously to create negated audio QA pairs. As a first attempt we +curate MAQA from AudioSet with this method, however it can be applied to other datasets contain- +ing annotated music audios. Next, we discuss the details of how we followed the 3 steps to create +MAQA from AudioSet. +Negative Attribute Sampling. We adopt negative sampling to create a balanced binary label distri- +bution. In particular, we sample hard negative attributes using sibling nodes in the ontology tree and +assign False label to the derived audio QA pair. Consider the example in Table 1, the audio clip is +tagged with Bass guitar and Acoustic guitar, which are both under the parent node Guitar. We sam- +ple hard negative attributes steel guitar and slide guitar from the sibling nodes, to create a negative +audio-attribute pair. This hard negative sampling approach encourages the model to differentiate +related but different music concepts. +Question Generation. We explore the following two approaches to generate questions from the +audio-attribute pair sampled from the first step. The first approach is template based, and it takes +advantage of the AudioSet ontology, where each music attribute is associated with one of the four +attribute types: genres, roles, instruments, and moods. We use type-specific templates to convert at- +tributes into a true-or-false question, e.g., “The of the song is .”. +The second approach leverages the few-shot text generation capability of PaLM (Chowdhery et al., +2022) to improve the diversity of generated questions. Similar to GPT-3 (Brown et al., 2020), when +prompted with an instruction, e.g., “Generate a sentence about music given the music attribute”, +PaLM learns from a few demonstrations and generates questions on unseen attributes. +Task Augmentation with Negation. The template-based approach convert a question to the nega- +tion form by inserting a modifier not before the noun, i.e., “The of the song is not +.” and the binary label is flipped. One of the limitation of this approach is that it +is attribute type specific and only modifies nouns. PaLM based method overcomes the limitation as +with few shot learning the model can generate different negation patterns by modifying both nouns +and verbs. For example, the negation patterns associated with the instrument attribute “guitar” in- +clude “no guitar”, “guitar is not played”, and “the song does not feature bass guitar”. For each music +attribute, we use PaLM to generate a few question candidates and manually pick the best one. Row +2 and 4 in Table 1 are example questions generated in this way. More example questions generated +by PaLM and the prompts used are shown in Appendix 8.3. +5 +Multimodal Modeling +Following the VQA literature (Changpinyo et al., 2022; Zhang et al., 2015), we treat the audio +QA as a binary classification task. +We adopt a multimodal T5 architecture similar to that +in (Changpinyo et al., 2022) to fuse the audio and text inputs, and we replace T5 with its multi- +lingual version MT5. Each music audio clip input is represented as a 128-dimensional embedding +obtained either from VGGish (Gemmeke et al., 2017)1, which uses a VGG ConvNet architecture, +or the transformer based MuLan model (Huang et al., 2022). The audio encoders are frozen when +we finetune the multimodal T5. The audio embeddings are projected to the text token embedding +1https://github.com/tensorflow/models/tree/master/research/ +audioset/vggish +4 + +space through a linear projection layer, which is initialized randomly at the beginning of finetuning. +Then, the audio token and text token are fed into the pre-trained multi-layer MT5 (Xue et al., 2021) +encoder as a sequence of vectors and the final multimodal representation is classified into the binary +classes. The multimodal code is based on the Flaxformer framework2. Training details can be found +in Appendix 8.1. +6 +Experiments and Results +We report experiment results on the ASBaseEval and ASNegationEval evaluation benchmark in Ta- +ble 3 and Table 4 respectively. The Audio QA task is formulated as a binary classification problem, +and we report the best AUC-ROC score and the corresponding accuracy in the positive class. To +evaluate model’s ability to generalize compositionally so that it can understand composed music +concepts like “no vocals” that involve negation, we split the data into train and test sets following +the design recommended by (Keysers et al., 2020). By design the music attributes or atoms are simi- +larly represented in the train and test sets, while the test set contains novel combinations of the atoms +that are not seen in the train set. Compound Divergence (CD) is used to measure quantitatively how +different is the compound distributions in the train and test split (Keysers et al., 2020), while in our +case CD is used as a qualitative measure (Tabel 5 in Appendix 8.2), and compound refers to the +QA pairs after applying compositional rules, e.g., negation to the atoms. For each split scenario, we +compare the performance of finetuned multimodal transformer with different audio feature extrac- +tors, as well as with different sized pre-trained MT5 model. Furthermore, we vary the types of QA +pairs (template-based or PaLM-based) used in training split and study how compound divergence +affects learning negation. +6.1 +Music Understanding +Table 3(a) shows the result for the first split scenario where the model is trained and evaluated on +non-negated QA pairs generated by PaLM. This Low CD experiment establish a fine-tuning base- +line on basic music concepts. The fine-tuned multimodal MT5 score over 90% AUC-ROC on the +ASBaseEval benchmark that features Audio QA tasks on music styles, moods, genres, instruments, +etc. Recall the random baseline is 50% for balanced binary classification tasks, this suggests mul- +timodal transformer learn to efficiently fuse audio and text signals through fine-tuning, even it is +warm started from a text-only checkpoint. Probing the model on different music attributes suggests +that music concepts like “Scary music”, “Children music” and popular percussion instruments like +“Cowbell” are easy for the fine-tuned model to pick up, while the model has a harder time to under- +stand electronic music genres such as “Drum and bass”, “Trance music”. +We further replace the training examples generated by PaLM with the template-generated QA exam- +ples resulting in the Medium CD setting. The model scores around 6% lower in the Medium CD +setting (Table 3(b)) compared to the Low CD setting. This suggest the model can still transfer most +of the music knowledge learned in a different linguistic context via compositional generalization. +For both split scenarios the best multimodal model is the MT5-XL with Mulan embedding as audio +features. +6.2 +Reasoning with Negation +For the third split scenario (Table 3(b)) we apply the same fine-tuning setup as in Table 3(a) but +evaluate on ASNegationEval, where the non-negated half is from ASBaseEval and the other half +contains their negation counterparts. As shown in Table 4(a), the multimodal MT5 fine-tuned on +non-negated audio QA pairs (ASBaseTrain) scores only 50% on the ASNegationEval benchmark +in this High CD setting. Although the model still scores around 80% on the non-negated QAs +(comparable to the accuracy on ASBaseEval in Table 3), it scores only around 20% on their negated +counterparts. The model does worse than the 50% random guess baseline on these negated questions +after fine-tuning. This shows that while the model is trained to answer the non-negated questions +correctly, they also learn to “ignore” the negation cue in the negated questions. We also show that +increasing the model size does not improve the AUC-ROC score, suggesting that even larger model +fail to generalize compositionally using the standard fine-tuning approach. +2https://github.com/google/flaxformer +5 + +Table 3: Accuracy on ASBaseEval for two different Compound Divergence (CD) settings. +Finetuning Details +ASBaseEval +Model +Train Data - QA Type +CD Type +AUC +Acc +a) +MT5-Base+VGGish +ASBaseTrain-PaLM +Low CD +0.905 +0.821 +MT5-XL+VGGish +ASBaseTrain-PaLM +Low CD +0.911 +0.827 +MT5-Base+MuLan +ASBaseTrain-PaLM +Low CD +0.913 +0.828 +MT5-XL+MuLan +ASBaseTrain-PaLM +Low CD +0.918 +0.832 +b) +MT5-Base+VGGish +ASBaseTrain-Temp +Med CD +0.847 +0.771 +MT5-XL+VGGish +ASBaseTrain-Temp +Med CD +0.850 +0.765 +MT5-Base+MuLan +ASBaseTrain-Temp +Med CD +0.845 +0.766 +MT5-XL+MuLan +ASBaseTrain-Temp +Med CD +0.851 +0.767 +Table 4: Accuracy on ASNegationEval for three different Compound Divergence (CD) settings. +Finetuning Details +ASNegationEval +Acc +Model +Train Data - QA Type +CD Type +AUC +Avg +Neg +NoNeg +a) +MT5-Base+VGGish +ASBaseTrain-PaLM +High CD +0.524 +0.513 +0.218 +0.803 +MT5-XL+VGGish +ASBaseTrain-PaLM +High CD +0.525 +0.525 +0.247 +0.802 +MT5-Base+MuLan +ASBaseTrain-PaLM +High CD +0.553 +0.541 +0.273 +0.802 +MT5-XL+MuLan +ASBaseTrain-PaLM +High CD +0.528 +0.520 +0.220 +0.819 +b) +MT5-Base+VGGish +ASNegationTrain-PaLM +Low CD +0.896 +0.814 +0.814 +0.813 +MT5-XL+VGGish +ASNegationTrain-PaLM +Low CD +0.903 +0.821 +0.821 +0.821 +MT5-Base+MuLan +ASNegationTrain-PaLM +Low CD +0.905 +0.821 +0.821 +0.822 +MT5-XL+MuLan +ASNegationTrain-PaLM +Low CD +0.907 +0.825 +0.825 +0.825 +c) +MT5-Base+VGGish +ASNegationTrain-Temp +Med CD +0.784 +0.715 +0.690 +0.741 +MT5-XL+VGGish +ASNegationTrain-Temp +Med CD +0.823 +0.743 +0.724 +0.763 +MT5-Base+MuLan +ASNegationTrain-Temp +Med CD +0.805 +0.739 +0.723 +0.755 +MT5-XL+MuLan +ASNegationTrain-Temp +Med CD +0.828 +0.750 +0.740 +0.759 +6.3 +Task Augmentation +We then apply task augmentation during training by augmenting ASBaseTrain with negated QA +example generated by PaLM (AsNegationTrain-PaLM),which lower the compound divergence. The +task augmentation proves to be an effective strategy for tackling negation. As shown in Table 4(b), +multimodal MT5 fined-tuned with task augmentation improves the baseline on ASNegationEval +as shown in Table 4(a) by nearly 40%, while obtaining similar performance on the non-negated QA +pairs (ASBaseEval). The AUC-ROC score and accuracy is on par with the scores on the non-negated +Audio QA pairs (ASBaseEval), suggesting that task augmentation can indeed help the model to learn +to answer the questions with negation correctly. The best result on ASNegationEval is obtained by +fine-tuned MT5-XL with MuLan audio embedding. +6.4 +Template versus PaLM +We further explore how different task augmentation strategy affects the learning outcome. As shown +in Table 4(c), we use template-based approach for composing QA pairs and task augmentation, and +compare with the fine-tuning results with PaLM-generated QA pairs. The Template-based fine- +tuning scores around 10% lower in AUC score compared to PaLM-based fine-tuning. The observed +gap can be explained by the increased compound divergence between the training data and the eval- +uation data. The accuracy difference on the non-negated split is around 7% while the difference +on the negated split is around 10% to 12%. Recall that the template-based approach only modifies +the noun for negation while Palm-based approach incorporates more variations, which can explain +why the template-based fine-tuning performs worse on the negated split. This experiment has high- +lighted the importance of composing augmented tasks with natural linguistic variations that match +human language used in production environment. However, even Template-based task augmentation +can improve negation understanding significantly, on average 30% higher than training without task +augmentation (Table 4(a)). +6 + +7 +Conclusion +In this work, we propose a new Binary Audio QA benchmark MAQA in the music domain to probe +the state-of-the-art multimodal models on understanding negation. MAQA fills in the gap of lack- +ing negation-focused evaluation benchmark in the multimodal setting. Our experiments show that +standard fine-tuning prevents the multimodal transformers from generalizing to new concepts that +are negation of the learned concepts. While increasing the model size or adopting a better audio +encoder doesn’t help with negation, task augmentation allows the model to reason with negation by +providing more fine-tuning examples that contain negation. And LLMs like PaLM can be used to +generate negated examples with more natural linguistic variations, which have a significant effect +on the learning outcome. 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Association for Computational +Linguistics. +Jiahui Yu, Yuanzhong Xu, Jing Yu Koh, Thang Luong, Gunjan Baid, Zirui Wang, Vi- +jay Vasudevan, Alexander Ku, Yinfei Yang, Burcu Karagol Ayan, Ben Hutchinson, Wei +Han, +Zarana Parekh, +Xin Li, +Han Zhang Jason Baldridge, +and Yonghui Wu. 2022. +Scaling autoregressive models for content-rich text-to-image generation. ArXiv. +Bowen +Zhang, +Hexiang +Hu, +Linlu +Qiu, +Peter +Shaw, +and +Fei +Sha. +2021. +Visually grounded concept composition. +In Findings of the Association for Computational +Linguistics: EMNLP 2021, pages 201–215, Punta Cana, Dominican Republic. Association for +Computational Linguistics. +Peng Zhang, Yash Goyal, Douglas Summers-Stay, Dhruv Batra, and Devi Parikh. 2015. +Yin and Yang: Balancing and Answering Binary Visual Questions. arXiv. +8 +Appendix +8.1 +Training Details +For the MT5-base encoder there are 12 Transformer encoder layers and the input embedding di- +mension is 768. For MT5-XL there are 24 Transformer encoder layers and the input embedding +dimension is 2048. MT5-XL has 3.7 billion parameters and MT5-base has 580 million parameters. +The batch size is 64 for MT5-base model and 128 for MT5-XL model. We use fixed a learning +rate among {10−3, 10−4, 5−5} and observe 1 × 10−3 works best in general. The model outputs 2- +dimensional logits as the Audio QA task is formulated as binary classification. We train all models +9 + +Attribute +Train Examples +Test Examples +CD Type +(a) Without Task Augmentation +Banjo +The song is played with a banjo. +The song is played with a banjo. +Low CD +(ASBaseTrain-PaLM) +(ASBaseEval-PaLM) +Blues +The [genre] of the song is [blues]. +The song is a blues song. +Med CD +(ASBaseTrain-Temp) +(ASBaseEval-PaLM) +Scary +The music is scary. +The music is scary. +High CD +The music is not scary. +(ASBaseTrain-PaLM) +(ASNegEval-PaLM) +(b) With Task Augmentation (ASNegationTrain) +Chant +The music is a Chant. +The music is a Chant. +Low CD +The music is not a Chant. +The music is not a Chant. +(ASNegTrain-PaLM) +(ASBaseTrain-PaLM) +Dance +The [music role] of the song is +The music is suitable for dancing. +Med CD +music +[Dance music]. +The [music role] of the song is not +The music is not suitable for dancing. +[Dance music]. +(ASNegTrain-Temp) +(ASBaseTrain-PaLM) +Table 5: The train and test split design of MAQA. For each of the 5 split scenarios we list a few +example questions included in the train and test split. All the QA examples or compounds are com- +posed from the seed music attributes or atoms via Template-based (Temp) or PaLM-based (PaLM) +approach. Compound Divergence (CD) Type is used to measure the difference between the train and +test compound distribution. Task Augmentation is applied during training for the ASNegationEval- +LowCD and ASNegationEval-MedCD split scenario. +with data parallelism using 16 Cloud TPU Pods. For all experiments we run for 50, 000 steps and +reports the AUCROC and Accuracy based on the best checkpoint measured by AUCROC. For each +experiment we pick the highest AUCROC of multiple runs. It takes around 1 hour for MT5-Base +and around 4 hours for MT5-XL. +8.2 +Compound Divergence +As PaLM-generated composition does not depend on rules to combine different atoms, hence it’s +hard to compute CD directly. Here we use Compound Divergence as a qualitative instead of quanti- +tative measure of the difference of composition in the train and test split as shown in Table 5. +8.3 +Text Prompts used for PaLM +After a few iterations, we’ve found the text prompts shown in Table 6 works best for generating +negated and non-negated QA pairs. The one-line instruction at the beginning of the prompt is crucial +to guide PaLM to generate desired sentences. Table 7 shows more PaLM-generated questions. The +total number of seed attributes used for text generation are 130, and for each attribute we pick one +non-negated and one negated question selected from the candidate questions generated from PaLM. +10 + +User Input (Music Tag): Trumpet +Model Input: +Generate a sentence about music and its negation given the music tag +music tag: drum +describe the music: The music is played with a drum +negate the description: drum is not played in the song +music tag: rock +describe the music: This is a rock song +negate the description: the genre for the song is not rock +music tag: happy +describe the music: The music makes one feel happy +negate the description: The mood of the song is not happy +music tag: Choir +describe the music: The song features a choir +negate the description: The song does not feature a choir +music tag: Tabla +describe the music: Tabla is played in the song +negate the description: Tabla is not played in the song +music tag: Trumpet +describe the music: +negate the description: +Model Output: +: A trumpet is played in this song +: A trumpet is not played in this song +Table 6: Text prompts used for PaLM to generate non-negated and negated sentences from a seed +music attribute. Here we show a example output for the attribute Trumpet. +11 + +Seed Attribute +Generated Questions +Generated Negated Questions +A capella +The song is sung without any musical +instruments +The song is not sung without any musical instru- +ments +Accordion +The musical instrument of the song is +accordion +accordion is not played in the song +Acoustic guitar +The song is played with an acoustic gui- +tar +The song is not played with an acoustic guitar +Ambient music +The song is ambient music +The song is not ambient music +Angry music +The song makes one feel angry +The mood of the song is not angry +Background music +The song is background music +The song is not background music +Bagpipes +The musical instrument of the song is +Bagpipes +Bagpipes is not played in the song +Banjo +The song is played with a banjo +The song is not played with a banjo +Bass drum +The musical instrument of the song is +Bass drum +Bass drum is not played in the song +Bass guitar +The musical instrument of the song is +Bass guitar +Bass guitar is not played in the song +Beatboxing +Beatboxing features in the song +Beatboxing does not feature in the song +Bell +The musical instrument of the song is +bell +bell is not played in the song +Bluegrass +The genre of the song is bluegrass +bluegrass is not the genre of the song +Blues +The song is a blues song +the song is not a blues song +Bowed string instrument +The musical instrument of the song is a +bowed string instrument +The music is not played with a bowed string in- +strument +Brass instrument +The musical instrument of the song is +brass +brass is not played in the song +Carnatic music +The music is Carnatic +The music is not Carnatic +Cello +The musical instrument of the song is +Cello +Cello is not played in the song +Chant +The music is a chant +The music is not a chant +Choir +The song features a choir +The song does not feature a choir +Christian music +It is Christian music +It is not Christian music +Christmas music +The music is Christmas music +The music is not Christmas music +Clarinet +The song is played by clarinet +clarinet is not played in the song +Classical music +The music is classical +The genre for the song is not classical +Country +The genre of the song is Country +This is not a Country song +Cowbell +The song has cowbell +The song does not have cowbell +Cymbal +The musical instrument of the song is +Cymbal +Cymbal is not played in the song +Dance music +The music is suitable for dancing +The music is not suitable for dancing +Didgeridoo +The music is played on a didgeridoo +The music is not played on a didgeridoo +Disco +The song belongs to the disco genre +The song does not belong to the disco genre +Double bass +The musical instrument of the song is +double bass +double bass is not played in the song +Drum and bass +The music is drum and bass +The song is not drum and bass +Table 7: More examples of PaLM-generated non-negated and negated questions. +12 + diff --git a/mtE1T4oBgHgl3EQfhQRC/content/tmp_files/load_file.txt b/mtE1T4oBgHgl3EQfhQRC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d2b065e214d3e0a1c1b0f7fa6e63ab040f861072 --- /dev/null +++ b/mtE1T4oBgHgl3EQfhQRC/content/tmp_files/load_file.txt @@ -0,0 +1,739 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf,len=738 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='03238v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='CL] 9 Jan 2023 MAQA: A Multimodal QA Benchmark for Negation Judith Yue Li Google Research judithyueli@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='com Aren Jansen Google Research arenjansen@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='com Qingqing Huang Google Research qqhuang@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='com Joonseok Lee Google Research & Seoul National University joonseok@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='com Ravi Ganti Google Research gmravi@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='com Dima Kuzmin Google Research gmravi@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='com Abstract Multimodal learning can benefit from the representation power of pretrained Large Language Models (LLMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' However, state-of-the-art transformer based LLMs of- ten ignore negations in natural language and there is no existing benchmark to quantitatively evaluate whether multimodal transformers inherit this weakness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' In this study, we present a new multimodal question answering (QA) benchmark adapted from labeled music videos in AudioSet (Gemmeke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2017) with the goal of systematically evaluating if multimodal transformers can perform complex reasoning to recognize new concepts as negation of previously learned concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' We show that with standard fine-tuning approach multimodal transformers are still incapable of correctly interpreting negation irrespective of model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' However, our experiments demonstrate that augmenting the original training task distribu- tions with negated QA examples allow the model to reliably reason with negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' To do this, we describe a novel data generation procedure that prompts the 540B- parameter PaLM model to automatically generate negated QA examples as com- positions of easily accessible video tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The generated examples contain more natural linguistic patterns and the gains compared to template-based task augmen- tation approach are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 1 Introduction Large language models (LLMs) have difficulty understanding negation in natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Pre- trained LLMs often ignore negation in cloze questions and give same prediction for negated ("Birds cannot [MASK]") and non-negated ("Birds can [MASK]") queries (Kassner and Schütze, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Hosseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Hossain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' (2022) analyzed the training corpora of state-of-the-art LLMs and found that negation is rarely present, leading to the poor handling of negation at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' State-of-the-art multimodal learning leverages pretrained LLMs for fusing different modali- ties (Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Oncescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Kilgour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Nagrani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Will the fine-tuned LLMs intended for multimodal applications inherit the negation problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' (2022) showed that the zero-shot performance on the text query based audio retrieval task degrades when the text query includes negation cues, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', "no vocals".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' (2022) showed that the text-to-image model generates items that are mentioned in the text prompt, even when the NeurIPS 2022 Workshop on Synthetic Data for Empowering ML Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' prompt suggests the absence of the item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' However, there is no benchmark for quantitatively evalua- tion of how well negation patterns in the text are handled in such multimodal settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' In this study, we created MAQA, a binary music audio question answering benchmark, to evalu- ate how well the multimodal transformers understand negation in music related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' This benchmark is created from labeled videos in the music-related portion of AudioSet (Gemmeke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' While the original benchmarks features 5000 hours of audios labeled with 527 audio event classes and only contains a handful of labels including negation, the proposed benchmark MAQA features a significant portion of negated questions that are generated programmatically from the orig- inal audio labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Our goal is to evaluate if multimodal transformer can be fine-tuned to understand new concepts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', “no vocals” as negation of the previously learned concept, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', “vocals” through compositional generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The main contributions of the paper are: (1) A compositional generalization experiment that demon- strates standard fine-tuning prevents our baseline model, a multimodal transformer modified from the multilingual T5 (MT5) (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2021) from generalizing to new concepts that are negation of learned concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' (2) A PaLM-based data generation approach that automat- ically generate negated QA examples from easily access video tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' (3) Two task augmentation strategies that lead to a significant boost of the model performance on portion of MAQA with text negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Section 2 provides relevant background and related work on negation, compositional generalization and multimodal learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Section 3 provides an overview of the MAQA dataset and its statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Section 4 details how we create the benchmark through data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The models and experiment results are presented in Section 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The paper closes with a discussion on the limitations, implications of our results and future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 2 Related Works Negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Despite improvements of LLMs in many NLP tasks such as natural language under- standing, reading comprehension, zero-shot text generation, negation remains a challenge for pre- trained LLMs (Kassner and Schütze, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Hosseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Data augmentation has been used to tackle negation in the NLP literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' For example, modification of the natural language understanding corpora by adding negation to the verb or adjective and reversing the labels was proposed in (Hossain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2020), and an unlikelihood loss for the span corruption pre-training tasks was proposed in (Hosseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Negation is also addressed in the meta-learning litera- ture (Murty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2021), where it is treated as one of the reasoning categories that requires additional few-shot classification tasks to augment the original task distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Compositional Generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Compositional Generalization refers to the ability to under- stand novel concept as compositions of previously learned concept or atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Negation can be thought as a form of composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' In the field of semantic parsing, several benchmarks have been proposed to evaluate compositional generalization (Lake and Baroni, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Keysers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Kim and Linzen, 2020), which have encouraged development of techniques and architec- tures to make LLMs better at solving compositional tasks (Furrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Ontanon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Csordás et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Several multimodal benchmarks have shown vi- sually grounded LLMs often struggle with compositional generalization in visual reasoning tasks (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2017), visual grounded command following tasks (Ruis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2020), text-to- image matching (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2021), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Our study focus on evaluating audio grounded LLMs on compositional tasks involves negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Multimodal QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Multimodal question answering benchmarks are used to probe the multimodal models to evaluate their perception and reasoning capability on different modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Visual Question Answering benchmarks (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Agrawal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2015) commonly consist of triplets of (image, a natural language question about the image, answer), and the task is to an- swer the question based on the visual cue in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' In the field of audio perception, au- dio QA benchmarks (Fayek and Johnson, 2020) are less common than audio classification bench- marks (Gemmeke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' In the music domain, most benchmarks are music information re- trieval tasks (Law and Von Ahn, 2009), where the text labels are usually in the format of short form music tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 2 Table 1: Examples of generated Binary Audio QA Pairs in MAQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The original AudioSet example is a music audio clip associated with the following tags: Bass guitar, Guitar, Acoustic guitar and Strum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Questions and their negated counterpart are generated from the sampled attributes with the PaLM based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Negative attributes steel guitar, slide guitar are sampled from the sibling nodes in the AudioSet ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Sampled Attributes Question Answer Negated Bass guitar (+) Q1)The musical instrument played TRUE No Bass guitar (+) in the song is Bass guitar TRUE No Bass guitar (+) Q2)Bass guitar is not played in the song FALSE Yes steel guitar, slide guitar (−) Q3)The song has steel guitar or slide guitar FALSE No steel guitar, slide guitar (−) Q4)The song does not have slide guitar or steel guitar TRUE Yes Table 2: Statistics on Music Audio QA (MAQA) Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Statistics on Music Audio QA (MAQA) Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Both evaluation sets ASBaseEval and ASNegationEval are generated by PaLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Each training set has a template-generated and a PaLM-generated version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' All the datasets have balanced binary label distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' ASNegationEval contains ASBaseEval and its negated counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The music attributes have a simialr distribution in training and evaluation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Label Stats # of QA Pairs # of mentions Data Version True Negated Non- negated Negated Genre Mood Instrument Role ASBaseEval 50% 0% 17,028 0 5574 730 9740 984 ASNegationEval 50% 50% 17,028 17,028 (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='7%) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='3%) (57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='2%) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='8%) ASBaseTrain 50% 0% 1,263,004 0 439,904 28,634 740,126 54,340 ASNegationTrain 50% 50% 1,263,004 1,263,004 (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='8%) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='3%) (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='6%) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='3%) Multimodal Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' A series of Transformer-based multimodal models (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Tan and Bansal, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2019), referred to as “Multimodal Transformers” in this study, ex- plored using Transformer encoder as a join encoder for multimodal fusion achieve state-of-the-art results on a range of multimodal QA tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Changpinyo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' (2022) proposed a multimodal version of T5 (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Given the image and the question in a VQA example, the multimodal T5 takes the global and regional image features generated by a pre-trained visual encoder and text tokens of the question as inputs, and solves a classification problem with pre-defined classes of an- swers for the VQA task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The parameters of the visual encoder are frozen during T5 fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' We follow the same approach but use pre-trained audio encoders (Gemmeke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2022) to extract global representation of the music audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' A detailed survey of audio representation can be found in Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 3 Music Audio Question Answering (MAQA) To evaluate the ability of multimodal models to reason with negation, we create a music au- dio QA benchmark (MAQA) which emphasizes on correct understanding of text negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The music audio QA pairs are generated programmatically from the music related portion of Au- dioSet (Gemmeke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2017), which contains music audio clips annotated with music attributes and an ontology describing their relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' There are 388, 262 and 4, 497 unique music audio clips in the train and evaluation split, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Each clip is labeled with one or more music tags out of the 141 unique music attributes covering music genres, music roles, music instruments and music moods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Table 1 presents an example in MAQA, which consists of four QA pairs generated from an example of music attribute labeled audio clip in AudioSet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Q1 and Q2 are questions generated from the same seed attribute, and essentially probe about the same musical skill, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', listen to a music audio and try to identify if a bass guitar is played.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Q2 is a negated form of Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' If a model answer Q1 correctly and fail on its negated counterpart Q2, it suggests that the model does not understand the negation logic in the question and unable to perform compositional generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' MAQA contains two evaluation sets ASBaseEval and ASNegationEval and two training sets AS- BaseTrain and ASNegationTrain as shown in Table 2 with balanced binary label distribution, featur- ing QA pairs about music moods, genre, instrument and roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' ASBaseTrain / ASBaseEval contains 3 non-negated QA pairs about music audio recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' ASNegationTrain / ASNegationEval is a su- perset of ASBaseTrain / ASBaseEval, and it also includes their negated counterparts of the QA pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' A multimodal model with strong music audio understanding capabilities should score high on AS- BaseEval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Moreover, to demonstrate its ability of reasoning about negation logic, it has to also score high on ASNegationEval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 4 Data Generation Since music descriptive text that involves negation rarely occur in the standard text corpora Hossain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' (2022), we propose the following 3-step approach to programmatically generate bi- nary audio QA pairs that involve text negation: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' For each music audio-attribute pair in the original dataset, we sample a negative attribute that is not associated with the audio clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Convert the positive and the negative audio-attribute pair into a binary AQA example in the format of a triplet (audio clip, question on the attribute, True / False label).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Perform a text negation on the question and flip the binary label simultaneously to create negated audio QA pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' As a first attempt we curate MAQA from AudioSet with this method, however it can be applied to other datasets contain- ing annotated music audios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Next, we discuss the details of how we followed the 3 steps to create MAQA from AudioSet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Negative Attribute Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' We adopt negative sampling to create a balanced binary label distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' In particular, we sample hard negative attributes using sibling nodes in the ontology tree and assign False label to the derived audio QA pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Consider the example in Table 1, the audio clip is tagged with Bass guitar and Acoustic guitar, which are both under the parent node Guitar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' We sam- ple hard negative attributes steel guitar and slide guitar from the sibling nodes, to create a negative audio-attribute pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' This hard negative sampling approach encourages the model to differentiate related but different music concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Question Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' We explore the following two approaches to generate questions from the audio-attribute pair sampled from the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The first approach is template based, and it takes advantage of the AudioSet ontology, where each music attribute is associated with one of the four attribute types: genres, roles, instruments, and moods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' We use type-specific templates to convert at- tributes into a true-or-false question, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', “The of the song is .”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The second approach leverages the few-shot text generation capability of PaLM (Chowdhery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2022) to improve the diversity of generated questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Similar to GPT-3 (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2020), when prompted with an instruction, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', “Generate a sentence about music given the music attribute”, PaLM learns from a few demonstrations and generates questions on unseen attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Task Augmentation with Negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The template-based approach convert a question to the nega- tion form by inserting a modifier not before the noun, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', “The of the song is not .” and the binary label is flipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' One of the limitation of this approach is that it is attribute type specific and only modifies nouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' PaLM based method overcomes the limitation as with few shot learning the model can generate different negation patterns by modifying both nouns and verbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' For example, the negation patterns associated with the instrument attribute “guitar” in- clude “no guitar”, “guitar is not played”, and “the song does not feature bass guitar”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' For each music attribute, we use PaLM to generate a few question candidates and manually pick the best one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Row 2 and 4 in Table 1 are example questions generated in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' More example questions generated by PaLM and the prompts used are shown in Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 5 Multimodal Modeling Following the VQA literature (Changpinyo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2015), we treat the audio QA as a binary classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' We adopt a multimodal T5 architecture similar to that in (Changpinyo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2022) to fuse the audio and text inputs, and we replace T5 with its multi- lingual version MT5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Each music audio clip input is represented as a 128-dimensional embedding obtained either from VGGish (Gemmeke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2017)1, which uses a VGG ConvNet architecture, or the transformer based MuLan model (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The audio encoders are frozen when we finetune the multimodal T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The audio embeddings are projected to the text token embedding 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='com/tensorflow/models/tree/master/research/ audioset/vggish 4 space through a linear projection layer, which is initialized randomly at the beginning of finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Then, the audio token and text token are fed into the pre-trained multi-layer MT5 (Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2021) encoder as a sequence of vectors and the final multimodal representation is classified into the binary classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The multimodal code is based on the Flaxformer framework2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Training details can be found in Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 6 Experiments and Results We report experiment results on the ASBaseEval and ASNegationEval evaluation benchmark in Ta- ble 3 and Table 4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The Audio QA task is formulated as a binary classification problem, and we report the best AUC-ROC score and the corresponding accuracy in the positive class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' To evaluate model’s ability to generalize compositionally so that it can understand composed music concepts like “no vocals” that involve negation, we split the data into train and test sets following the design recommended by (Keysers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' By design the music attributes or atoms are simi- larly represented in the train and test sets, while the test set contains novel combinations of the atoms that are not seen in the train set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Compound Divergence (CD) is used to measure quantitatively how different is the compound distributions in the train and test split (Keysers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', 2020), while in our case CD is used as a qualitative measure (Tabel 5 in Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='2), and compound refers to the QA pairs after applying compositional rules, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=', negation to the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' For each split scenario, we compare the performance of finetuned multimodal transformer with different audio feature extrac- tors, as well as with different sized pre-trained MT5 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Furthermore, we vary the types of QA pairs (template-based or PaLM-based) used in training split and study how compound divergence affects learning negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='1 Music Understanding Table 3(a) shows the result for the first split scenario where the model is trained and evaluated on non-negated QA pairs generated by PaLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' This Low CD experiment establish a fine-tuning base- line on basic music concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The fine-tuned multimodal MT5 score over 90% AUC-ROC on the ASBaseEval benchmark that features Audio QA tasks on music styles, moods, genres, instruments, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Recall the random baseline is 50% for balanced binary classification tasks, this suggests mul- timodal transformer learn to efficiently fuse audio and text signals through fine-tuning, even it is warm started from a text-only checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Probing the model on different music attributes suggests that music concepts like “Scary music”, “Children music” and popular percussion instruments like “Cowbell” are easy for the fine-tuned model to pick up, while the model has a harder time to under- stand electronic music genres such as “Drum and bass”, “Trance music”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' We further replace the training examples generated by PaLM with the template-generated QA exam- ples resulting in the Medium CD setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The model scores around 6% lower in the Medium CD setting (Table 3(b)) compared to the Low CD setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' This suggest the model can still transfer most of the music knowledge learned in a different linguistic context via compositional generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' For both split scenarios the best multimodal model is the MT5-XL with Mulan embedding as audio features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='2 Reasoning with Negation For the third split scenario (Table 3(b)) we apply the same fine-tuning setup as in Table 3(a) but evaluate on ASNegationEval, where the non-negated half is from ASBaseEval and the other half contains their negation counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' As shown in Table 4(a), the multimodal MT5 fine-tuned on non-negated audio QA pairs (ASBaseTrain) scores only 50% on the ASNegationEval benchmark in this High CD setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Although the model still scores around 80% on the non-negated QAs (comparable to the accuracy on ASBaseEval in Table 3), it scores only around 20% on their negated counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The model does worse than the 50% random guess baseline on these negated questions after fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' This shows that while the model is trained to answer the non-negated questions correctly, they also learn to “ignore” the negation cue in the negated questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' We also show that increasing the model size does not improve the AUC-ROC score, suggesting that even larger model fail to generalize compositionally using the standard fine-tuning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='com/google/flaxformer 5 Table 3: Accuracy on ASBaseEval for two different Compound Divergence (CD) settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Finetuning Details ASBaseEval Model Train Data - QA Type CD Type AUC Acc a) MT5-Base+VGGish ASBaseTrain-PaLM Low CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='905 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='821 MT5-XL+VGGish ASBaseTrain-PaLM Low CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='827 MT5-Base+MuLan ASBaseTrain-PaLM Low CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='828 MT5-XL+MuLan ASBaseTrain-PaLM Low CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='832 b) MT5-Base+VGGish ASBaseTrain-Temp Med CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='771 MT5-XL+VGGish ASBaseTrain-Temp Med CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='765 MT5-Base+MuLan ASBaseTrain-Temp Med CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='766 MT5-XL+MuLan ASBaseTrain-Temp Med CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='851 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='767 Table 4: Accuracy on ASNegationEval for three different Compound Divergence (CD) settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Finetuning Details ASNegationEval Acc Model Train Data - QA Type CD Type AUC Avg Neg NoNeg a) MT5-Base+VGGish ASBaseTrain-PaLM High CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='524 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='803 MT5-XL+VGGish ASBaseTrain-PaLM High CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='802 MT5-Base+MuLan ASBaseTrain-PaLM High CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='802 MT5-XL+MuLan ASBaseTrain-PaLM High CD 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='763 MT5-Base+MuLan ASNegationTrain-Temp Med CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='739 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='755 MT5-XL+MuLan ASNegationTrain-Temp Med CD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='828 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='740 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='759 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='3 Task Augmentation We then apply task augmentation during training by augmenting ASBaseTrain with negated QA example generated by PaLM (AsNegationTrain-PaLM),which lower the compound divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The task augmentation proves to be an effective strategy for tackling negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' As shown in Table 4(b), multimodal MT5 fined-tuned with task augmentation improves the baseline on ASNegationEval as shown in Table 4(a) by nearly 40%, while obtaining similar performance on the non-negated QA pairs (ASBaseEval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The AUC-ROC score and accuracy is on par with the scores on the non-negated Audio QA pairs (ASBaseEval), suggesting that task augmentation can indeed help the model to learn to answer the questions with negation correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The best result on ASNegationEval is obtained by fine-tuned MT5-XL with MuLan audio embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='4 Template versus PaLM We further explore how different task augmentation strategy affects the learning outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' As shown in Table 4(c), we use template-based approach for composing QA pairs and task augmentation, and compare with the fine-tuning results with PaLM-generated QA pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The Template-based fine- tuning scores around 10% lower in AUC score compared to PaLM-based fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The observed gap can be explained by the increased compound divergence between the training data and the eval- uation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The accuracy difference on the non-negated split is around 7% while the difference on the negated split is around 10% to 12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Recall that the template-based approach only modifies the noun for negation while Palm-based approach incorporates more variations, which can explain why the template-based fine-tuning performs worse on the negated split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' This experiment has high- lighted the importance of composing augmented tasks with natural linguistic variations that match human language used in production environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' However, even Template-based task augmentation can improve negation understanding significantly, on average 30% higher than training without task augmentation (Table 4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 6 7 Conclusion In this work, we propose a new Binary Audio QA benchmark MAQA in the music domain to probe the state-of-the-art multimodal models on understanding negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' MAQA fills in the gap of lack- ing negation-focused evaluation benchmark in the multimodal setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Our experiments show that standard fine-tuning prevents the multimodal transformers from generalizing to new concepts that are negation of the learned concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' While increasing the model size or adopting a better audio encoder doesn’t help with negation, task augmentation allows the model to reason with negation by providing more fine-tuning examples that contain negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' And LLMs like PaLM can be used to generate negated examples with more natural linguistic variations, which have a significant effect on the learning outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' With the MAQA benchmark, we hope to encourage multimodal research community to develop new modeling frameworks or algorithms to handle complex natural language instructions that involves negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' We plan to release the MAQA dataset on Github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' References Aishwarya Agrawal, Jiasen Lu, Stanislaw Antol, Margaret Mitchell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Lawrence Zitnick, Devi Parikh, and Dhruv Batra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Vqa: Visual question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' International Journal of Com- puter Vision, 123:4–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Tom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhari- wal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Henighan, Rewon Child, Aditya Ramesh, Daniel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Ziegler, Jeff Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Language models are few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' ArXiv, abs/2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='14165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Soravit Changpinyo, Doron Kukliansky, Idan Szpektor, Xi Chen, Nan Ding, and Radu Soricut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' All you may need for vqa are image captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' ArXiv, abs/2205.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' A benchmark for systematic generalization in grounded language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Advances in neural information processing systems, 33:19861–19872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Chen Sun, Austin Myers, Carl Vondrick, Kevin Murphy, and Cordelia Schmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Videobert: A joint model for video and language representation learning.' metadata={'source': 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+page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Peng Zhang, Yash Goyal, Douglas Summers-Stay, Dhruv Batra, and Devi Parikh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Yin and Yang: Balancing and Answering Binary Visual Questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 8 Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='1 Training Details For the MT5-base encoder there are 12 Transformer encoder layers and the input embedding di- mension is 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' For MT5-XL there are 24 Transformer encoder layers and the input embedding dimension is 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' MT5-XL has 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='7 billion parameters and MT5-base has 580 million parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The batch size is 64 for MT5-base model and 128 for MT5-XL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' We use fixed a learning rate among {10−3, 10−4, 5−5} and observe 1 × 10−3 works best in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The model outputs 2- dimensional logits as the Audio QA task is formulated as binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' We train all models 9 Attribute Train Examples Test Examples CD Type (a) Without Task Augmentation Banjo The song is played with a banjo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The song is played with a banjo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Low CD (ASBaseTrain-PaLM) (ASBaseEval-PaLM) Blues The [genre] of the song is [blues].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The song is a blues song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Med CD (ASBaseTrain-Temp) (ASBaseEval-PaLM) Scary The music is scary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The music is scary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' High CD The music is not scary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' (ASBaseTrain-PaLM) (ASNegEval-PaLM) (b) With Task Augmentation (ASNegationTrain) Chant The music is a Chant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The music is a Chant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Low CD The music is not a Chant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The music is not a Chant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' (ASNegTrain-PaLM) (ASBaseTrain-PaLM) Dance The [music role] of the song is The music is suitable for dancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Med CD music [Dance music].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The [music role] of the song is not The music is not suitable for dancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' [Dance music].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' (ASNegTrain-Temp) (ASBaseTrain-PaLM) Table 5: The train and test split design of MAQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' For each of the 5 split scenarios we list a few example questions included in the train and test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' All the QA examples or compounds are com- posed from the seed music attributes or atoms via Template-based (Temp) or PaLM-based (PaLM) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Compound Divergence (CD) Type is used to measure the difference between the train and test compound distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Task Augmentation is applied during training for the ASNegationEval- LowCD and ASNegationEval-MedCD split scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' with data parallelism using 16 Cloud TPU Pods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' For all experiments we run for 50, 000 steps and reports the AUCROC and Accuracy based on the best checkpoint measured by AUCROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' For each experiment we pick the highest AUCROC of multiple runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' It takes around 1 hour for MT5-Base and around 4 hours for MT5-XL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='2 Compound Divergence As PaLM-generated composition does not depend on rules to combine different atoms, hence it’s hard to compute CD directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Here we use Compound Divergence as a qualitative instead of quanti- tative measure of the difference of composition in the train and test split as shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='3 Text Prompts used for PaLM After a few iterations, we’ve found the text prompts shown in Table 6 works best for generating negated and non-negated QA pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The one-line instruction at the beginning of the prompt is crucial to guide PaLM to generate desired sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Table 7 shows more PaLM-generated questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' The total number of seed attributes used for text generation are 130, and for each attribute we pick one non-negated and one negated question selected from the candidate questions generated from PaLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='User Input (Music Tag): Trumpet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Model Input: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Generate a sentence about music and its negation given the music tag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='music tag: drum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='describe the music: The music is played with a drum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='negate the description: drum is not played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='music tag: rock ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='describe the music: This is a rock song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='negate the description: the genre for the song is not rock ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='music tag: happy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='describe the music: The music makes one feel happy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='negate the description: The mood of the song is not happy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='music tag: Choir ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='describe the music: The song features a choir ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='negate the description: The song does not feature a choir ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='music tag: Tabla ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='describe the music: Tabla is played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='negate the description: Tabla is not played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='music tag: Trumpet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='describe the music: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='negate the description: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Model Output: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=': A trumpet is played in this song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=': A trumpet is not played in this song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Table 6: Text prompts used for PaLM to generate non-negated and negated sentences from a seed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='music attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' Here we show a example output for the attribute Trumpet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Seed Attribute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Generated Questions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Generated Negated Questions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='A capella ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is sung without any musical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='instruments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is not sung without any musical instru- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='ments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Accordion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The musical instrument of the song is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='accordion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='accordion is not played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Acoustic guitar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is played with an acoustic gui- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='tar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is not played with an acoustic guitar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Ambient music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is ambient music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is not ambient music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Angry music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song makes one feel angry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The mood of the song is not angry ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Background music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is background music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is not background music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Bagpipes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The musical instrument of the song is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Bagpipes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Bagpipes is not played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Banjo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is played with a banjo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is not played with a banjo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Bass drum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The musical instrument of the song is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Bass drum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Bass drum is not played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Bass guitar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The musical instrument of the song is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Bass guitar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Bass guitar is not played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Beatboxing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Beatboxing features in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Beatboxing does not feature in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Bell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The musical instrument of the song is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='bell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='bell is not played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Bluegrass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The genre of the song is bluegrass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='bluegrass is not the genre of the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Blues ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is a blues song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='the song is not a blues song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Bowed string instrument ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The musical instrument of the song is a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='bowed string instrument ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is not played with a bowed string in- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='strument ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Brass instrument ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The musical instrument of the song is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='brass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='brass is not played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Carnatic music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is Carnatic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is not Carnatic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Cello ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The musical instrument of the song is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Cello ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Cello is not played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Chant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is a chant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is not a chant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Choir ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song features a choir ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song does not feature a choir ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Christian music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='It is Christian music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='It is not Christian music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Christmas music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is Christmas music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is not Christmas music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Clarinet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is played by clarinet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='clarinet is not played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Classical music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is classical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The genre for the song is not classical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Country ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The genre of the song is Country ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='This is not a Country song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Cowbell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song has cowbell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song does not have cowbell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Cymbal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The musical instrument of the song is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Cymbal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Cymbal is not played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Dance music ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is suitable for dancing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is not suitable for dancing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Didgeridoo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is played on a didgeridoo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is not played on a didgeridoo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Disco ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song belongs to the disco genre ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song does not belong to the disco genre ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Double bass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The musical instrument of the song is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='double bass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='double bass is not played in the song ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Drum and bass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The music is drum and bass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='The song is not drum and bass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content='Table 7: More examples of PaLM-generated non-negated and negated questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf'} diff --git a/o9AzT4oBgHgl3EQfqv2W/content/tmp_files/2301.01634v1.pdf.txt b/o9AzT4oBgHgl3EQfqv2W/content/tmp_files/2301.01634v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff2dcfc99b54a07930aaaee4c65e466bf5117f39 --- /dev/null +++ b/o9AzT4oBgHgl3EQfqv2W/content/tmp_files/2301.01634v1.pdf.txt @@ -0,0 +1,858 @@ +arXiv:2301.01634v1 [math.FA] 4 Jan 2023 +JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY +RONGWEI YANG +ABSTRACT. The first half of this mostly expository note reviews some notions of +joint spectrum of linear operators, and it gives a new characterization of amenable +groups in terms of projective spectrum. The second half revisits an application +of projective spectrum to the study of self-similar group representations made in +[16]. In the case π is the Koopman representation of the infinite dihedral group +D∞ on the binary tree, it shows that the projective spectrum of D∞ coincides with +the Julia set of a rational map Fπ : P2 → P2 derived from the self-similarity of π. +This improves the main result in [16]. +1. INTRODUCTION +Given a bounded linear operator A on a Hilbert space H, its spectrum σ(A) +plays a fundamental role in classical operator theory. For several linear operators +A1, A2, ..., An, however, it is not at all clear how to make a good definition of their +joint spectrum, and the problem is nontrivial even for matrices. However, when the +operators are commuting, i.e., AiAj = AjAi, 0 ≤ i, j ≤ n, two influential notions +of joint spectrum have been introduced in the 1970s. +Definition 1.1 (H¨ormander [18]). For a tuple A = (A1, ..., An) of elements in an +abelian Banach algebra B, their joint spectrum Sp(A) is the collection of λ = +(λ1, ..., λn) ∈ Cn such that the ideal generated by A1 − λ1I, ..., An − λnI is proper +in B. +2010 Mathematics Subject Classification: Primary 47A13; Secondary 20E08 and 20Cxx. +Key words and phrases: projective spectrum, dihedral group, C∗-algebra, weak containment, +self-similar representation, Fatou set, Julia set +1 + +2 +R. YANG +In other words, λ is not in Sp(A) if and only if there are elements B1, ..., Bn in B +such that (A1 − λ1I)B1 + · · · + (An − λnI)Bn = I. +Let {e1, ..., en} be a basis of a complex vector space V equipped with wedge +product ∧. For 1 ≤ p ≤ n, the space of p-forms, which we denote by Λp(V ). +The Taylor spectrum is defined through the following Koszul complex E(H, A) of +cochains: +0 +d−1 +−→ H ⊗ Λ0 +d0 +−→ H ⊗ Λ1 +d1 +−→ · · · +dn−1 +−→ H ⊗ Λn +dn +−→ 0, +(1.1) +where d−1 = dn = 0, and dp : H ⊗ Λp(V ) → H ⊗ Λp+1(V ), 0 ≤ p ≤ n − 1, is +induced by the map +dp(x ⊗ ω) = +n +� +i=1 +Aix ⊗ (ei ∧ ω), +x ∈ H, ω ∈ Λp(V ). +The commutativity of the operators implies that dp+1dp = 0, 0 ≤ p ≤ n − 1. For a +vector λ = (λ1, ..., λn) ∈ Cn, we let A − λ stand for (A − λ1I, ..., An − λnI). +Definition 1.2 (Taylor [23]). The Taylor spectrum of the tuple A is defined as +σT (A) = {λ ∈ Cn | E(H, A − λ) is not exact}. +Indeed, Taylor spectrum has been one of the main themes in multivariable operator +theory. Unlike the two spectra above, the notion of Harte spectrum is valid for +noncommuting operators. +Definition 1.3 (Harte [17]). For a tuple A of operators on H that is not necessarily +commuting, the joint spectrum σH(A) is the set of vectors λ ∈ Cn such that at least +one of the following two conditions holds: +1) there exists a sequence of unit vectors xk ∈ H such that +lim +k→∞ ∥(Aj − λj)xk∥ = 0, ∀1 ≤ j ≤ n; +(1.2) +2) the sum of vector spaces (A1 − λ1)H + · · · + (An − λn)H is not equal to H. + +JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY +3 +Each of the spectra above is good in its own right, and they all behave well with +respect to function calculus. In particular, spectral mapping theorems hold. For +instance, if Ω is a bounded domain that contains the Taylor spectrum σT(A), then +for every holomorphic map f : Cn → Cm we have σT(f(A)) = f(σT(A)). +1.1. Projective Spectrum. Unfortunately, the joint spectra Sp(A) and σT(A) don’t +have a natural generalization to noncommuting operators. And the Harte spectrum +is difficult to compute for noncommuting operators. These difficulties motivated +a drastically different approach. In [24], the notion of projective spectrum was +introduced as follows. +Definition 1.4. For several elements A0, A1, ..., An in a unital Banach algebra B, +their projective spectrum p(A) is the collection of z ∈ Pn such that the multiparam- +eter linear pencil A(z) = z0A0 + z1A1 + · · · + znAn is not invertible in B. +The letter “A” in p(A) refers to the pencil A(z). Apparently, this definition does +not take the unit I as a base point, and it treats the linears operators in an equal foot- +ing. This conspicuously simple and symmetric definition enables the computation +of many examples which form a basis for an extensive study involving a wide range +of fields. This paper concerns with an application of projective spectrum to group +representations. +Consider a finitely generated group G = ⟨g1, g2, · · · , gn⟩ and a unitary repre- +sentation (π, H). Since the unit 1 is a special element in G, it makes good sense +to consider the pencil Aπ(z) = z0I + z1π(g1) + · · · + znπ(gn). Although projec- +tive spectrum p(Aπ) depends on the choice of the generating set {g1, ..., gn}, it is +capable of capturing intrinsic properties of G and π. If π is a finite dimensional +representation , then the determinant Qπ(z) = det Aπ(z) is called the characteristic +polynomial of G with respect to π, in which case p(Aπ) is the projective hypersur- +face {Qπ(z) = 0}. Computation of Qπ for finite groups can be traced back to the +work of Dedekind and Frobenius in the late 19th century. In fact, their work is the + +4 +R. YANG +starting point of group representation theory ([10]). More recent work on the joint +spectrum of groups can be found in [3, 14]. Definition 1.4 has motivated some new +lines of research on groups, and we refer the readers to [1, 6, 15, 16, 19, 21] for +details. +This paper concerns mostly with the Koopman representation and the left regular +representation. If a locally compact group G has a measure-preserving action on a +measure space (X, µ), then the Koopman representation π : G → U +� +L2(X, µ) +� +is +defined by +π(g)f(x) = f(g−1x), x ∈ X, g ∈ G. +In the case X = G and µ is the Haar measure, the Koopman representation is the +left regular representation of G, and it is often denoted by λG or simply λ. +1.2. Examples and Main Theorems. We give three examples that are relevant to +the study here. +Example 1.5. If 1G is the trivial representation of G, then A1G(z) = z0 + · · · + zn +and hence p(A1G) is the hyperplane Hs := {z ∈ Pn | z0 + · · · + zn = 0}. +Example 1.6. The infinite dihedral group D∞ = ⟨a, t|a2 = t2 = 1⟩ is isomorphic +to the free product Z2 ∗ Z2. Although it is probably the simplest nonabelian group, +it plays important roles in group theory and in Lie algebras. We consider the pencil +Aλ(z) = z0I + z1λ(a) + z2λ(t). It is shown in [15] that +p(Aλ) = +� +−1≤x≤1 +{z ∈ P2 | z2 +0 − z2 +1 − z2 +2 − 2z1z2x = 0}. +Example 1.7. Consider the free group Fn = ⟨g1, ..., gn⟩ with n ≥ 2. It follows +from [1] Proposition 3.2 that the pencil Aλ(z) = z0I + z1λ(g1) + · · ·+ znλ(gn) has +projective spectrum p(Aλ) = ∩n +j=0Rj, where +Rj = {z ∈ Pn | 2|zj|2 ≤ ∥z∥2}, +j = 0, 1, ..., n. + +JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY +5 +To be more explicit, for the case n = 2 the spectrum p(Aλ) is equal to +{|z0|2 ≤ |z1|2 + |z2|2} ∩ {|z1|2 ≤ |z0|2 + |z2|2} ∩ {|z2|2 ≤ |z0|2 + |z1|2}. +In Section 2, based on the connection between amenability and weak contain- +ment, we prove the following result. +Theorem 1.8. A finitely generated group G = ⟨g1, g2, · · · , gn⟩ is amenable if and +only if Hs ⊂ p(Aλ). +Similar descriptions of Haagerup property and Kazhdan’s property (T) of groups +are also given. +The Koopman representation of D∞ is self-similar, and this fact gives rise to a +rational map Fπ(z) = [2τ(z)z0 : z1 : 2τ(z)z2 + z1] on P2, where τ : C3 → ˆC = +C ∪ {∞} is defined by +τ(z) = + + + + + +0, +if z2 +0 − z2 +1 − z2 +2 = 0; +z2 +0 − z2 +1 − z2 +2 +2z1z2 +, +otherwise. +Observe that z ∈ p(Aλ) if and only if τ(z) ∈ [−1, 1] (Example 1.6). Section 3 +studies the Julia set J (Fπ) of the map Fπ and relates it to the projective spectrum +p(Aλ) in Example 1.6. +Theorem 1.9. J (Fπ) = p(Aλ). +This improves the main result in [16]. It is a rare example of a nontrivial rational +map in several variables whose Julia set can be completely described. +2. AMENABILITY +The Banach-Tarski paradox asserts that there exists a decomposition of a solid +ball in R3 into several disjoint pieces such that they can be reassembled, by means of +rotation and shift, to form two solid balls identical to the original one. The construc- +tion of the decomposition is achieved through a decomposition of the free group F2. + +6 +R. YANG +But lying at the core of this unintuitive phenomenon is the Axiom of Choice which +allows for the construction of nonmeasurable sets. The notion of amenability was +introduced by von Neumann [22] to describe groups that do not give rise to the +Banach-Tarski paradox. A finitely generated group G = ⟨g1, g2, · · · , gn⟩ is said to +be amenable if there exists a G-invariant mean, namely a state on the C∗-algebra +L∞(G) satisfying +φ(gf) = φ(f), +f ∈ L∞(G), g ∈ G, +where gf(x) = f(g−1x), x ∈ G. Amenability has several equivalent character- +izations, one of which involves the notion of weak containment of unitary group +representations. +Definition 2.1. Consider two representations (π, H) and (ρ, K) of a discrete group +G. We say π is weakly contained in ρ (denoted by π ≺ ρ) if for every x ∈ H, every +finite subset F ⊂ G, and every ǫ > 0, there exist y1, y2, · · · , yn in K such that for +all g ∈ F +|⟨π(g)x, x⟩ − +n +� +i=1 +⟨ρ(g)yi, yi⟩| < ǫ. +Moreover, two representations π and ρ are said to be weakly equivalent if π ≺ ρ +and ρ ≺ π, in which case we write π ∼ ρ. Weak containment provides a partial +order on the set of unitary representations of G, and there are several equivalent +statements. In particular, for finite groups π ≺ ρ if and only if π is contained in ρ, +i.e., π < ρ. We refer readers to [4, 9] for details. +For a unitary representation (ρ, H) of G = ⟨g1, g2, · · · , gn⟩, we let C∗ +ρ(G) de- +note the C∗-algebra generated by the unitaries ρ(gi), 1 ≤ i ≤ n. Let µ be a positive +regular Borel measure on G, and let L1(G, µ) be the set of integrable complex func- +tions on G. For a unitary representation π, we define +π(m) = +� +G +m(g)π(g)dµ(g), + +JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY +7 +where the convergence of the integral is with respect to the weak operator topology. +Since each π(g) is a unitary, we have +∥π(m)∥ ≤ +� +G +|m(g)|dµ(g) = ∥m∥1. +Further, if the the constant function 1 ∈ L1(G, µ), we write π(1) as π(µ) to avoid +confusion. Clearly, for a discrete group G with counting measure and an arbitrary +element m = z1g1 + · · · + zkgk ∈ C[G], we have π(m) = z1π(g1) + · · · + zkπ(gk). +Other measures on a countable group is also useful. +Example 2.2. Consider the measure µ on group G generated by the set S = +{g1, ..., gn} such that µ{gi} = +1 +n, j = 1, 2, · · · , n, and µ{g} = 0 for all g /∈ S. +Clearly, µ is a probability measure that is absolutely continuous with respect to the +Haar measure on G, and µ’s support is S. In this case +π(µ) = +� +G +π(g)dµ(g) = 1 +n(π(g1) + π(g2) + · · · π(gn)), +and it is called the Markov operator with respect to the generating set S and the +representation π, for which we denote by Mπ. Since each π(gi) is a unitary, the +triangular inequality implies ∥Mπ∥ ≤ 1. +As we shall soon see, the spectrum of Mπ holds important information about amenabil- +ity. Weak containment has a useful description in terms of C∗-algebras. +Proposition 2.3. Let ρ and π be two unitary representations of a countable group +G. Then the following statements are equivalent: +(a) π ≺ ρ; +(b) σ(π(m)) ⊂ σ(ρ(m)) for each m ∈ ℓ1(G); +(c) ∥π(m)∥ ≤ ∥ρ(m)∥ for each positive m ∈ C[G]; +(d) the map φ : C∗ +ρ(G) −→ C∗ +π(G) defined by φ(ρ(g)) = π(g), g ∈ G extends to +a surjective homomorphisim. +The following theorem is [4] Theorem G 3.2. + +8 +R. YANG +Theorem 2.4 (Hulanicki-Reiter). Let G be a locally compact group. The following +properties are equivalent: +(i) G is amenable; +(ii) 1G ≺ λ; +(iii) π ≺ λ for every unitary representation π of G. +In other words, the group G is amenable if and only if its left regular representa- +tion is maximal with respect to weak containment. A spectral characterization of +amenability was given by Kesten [20]. +Theorem 2.5. Let G be a locally compact group and µ be a probability measure on +G. Assume that µ is absolutely continuous with respect to the Haar measure on G +and supp(µ) generates a dense subgroup of G. Then the following are equivalent. +(a) G is amenable. +(b) 1 ∈ σ(λ(µ)). +(c) The spectral radius of λ(µ) is 1. +If µ is the probability measure in Example 2.2, then λ(µ) = Mλ. Hence a finitely +generated group G is amenable if and only if 1 ∈ σ(Mλ). +2.1. Projective Spectrum and Amenability. Two representations (ρ1, H1) and +(ρ2, H2) are said to be equivalent if there is a unitary map U : H1 → H2 such +that ρ2(g) = Uρ1(g)U−1, g ∈ G. It is obvious that in this case for any elements +g1, ..., gn ∈ G, we have Aρ2(z) = UAρ1(z)U−1 and hence p(Aρ1) = p(Aρ2). +This indicates that projective spectrum is a unitary invariant for group represen- +tations. Moreover, if π ≺ ρ, then Proposition 2.3 (d) implies that if ρ(m) is in- +vertible in C∗ +ρ(G) then π(m) is invertible in C∗ +π(G). Since Aπ(z) = π(m(z)) and +Aρ(z) = ρ(m(z)), where m(z) = z01 + z1g1 + · · · + zngn ∈ C[G], the following is +immediate. + +JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY +9 +Lemma 2.6. Consider two unitary represntations π and ρ of a discrete group G. If +π ≺ ρ, then p(Aπ) ⊂ p(Aρ). +It follows that if π ∼ ρ then p(Aπ) = p(Aρ). In other words, the projective spectrum +p(Aπ) is also an invariant of π with respect to weak equivalence. It is not clear +whether the converse of Lemma 2.6 holds. At least, it holds when π = 1G. +Lemma 2.7. Suppose group G is generated by a finite set S and π is a unitary +representation. Then 1G ≺ π if and only if Hs ⊂ p(Aπ). +Proof. The necessity has been observed in Lemma 2.6. For the other direction, we +assume S = {g1, ..., gn}. Since Hs ⊂ p(Aπ), we have (−1, 1 +n, ..., 1 +n) ∈ p(Aπ), i.e., +1 ∈ σ(Mπ), where Mπ is the associated Markov operator. Thus, either Mπ or M∗ +π is +not bounded below. Without loss of generality, we assume the former occurs. Then +there exists a sequence of unit vectors ξk ∈ H such that ∥Mπξk − ξk∥ → 0 (and +hence ∥Mπξk∥ → 1). It follows that +n +� +i=1 +∥π(gi)ξk − ξk∥2 = n(2 − 2 Re⟨Mπξk, ξk⟩) += n(∥Mπξk − ξk∥2 + 1 − ∥Mπξk∥2) → 0, +which implies ∥π(gi)ξk − ξk∥ = ∥ξk − π(g−1 +i )ξk∥ → 0 for each i. We let S−1 stand +for the set {g−1 +1 , ..., g−1 +n }. Then G = ∪∞ +m=0(S ∪ S−1)m. Since G is countable, its +subset Q is compact if and only if it is finite, in which case there exists an integer +M such that Q ⊂ ˆS := ∪M +m=0(S ∪ S−1)m. Given any ǫ > 0, we let ξ be such that +∥π(gi)ξ − ξ∥ = ∥ξ − π(g−1 +i )ξ∥ < ǫ +M , +1 ≤ i ≤ n. + +10 +R. YANG +Then for each x = x1 · · · xm ∈ ˆS, where xi ∈ S ∪ S−1, we have +∥π(x1 · · · xm)ξ − ξ∥ +=∥π(x1 · · · xm)ξ − π(x1 · · · xm−1)ξ + · · · + π(x1x2)ξ − π(x1)ξ + π(x1)ξ − ξ∥ +≤∥π(x1 · · · xm−1)(π(xm)ξ − ξ)∥ + · · · + ∥π(x1)(π(x2)ξ − ξ)∥ + ∥π(x1)ξ − ξ∥ +=∥π(xm)ξ − ξ∥ + · · · + ∥π(x2)ξ − ξ∥ + ∥π(x1)ξ − ξ∥ + 0 and every x, y ∈ H the set +{g ∈ G : |⟨π(g)x, y⟩| ≥ ǫ} + +JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY +11 +is compact. +Clearly, π contains 1G if and only if π possesses an invariant vector x. In this case +the above set is then equal to G when 0 < ǫ < |⟨x, y⟩|, and hence π is not of C0. It is +known that the left regular representation λG is of C0 ([9]). A locally compact group +G is said to have Haagerup property or (a-T-menability) if there is a C0 represen- +tation of G that weakly contains the trivial representation 1G. Then it immediately +follows from Lemma 2.7 that if G is a group with generators {g1, g2, · · · , gn}, then +it has Haagerup property if and only if there is a C0 representation π of G such that +Hs ⊂ p(Aπ). +Definition 2.10. A topological group G is said to have Kazhdan’s Property (T) if +for every unitary representation π of G, the weak containment 1G ≺ π implies the +containment 1G < π. +It is easy to see that a representation (π, H) contains 1G if and only if there exists +a non-zero vector x ∈ H such that π(g)x = x for all g ∈ G. Such x is called an +invariant vector for π. Since for finite groups 1G ≺ π implies 1G < π, every finite +group has Kazhdan’s Property (T). A nontrivial example of group with Kazhdan’s +Property (T) is the special linear group SL(3, Z). On the other hand, no infinite +countable amenable group G has Kazhdan’s Property (T) because, if it were, then +Theorem 2.4 implies 1G < λ, i.e., the regular representation λ has an invariant +vector f ∈ L2(G). In other words, f is a nontrivial constant function in L2(G). +This would imply G must be finite which is a contradiction. +In view of Lemma 2.7, the above observations can be summarized as follows. +Proposition 2.11. Assume group G is finitely generated. Then the following state- +ments hold. +(a) G has Haagerup property if and only if there is a C0 representation π of G +such that Hs ⊂ p(Aπ). + +12 +R. YANG +(b) G has Kazhdan’s property (T) if and only if for any unitary representation π +the inclusion Hs ⊂ p(Aπ) implies 1G < π. +Theorem 1.8 and Proposition 2.11 serve to indicate that projective spectrum is +able to capture some intrinsic properties of group G. +3. SELF-SIMILARITY AND JULIA SET +Another important line of study on the joint spectra of groups was carried out +by Grigorchuk and his collaborators on the group G of intermediate growth, for +instance in [3, 13, 14]. Among other things, they discovered that G, as well as +some other groups including the dihedral group D∞ and the lamplighter group, has +a self-similarity property when they act on the rooted binary tree. +T +T0 +T00 +... +... +T01 +... +... +T1 +T10 +... +... +T11 +... +... +FIGURE 1. A rooted binary tree +Clearly, the tree T consists of two subtrees T0 and T1, each of which also con- +sists of two subtrees: T00 and T01, and T10 and T11, respectively, etc. The boundary +∂T of the tree T is the collection of all infinite sequences of directed arrows from +the vertex T down the tree. The uniform Bernoulli measure µ on ∂T is defined +by µ(∂Ti1···ip) = +1 +2p, where ik ∈ {0, 1} for each k. In other words, the measure µ +distributes evenly on the subtrees at every level. Since every element in ∂T corre- +sponds to an infinite sequence of directed arrows in T, it corresponds to a unique + +JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY +13 +sequence of 0s and 1s. Hence there is a natural bijection from ∂T to the interval +[0, 1] (expressed in binary numbers). And this bijection also identifies the measure +µ with the Lebesgue measure on [0, 1]. +Define the Hilbert space H = L2(∂T, µ). Let µi = 2µ, i = 0, 1 be the nor- +malized restrictions of µ on the boundary of the subtrees ∂T0, ∂T1, and define +Hi = L2(∂Ti, µi), i = 0, 1. Then each Hi can be identified with H and hence +H = H0 ⊕ H1 can be identified with H ⊕ H by a unitary W. +3.1. The Koopman Representation of D∞. Koopman representations of some +groups on the binary tree share an interesting self-similarity property. This was +discovered in +Definition 3.1. Given an integer d ≥ 2, a unitary representation(π, H) of a group +G is said to be d-similar if there exists a unitary operator W : H → Hd such that +for every g ∈ G the d × d block matrix ˆπ(g) = Wπ(g)W ∗ has all of its entries +either equal to 0 or of the form π(g), g ∈ G. +In this case, it is clear that ˆπ is a unitary representationof G on Hd. Since ˆπ(g) +itself and each of its nonzero entries are unitaries, every row or column of ˆπ(g) has +precisely one nonzero entry. +The Koopman representation π of D∞ = ⟨a, t | a2 = t2 = 1⟩ on the tree T is +realized by the following action of D∞ on the tree T: a swaps T0 and T1; t acts +on T0 like a acting on T, while it acts on T1 like what it does on T. This action is +described by the automaton in Figure 2, where a and t satisfy the recursive relation +a = σ, t ∼= a⊕t. Here, “0/1” means: given input 0, the output is 1; and given input +1, the output is 0. Thus, the Koopman representation of D∞ on H = L2(∂T, µ) is +2-similar, and the identification W : H = H0 ⊕ H1 → H ⊕ H mentioned earlier + +14 +R. YANG +e +t +σ +a +e +id +1 +0 +0/1 +0,1 +FIGURE 2. Automaton of the group D∞ +gives rise to the unitary equivalence +ˆπ(a) = + +0 +I +I +0 + + , +ˆπ(t) = + +π(a) +0 +0 +π(t) + + . +(3.1) +It is shown in [15] that the Koopman representationof D∞ is weakly equivalent +to the regular representationλD∞. Thus the following fact is an immediate conse- +quence of Example 1.6 and Lemma 2.6. +Corollary 3.2. With respect to the Koopman representation π of D∞, we have +p(Aλ) = +� +−1≤x≤1 +{z ∈ C3 : z2 +0 − z2 +1 − z2 +2 − 2z1z2x = 0}. +3.2. The Renormalization Map. Based on (3.1), we have +Aπ(z) = z0 + z1π(a) + z2π(t) ∼= Aˆπ(z) = + +z0 + z2π(a) +z1 +z1 +z0 + z2π(t) + + . (3.2) +Therefore, Aπ(z) is invertible if and only if the 2 × 2 block matrix on the righ-hand +side of (3.2) is invertible. For convenience, we often shall write π(g) simply as g in +the subsequent computations in this section. In the case z2 +0 ̸= z2 +2, the pencil z0 +z2a +is invertible and its inverse is (z0 − z2a)(z2 +0 − z2 +2)−1. In this case, the block matrix +Aˆπ(z) is invertible if and only if the Schur complement z0 +z2t−z2 +1(z0 −z2a)(z2 +0 − +z2 +2)−1 is invertible, or if and only if the rational pencil +z0(z2 +0 − z2 +1 − z2 +2) +z2 +0 − z2 +2 ++ +z2 +1z2 +z2 +0 − z2 +2 +a + z2t + +JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY +15 +is invertible. This gives rise to the following polynomial map: +F(z0, z1, z2) = +� +z0(z2 +0 − z2 +1 − z2 +2), z2 +1z2, z2(z2 +0 − z2 +2) +� +. +(3.3) +And it is known that F maps p(Aπ) into itself. Since F is homogeneous of degree +2, it induces a map in the projective space P2. For convenience, we shall denote this +map also by F, namely, +F([z0 : z1 : z2]) = [z0(z2 +0 − z2 +1 − z2 +2) : z2 +1z2 : z2(z2 +0 − z2 +2)], +z ∈ P2. +(3.4) +We should be aware that the map F is not well-defined on P2 at the common zeros +of F0, F1 and F2 because P2 contains no origin. For k = 0, 1, ..., we write the kth +iteration of F as F k([z0 : z1 : z2]) = [F k +0 : F k +1 : F k +2 ] and set +Ik = {z ∈ P2 | F k +j (z) = 0, j = 0, 1, 2}. +It is clear that Ik−1 ⊂ Ik for each k. The set Ik is called the indeterminacy set of +F k, and the closure E = ∪∞ +k=1In is the extended indeterminacy set of F. +Example 3.3. To determine the indeterminacy set I1, we solve the system of equa- +tions F0 = F1 = F2 = 0 in two cases: z1 = 0 or z2 = 0, and easily obtain +I1 = {[±1 : 1 : 0], [0 : 1 : 0], [±1 : 0 : 1]}. +(3.5) +Observe that I1 reflects the spectral property σ(π(a)) = σ(π(t)) = {±1}. +With a bit more efforts, I2 can be determined as well. However, it is not feasible to +fully determine Ik when k gets larger, not mentioning I∞(F) or the extended inde- +terminacy set E. This fact adds difficulty to the study of F ′s dynamical properties. +To simplify the situation, we consider the map τ : P2 → ˆC defined by +τ(z) = + + + + + +0, +if z2 +0 − z2 +1 − z2 +2 = 0; +z2 +0 − z2 +1 − z2 +2 +2z1z2 +, +otherwise. +(3.6) + +16 +R. YANG +Then, according to Corollary 3.2, τ(z) ∈ [−1, 1] if and only if z ∈ p(Aπ). This +fact, in particular, implies that pc(Aπ) = τ −1(C \ [−1, 1]) is a dense open subset of +P2. Further, Fπ(z) = F(z) whenever z1z2 ̸= 0. Using the function τ, we can write +F(z) = [2τ(z)z0z1z2 : z2 +1z2 : z1z2(2τ(z)z2 + z1)], z ∈ P2 \ I1. +This indicates that the complications of the indeterminacy sets Ik is largely due to +the common factor z1z2. Therefore, in order to avoid this unnecessary complica- +tion, we consider the renormalization map of D∞ associated with the Koopman +representation π as +Fπ(z) = [2τ(z)z0 : z1 : 2τ(z)z2 + z1], z ∈ P2 \ I1(Fπ), +(3.7) +which is display (5.1) in [16]. It is important to observe that, since τ is homoge- +neous of degree 0, the map Fπ on P2 is in fact homogeneous of degree 1. This +lends great convenience to the study of Fπ’s dynamical property. We denote the +indeterminacy sets of Fπ by Ik(Fπ), k = 0, 1, ... and EFπ. +Lemma 3.4. E(Fπ) = I1(Fπ) = {[±1 : 0 : 1]}. +Proof. It is sufficient to check that I2 = I1. For a point z ∈ P2 to be in I1(Fπ), we +must have z1 = 0, and 2τ(z)z0 = 2τ(z)z2 = 0. Since z0 and z0 are not both 0, +we have τ(z) = 0. It follows from the definition (3.6) that z2 +0 = z2 +2, and therefore +I1(Fπ) = {[±1 : 0 : 1]}. +To determine I2(Fπ), it remains to find points z ∈ P2 such that Fπ(z) ∈ I1(Fπ) +which means z1 = 0 and 2τ(z)z0 = ±2τ(z)z2 ̸= 0. This implies z0 = ±z2 ̸= 0 +and hence z ∈ I1(Fπ). +□ +Observe that E(Fπ) ⊂ p(Aπ). +3.3. The Julia Set of Fπ. For a rational map H : Pn → Pn, the notions of Fatou +set and Julia set is defined as follows ([11]). + +JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY +17 +Definition 3.5. A point p ∈ Pn is said to be a Fatou point of H if it has a neighbor- +hood U ⊂ Pn on which the sequence of iterations {Hk | k = 1, 2, ...} is equicon- +tinuous. The Fatou set F(H) is the set of Fatou points of H, and the Julia set J (H) +is the complement Pn \ F(H). +Clearly, the extended indeterminacy set E(H) is a subset of F(H). The follow- +ing one variable example is crucial for the subsequent discussion. +Example 3.6. We identify P with the extended complex plane ˆC. Consider the +Tchebyshev polynomial T(x) = 2x2 − 1, x ∈ ˆC. It is known, for instance see [2], +that its Julia set J (T) = [−1, 1]. Moreover, the iteration sequence {T n} converges +to ∞ uniformly on every compact subset in F(T). +Somewhat surprisingly, the Julia set of the map F in (3.3) was shown to be +closely related to the projective spectrum in [16] Corollary 3.2: J (F)∪E = p(Aπ). +However, this result is not entirely satisfactory due to the lack of a clear picture of +E. A careful review of the proof made us realize that the map Fπ in fact has a much +simpler extended indeterminacy set, and therefore a cleaner theorem holds. +Theorem 3.7. J (Fπ) = p(Aπ). +The proof follows the same line as that in [16], and only some small modifications +are needed to suit the change from F to Fπ. For the readers’ convenience, we +include all the necessary steps here but leave out the details which can be found in +[16]. We start with the following lemma. +Lemma 3.8. For k ≥ 1 the following diagram is commutative: +P2 \ E(Fπ) +P2 \ E(Fπ) +ˆC +ˆC. +F k +π +τ +τ +T k + +18 +R. YANG +Proof. It is sufficient to check that τ(Fπ(z)) = T(τ(z)), z /∈ E(Fπ), and it can be +easily done by a direct computation. +□ +This connection is pivotal because the iterations of Fπ can now be studied through +that of T, and thus Example 3.6 can be used to study the dynamical property of Fπ. +For n = 1, 2, ..., proof by induction gives [16] Lemma 5.8, namely, +F n +π (z) += +� +2nz0 +n−1 +� +k=0 +T k(τ) : z1 : 2nz2 +n−1 +� +k=0 +T ◦k(τ) + z1 +� +1 + +n−1 +� +k=1 +2k +k +� +i=1 +T (n−i)(τ) +�� +. (3.8) +To simplify subsequent calculations, we define functions +pn(z) = 2n +n−1 +� +k=0 +T k(τ(z)), z ∈ P2, n = 1, 2, ... +(3.9) +Observe that if z ∈ pc(Aπ), then τ(z) /∈ [−1, 1], and Example 3.6 implies T k(τ(z)) ̸= +0 for every k ≥ 0. Thus, as in [16] display (5.2), we can set +fn(z) = +n +� +j=1 +1 +pj(z), n ≥ 2, z ∈ pc(Aπ). +(3.10) +Since T(x) is holomorphic on ˆC and τ is holomorphic on the set P2\{z2 +0−z2 +1−z2 +2 = +0} which contains pc(Aπ), the function fn is holomorphic on pc(Aπ). Then, F n +π (z) +can be simplified as +F n +π (z) = +� +z0 : +z1 +pn(z) : z2 + z1fn(z) +� +, z ∈ pc(Aπ), +(3.11) +which is Lemma 5.8 in [16]. And the following lemma holds as well. +Lemma 3.9. pc(Aπ) ⊂ F(Fπ). +Proof. Since τ(z) is holomorphic on pc(Aπ), for every compact subset K ⊂ pc(Aπ) +the image τ(K) is compact in ˆC \ [−1, 1]. Thus {T k(τ(z))} converges uniformly + +JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY +19 +to ∞ on K by Example 3.6. The definition (3.9) thus permits the existence of a +N ∈ N such that +1 +|pn(z)| ≤ 2−n for every n ≥ N. This implies that the series +f(z) = lim +n→∞ fn(z) = +1 +p1(z) + +1 +p2(z) + · · · +(3.12) +converges uniformly on K. Equation 3.11 then implies that F n +π (z) converges nor- +mally on pc(Aπ) to the map +F∗([z0 : z1 : z2]) = [z0 : 0 : z2 + z1f(z)], +z ∈ pc(Aπ). +(3.13) +□ +The fact that p(Aπ) ⊂ J (Fπ) is already shown in the proof of [16] Theorem +5.11. The proof here is simpler and it uses the density of pc(Aπ) in P2. Observe that +if ξ ∈ p(Aπ) then τ(ξ) ∈ [−1, 1], and hence we can write τ(ξ) = cos θ, for some +θ ∈ [0, π]. Then T n(τ(ξ)) = cos(2nθ), n = 1, 2, .... Suppose ξ ∈ p(Aπ) is such +that θ +π is non-dyadic, i.e., 2n θ +π /∈ Z for any integer n ≥ 0. Then for every n, +pn(ξ) = 2n +n−1 +� +k=0 +cos(2kθ) = sin(2nθ) +sin θ +̸= 0, +(3.14) +and it follows from (3.8) that +F n +π (ξ) = +� +ξ0 : ξ1 +sin θ +sin(2nθ) : ξ2 + ξ1 sin θ +n +� +k=1 +(sin(2kθ))−1 +� +. +If ξ were a Fatou point, then there would exist a neighborhood V of ξ and a sub- +sequence F nk +π +that converges normally to a holomorphic function ˆF∗ on V . Since +pc(Aπ) is dense in P2, the limit ˆF∗ must be an holomorphic extension of the func- +tion F∗ in (3.13) from V ∩ pc(Aπ) to V . But due to the fact | +sin θ +sin(2nθ)| ≥ sin θ > 0 +for all n, ˆF∗(z) is not continuous at ξ. This is a contradiction. Theorem 3.7 is thus +established. + +20 +R. YANG +Using the same method as in the proof of [16] Theorem 5.14, the limit function +f can be explicitly determined as +f(z) = τ(z) − +� +τ 2(z) − 1, +z ∈ pc(Aπ), +where τ is as defined in (3.6). +REFERENCES +[1] J. Bannon, P. Cade and R. Yang, On the spectrum of operator-valued entire functions. Illinois J. +of Mathematics 55 No.4 (2011). +[2] A. F. BEARDON, Iteration of Rational Functions, Graduate Text in Mathematics 132, Springer- +Verlag, New York, 1991. +[3] L. Bartholdi and R. Grigorchuk, On the spectrum of Hecke type operators related to some fractal +groups, Tr. Mat. Inst. Steklova 231 (2000), Din. Sist., Avtom. i Beskon. Gruppy, 5–45; transla- +tion in Proc. Steklov Inst. Math. 2000, no. 4 (231), 1’ +¨A`ı41. +[4] B. Bekka, de la Harpe and A. Valette, Kazhdan’s property (T), New Mathematical Monographs +11, Cambridge University Press, Cambridge, 2008. +[5] P-A. Cherix, M. Cowling, P. Jolissaint, P. Julg and A. Valette, Groups with the Haagerup prop- +erty, Progress in Math. 197, Birkh¨auser Verlag, 2001. +[6] ˇZ. ˇCuˇckovi´c, M. Stessin and A. Tchernev, Determinantal hypersurfaces and representations of +Coxeter groups, Pacific J. Math. 313 (2021), no. 1, 103-135. +[7] A. Dudko and R. Grigorchuk, On spectra of Koopman, groupoid and quasi-regular representa- +tions, arxiv: 1510.00897v3, 2016. +[8] B. Dang, R. Grigorchuk and M. Lyubich, Self-similar groups and holomorphic dynamics: +Renormalization, integrability, and spectrum, arXiv:2010.00675. +[9] J. Dixmier, C∗-algebras (a translation of Les C∗-alg`ebres et leurs repr´esentations), North- +Holland Publishing Company, 1977. +[10] L. E. Dickson, An elementary exposition of Frobenius theory of group characters and group- +determinants, Ann. of Math. 4 (1902), 25-49; Mathematical Papers, Vol. II. Chelsea, New York, +1975, 737-761. +[11] J. Fornaess, Dynamics in Several Complex Variables, CBMS Regional Conference Series in +Mathematics 87, the American Mathematical Society, Providence, RI, 1996. + +JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY +21 +[12] R. I. Grigorchuk, D. Lenz and T. Smirnova-Nagnibeda, Spectra of Schreier graphs of +Grigorchuk’ +¨Aˆos group and Schroedinger operators with aperiodic order, arXiv:1412.6822. +[13] R. Grigorchuk and V. Nekrashevych, Self-similar Groups, Operator Algebras and Schur Com- +plement, J. Mod. Dyn. 1 3 (2007), 323-370. +[14] R. Grigorchuk and Z. ˇSuni´c, Schreier spectrum of the Hanoi Towers group on three pegs, Proc. +of Symposia in Pure Math. Vol. 77, 2008. +[15] R. Grigorchuk and R. Yang, Joint spectrum and infinite dihedral group, Proc. of the Steklov +Institute of Math., 2017, Vol. 297, 145-178. +[16] B. Goldberg and R. Yang, Self-similarity and spectral dynamics, J. Operator Theory 87 (2022), +no. 2, 355-388. +[17] R. Harte, Spectral mapping theorems, Proc. Royal Irish Acad., 72 A (1972), 89-107. +[18] L. H¨ormander, An introduction to complex analysis in several variables, 3rd ed., North Hol- +land, Amsterdam, 1990. +[19] Z. Hu and R. Yang, On the characteristic polynomials of multiparameter pencils, Linear Alg. +and its Appl. 558 (2018), 250-263. +[20] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 22 (1959), 336-354. +[21] I. Klep and J. Volˇciˇc, A note on group representations, determinantal hypersurfaces and their +quantizations, Operator theory, functional analysis and applications, 393-402, Oper. Theory +Adv. Appl., 282, Birkh´’auser/Springer, Cham, 2021. +[22] J. von Neumann, Zur allgemeinen theorie des masses, Fund. Math., 13 (1929), 73-116. +[23] J. L. Taylor, A joint spectrum for several commutative operators, J. Funct. Analy. 6 (1970), +172-191. +[24] R. Yang, Projective spectrum in Banach algebras, J. Topol. and Analy. 1 (2009), No. 3, 289- +306. +RONGWEI YANG: DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY AT AL- +BANY, THE STATE UNIVERSITY OF NEW YORK, ALBANY, NY 12222, U.S.A. +Email address: ryang@albany.edu + diff --git a/o9AzT4oBgHgl3EQfqv2W/content/tmp_files/load_file.txt b/o9AzT4oBgHgl3EQfqv2W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8e8f0ddb725c2ab6f38287c734c7aca4124103bb --- /dev/null +++ b/o9AzT4oBgHgl3EQfqv2W/content/tmp_files/load_file.txt @@ -0,0 +1,640 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf,len=639 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='01634v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='FA] 4 Jan 2023 JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY RONGWEI YANG ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The first half of this mostly expository note reviews some notions of joint spectrum of linear operators, and it gives a new characterization of amenable groups in terms of projective spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The second half revisits an application of projective spectrum to the study of self-similar group representations made in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' In the case π is the Koopman representation of the infinite dihedral group D∞ on the binary tree, it shows that the projective spectrum of D∞ coincides with the Julia set of a rational map Fπ : P2 → P2 derived from the self-similarity of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' This improves the main result in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' INTRODUCTION Given a bounded linear operator A on a Hilbert space H, its spectrum σ(A) plays a fundamental role in classical operator theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' For several linear operators A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', An, however, it is not at all clear how to make a good definition of their joint spectrum, and the problem is nontrivial even for matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' However, when the operators are commuting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', AiAj = AjAi, 0 ≤ i, j ≤ n, two influential notions of joint spectrum have been introduced in the 1970s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='1 (H¨ormander [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' For a tuple A = (A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', An) of elements in an abelian Banach algebra B, their joint spectrum Sp(A) is the collection of λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', λn) ∈ Cn such that the ideal generated by A1 − λ1I, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', An − λnI is proper in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' 2010 Mathematics Subject Classification: Primary 47A13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Secondary 20E08 and 20Cxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Key words and phrases: projective spectrum, dihedral group, C∗-algebra, weak containment, self-similar representation, Fatou set, Julia set 1 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' YANG In other words, λ is not in Sp(A) if and only if there are elements B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', Bn in B such that (A1 − λ1I)B1 + · · · + (An − λnI)Bn = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Let {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', en} be a basis of a complex vector space V equipped with wedge product ∧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' For 1 ≤ p ≤ n, the space of p-forms, which we denote by Λp(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The Taylor spectrum is defined through the following Koszul complex E(H, A) of cochains: 0 d−1 −→ H ⊗ Λ0 d0 −→ H ⊗ Λ1 d1 −→ · · · dn−1 −→ H ⊗ Λn dn −→ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='1) where d−1 = dn = 0, and dp : H ⊗ Λp(V ) → H ⊗ Λp+1(V ), 0 ≤ p ≤ n − 1, is induced by the map dp(x ⊗ ω) = n � i=1 Aix ⊗ (ei ∧ ω), x ∈ H, ω ∈ Λp(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The commutativity of the operators implies that dp+1dp = 0, 0 ≤ p ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' For a vector λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', λn) ∈ Cn, we let A − λ stand for (A − λ1I, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', An − λnI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='2 (Taylor [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The Taylor spectrum of the tuple A is defined as σT (A) = {λ ∈ Cn | E(H, A − λ) is not exact}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Indeed, Taylor spectrum has been one of the main themes in multivariable operator theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Unlike the two spectra above, the notion of Harte spectrum is valid for noncommuting operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='3 (Harte [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' For a tuple A of operators on H that is not necessarily commuting, the joint spectrum σH(A) is the set of vectors λ ∈ Cn such that at least one of the following two conditions holds: 1) there exists a sequence of unit vectors xk ∈ H such that lim k→∞ ∥(Aj − λj)xk∥ = 0, ∀1 ≤ j ≤ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='2) 2) the sum of vector spaces (A1 − λ1)H + · · · + (An − λn)H is not equal to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY 3 Each of the spectra above is good in its own right, and they all behave well with respect to function calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' In particular, spectral mapping theorems hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' For instance, if Ω is a bounded domain that contains the Taylor spectrum σT(A), then for every holomorphic map f : Cn → Cm we have σT(f(A)) = f(σT(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Projective Spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Unfortunately, the joint spectra Sp(A) and σT(A) don’t have a natural generalization to noncommuting operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' And the Harte spectrum is difficult to compute for noncommuting operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' These difficulties motivated a drastically different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' In [24], the notion of projective spectrum was introduced as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' For several elements A0, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', An in a unital Banach algebra B, their projective spectrum p(A) is the collection of z ∈ Pn such that the multiparam- eter linear pencil A(z) = z0A0 + z1A1 + · · · + znAn is not invertible in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The letter “A” in p(A) refers to the pencil A(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Apparently, this definition does not take the unit I as a base point, and it treats the linears operators in an equal foot- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' This conspicuously simple and symmetric definition enables the computation of many examples which form a basis for an extensive study involving a wide range of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' This paper concerns with an application of projective spectrum to group representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Consider a finitely generated group G = ⟨g1, g2, · · · , gn⟩ and a unitary repre- sentation (π, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Since the unit 1 is a special element in G, it makes good sense to consider the pencil Aπ(z) = z0I + z1π(g1) + · · · + znπ(gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Although projec- tive spectrum p(Aπ) depends on the choice of the generating set {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', gn}, it is capable of capturing intrinsic properties of G and π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' If π is a finite dimensional representation , then the determinant Qπ(z) = det Aπ(z) is called the characteristic polynomial of G with respect to π, in which case p(Aπ) is the projective hypersur- face {Qπ(z) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Computation of Qπ for finite groups can be traced back to the work of Dedekind and Frobenius in the late 19th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' In fact, their work is the 4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' YANG starting point of group representation theory ([10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' More recent work on the joint spectrum of groups can be found in [3, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='4 has motivated some new lines of research on groups, and we refer the readers to [1, 6, 15, 16, 19, 21] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' This paper concerns mostly with the Koopman representation and the left regular representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' If a locally compact group G has a measure-preserving action on a measure space (X, µ), then the Koopman representation π : G → U � L2(X, µ) � is defined by π(g)f(x) = f(g−1x), x ∈ X, g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' In the case X = G and µ is the Haar measure, the Koopman representation is the left regular representation of G, and it is often denoted by λG or simply λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Examples and Main Theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' We give three examples that are relevant to the study here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' If 1G is the trivial representation of G, then A1G(z) = z0 + · · · + zn and hence p(A1G) is the hyperplane Hs := {z ∈ Pn | z0 + · · · + zn = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The infinite dihedral group D∞ = ⟨a, t|a2 = t2 = 1⟩ is isomorphic to the free product Z2 ∗ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Although it is probably the simplest nonabelian group, it plays important roles in group theory and in Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' We consider the pencil Aλ(z) = z0I + z1λ(a) + z2λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' It is shown in [15] that p(Aλ) = � −1≤x≤1 {z ∈ P2 | z2 0 − z2 1 − z2 2 − 2z1z2x = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Consider the free group Fn = ⟨g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', gn⟩ with n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' It follows from [1] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='2 that the pencil Aλ(z) = z0I + z1λ(g1) + · · ·+ znλ(gn) has projective spectrum p(Aλ) = ∩n j=0Rj, where Rj = {z ∈ Pn | 2|zj|2 ≤ ∥z∥2}, j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY 5 To be more explicit, for the case n = 2 the spectrum p(Aλ) is equal to {|z0|2 ≤ |z1|2 + |z2|2} ∩ {|z1|2 ≤ |z0|2 + |z2|2} ∩ {|z2|2 ≤ |z0|2 + |z1|2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' In Section 2, based on the connection between amenability and weak contain- ment, we prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' A finitely generated group G = ⟨g1, g2, · · · , gn⟩ is amenable if and only if Hs ⊂ p(Aλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Similar descriptions of Haagerup property and Kazhdan’s property (T) of groups are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The Koopman representation of D∞ is self-similar, and this fact gives rise to a rational map Fπ(z) = [2τ(z)z0 : z1 : 2τ(z)z2 + z1] on P2, where τ : C3 → ˆC = C ∪ {∞} is defined by τ(z) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0, if z2 0 − z2 1 − z2 2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' z2 0 − z2 1 − z2 2 2z1z2 , otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Observe that z ∈ p(Aλ) if and only if τ(z) ∈ [−1, 1] (Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Section 3 studies the Julia set J (Fπ) of the map Fπ and relates it to the projective spectrum p(Aλ) in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' J (Fπ) = p(Aλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' This improves the main result in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' It is a rare example of a nontrivial rational map in several variables whose Julia set can be completely described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' AMENABILITY The Banach-Tarski paradox asserts that there exists a decomposition of a solid ball in R3 into several disjoint pieces such that they can be reassembled, by means of rotation and shift, to form two solid balls identical to the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The construc- tion of the decomposition is achieved through a decomposition of the free group F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' 6 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' YANG But lying at the core of this unintuitive phenomenon is the Axiom of Choice which allows for the construction of nonmeasurable sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The notion of amenability was introduced by von Neumann [22] to describe groups that do not give rise to the Banach-Tarski paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' A finitely generated group G = ⟨g1, g2, · · · , gn⟩ is said to be amenable if there exists a G-invariant mean, namely a state on the C∗-algebra L∞(G) satisfying φ(gf) = φ(f), f ∈ L∞(G), g ∈ G, where gf(x) = f(g−1x), x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Amenability has several equivalent character- izations, one of which involves the notion of weak containment of unitary group representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Consider two representations (π, H) and (ρ, K) of a discrete group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' We say π is weakly contained in ρ (denoted by π ≺ ρ) if for every x ∈ H, every finite subset F ⊂ G, and every ǫ > 0, there exist y1, y2, · · · , yn in K such that for all g ∈ F |⟨π(g)x, x⟩ − n � i=1 ⟨ρ(g)yi, yi⟩| < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Moreover, two representations π and ρ are said to be weakly equivalent if π ≺ ρ and ρ ≺ π, in which case we write π ∼ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Weak containment provides a partial order on the set of unitary representations of G, and there are several equivalent statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' In particular, for finite groups π ≺ ρ if and only if π is contained in ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', π < ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' We refer readers to [4, 9] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' For a unitary representation (ρ, H) of G = ⟨g1, g2, · · · , gn⟩, we let C∗ ρ(G) de- note the C∗-algebra generated by the unitaries ρ(gi), 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Let µ be a positive regular Borel measure on G, and let L1(G, µ) be the set of integrable complex func- tions on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' For a unitary representation π, we define π(m) = � G m(g)π(g)dµ(g), JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY 7 where the convergence of the integral is with respect to the weak operator topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Since each π(g) is a unitary, we have ∥π(m)∥ ≤ � G |m(g)|dµ(g) = ∥m∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Further, if the the constant function 1 ∈ L1(G, µ), we write π(1) as π(µ) to avoid confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Clearly, for a discrete group G with counting measure and an arbitrary element m = z1g1 + · · · + zkgk ∈ C[G], we have π(m) = z1π(g1) + · · · + zkπ(gk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Other measures on a countable group is also useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Consider the measure µ on group G generated by the set S = {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', gn} such that µ{gi} = 1 n, j = 1, 2, · · · , n, and µ{g} = 0 for all g /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Clearly, µ is a probability measure that is absolutely continuous with respect to the Haar measure on G, and µ’s support is S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' In this case π(µ) = � G π(g)dµ(g) = 1 n(π(g1) + π(g2) + · · · π(gn)), and it is called the Markov operator with respect to the generating set S and the representation π, for which we denote by Mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Since each π(gi) is a unitary, the triangular inequality implies ∥Mπ∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' As we shall soon see, the spectrum of Mπ holds important information about amenabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Weak containment has a useful description in terms of C∗-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Let ρ and π be two unitary representations of a countable group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Then the following statements are equivalent: (a) π ≺ ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' (b) σ(π(m)) ⊂ σ(ρ(m)) for each m ∈ ℓ1(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' (c) ∥π(m)∥ ≤ ∥ρ(m)∥ for each positive m ∈ C[G];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' (d) the map φ : C∗ ρ(G) −→ C∗ π(G) defined by φ(ρ(g)) = π(g), g ∈ G extends to a surjective homomorphisim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The following theorem is [4] Theorem G 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' 8 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' YANG Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='4 (Hulanicki-Reiter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Let G be a locally compact group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The following properties are equivalent: (i) G is amenable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' (ii) 1G ≺ λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' (iii) π ≺ λ for every unitary representation π of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' In other words, the group G is amenable if and only if its left regular representa- tion is maximal with respect to weak containment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' A spectral characterization of amenability was given by Kesten [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Let G be a locally compact group and µ be a probability measure on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Assume that µ is absolutely continuous with respect to the Haar measure on G and supp(µ) generates a dense subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' (a) G is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' (b) 1 ∈ σ(λ(µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' (c) The spectral radius of λ(µ) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' If µ is the probability measure in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='2, then λ(µ) = Mλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Hence a finitely generated group G is amenable if and only if 1 ∈ σ(Mλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Projective Spectrum and Amenability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Two representations (ρ1, H1) and (ρ2, H2) are said to be equivalent if there is a unitary map U : H1 → H2 such that ρ2(g) = Uρ1(g)U−1, g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' It is obvious that in this case for any elements g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', gn ∈ G, we have Aρ2(z) = UAρ1(z)U−1 and hence p(Aρ1) = p(Aρ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' This indicates that projective spectrum is a unitary invariant for group represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Moreover, if π ≺ ρ, then Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='3 (d) implies that if ρ(m) is in- vertible in C∗ ρ(G) then π(m) is invertible in C∗ π(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Since Aπ(z) = π(m(z)) and Aρ(z) = ρ(m(z)), where m(z) = z01 + z1g1 + · · · + zngn ∈ C[G], the following is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' JOINT SPECTRUM IN AMENABILITY AND SELF-SIMILARITY 9 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Consider two unitary represntations π and ρ of a discrete group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' If π ≺ ρ, then p(Aπ) ⊂ p(Aρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' It follows that if π ∼ ρ then p(Aπ) = p(Aρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' In other words, the projective spectrum p(Aπ) is also an invariant of π with respect to weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' It is not clear whether the converse of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='6 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' At least, it holds when π = 1G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Suppose group G is generated by a finite set S and π is a unitary representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Then 1G ≺ π if and only if Hs ⊂ p(Aπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' The necessity has been observed in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' For the other direction, we assume S = {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', gn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Since Hs ⊂ p(Aπ), we have (−1, 1 n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', 1 n) ∈ p(Aπ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', 1 ∈ σ(Mπ), where Mπ is the associated Markov operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Thus, either Mπ or M∗ π is not bounded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Without loss of generality, we assume the former occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Then there exists a sequence of unit vectors ξk ∈ H such that ∥Mπξk − ξk∥ → 0 (and hence ∥Mπξk∥ → 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' It follows that n � i=1 ∥π(gi)ξk − ξk∥2 = n(2 − 2 Re⟨Mπξk, ξk⟩) = n(∥Mπξk − ξk∥2 + 1 − ∥Mπξk∥2) → 0, which implies ∥π(gi)ξk − ξk∥ = ∥ξk − π(g−1 i )ξk∥ → 0 for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' We let S−1 stand for the set {g−1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=', g−1 n }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Then G = ∪∞ m=0(S ∪ S−1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Since G is countable, its subset Q is compact if and only if it is finite, in which case there exists an integer M such that Q ⊂ ˆS := ∪M m=0(S ∪ S−1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' Given any ǫ > 0, we let ξ be such that ∥π(gi)ξ − ξ∥ = ∥ξ − π(g−1 i )ξ∥ < ǫ M , 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' 10 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AzT4oBgHgl3EQfqv2W/content/2301.01634v1.pdf'} +page_content=' YANG Then for each x = x1 · · · xm ∈ ˆS, where xi ∈ S ∪ S−1, we have ∥π(x1 · · · xm)ξ − ξ∥ =∥π(x1 · · · xm)ξ − π(x1 · · · xm−1)ξ + · · · + π(x1x2)ξ − π(x1)ξ + π(x1)ξ − ξ∥ ≤∥π(x1 · · · xm−1)(π(xm)ξ − ξ)∥ + · · · + ∥π(x1)(π(x2)ξ − ξ)∥ + ∥π(x1)ξ − ξ∥ =∥π(xm)ξ − ξ∥ + · · · + ∥π(x2)ξ − ξ∥ + ∥π(x1)ξ − ξ∥ 0, an exponential solution for the scale factor. +Using the definition of the Ricci scalar, +R = −6 +� +˙H + 2H2 +� +, +(14) + +4 +the solution implies R constant and negative, in agreement with a maximally symmetric de Sitter space-time. The +solution for the scale factor is given by, +a ∝ e±Ht, +(15) +describing either an exponentially expanding or a contracting universe. +For the applications for the inflationary +universe, only the expanding solution is used. Remark that the Minkowski case corresponds to Λ = 0. Hence, the +Minkowski and the de Sitter are the possible space-times in the vacuum cosmological solutions of the GR theory in +presence of a cosmological constant, Minkowski being the trivial one. +For UG, the vacuum case leads to two simple equations: +˙H = 0, +(16) +˙R = 0. +(17) +The last one implies R = −2ΛU, where now ΛU is an integration constant which be related to the cosmological +constant in the GR context. The factor −2 has been introduced just to make this connection easier. Again, the +solution of (17) implies, +H = ± +� +ΛU +3 +→ +a ∝ e± +� +ΛU +3 . +(18) +The constant ΛU must be positive or zero, the last case leading to the trivial Minkowski space-time. The Ricci scalar +leads to the equation, +H2 = ΛU +3 , +(19) +that is the Friedmann equation in GR for the vacuum case in presence of a cosmological constant. But the origin +of this cosmological constant is quite different: it does not come from the Lagrangian with a cosmological term, but +as an integration constant. In general it is argued that this (important) formal difference between the Friedmann +equation in GR and UG alleviates the cosmological constant problem. +What happens if matter is introduced? The main point is that, as it was already briefly discussed above, it is +impossible to solve the UG cosmological equations in this case since there is just equation (11) for two variables ρ and +a: the equation can not be solved without an extra assumption. If the conservation of the energy-momentum tensor is +imposed, we obtain the same solutions as GR in presence of a cosmological constant, the cosmological term appearing, +as in the vacuum case, as an integration constant. Another path has been followed in Ref. [8], exploring the traceless +character of the field equations in UG, which implies that the matter sector must be also traceless indicating that a +radiative fluid is the natural choice for the matter sector. In this case, the usual radiative solutions in presence of a +cosmological constant are recovered. However, the perturbative behavior strongly differs from the GR case. +Now, we turn to the perturbative analysis of the vacuum solution, introducing some comments for the case matter +is present. +IV. +PERTURBATIVE ANALYSIS +At perturbative level the differences between GR an UG becomes more evident. First of all, while in GR the general +diffeomorphism invariance allow to fix a coordinate condition or to use a gauge invariant set of variables when the +perturbative analysis is made, in UG the choices are much more restricted due to the invariance by the TD. This can +been seen by considering the unimodular constraint, +√−g = ξ. +(20) +As already discussed, ξ is a fixed external quantity. Perturbing the metric, +˜gµν = gµν + hµν, +(21) +and preserving the unimodular constraint, we are led to the relation, +h = hρ +ρ = 0. +(22) + +5 +Let us now consider the general perturbed metric restricted to the scalar sector: +ds2 = a2 +� +(1 + 2φ)dη2 − 2B,idxidη− +� +(1 − 2ψ)δij + 2E,i,j +� +dxidxj +� +. +(23) +Here on we follow closely the notation of Ref. [10]. The condition h = 0, implies, +φ − 3ψ − ∇2E = 0. +(24) +The newtonian gauge is obtained by fixing B = E = 0, implying φ = 3ψ. This condition contradicts the other +condition obtained from the perturbed equations when anisotropic pressure is absent, φ = ψ, leading to φ = φ = 0 +and no perturbation is present. The situation with the synchronous coordinate condition is more involved, since +this condition implies φ = B = 0, leading to ∇2E = −3ψ, which can be re-expressed as hkk = 0. However, if the +conservation of the energy-momentum is imposed, hkk is directly related to the matter perturbation: hkk being zero, +there is no matter perturbation also. The situation changes when the conservation of the energy-momentum tensor +is not imposed, as we will se latter. The gauge invariant formalism [10] can always be used, but with the additional +condition (24). +A. +The perturbed equations in the gauge invariant formalism +The perturbed field equations in GR using the gauge invariant formalism, with the hydrodynamical approach, read +[10]: +−3H(Ψ′ + HΦ) + ∇2Ψ = 4πGa2δ¯ρ, +(25) +� +Ψ′ + HΦ +� +,i += −4πG(ρ + p)a3δ¯ui, +(26) +� +Ψ′′ + H(2Ψ′ + Φ′) + (2H′ + H2)Ψ + 1 +2D +� +δij − 1 +2D,i,j = 4πGa2δ¯pδij, +(27) +where D = Φ − Ψ. Moreover, H = a′/a, the primes indicating derivative with respect to the conformal time. +No anisotropic pressure is considered. The non-diagonal terms of (27) i ̸= j lead to D = 0, implying Φ = Ψ. The +resulting equations are, +−3H(Φ′ + HΦ) + ∇2Φ = 4πGa2δ¯ρ, +(28) +� +Φ′ + HΦ +� +,i += 4πG(ρ + p)a3δ¯ui, +(29) +Φ′′ + 3HΦ′ + (2H′ + H2)Φ = 4πGa2δ¯p. +(30) +The bars in the perturbed fluid quantities indicate that we are using the gauge invariant expressions. +In UG we must perturb the equations, +Eµν = 8πGτµν, +(31) +with the definitions, +Eµν = Rµν − 1 +4gµνR, +(32) +τµν = Tµν − 1 +4gµνT. +(33) +The perturbed equations of the UG equations (32,33) coupled to a fluid, using the gauge invariant formalism are: +Φ′′ + 2(H′ − H2)Φ + ∇2Φ = 4πGa2(δ¯ρ + δ¯p), +(34) +� +Φ′ + HΦ +� +,i += −4πGa3δ¯ui. +(35) +In obtaining these last expressions we have already used the fact that Φ = Ψ. There are two important remarks on +the equations (34,35). First, there are two equations for three functions, Φ, δ˜ρ = δ¯ρ + δ¯p, and δ¯ui. We will comment +more on this issue later. The second important remarks is the the term ∇2Φ appears with the ”wrong” sign compared +with the GR case. We will also discuss more this fact later. + +6 +B. +Vacuum case: perturbations +For vacuum, δ¯ρ, δ¯p and δ¯ui are absent. The GR perturbed equations become, +−3H(Φ′ + HΦ) + ∇2Φ = 0, +(36) +� +Φ′ + HΦ +� +,i += 0, +(37) +Φ′′ + 3HΦ′ + (2H′ + H2)Φ = 0. +(38) +On the other hand, the corresponding equations for UG are, +Φ′′ + 2(H′ − H2)Φ + ∇2Φ = 0, +(39) +� +Φ′ + HΦ +� +,i += 0. +(40) +Let us first consider the de Sitter solution, for which, in the conformal time, a ∝ 1 +η . Equations (37) and (40) are +the same, and it is satisfied in two cases: either Φ ∝ 1/a or the perturbed quantities are spatial independent. Both +hypothesis are consistent with each other. Hence, in both GR and UG cases, the solution of the perturbed equations +are, +Φ = Φ0 +a , +(41) +Φ0 being a constant. The metric perturbation decreases as the universe expands, in agreement with the structure of +the de Sitter space-time. +If now the Minkowski vacuum solution is inserted in the perturbed equations, Φ′′ = 0, leading to Φ ∝ η + constant, +both for GR and UG. Since the conformal time is, for the Minkowski case, equivalent to the cosmic time, the solution +represents a growing mode. +C. +Introducing matter fields +When matter is present many new features appear. +First of all, many aspects of the problem depend if the +conservation of the energy-momentum tensor is imposed or not. If the energy-momentum tensor conserves as in GR +one of the first consequence is that the synchronous coordinate condition can not be use. The reason is the following. +The unimodular constraint implies, +hρ +ρ = 0. +(42) +If the synchronous coordinate condition hµ0 = 0 is imposed, the unimodular constraint reduces to hkk = 0 (a sum on +the indice k is understood). Using the conservation of the energy-momentum tensor, the UG equations reduce to the +GR in presence of a cosmological constant. The perturbation of the field equations lead to the perturbed equation +[11], +¨˜h + 2H ˙˜h = 8πGδρ, +(43) +with ˜h = hkk/a2. If hkk = 0 then δρ = 0 and no perturbation is present. +Due to this property, a possibility is to use the gauge invariant formalism. +This has been done in Ref. +[12]. +There, they found essentially the same equations of GR but with a new ingredient, a relation between the perturbed +quantities due to the unimodular constraint. Hence, at perturbative level, even imposing the conservation of the +energy-momentum tensor, UG has some distinguishing features. +If the conservation of the energy-momentum tensor is not imposed, the situation becomes more complex. The +restriction to the use of the synchronous coordinate condition does not exist any more, but even so hkk = 0. However, +the density perturbation becomes connected to another metric perturbation f = hik,i,k/a2. One important remark +is that now there is no residual coordinate freedom associated to the synchronous coordinate condition. In fact, in +GR the synchronous coordinate condition does not fix completely the coordinate system, and a residual, non-physical +mode remains [13]. This fact is reflected in the third order (instead of a second order) differential equation for the + +7 +density perturbation. However, in UG the unimodular condition eliminates this non-physical mode, and we end up +with second order differential equations. +Of course, the gauge invariant formalism can always be used in UG, even with the modified conservation laws. +However, there is a technical issue. As we can inspect from equations (34,35), in the perturbed UG field equations +there are two equations for three unknown functions; a new independent equation is needed. This new equation comes +from the modified conservation law [8]. Using the gauge invariant formalism, to determine this new independent +equation is a quite involved technical issue, while it is somehow direct using the synchronous coordinate condition. +This has been done in Ref. [8], where it was obtained the equation, +¨f + 3H ˙f − k2 +3a2 f = 0. +(44) +In this equation, k is the wavenumber associated with the perturbation.. +The final solution in terms of the conformal time (η ∝ t1/2) reads +f = A +sinh +k +√ +3η +kη ++ B +cosh +k +√ +3η +kη +. +(45) +This solution reveals an exponential growth of the perturbations even if the background corresponds to the radiative +phase. This is due to the ”wrong” sign with the k-dependent term in (44) which is related with the Laplacian operator. +We have already remarked that in the gauge invariant formalism such ”wrong sign” of the Laplacian operator also +appears, see (34), and a similar behavior can be expected. We are currently analysing this issue. +In Ref. [8] it has been also shown that a possible viable cosmological model can be obtained in UG when the +modified energy-momentum tensor conservation is retained. This model must be refined in many ways, but in general +lines, the age of the universe, the CMB radiation, the present accelerated phase, and the origin of the structures +resulting from the gravitational collapse out of a homogeneous and isotropic universe are well predicted by this model. +V. +AN EXTENSION OF UNIMODULAR GRAVITY: INCLUDING SCALAR FIELDS +The most direct extension of GR is by including scalar fields. It can be a self-interacting field representing the +matter sector. In this case, the we modify only the right hand side of the field equations. However, it can also be +implemented by a non-trivial coupling with the geometric sector and, in this case, the implications are more profound. +A paradigmatic example is the Brans-Dicke (BD) theory, whose field equations, in presence of a cosmological constant, +is given by[14], +Rµν − 1 +2gµνR = 8π +φ Tµν + ω +φ2 +� +φ;µφ;ν − 1 +2gµνφ;ρφ;ρ +� ++ 1 +φ +� +φ;µν − gµν□φ +� ++gµνΛ, +(46) +□φ = +8πT +3 + 2ω + +4 +3 + 2ωΛ, +(47) +T µν;µ = 0. +(48) +In these equations, ω is a free coupling parameter. GR is recovered when ω → ∞. The present estimations indicate +a very high value for ω [15]. Even though, BD remain an intensive object of studies, and it can be connected with +many other fundamental theories, like string theories[16]. +The unimodular version of the Brans-Dicke (UBD) theory has been proposed in Ref. [9]. The deduction of the field +equations follows closely the RG case, introducing the unimodular constraint through Lagrangian multipliers. The +final equations read, +Rµν − 1 +4gµνR = 8π +φ +� +Tµν − 1 +4gµνT +� ++ ω +φ2 (φ;µφ;ν − 1 +4gµνφ;ρφ;ρ) ++ 1 +φ(φ;µν − 1 +4gµν□φ), +(49) +□φ = 1 +2 +φ;ρφ;ρ +φ +− φ +2ωR, +(50) +(φR);ν = ω +�φ;ρφ;ρ +φ +�;ν ++32π +� +T µν;µ − 1 +4T ;ν +� ++3(□φ);ν. +(51) + +8 +In this case, as in the GR one, the usual conservation of the energy-momentum tensor has not been imposed. If the +usual conservation laws are introduced, the BD equations in presence of a cosmological constant are recovered. +The UBD has many new features in comparison with the traditional BD theory. We will comment just one of them. +In Ref.[17] an extensive perturbative analysis of cosmological models obtained from the BD theory was carried out. +The vacuum cosmological solutions in UBD coincide with the BD cosmological solutions in presence of a cosmological +constant, as it happens with the corresponding solutions in GR. In Ref. [17], the vacuum solutions in presence of +a cosmological term in the BD have been shown to be stable. However, in the UBD case the vacuum solution are +unstable in the interval −1/2 < ω < 3/2. This is an important difference, pointing out that, even if the background +UBD solutions can be mapped in the BD solutions in presence of a cosmological constant (something we could +expect from the experience with GR and UG), the perturbative behavior is sensitively different. The inequivalence +at perturbative level of BD and UBD theories is more evident than in the GR and UG case. This seems to be due to +the presence of the scalar field non-minimally coupled to the gravity sector. +In UBD the scalar perturbations in the vacuum case, using the synchronous coordinate condition (which is now +allowed), may be expressed through the master equation, +(3 − 2ω)δ ¨φ− +� +3(1 + 2ω)H − 8ω +˙φ +φ +� +δ ˙φ+ +� +12( ˙H + H2) − 4ω +˙φ2 +φ2 +� +δφ + 1 + 2ω +a2 +∇2δφ = 0. +(52) +The existence of instability in the UBD case, at least for some values of ω, can be directly seen from (52) by inspecting +the relative sign of the second derivative and the Laplacian terms of δφ. In some sense, it is connected with the sound +speed in the perturbations which, in (52), becomes imaginary for −1/2 < ω < 3/2. +VI. +CONCLUSIONS +Unimodular Gravity has been extensively discussed in the literature due to the possibility of giving new insights +to some of the most important problems appearing in General Relativity. In particular, in UG the cosmological +constant is somehow hidden in the general structure of the theory and, as far as the usual energy-momentum tensor +conservation is imposed, it appears explicitly as an integration constant. This is generally considered as, at least, an +alleviation of the cosmological constant problem that plagues GR. At quantum level also, it has been argued that UG +can shed some light in the issues that appear in GR[6]. +The discussion of the possible equivalence between UG (without cosmological constant) and GR (in presence of a +cosmological constant) is intensive, see Refs. [4–6] for example. At the cosmological background level, the equivalence +seems to be clearly set out. But, the situation is less clear at the level of cosmological perturbations. In our point +of view, the main aspect to be stressed is the invariance of UG to the more restricted transverse diffeomorphic +transformation, while GR is invariant by the full diffeomorphism group. +We have discussed the background and perturbative issues in UG comparing them to GR (always with a cosmological +constant). In vacuum, UG provides the same results as GR also at perturbative level, even if following a different +path. In presence of matter, however, the situation is much more complex, depending first if the usual conservation +laws are retained or not. If not, the configuration is clearly different [8], as it could be expected, but even if the usual +conservation laws are preserved, some new features appear. The difference becomes stronger and more pronounced +if a non-minimally coupled escalar field is introduced. In doing so, we can generalize the Brans-Dicke theory to the +unimodular Brans-Dicke theory. An example of the this strong difference is the appearance of unstable modes in UBD +which do not exist in the BD case. +Many aspects of the perturbative analysis presented here must be extended. One example is the use of the full gauge +invariant formalism for the cases where matter is present and the conservation of the canonical energy-momentum +tensor is not retained. But, the results exposed here seem to reinforce that the equivalence between UG and RG, even +at classical level, is not complete, mainly when the perturbations are included. +Acknowledgments: We thank CNPq, FAPES and CAPES for partial financial support. We thank also Luiz Filipe +Guimar˜aes for his careful reading of the text. +[1] A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 349–356 (1919). +[2] W. Pauli, Theory of Relativity, Dover, New York (1981). +[3] J. J. Lopez-Villarejo, JCAP 11, 002(2011). +[4] S. Weinberg, Rev. Mod. Phys. 61, 1(1989). + +9 +[5] G. P. de Brito, O. Melichev, R. Percacci and A. D. Pereira, JHEP 12, 090(2021). +[6] R. Carballo-Rubio, L.J. Garay and G. Garc´ıa-Moreno, Class. Quantum Grav. 39, 243001(2022). +[7] R.M. Wald, General Relativity, Chicago University Press, Chicago (1984). +[8] J.C. Fabris, M.H. Alvarenga, M. Hamani-Daouda and H. Velten, Eur. Phys. J. C82, 522(2022). +[9] A.M.R. Almeida, J.C. Fabris, M. Hamani Daouda, R. Kerner, H. Velten and W.S. Hip´olito-Ricaldi, Universe 8, 429 (2022). +[10] V.F. Mukhanov, H.A. Feldman and R.H. Brandenberger, Phys. Rep. 215, 203(1992). +[11] S. Weinberg, Gravitation and Cosmology, Wiley, New York(1972). +[12] C. Gao, R.H. Brandenberger, Y. Cai and P. Chen, JCAP 09, 021(2014). +[13] P.J.E. Peebles, The large scale structure of the universe, Princeton university press, Princeton(2020). +[14] C. Brans and R.H. Dicke, Phys. Rev. 124, 925(1962). +[15] C.M. Will, Living Rev. Relativity, 17, 4(2014). +[16] J. E. Lidsey, D. Wands and E. J. Copeland, Phys. Rep. 337, 343 (2000). +[17] J. P. Baptista, J. C. Fabris and S. V. B. Goncalves, Astrophys. Space Sci. 246, 315(1996). + diff --git a/o9FMT4oBgHgl3EQf7jFB/content/tmp_files/load_file.txt b/o9FMT4oBgHgl3EQf7jFB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6346ad3adceec151a78ada38572567c279fed1d2 --- /dev/null +++ b/o9FMT4oBgHgl3EQf7jFB/content/tmp_files/load_file.txt @@ -0,0 +1,404 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf,len=403 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='12464v1 [gr-qc] 29 Jan 2023 APS/123-QED Using cosmological perturbation theory to distinguish between General Relativity and Unimodular Gravity Marcelo H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Alvarenga∗ N´ucleo Cosmo-ufes & Departamento de F´ısica, UFES, Vit´oria, ES, Brazil J´ulio C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Fabris† N´ucleo Cosmo-ufes & Departamento de F´ısica, UFES, Vit´oria, ES, Brazil National Research Nuclear University MEPhI, Kashirskoe sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' 31, Moscow 115409, Russia Hermano Velten‡ Departamento de F´ısica, Universidade Federal de Ouro Preto (UFOP), Campus Universit´ario Morro do Cruzeiro, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='400-000, Ouro Preto, Brazil (Dated: January 31, 2023) Unimodular Gravity is one of the oldest geometric gravity theory alternative to General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Essentially, it is based on the Einstein-Hilbert Lagrangian with an additional constraint on the determinant of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' It can be explicitly shown that Unimodular Gravity can be recast as General Relativity in presence of a cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This fact has led to many discussions on the equivalence of both theories at classical and quantum levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Here we present an analysis focused on the classical scalar perturbations around a cosmological background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The discussion is extended for the case where a non-minimal coupled scalar field is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Our results indicate that the equivalence is not verified completely at perturbative level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' PACS numbers: 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='Cv, 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='-m, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='-k I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' INTRODUCTION General Relativity (GR) theory aims to describe gravity as a manifestation of the geometry of the space-time in four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' It is based on the mathematics of differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The theory is invariant by the full group of diffeomorphism transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' It has led to important previsions like the existence of black holes and gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' At same time, it admits very successful applications for the description of the universe as a whole, leading to the standard cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The latter explains in a quite simple way all observational data, even at the price of introducing a dark sector composed of two exotic components (dark matter and dark energy), not detected directly until now, and also a primordial inflationary phase driven by a new field, the inflation, whose nature is still a matter of debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' On the other hand, GR is plagued by the presence of singularities and, at the same time, it has not been quantized until now in a fully consistent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' These last problems led to an intensive search for an alternative description of the gravitational phenomena, in general keeping the geometrical approach, which may cope with the existence of the dark sector, the inflationary phase, being free of singularities and admitting a consistent quantum version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' GR has been proposed in 1915 and soon after alternative formulations appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In 1919, the unimodular con- strained version of GR has been formulated, leading to what is now known as Unimodular Gravity (UG) [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In UG the determinant of the metric is fixed as a constant, in occurrence, equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Whereas this can be viewed as a choice of the coordinate system, it has some important consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' First of all, UG is invariant by a subclass of transformations, called transverse diffeomorphism (TD) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Moreover, the field equations are traceless implying the absence of information about geometrical quantities like e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=', the Ricci scalar R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Matter can be coupled in a particular way to the geometrical sector, preserving the traceless character of the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' If the conservation of the canonical energy-momentum tensor is imposed, the UG field equations imply the GR equations in presence of a cosmological constant Λ, which appears as an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This is generally viewed as an advantage with respect to GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Moreover, it is generally argued that UG has some improvements over GR at quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Many discussions exist, on the other hand, on the possible equivalence between GR e UG, but the restriction of the invariance of UG to the ∗Electronic address: marcelo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='alvarenga@edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ufes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='br †Electronic address: julio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='fabris@cosmo-ufes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='org ‡Electronic address: hermano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='velten@ufop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='br 2 TD group seems to indicate that this equivalence is not complete, even if it may appear in some contexts (see, for example, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [4–6] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' One of the possible reasons for the non-equivalence between GR and UG comes from the conservation of the canonical energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In GR the conservation laws are direct consequences of the invariance with respect to general diffeomorphism transformation [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This property is reflected in the use of the Bianchi identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' For UG the invariance with respect to the TD does not allow to obtain the same conservation laws as in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Moreover, the UG field equations are traceless and the application of the Bianchi identities leads to a relation where the divergence of the canonical energy-momentum tensor is not necessarily zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' However, we can impose by hand the same conservation laws as in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In doing so, UG leads to the GR field equations in presence of a integration constant (identified as the cosmological constant) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Does this means an equivalence between GR and UG?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' First we must remember that the usual conservations constitute a choice in UG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Moreover, in UG, even with this choice, the invariance remains dictated by the restricted TD, instead of the full diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In this text, we will discuss the issue of the perturbative features of UG in comparison to GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' It will be shown that, while for the vacuum case there seems to be an equivalence, the presence of matter may modify this conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This perturbative non-equivalence can be extended even for the vacuum solutions when an extension of the unimodular structure to non-minimal coupled scalar field is implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In next section, we settle out both the fundamental equations for the cosmological background both for GR and UG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' We turn to the the specific cosmological background solutions in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In section IV the perturbative issue is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' It is shown the equivalence of the results in both context in the vacuum case, and it is discussed the case when matter is present, for which the equivalence may not be verified anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The unimodular version of the Brans-Dicke theory is discussed in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In section VI we present our final considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' BACKGROUND COSMOLOGICAL STRUCTURE The GR equations in the presence of a cosmological constant and of a matter sector, as deduced from the Einstein- Hilbert Lagrangian, are, Rµν − 1 2gµνR = 8πGTµν + gµνΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (1) The application of the Bianchi identities leads to the energy-momentum tensor T µν conservation: T µν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (2) The conservation laws related to the energy-momentum tensor can be alternatively deduced from the invariance of the Einstein-Hilbert Lagrangian by diffeomorphic transformations [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The UG equations can also be deduced from the Einstein-Hilbert Lagrangian but through the introduction of a constraint on the determinant of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' For details, see [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' With this procedure, the UG field equations read, Rµν − 1 4gµνR = 8πG � Tµν − 1 4gµνT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (3) Equations (3) are valid even if we introduce a cosmological constant in the action, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=', after solving the constraints represented by the Lagrangian multiplier, the cosmological term disappears from the final equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The application of the Bianchi identities leads to, R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ν 4 = 8πG � T µν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='µ − T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ν 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (4) Equation (4) deserves some comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In opposition to what happens in GR, the TD, on which is based UG, does not lead necessarily to the conservation equations (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Instead, it predicts a non-vanishing divergence of the energy-momentum tensor given by, T µν ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='µ = Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ν, (5) Θ being, in principle, an unknown scalar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Nothing forbids to impose the extra condition given by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' If this extra condition is imposed, (4) can be integrated leading to (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' However, in doing so, Λ appears as an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This is the basis of the usual remark that UG may alleviate the cosmological constant problem, since Λ now is not associated with the vacuum energy or to a geometric zero order Lovelock invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' 3 In fact, if the conservation of the energy-momentum tensor is imposed, by vanishing the right hand side of (5), then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (4) implies, R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ν = −8πGT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ν ⇒ R + 8πGT = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (6) Identifying the constant in the right hand side of the above result with −4Λ, the GR equations with a cosmological term are recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Of course, we can also keep (4), since it results from the application of the Bianchi identities via the identification Θ = R 4 + 2πGT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (7) But, again, this is a choice and many other possible ones are admitted since Θ is not determined from the unimodular construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Let us for the moment make the choice (7) in order to verify the consequences in a specific cosmological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In any case, the choice Θ = 0 implies GR with a cosmological constant, with the remaining issue of the interpretation of this constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' For sake of simplicity, let us consider the flat FLRW metric given by, ds2 = dt2 − a2(t)(dx2 + dy2 + dz2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (8) In our approach to the UG, the unimodular condition is √−g = ξ where ξ is an arbitrary reference (external, if one prefers) tensorial density, allowing to choose freely the coordinate system, in opposition to the original choice √−g = 1, which fixes the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Applying the metric (8) on the GR equations we obtain the following equations of motion: H2 = 8πG 3 ρ + Λ 3 , (9) ˙ρ + 3H(ρ + p) = 0, (10) with H = ˙a/a, the Hubble function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' We must specify an equation of state connecting ρ and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In doing so, we are left with two equations for two variables, ρ and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' On the other hand, the same metric applied to (3,4) implies, ˙H = −4πG(ρ + p), (11) ¨H + 4H ˙H = −4πG[ ˙ρ + ˙p + 4H(ρ + p)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (12) The main point here is that, inserting (11) into (12) we obtain an identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Hence, both equations have the same content: the system is underdetermined, having more unknown functions than equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This is a consequence of the fact that the UG equations (3) are traceless, leading to no information on the Ricci scalar, in opposition to the GR equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Moreover, equation (11), as (12), is sensitive only to the combination ρ + p which is the enthalpy density of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [8] it has been made the choice that matter behaves as radiation, as suggested by the traceless character of the field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This choice has led to many drastic implications at perturbative level, as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In particular, as we will discuss later, there is a direct transition from the radiation era to the de Sitter era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' However, due to the structure of UG, fluctuations grow strongly even during the radiation phase, what may assure the formation of structures in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In what follows, we will focus first on the vacuum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This will allow us to verify, for a specific configuration, to which extent the UG is, at least classically, equivalent to GR as frequently evoked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' UG AND GR VACUUM COSMOLOGICAL SOLUTIONS In the vacuum case, ρ = p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Notice however that Λ may remain different from zero in GR, leading to, H2 = Λ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (13) The first remark is that only Λ ≥ 0 is possible, that is, a de Sitter or Minkowski solution, excluding the Anti-de Sitter case (Λ < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The Hubble function is constant, implying, for Λ > 0, an exponential solution for the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Using the definition of the Ricci scalar, R = −6 � ˙H + 2H2 � , (14) 4 the solution implies R constant and negative, in agreement with a maximally symmetric de Sitter space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The solution for the scale factor is given by, a ∝ e±Ht, (15) describing either an exponentially expanding or a contracting universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' For the applications for the inflationary universe, only the expanding solution is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Remark that the Minkowski case corresponds to Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Hence, the Minkowski and the de Sitter are the possible space-times in the vacuum cosmological solutions of the GR theory in presence of a cosmological constant, Minkowski being the trivial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' For UG, the vacuum case leads to two simple equations: ˙H = 0, (16) ˙R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (17) The last one implies R = −2ΛU, where now ΛU is an integration constant which be related to the cosmological constant in the GR context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The factor −2 has been introduced just to make this connection easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Again, the solution of (17) implies, H = ± � ΛU 3 → a ∝ e± � ΛU 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (18) The constant ΛU must be positive or zero, the last case leading to the trivial Minkowski space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The Ricci scalar leads to the equation, H2 = ΛU 3 , (19) that is the Friedmann equation in GR for the vacuum case in presence of a cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' But the origin of this cosmological constant is quite different: it does not come from the Lagrangian with a cosmological term, but as an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In general it is argued that this (important) formal difference between the Friedmann equation in GR and UG alleviates the cosmological constant problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' What happens if matter is introduced?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The main point is that, as it was already briefly discussed above, it is impossible to solve the UG cosmological equations in this case since there is just equation (11) for two variables ρ and a: the equation can not be solved without an extra assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' If the conservation of the energy-momentum tensor is imposed, we obtain the same solutions as GR in presence of a cosmological constant, the cosmological term appearing, as in the vacuum case, as an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Another path has been followed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [8], exploring the traceless character of the field equations in UG, which implies that the matter sector must be also traceless indicating that a radiative fluid is the natural choice for the matter sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In this case, the usual radiative solutions in presence of a cosmological constant are recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' However, the perturbative behavior strongly differs from the GR case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Now, we turn to the perturbative analysis of the vacuum solution, introducing some comments for the case matter is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' PERTURBATIVE ANALYSIS At perturbative level the differences between GR an UG becomes more evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' First of all, while in GR the general diffeomorphism invariance allow to fix a coordinate condition or to use a gauge invariant set of variables when the perturbative analysis is made, in UG the choices are much more restricted due to the invariance by the TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This can been seen by considering the unimodular constraint, √−g = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (20) As already discussed, ξ is a fixed external quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Perturbing the metric, ˜gµν = gµν + hµν, (21) and preserving the unimodular constraint, we are led to the relation, h = hρ ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (22) 5 Let us now consider the general perturbed metric restricted to the scalar sector: ds2 = a2 � (1 + 2φ)dη2 − 2B,idxidη− � (1 − 2ψ)δij + 2E,i,j � dxidxj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (23) Here on we follow closely the notation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The condition h = 0, implies, φ − 3ψ − ∇2E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (24) The newtonian gauge is obtained by fixing B = E = 0, implying φ = 3ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This condition contradicts the other condition obtained from the perturbed equations when anisotropic pressure is absent, φ = ψ, leading to φ = φ = 0 and no perturbation is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The situation with the synchronous coordinate condition is more involved, since this condition implies φ = B = 0, leading to ∇2E = −3ψ, which can be re-expressed as hkk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' However, if the conservation of the energy-momentum is imposed, hkk is directly related to the matter perturbation: hkk being zero, there is no matter perturbation also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The situation changes when the conservation of the energy-momentum tensor is not imposed, as we will se latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The gauge invariant formalism [10] can always be used, but with the additional condition (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The perturbed equations in the gauge invariant formalism The perturbed field equations in GR using the gauge invariant formalism, with the hydrodynamical approach, read [10]: −3H(Ψ′ + HΦ) + ∇2Ψ = 4πGa2δ¯ρ, (25) � Ψ′ + HΦ � ,i = −4πG(ρ + p)a3δ¯ui, (26) � Ψ′′ + H(2Ψ′ + Φ′) + (2H′ + H2)Ψ + 1 2D � δij − 1 2D,i,j = 4πGa2δ¯pδij, (27) where D = Φ − Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Moreover, H = a′/a, the primes indicating derivative with respect to the conformal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' No anisotropic pressure is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The non-diagonal terms of (27) i ̸= j lead to D = 0, implying Φ = Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The resulting equations are, −3H(Φ′ + HΦ) + ∇2Φ = 4πGa2δ¯ρ, (28) � Φ′ + HΦ � ,i = 4πG(ρ + p)a3δ¯ui, (29) Φ′′ + 3HΦ′ + (2H′ + H2)Φ = 4πGa2δ¯p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (30) The bars in the perturbed fluid quantities indicate that we are using the gauge invariant expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In UG we must perturb the equations, Eµν = 8πGτµν, (31) with the definitions, Eµν = Rµν − 1 4gµνR, (32) τµν = Tµν − 1 4gµνT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (33) The perturbed equations of the UG equations (32,33) coupled to a fluid, using the gauge invariant formalism are: Φ′′ + 2(H′ − H2)Φ + ∇2Φ = 4πGa2(δ¯ρ + δ¯p), (34) � Φ′ + HΦ � ,i = −4πGa3δ¯ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (35) In obtaining these last expressions we have already used the fact that Φ = Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' There are two important remarks on the equations (34,35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' First, there are two equations for three functions, Φ, δ˜ρ = δ¯ρ + δ¯p, and δ¯ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' We will comment more on this issue later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The second important remarks is the the term ∇2Φ appears with the ”wrong” sign compared with the GR case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' We will also discuss more this fact later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Vacuum case: perturbations For vacuum, δ¯ρ, δ¯p and δ¯ui are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The GR perturbed equations become, −3H(Φ′ + HΦ) + ∇2Φ = 0, (36) � Φ′ + HΦ � ,i = 0, (37) Φ′′ + 3HΦ′ + (2H′ + H2)Φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (38) On the other hand, the corresponding equations for UG are, Φ′′ + 2(H′ − H2)Φ + ∇2Φ = 0, (39) � Φ′ + HΦ � ,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (40) Let us first consider the de Sitter solution, for which, in the conformal time, a ∝ 1 η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Equations (37) and (40) are the same, and it is satisfied in two cases: either Φ ∝ 1/a or the perturbed quantities are spatial independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Both hypothesis are consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Hence, in both GR and UG cases, the solution of the perturbed equations are, Φ = Φ0 a , (41) Φ0 being a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The metric perturbation decreases as the universe expands, in agreement with the structure of the de Sitter space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' If now the Minkowski vacuum solution is inserted in the perturbed equations, Φ′′ = 0, leading to Φ ∝ η + constant, both for GR and UG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Since the conformal time is, for the Minkowski case, equivalent to the cosmic time, the solution represents a growing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Introducing matter fields When matter is present many new features appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' First of all, many aspects of the problem depend if the conservation of the energy-momentum tensor is imposed or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' If the energy-momentum tensor conserves as in GR one of the first consequence is that the synchronous coordinate condition can not be use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The reason is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The unimodular constraint implies, hρ ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (42) If the synchronous coordinate condition hµ0 = 0 is imposed, the unimodular constraint reduces to hkk = 0 (a sum on the indice k is understood).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Using the conservation of the energy-momentum tensor, the UG equations reduce to the GR in presence of a cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The perturbation of the field equations lead to the perturbed equation [11], ¨˜h + 2H ˙˜h = 8πGδρ, (43) with ˜h = hkk/a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' If hkk = 0 then δρ = 0 and no perturbation is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Due to this property, a possibility is to use the gauge invariant formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This has been done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' There, they found essentially the same equations of GR but with a new ingredient, a relation between the perturbed quantities due to the unimodular constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Hence, at perturbative level, even imposing the conservation of the energy-momentum tensor, UG has some distinguishing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' If the conservation of the energy-momentum tensor is not imposed, the situation becomes more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The restriction to the use of the synchronous coordinate condition does not exist any more, but even so hkk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' However, the density perturbation becomes connected to another metric perturbation f = hik,i,k/a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' One important remark is that now there is no residual coordinate freedom associated to the synchronous coordinate condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In fact, in GR the synchronous coordinate condition does not fix completely the coordinate system, and a residual, non-physical mode remains [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This fact is reflected in the third order (instead of a second order) differential equation for the 7 density perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' However, in UG the unimodular condition eliminates this non-physical mode, and we end up with second order differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Of course, the gauge invariant formalism can always be used in UG, even with the modified conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' However, there is a technical issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' As we can inspect from equations (34,35), in the perturbed UG field equations there are two equations for three unknown functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' a new independent equation is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This new equation comes from the modified conservation law [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Using the gauge invariant formalism, to determine this new independent equation is a quite involved technical issue, while it is somehow direct using the synchronous coordinate condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This has been done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [8], where it was obtained the equation, ¨f + 3H ˙f − k2 3a2 f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (44) In this equation, k is the wavenumber associated with the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='. The final solution in terms of the conformal time (η ∝ t1/2) reads f = A sinh k √ 3η kη + B cosh k √ 3η kη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (45) This solution reveals an exponential growth of the perturbations even if the background corresponds to the radiative phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This is due to the ”wrong” sign with the k-dependent term in (44) which is related with the Laplacian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' We have already remarked that in the gauge invariant formalism such ”wrong sign” of the Laplacian operator also appears, see (34), and a similar behavior can be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' We are currently analysing this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [8] it has been also shown that a possible viable cosmological model can be obtained in UG when the modified energy-momentum tensor conservation is retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This model must be refined in many ways, but in general lines, the age of the universe, the CMB radiation, the present accelerated phase, and the origin of the structures resulting from the gravitational collapse out of a homogeneous and isotropic universe are well predicted by this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' AN EXTENSION OF UNIMODULAR GRAVITY: INCLUDING SCALAR FIELDS The most direct extension of GR is by including scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' It can be a self-interacting field representing the matter sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In this case, the we modify only the right hand side of the field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' However, it can also be implemented by a non-trivial coupling with the geometric sector and, in this case, the implications are more profound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' A paradigmatic example is the Brans-Dicke (BD) theory, whose field equations, in presence of a cosmological constant, is given by[14], Rµν − 1 2gµνR = 8π φ Tµν + ω φ2 � φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='µφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ν − 1 2gµνφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ρφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ρ � + 1 φ � φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='µν − gµν□φ � +gµνΛ, (46) □φ = 8πT 3 + 2ω + 4 3 + 2ωΛ, (47) T µν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (48) In these equations, ω is a free coupling parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' GR is recovered when ω → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The present estimations indicate a very high value for ω [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Even though, BD remain an intensive object of studies, and it can be connected with many other fundamental theories, like string theories[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The unimodular version of the Brans-Dicke (UBD) theory has been proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The deduction of the field equations follows closely the RG case, introducing the unimodular constraint through Lagrangian multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The final equations read, Rµν − 1 4gµνR = 8π φ � Tµν − 1 4gµνT � + ω φ2 (φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='µφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ν − 1 4gµνφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ρφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ρ) + 1 φ(φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='µν − 1 4gµν□φ), (49) □φ = 1 2 φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ρφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ρ φ − φ 2ωR, (50) (φR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ν = ω �φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ρφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ρ φ �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ν +32π � T µν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='µ − 1 4T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ν � +3(□φ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content='ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (51) 8 In this case, as in the GR one, the usual conservation of the energy-momentum tensor has not been imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' If the usual conservation laws are introduced, the BD equations in presence of a cosmological constant are recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The UBD has many new features in comparison with the traditional BD theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' We will comment just one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [17] an extensive perturbative analysis of cosmological models obtained from the BD theory was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The vacuum cosmological solutions in UBD coincide with the BD cosmological solutions in presence of a cosmological constant, as it happens with the corresponding solutions in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [17], the vacuum solutions in presence of a cosmological term in the BD have been shown to be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' However, in the UBD case the vacuum solution are unstable in the interval −1/2 < ω < 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This is an important difference, pointing out that, even if the background UBD solutions can be mapped in the BD solutions in presence of a cosmological constant (something we could expect from the experience with GR and UG), the perturbative behavior is sensitively different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The inequivalence at perturbative level of BD and UBD theories is more evident than in the GR and UG case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This seems to be due to the presence of the scalar field non-minimally coupled to the gravity sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In UBD the scalar perturbations in the vacuum case, using the synchronous coordinate condition (which is now allowed), may be expressed through the master equation, (3 − 2ω)δ ¨φ− � 3(1 + 2ω)H − 8ω ˙φ φ � δ ˙φ+ � 12( ˙H + H2) − 4ω ˙φ2 φ2 � δφ + 1 + 2ω a2 ∇2δφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' (52) The existence of instability in the UBD case, at least for some values of ω, can be directly seen from (52) by inspecting the relative sign of the second derivative and the Laplacian terms of δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In some sense, it is connected with the sound speed in the perturbations which, in (52), becomes imaginary for −1/2 < ω < 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' CONCLUSIONS Unimodular Gravity has been extensively discussed in the literature due to the possibility of giving new insights to some of the most important problems appearing in General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In particular, in UG the cosmological constant is somehow hidden in the general structure of the theory and, as far as the usual energy-momentum tensor conservation is imposed, it appears explicitly as an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' This is generally considered as, at least, an alleviation of the cosmological constant problem that plagues GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' At quantum level also, it has been argued that UG can shed some light in the issues that appear in GR[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The discussion of the possible equivalence between UG (without cosmological constant) and GR (in presence of a cosmological constant) is intensive, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [4–6] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' At the cosmological background level, the equivalence seems to be clearly set out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' But, the situation is less clear at the level of cosmological perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In our point of view, the main aspect to be stressed is the invariance of UG to the more restricted transverse diffeomorphic transformation, while GR is invariant by the full diffeomorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' We have discussed the background and perturbative issues in UG comparing them to GR (always with a cosmological constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In vacuum, UG provides the same results as GR also at perturbative level, even if following a different path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In presence of matter, however, the situation is much more complex, depending first if the usual conservation laws are retained or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' If not, the configuration is clearly different [8], as it could be expected, but even if the usual conservation laws are preserved, some new features appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' The difference becomes stronger and more pronounced if a non-minimally coupled escalar field is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' In doing so, we can generalize the Brans-Dicke theory to the unimodular Brans-Dicke theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' An example of the this strong difference is the appearance of unstable modes in UBD which do not exist in the BD case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Many aspects of the perturbative analysis presented here must be extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' One example is the use of the full gauge invariant formalism for the cases where matter is present and the conservation of the canonical energy-momentum tensor is not retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' But, the results exposed here seem to reinforce that the equivalence between UG and RG, even at classical level, is not complete, mainly when the perturbations are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Acknowledgments: We thank CNPq, FAPES and CAPES for partial financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' We thank also Luiz Filipe Guimar˜aes for his careful reading of the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Einstein, Sitzungsber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Preuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Berlin (Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9FMT4oBgHgl3EQf7jFB/content/2301.12464v1.pdf'} +page_content=' ), 349–356 (1919).' 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@@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0ea4abdca4f19b372375b18a7fc3eb06213f7115fe687537091fe0e557cb0353 +size 162155 diff --git a/p9E2T4oBgHgl3EQf0ggt/content/tmp_files/2301.04141v1.pdf.txt b/p9E2T4oBgHgl3EQf0ggt/content/tmp_files/2301.04141v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..67a88df795f47ad376bb8f9c1541bbc3f175d0f0 --- /dev/null +++ b/p9E2T4oBgHgl3EQf0ggt/content/tmp_files/2301.04141v1.pdf.txt @@ -0,0 +1,7629 @@ +APPLICATION OF MACHINE +LEARNING TO GAS +FLARING +by +Rong Lu +arXiv:2301.04141v1 [cs.LG] 11 Jan 2023 + +© Copyright by Rong Lu, 2020 +All Rights Reserved + +A thesis submitted to the Faculty and the Board of Trustees of the Colorado School +of Mines in partial fulfillment of the requirements for the degree of Doctor of Philosophy +(Petroleum Engineering). +Golden, Colorado +Date +Signed: +Rong Lu +Signed: +Dr. Jennifer L. Miskimins +Thesis Advisor +Golden, Colorado +Date +Signed: +Dr. Jennifer L. Miskimins +Professor and Head +Department of Petroleum Engineering +ii + +ABSTRACT +Currently in the petroleum industry, operators often flare the produced gas instead of +commodifying it. The flaring magnitudes are large in some states, which constitute problems +with energy waste and CO2 emissions. In North Dakota, operators are required to estimate +and report the volume flared. The questions are, how good is the quality of this reporting, +and what insights can be drawn from it? +Apart from the company-reported statistics, which are available from the North Dakota +Industrial Commission (NDIC), flared volumes can be estimated via satellite remote sensing, +serving as an unbiased benchmark. Since interpretation of the Landsat 8 imagery is hindered +by artifacts due to glow, the estimated volumes based on the Visible Infrared Imaging +Radiometer Suite (VIIRS) are used. Reverse geocoding is performed for comparing and +contrasting the NDIC and VIIRS data at different levels, such as county and oilfield. +With all the data gathered and preprocessed, Bayesian learning implemented by Markov +chain Monte Carlo methods is performed to address three problems: county level model +development, flaring time series analytics, and distribution estimation. +First, there is +heterogeneity among the different counties, in the associations between the NDIC and VIIRS +volumes. In light of such, models are developed for each county by exploiting hierarchical +models. Second, the flaring time series, albeit noisy, contains information regarding trends +and patterns, which provide some insights into operator approaches. Gaussian processes are +found to be effective in many different pattern recognition scenarios. Third, distributional +insights are obtained through unsupervised learning. The negative binomial and Gaussian +mixture models are found to effectively describe the oilfield flare count and flared volume +distributions, respectively. Finally, a nearest-neighbor-based approach for operator level +monitoring and analytics is introduced. +iii + +TABLE OF CONTENTS +ABSTRACT +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii +LIST OF FIGURES +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii +LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii +LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +xiii +LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii +ACKNOWLEDGMENTS +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix +DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi +CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 +1.1 +Research Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 +1.2 +Dissertation Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 +1.3 +Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 +CHAPTER 2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 +2.1 +Satellite Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 +2.2 +Bayesian Inference +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 +2.3 +Markov Chain Monte Carlo +. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 +2.4 +Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 +2.5 +Analytics Toolset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 +CHAPTER 3 DATA PREPROCESSING AND EXPLORATORY DATA ANALYSIS . 17 +3.1 +Data Gathering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 +3.1.1 +Landsat 8 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 +iv + +3.1.2 +VIIRS Estimated Volumes . . . . . . . . . . . . . . . . . . . . . . . . . 18 +3.1.3 +NDIC Monthly Production Reports . . . . . . . . . . . . . . . . . . . . 18 +3.1.4 +NDIC Shapefiles +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 +3.2 +Satellite Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 +3.3 +Reverse Geocoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 +3.4 +Correlational Analysis +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 +3.5 +State Level Flaring Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 +CHAPTER 4 COUNTY LEVEL FLARING MODEL +. . . . . . . . . . . . . . . . . . 32 +4.1 +Learning the Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 +4.2 +Hierarchical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 +4.3 +Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 +4.4 +Model Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 +4.5 +Model Reparameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 +4.6 +Model Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 +4.7 +Model Extensibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 +CHAPTER 5 FLARING TIME SERIES ANALYTICS . . . . . . . . . . . . . . . . . . 52 +5.1 +Learning the Flaring Pattern and Behavior . . . . . . . . . . . . . . . . . . . . 52 +5.2 +Gaussian Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 +5.2.1 +Mean Function +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 +5.2.2 +Covariance Function +. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 +5.2.3 +Inference and Model Reparameterization . . . . . . . . . . . . . . . . . 56 +5.3 +Suite of Models for Pattern Recognition +. . . . . . . . . . . . . . . . . . . . . 57 +5.3.1 +Modeling Proportion of Gas Flared . . . . . . . . . . . . . . . . . . . . 57 +v + +5.3.2 +Modeling Proportion of Wells Flaring . . . . . . . . . . . . . . . . . . . 63 +5.3.3 +Modeling Flare Detection Count . . . . . . . . . . . . . . . . . . . . . . 66 +5.3.4 +Modeling Proportion of Oil Flared +. . . . . . . . . . . . . . . . . . . . 70 +5.3.5 +Modeling Scale Factor between VIIRS and NDIC +. . . . . . . . . . . . 74 +5.3.6 +Predicting NDIC Flared Volume . . . . . . . . . . . . . . . . . . . . . . 80 +5.3.7 +A Look Back at the Prior Choices . . . . . . . . . . . . . . . . . . . . . 81 +CHAPTER 6 UNSUPERVISED LEARNING FROM MULTIPLE PERSPECTIVES . 83 +6.1 +Learning the Distribution +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 +6.2 +Probability Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 +6.3 +Modeling VIIRS Detection Count . . . . . . . . . . . . . . . . . . . . . . . . . 85 +6.4 +Modeling Flared Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 +6.4.1 +Model Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 +6.4.2 +Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 +6.4.3 +Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 +CHAPTER 7 DISCUSSION +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +102 +7.1 +Operator Level Monitoring and Analytics . . . . . . . . . . . . . . . . . . . . +102 +7.2 +Warnings Regarding Inconsistencies . . . . . . . . . . . . . . . . . . . . . . . +106 +7.3 +Caveats in Petroleum Data Analytics . . . . . . . . . . . . . . . . . . . . . . +111 +CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . +117 +8.1 +Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +117 +8.2 +Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +119 +vi + +LIST OF FIGURES +Figure 1.1 +Top 30 countries ranked by flared gas volume in 2018. . . . . . . . . . . . . 1 +Figure 1.2 +The time series show the trend of gas flaring for the top two states in the +United States (EIA 2019a). . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 +Figure 1.3 +This Google Earth imagery shows gas flaring being conducted on a well +location in North Dakota (Google Earth 2019). . . . . . . . . . . . . . . . . 3 +Figure 1.4 +Part of the original poster (Earth Observation Group at Payne Institute +2019) which uses one year accumulation of VIIRS low light imaging data +to showcase human activities, e.g., gas flaring, fishing, and city lights. +. . . 4 +Figure 2.1 +Landsat 8’s spatial resolution (NASA 2020). +. . . . . . . . . . . . . . . . . 8 +Figure 2.2 +The evolution of four random walk Metropolis Markov chains (Carpenter +2020), each started in a different location. . . . . . . . . . . . . . . . . . . 14 +Figure 2.3 +The flowchart adapted from (Betancourt 2020) shows a principled +Bayesian workflow. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 +Figure 3.1 +A screenshot of the top ∼50 rows in the October 2018 production report. +Each row corresponds to a well. There are in total 17,135 rows in this +spreadsheet, with the first row being the header. . . . . . . . . . . . . . . 19 +Figure 3.2 +The nighttime combustion source detection limits of VIIRS (top) and L8 +(bottom). For natural gas flaring whose temperature is generally greater +than 1500 K, L8 detected flares show source areas (around 10−2 m2) +orders of magnitude less than that of VIIRS (around 1 m2). . . . . . . . . 21 +Figure 3.3 +A count plot showing the distribution of cluster sizes: clearly there are a +certain number of large clusters (as shown by the tail to the right). . . . . 22 +Figure 3.4 +A large flare consisting of many hot pixels (detections), which is found +by running the nightfire algorithm on L8 images. Both the Band 6 +(grayscale image) and the KMZ view are shown and provided by +Christopher D. Elvidge (personal communication). . . . . . . . . . . . . . 22 +Figure 3.5 +A heat map showing the pairwise Spearman correlations between the +original time series’ monthly observations. . . . . . . . . . . . . . . . . . . 25 +vii + +Figure 3.6 +A heat map showing the pairwise Spearman correlations between the +time series after applying the first differences. . . . . . . . . . . . . . . . . 26 +Figure 3.7 +Visualizations of both the NDIC and VIIRS reportings. . . . . . . . . . . 27 +Figure 3.8 +Posterior distributions (left column) and trace plots (right column) for +the state level flaring model. . . . . . . . . . . . . . . . . . . . . . . . . . 29 +Figure 3.9 +Intervals are constructed using posterior predictive samples. . . . . . . . . 30 +Figure 4.1 +Scatterplots of NDIC and VIIRS reportings for different counties. . . . . . 36 +Figure 4.2 +Scatterplots of NDIC and VIIRS reportings for different counties, +without sharing neither x- nor y-axis for all the subplots. +. . . . . . . . . 38 +Figure 4.3 +LKJcorr(η = eta) probability density. . . . . . . . . . . . . . . . . . . . . 40 +Figure 4.4 +Posterior distributions and trace plots of the slopes for each county. +. . . 43 +Figure 4.5 +Posterior distributions and trace plots of the intercepts for each county. . 44 +Figure 4.6 +A forest plot showing the uncertainties around each county’s slope +estimate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 +Figure 4.7 +A forest plot showing the uncertainties around each county’s intercept +estimate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 +Figure 4.8 +Correlation between the intercepts and slopes. +. . . . . . . . . . . . . . . 48 +Figure 5.1 +Posterior distributions and trace plots for the Blue Buttes Oilfield gas +flaring proportion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 +Figure 5.2 +Posterior predictive samples showing the gas flaring proportion variations +at the Blue Buttes Oilfield. . . . . . . . . . . . . . . . . . . . . . . . . . . 60 +Figure 5.3 +Posterior distributions and trace plots for the Operator A gas flaring +proportion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 +Figure 5.4 +Posterior predictive samples showing the gas flaring proportion variations +of Operator A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 +Figure 5.5 +Posterior distributions and trace plots for the Blue Buttes Oilfield well +flaring proportion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 +viii + +Figure 5.6 +Posterior predictive samples showing the well flaring proportion +variations at the Blue Buttes Oilfield. . . . . . . . . . . . . . . . . . . . . 64 +Figure 5.7 +Posterior distributions and trace plots for the Operator A well flaring +proportion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 +Figure 5.8 +Posterior predictive samples showing the well flaring proportion +variations of Operator A. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 +Figure 5.9 +Posterior distributions and trace plots for the Blue Buttes Oilfield flare +count model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 +Figure 5.10 +Posterior predictive samples showing the flare count variations at the +Blue Buttes Oilfield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 +Figure 5.11 +Posterior distributions and trace plots for the North Dakota flare count +model. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 +Figure 5.12 +Posterior predictive samples showing the flare count variations in North +Dakota. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 +Figure 5.13 +Posterior distributions and trace plots for the Blue Buttes Oilfield BOE +flaring proportion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 +Figure 5.14 +Posterior predictive samples showing the BOE flaring proportion +variations at the Blue Buttes Oilfield. . . . . . . . . . . . . . . . . . . . . 72 +Figure 5.15 +Posterior distributions and trace plots of the BOE flaring proportion +model for Operator A. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 +Figure 5.16 +Posterior predictive samples showing the BOE flaring proportion +variations of Operator A. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 +Figure 5.17 +Posterior distributions and trace plots for the North Dakota +VIIRS-NDIC scale factor model. . . . . . . . . . . . . . . . . . . . . . . . 77 +Figure 5.18 +Posterior predictive samples showing the scale factor variations of North +Dakota. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 +Figure 5.19 +Posterior distributions and trace plots for the Blue Buttes Oilfield +VIIRS-NDIC scale factor model. . . . . . . . . . . . . . . . . . . . . . . . 79 +Figure 5.20 +Posterior predictive samples showing the scale factor variations in the +Blue Buttes Oilfield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 +ix + +Figure 5.21 +Posterior predictive samples showing predictions of the scale factor for +the next six months. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 +Figure 6.1 +Effective usage of histograms can be surprisingly subtle. . . . . . . . . . . 84 +Figure 6.2 +A histogram for the distribution of the oilfield detection counts from +October 2018. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 +Figure 6.3 +Posterior distributions and trace plots for the oilfield detection counts +distribution, fitted with the data from October 2018. . . . . . . . . . . . . 88 +Figure 6.4 +Histograms for the distribution of the oilfield detection counts from +October 2018. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 +Figure 6.5 +Histograms for the distribution of the oilfield detection counts from +October 2018, with the y-axis clipped to better present those counts +which are greater than zero. +. . . . . . . . . . . . . . . . . . . . . . . . . 90 +Figure 6.6 +Histogram for the distribution of the oilfield flared volumes from Q4 2018. . 92 +Figure 6.7 +Distribution of the oilfield flared volume magnitudes from Q4 2018. +. . . 93 +Figure 6.8 +Ten random draws from a Dirichlet prior with α = (6, 6, 6, 6, 6, 6, 6). . . . 95 +Figure 6.9 +GMM inference results with different K’s. . . . . . . . . . . . . . . . . . . 97 +Figure 6.10 +WAIC values with different K’s. . . . . . . . . . . . . . . . . . . . . . . . 98 +Figure 6.11 +A scatterplot of oil production and flared gas volumes for different +oilfields in Q4 2018. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +100 +Figure 7.1 +Examples of good fits between the NDIC and VIIRS reported volumes, +at the operator level. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +105 +Figure 7.2 +Examples of poor fits between the NDIC and VIIRS reported volumes, +at the operator level. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +106 +Figure 7.3 +Time series of the two example operators whose reporting did not quite +align with the VIIRS detected trends/patterns. The points or periods in +time for which the company-reported data were significantly different +from the satellite detections are annotated. . . . . . . . . . . . . . . . . +107 +Figure 7.4 +A more comprehensive time series plot for Operator D. +. . . . . . . . . +109 +Figure 7.5 +A more comprehensive time series plot for Operator E. +. . . . . . . . . +110 +x + +Figure 7.6 +A neural network designed for the hypothetical well performance +classification problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +115 +xi + +LIST OF TABLES +Table 2.1 +Resolutions of Landsat 8 and VIIRS . . . . . . . . . . . . . . . . . . . . . . . 7 +Table 3.1 +Parameter Estimates of State Level Flaring Model . . . . . . . . . . . . . . 28 +Table 4.1 +North Dakota County Abbreviations . . . . . . . . . . . . . . . . . . . . . . 37 +Table 4.2 +Parameter Estimates of County Level Flaring Model . . . . . . . . . . . . . 49 +Table 6.1 +Parameter Estimates of Oilfield Detection Count Distribution . . . . . . . . 88 +Table 7.1 +Units for Operator Time Series in Figures 7.4 and 7.5 +. . . . . . . . . . . +111 +Table 7.2 +Publication Count Rise on OnePetro . . . . . . . . . . . . . . . . . . . . . +112 +Table 8.1 +Models Developed in this Dissertation . . . . . . . . . . . . . . . . . . . . +119 +xii + +LIST OF SYMBOLS +Vectors and matrices are in bold type. A subscript asterisk, such as in y∗, indicates reference +to a test set quantity or a prediction. +General Nomenclature +ℓ2 norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∥·∥ +Data set: D = {(xi, yi) | i = 1, . . . , n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . D +Natural numbers with zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N0 +Pi (italic) representing a variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +π +Pi (upright) denoting the transcendental constant (3.14159 · · · ) +. . . . . . . . . . . . . +π +Prediction for a test input x∗ +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . y∗ +Proportional to; e.g., p(x | y) ∝ p(x, y) means that p(x | y) is +equal to p(x, y) times a factor which is independent of x +. . . . . . . . . . . . . . . ∝ +Real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R +Universal quantifier: for all x +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∀x +a is defined as b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a := b +b is defined as a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a =: b +n-dimensional vector space of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . Rn +Probability and Statistics +Conditional probability density function . . . . . . . . . . . . . . . . . . . . . . . . p(· | ·) +Expectation; expectation of g(x) when x ∼ p . . . . . . . . . . . . . . . . . . E or Ep[g(x)] +Probability density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +p(·) +xiii + +Probability mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P(·) +Random variable X is distributed according to p +. . . . . . . . . . . . . . . . . . . X ∼ p +Variance; variance of g(x) when x ∼ p +. . . . . . . . . . . . . . . . . . . . . V or Vp[g(x)] +Probability Distributions +Binomial distribution with parameters n, p . . . . . . . . . . . . . . . . . . Binomial(n, p) +Categorical distribution with parameter p . . . . . . . . . . . . . . . . . . +Categorical(p) +Continuous uniform distribution with parameters a, b . . . . . . . . . . . . . Uniform(a, b) +Dirichlet distribution with parameter α +. . . . . . . . . . . . . . . . . . . . . Dirichlet(α) +Distribution over Cholesky decomposed covariance +matrices with parameters η, σ . . . . . . . . . . . . . . . . . . +LKJCholeskyCov(η, σ) +Exponential distribution with parameter λ . . . . . . . . . . . . . . . . . . Exponential(λ) +Gamma distribution with parameters α, β . . . . . . . . . . . . . . . . . . . Gamma(α, β) +Half-Cauchy distribution with parameter γ . . . . . . . . . . . . . . . . . . Half-Cauchy(γ) +Half-Normal distribution with parameter σ . . . . . . . . . . . . . . . . . . Half-Normal(σ) +LKJ distribution with parameter η . . . . . . . . . . . . . . . . . . . . . . . . LKJcorr(η) +Multivariate Gaussian distribution with parameters µ, Σ . . . . . . . . +MVNormal(µ, Σ) +Negative binomial distribution with parameters µ, φ . . . . . . . . . . . NegBinomial(µ, φ) +Poisson distribution with parameter λ +. . . . . . . . . . . . . . . . . . . . . . +Poisson(λ) +Student’s t-distribution with parameters ν, µ, σ . . . . . . . . . . . . . +Student-t(ν, µ, σ) +Univariate Gaussian distribution with parameters µ, σ +. . . . . . . . . . . . . . . N(µ, σ) +Gaussian Processes +Covariance function evaluated at x and x′ . . . . . . . . . . . . . . . . . . . . . . +k(x, x′) +xiv + +Gaussian process: f ∼ GP(m(x), k(x, x′)), the function f is +distributed as a Gaussian process . . . . . . . . . . . . . . . . . . . . . . . . . . . +GP +Mean function evaluated at x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +m(x) +Vector of latent function values, f = (f(x1), . . . , f(xn))⊤ +. . . . . . . . . . . . . . . . . . f +Vectors and Matrices +Cholesky decomposition: L is a lower triangular matrix +such that L · L⊤ = K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cholesky(K) +Identity matrix of size n × n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In +Transpose of matrix L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L⊤ +Vector of all 0’s of length n +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0n +Vector of all 1’s of length n +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1n +Vector of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +θ +Standard Functions +Inverse-logit function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +logit−1(·) +Natural exponential function +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . exp(·) +Natural logarithm function +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +log(·) +Units +barrels of oil equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +BOE +barrels per day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . bbl/day +billion cubic meter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . bcm +day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d +dollars per barrel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $/bbl +dollars per million Btu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $/MMBtu +xv + +kelvin +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +K +kilometer +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +km +megawatt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MW +meter +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m +million cubic feet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MMcf +thousand cubic feet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mcf +thousand cubic feet per barrel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mcf/bbl +xvi + +LIST OF ABBREVIATIONS +Autoregressive integrated moving average +. . . . . . . . . . . . . . . . . . . . . . ARIMA +Credible interval +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CI +Deep learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +DL +Density-Based Spatial Clustering of Applications +with Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DBSCAN +Energy Information Administration . . . . . . . . . . . . . . . . . . . . . . . . . . . . EIA +Exempli gratia (Latin: for example) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . e.g. +Gas oil ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +GOR +Gaussian mixture model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GMM +Gaussian process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +GP +Hamiltonian Monte Carlo +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMC +Hierarchical Density-Based Spatial Clustering +of Applications with Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . HDBSCAN +Highest density interval +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HDI +Id est (Latin: that is) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i.e. +Independent and identically distributed . . . . . . . . . . . . . . . . . . . . . . . . . . i.i.d. +Interquartile range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IQR +Kernel density estimation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +KDE +Landsat 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L8 +Long short-term memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LSTM +Markov chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MCMC +xvii + +Maximum a posteriori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +MAP +Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +MLE +National Aeronautics and Space Administration . . . . . . . . . . . . . . . . . . . . NASA +National Oceanic and Atmospheric Administration +. . . . . . . . . . . . . . . . . +NOAA +North American Datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +NAD +North Dakota Industrial Commission . . . . . . . . . . . . . . . . . . . . . . . . . . NDIC +Prediction interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PI +Radiant heat +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +RH +Right-hand side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RHS +Short-wave infrared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SWIR +VIIRS Nightfire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VNF +Visible Infrared Imaging Radiometer Suite . . . . . . . . . . . . . . . . . . . . . . . VIIRS +World Geodetic System +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +WGS +Zero-inflated Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ZIP +Zero-inflated negative binomial +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ZINB +xviii + +ACKNOWLEDGMENTS +In the very first place, I want to express my deepest appreciation to my advisor, Dr. Jennifer +L. Miskimins. Dr. Miskimins has been my MS/PhD advisor and mentor since 2011. Since I +returned to Mines to start my PhD in 2017, Dr. Miskimins has been providing me with the +best guidance, the greatest support, and the most opportunities that I could imagine. During +the first semester, I worked as a lab assistant in the High Bay; in a later semester, I worked as +a teaching assistant in her well stimulation course; ever since I started to become interested +in machine learning, she has provided me with a huge number of opportunities to connect +with different groups of people, for brainstorming and pursuing my research interest. To a +certain extent, I feel like I finally become a “qualified” FAST student member, thanks to all +of these precious experience. What I have achieved, including this dissertation, would have +never been possible without the guidance and support from Dr. Miskimins. Her world-class +technical expertise, attitudes toward work/life, and art of managing different teams at various +levels are what I hope I can learn from in my career and personal life. +I am deeply grateful to my dissertation committee members: Dr. Soutir Bandyopadhyay, +Dr. Alfred W. Eustes III, Dr. Yilin Fan, and Prof. Jim Crompton. My competency in my +research field, as well as the shape of this dissertation are built with the help of those fruitful +discussions and insightful comments from them. +I am indebted to my mentors, colleagues, and friends from the Payne Institute for Public +Policy. Especially, I want to thank Dr. Mikhail N. Zhizhin, Dr. Christopher D. Elvidge, and +Dr. Morgan D. Bazilian. It is such an eye-opening experience for me to work with these +world-class experts in remote sensing and satellite imagery. I would particularly like to thank +Dr. Zhizhin for his help, insights, and time. +I am really grateful to Dr. Bandyopadhyay and Dr. Luis Tenorio from the AMS Department, +for their fantastic teaching, knowing me personally and motivating me to work hard. Looking +xix + +back at what I have learned in machine learning which makes this dissertation possible, +taking their classes are definitely the most important resources for myself (excuse me for not +being a probabilist at this moment). By taking their statistical methods classes, I started +to appreciate what really is machine learning, and falling in love with mathematics, more +specifically, probability theory and statistical modeling. +The TA experience at Mines makes me a better PhD student. What I have learned, tech- +nical or non-technical, made their way into this dissertation. I want to thank Prof. Crompton, +Dr. Eustes, Dr. Mark G. Miller, Dr. Linda A. Battalora, and Dr. Miskimins for providing me +with those valuable TA opportunities. I am grateful to all of my students for their support +and feedback. +I want to thank Dr. Yu-Shu Wu, Dr. Xiaolong Yin, and Dr. Yilin Fan for their care, +support, and encouragement throughout my PhD study. +I would like to thank Denise +Winn-Bower, Rachel McDonald, and Joe Chen for their help. +I really appreciate the feedback from the FAST member companies’ representatives. A lot +of the discussions and the reflections following those were incorporated into this dissertation. +Especially, I want to thank Ty Woodworth for his time and help, in the process of collecting +plunger lift data for me. I got very warm welcomes every time I visited their Windsor office +in Northern Colorado. Ty kindly introduced me to the team he led, and I got the great +opportunities to ask questions and discuss with many field experts in different areas. Those +discussions helped me tremendously. +Special thanks go to the open source community. In the process of conducting this +research and typesetting this dissertation, I benefited a lot from the ecosystems around +Linux/GNU, TEX/LATEX, and Python. Especially, I want to thank the people behind PyMC3, +a probabilistic programming language that this dissertation is heavily dependent upon. +Last but not least, I would like to thank my family and friends. Thank you to my beloved +wife Xiaodan, for all her love, support, and delicious dishes. I also want to thank my parents +and parents-in-law for their support, encouragement, and understanding. +xx + +I dedicate this work to my mother, Dr. Lingying Ni, and my father, Mr. Honggang Lu. +谁言寸草心，报得三春晖。 +xxi + +CHAPTER 1 +INTRODUCTION +Currently in the petroleum industry, for wells which produce both crude oil and natural +gas, operators often choose to flare the produced gas instead of commodifying it. The +rationales behind such decisions are multifold. Variations in natural gas price can be an +important factor, especially when the processing and transportation cost is higher than +the value of gas (Srivastava et al. 2019). The amount of gas being flared each year on a +national level is huge, and an increasing trend can be observed for the top flaring countries +(Figure 1.1). +Source: NOAA, Colorado School of Mines, GGFR +The new ranking – top 30 flaring countries +(2014 – 2018) +Ranked by 2018 flare volume +Million m3 +gas/year +flared +Public Disclosure Authorized +Public Disclosure Authorized +Public Disclosure Authorized +Public Disclosure Authorized +Figure 1.1: Top 30 countries ranked by flared gas volume in 2018. United States ranks No. 4 +and has a large increase from 2017 to 2018 (World Bank 2019). +Due to the boom of unconventional resources (e.g., shale gas reservoirs) development +in the recent decade, the United States has been among the top flaring countries in terms +1 + +24,000 +22,000 +20,000 +18,000 +16,000 +14,000 +12,000 +10,000 +8,000 +6,000 +4,000 +2,000 +0 +2014 +■2015 +■2016 +2017 +■2018of total volume flared. +This is backed by the data from the U.S. Energy Information +Administration (EIA) (2019) showing North Dakota, which is underlain by the Bakken +Formation, and Texas, which houses the Permian Basin and the Eagle Ford Shale, are the +top two flaring states since 2013. The two states’ annual flaring volume time series are +shown in Figure 1.2. Some flaring sites can be clearly identified from Google Earth’s imagery +(Figure 1.3). +1995 +2000 +2005 +2010 +2015 +Date +0 +20000 +40000 +60000 +80000 +100000 +120000 +Gas Vented and Flared (MMcf) +North Dakota +Texas +Figure 1.2: The time series show the trend of gas flaring for the top two states in the United +States (EIA 2019a). Texas regained the lead in 2015. +Natural gas flaring constitutes a problem of energy waste and CO2 emissions. In recent +years, various organizations and government agencies have advocated reducing or eliminating +routine gas flaring. For example, the North Dakota Industrial Commission (NDIC) introduced +a gas flaring regulatory policy (Order 24665) in 2014, with goals of reducing flaring in different +aspects (e.g., volume of gas flared). The World Bank launched the “Zero Routine Flaring +by 2030” initiative in 2015. To monitor and benchmark flaring activity’s magnitude, a +precise and accurate method to obtain quantitative flaring information is desirable. However, +in certain situations, this information is only available through self-reporting mechanisms. +2 + +Figure 1.3: This Google Earth imagery shows gas flaring being conducted on a well location +in North Dakota (Google Earth 2019). +Inaccuracies might be introduced either intentionally or unintentionally. +Satellite remote sensing is one unbiased approach for solving this problem. It can help +detect active flares especially during nighttime and can be used to calibrate the estimation +for flared gas volume. For this work, two different types of sensors are considered, including +the Landsat 8 (L8)’s Operational Land Imager (OLI) and Thermal Infrared Sensor (TIRS), +as well as the Visible Infrared Imaging Radiometer Suite (VIIRS) that is on the Suomi +National Polar-orbiting Partnership (NPP) and NOAA-20 satellites. In the remainder of this +dissertation, they are referred to as L8 and VIIRS, respectively. An example of detecting +flaring with VIIRS low light imaging data is shown in Figure 1.4. +1.1 +Research Goal +This research is undertaken to achieve the following goals: +• Evaluate the methodology for estimating flared gas volume leveraging satellite imagery; +and, +3 + +Figure 1.4: Part of the original poster (Earth Observation Group at Payne Institute 2019) +which uses one year accumulation of VIIRS low light imaging data to showcase human +activities, e.g., gas flaring, fishing, and city lights. As annotated, North Dakota’s flaring +activities are very visible from space at night. +• Find insights into operators’ gas flaring behavior. +1.2 +Dissertation Objectives +To achieve the goals outlined in Section 1.1, more specific objectives are listed below: +1. Compare and contrast the flaring data from VIIRS and NDIC. +• Compare the VIIRS flared volumes to the NDIC, using the NDIC as a benchmark. +2. Evaluate the effectiveness of using Landsat 8 nighttime images to improve flare detection +and volume estimation. +• Determine the detection limits of Landsat 8 and compare it with VIIRS’ capabili- +ties. +3. Investigate operator approaches for gas flaring. +4 + +Shale Oil +The boom of shale oil +production is clearly depicted +by the number of gas flares +ected-inNorthDakota• Determine the correlation between gas price / oil price / oil production and flared +gas volume. +• Evaluate if the North Dakota regulatory policy (Order 24665) achieved its goals. +• Develop a model that can predict flared gas volume at a state level. +4. Find any hidden structure/clusters from all the producing entities. +1.3 +Outline and Contributions +The main contribution of this dissertation is demonstrating that Bayesian learning +implemented by Markov chain Monte Carlo methods is very effective in flaring data analytics. +A series of parametric and nonparametric machine learning models are developed for various +analytics goals and granularities, providing direct guidance for future modeling endeavors. +To demonstrate the effectiveness and robustness, they are all tested with real data. The +superiority of this approach is based on the fact that the inference stage is entirely probabilistic, +in that the parametric uncertainties arising from probable models as well as the stochastic +uncertainties arising from noisy observations are all properly characterized and quantified. It +makes the extracted insights robust and interpretable for decision- and policy-making by, for +example, a state government. +In Chapter 2, a literature review is given for the state of the art in satellite imagery +processing, Bayesian inference, Markov chain Monte Carlo methods, and machine learning. +In Chapter 3, the data gathering processes are discussed. Results from some exploratory +data analysis are presented. +In Chapter 4, county level models are built to study the correlations between VIIRS and +NDIC, and to explore the heterogeneity among the counties in North Dakota. +In Chapter 5, flaring time series analytics is presented for the purposes of revealing trends +and patterns at different levels. +In Chapter 6, unsupervised learning is applied on flaring data to characterize the latent +structures. +5 + +In Chapter 7, a method of operator level monitoring and analytics is introduced, and +some discussions about applying Bayesian learning are given. +In Chapter 8, major conclusions drawn are presented. Recommendations based on this +work are given. A number of future research areas are outlined. +6 + +CHAPTER 2 +LITERATURE REVIEW +In the 1990s, the World Bank started gathering nighttime satellite images, from which +big cities and oilfields were both bright and needed to be sorted using extra information. The +situation changed in 2012 when infrared data became available from VIIRS (Rassenfoss and +Zborowski 2018). One of the data products, VIIRS Nightfire (VNF) specializes in natural +gas flaring observation and is even able to distinguish between biomass burning and gas +flaring (Elvidge et al. 2017). +VNF’s development was based upon VIIRS imagery. To improve the performance of +flare detection and gas volume estimation, other sources of information, such as L8 imagery, +can be leveraged. Table 2.1 presents a comparison of L8 and VIIRS spatial and temporal +resolutions (NASA 2019; Wikipedia 2019). Figure 2.1 illustrates L8’s spatial resolution. In +addition, L8 collects data in 11 different spectral bands of the electromagnetic spectrum. +VIIRS has 22 bands. Both L8 and VIIRS are in near-polar orbits of the earth and can reveal +rich features in the landscape. Therefore, L8 should be able to identify smaller gas flares +compared to VIIRS’ capability, although its longer satellite revisit time poses a challenge to +identify less persistent flares. More details on the processing steps of VNF are discussed in +Section 2.1, the essence of which will be applied to L8. +Table 2.1: Resolutions of Landsat 8 and VIIRS +Resolution Type +Spatial [m] +Temporal [d] +Landsat 8 +15 to 100† +16.0‡ +VIIRS +375 to 750† +0.5 +† Depends on the band of the electro- +magnetic spectrum +‡ For daytime mode +7 + +Figure 2.1: Landsat 8’s spatial resolution (NASA 2020). Each Landsat pixel (30 by 30 meter +area) is roughly the size of a baseball diamond. +Nowadays, one resource which is more than abundant is data. For a certain discipline or +research field, new sources of data bring in new dimensions of information, such as satellite +images are now playing a role in gas flaring analytics. How to analyze data effectively and +intelligently to gain insights is a central problem. In the petroleum engineering domain, for +example, data driven approaches have been proposed to analyze stimulation treatments (Kaza- +kov and Miskimins 2011) and predict screenouts (Yu et al. 2020). Machine learning is a +powerful tool for this purpose. It is at the core of artificial intelligence and data science, +and lies at the intersection of statistics and computer science (Jordan and Mitchell 2015). +Frameworks in computational learning theory, such as the PAC learning proposed by Valiant +(1984), help provide a theoretical backbone for some learning algorithms. +One subset of machine learning, deep learning (DL), had its debut in 2006 when Hinton and +Salakhutdinov introduced Deep Belief Networks (DBN), but it did not gain wide acceptance +until 2012 when AlexNet showed the breakthrough performance on classification accuracy +in the ImageNet competition (Krizhevsky et al. 2012). AlexNet is a DL-based model (more +8 + +Landsat 8's Spatial Resolution +30 m +15 m +100m +Vis-NIR-SWIR = 30 m +Panchromatic=15m +Thermal IR = 100 m +(Resampledto30m)specifically a convolutional neural network) and achieved an error rate of 15.3 %, which +is more than 10 % lower than the runner-up. DL dominated the competition thereafter, +and DL-based models finally surpassed human performance on the classification data set in +2015 (He et al. 2015). +Although neural network-based models have gained much success in recent years, it should +be noted that no one type of model can always be the best candidate for all problems. This +has been formally shown by Wolpert (1996), and is usually referred to as the “no free lunch” +(NFL) theorem. More recently, Olson et al. (2017) empirically assessed 13 classification +algorithms on 165 different problem sets, and the results aligned with the theorem: even the +union of the top five best performing algorithms cannot dominate all of the problem sets. +In the following sections, a detailed review is given for the aspects below, which serve as +the foundation and inspiration for this work: +1. Satellite image processing +2. Bayesian inference +3. Markov chain Monte Carlo +4. Machine learning +5. Analytics toolset +2.1 +Satellite Image Processing +Satellite images are utilized to estimate flared gas volume. The fire detection algorithm +based on Planck curve fitting and physical laws, known as VIIRS Nightfire (VNF) due to +Elvidge et al. (2013), serves as a starting point for analyzing L8 images in this research. The +method consists of several major steps: +1. Detection of hot pixels +During nighttime, the sensors mainly record instrument noise which approximately +follows a Gaussian distribution, except for the few pixels that contain an infrared +9 + +emitter such as a gas flare. Therefore hot pixels can be identified by setting a cutoff on +the tail of the distribution, e.g., those pixels with digital numbers exceeding the mean +plus four standard deviations. +2. Noise filtering +Hot pixels that are detected in only one spectral band are treated as noise and filtered +out. +3. Atmospheric correction +Losses in radiance due to scattering and absorption effects can be corrected. MOD- +TRAN ® 5 (Berk et al. 2006), parameterized with atmospheric water vapor and tem- +perature profiles, is used to derive the correction coefficients for each spectral band. +4. Planck curve fitting +Planck curves are modeled for gas flares, which appear as gray bodies because they are +sub-pixel sources. Therefore the output of the fitting is an estimate of the temperature +and an emission scaling factor (the emissivity term in the Planck function). The latter +is used subsequently to estimate the source area. +5. Calculation of source area +The source area S is calculated using +S = εA , +(2.1) +where ε is the emission scaling factor and A is the size of the pixel footprint. +6. Calculation of radiant heat +The radiant heat is calculated using the Stefan–Boltzmann law: +RH = σT 4S , +(2.2) +10 + +where RH is the radiant heat in MW, σ is the Stefan–Boltzmann constant, T is the +temperature in K, and S is the source area in m2. +Once RH is obtained, previous work by Elvidge et al. (2015) developed a calibration for +estimating flared gas volume, utilizing nation-level flaring reporting provided by Cedigaz +(2015) and state-level reporting from Texas and North Dakota. The developed calibration +can then be applied to each individual flaring site worldwide for estimation of flared gas +volume, etc. +2.2 +Bayesian Inference +Bayesian inference leverages conditional probability theory to establish a formal procedure +for learning from data (Betancourt 2018). Bayesian models provide full joint probability +distributions p(D, θ) over observable data D and unobservable model parameters θ. The +essence of Bayesian analysis is to obtain the posterior distribution p(θ | D), which characterizes +the conditional probability of parameters θ given some data D. It can be derived through +Bayes’ theorem: +p(θ | D) = p(D | θ) p(θ) +p(D) +(2.3a) += +p(D | θ) p(θ) +� +p(D | θ′) p(θ′) dθ′ +(2.3b) +∝ p(D | θ) p(θ) , +(2.3c) +where p(D | θ) is the likelihood (also referred to as the observation model) which denotes +how likely the data is given a certain set of parameters, and p(θ) is the prior which models +the probability of the parameters before observing any data. The prior encodes domain +expertise. Once some observations are given, it is updated into a posterior which quantifies +how consistent the model configurations are with both the domain knowledge and the observed +data (Betancourt 2018). After the posterior is obtained, most if not all inferential questions +can then be answered with posterior expectation values of certain functions (Betancourt +11 + +2019): +Ep[g(θ)] = +� +g(θ) p(θ | D) dθ , +(2.4) +where g(θ) is the function encoding some inferential question (e.g., where in the model +configuration space the posterior concentrates). +Predictions can be made in the form of a posterior predictive distribution: +p(y∗ | x∗, D) = +� +p(y∗ | θ, x∗) p(θ | D) dθ , +(2.5) +where y∗ is the predictions based on the training set D for a test input x∗. Essentially this +is integrating the prediction p(y∗ | θ, x∗) over the posterior distribution of parameters (Ras- +mussen and Williams 2006). Note that by giving the final results in terms of a probability +distribution, richer information and more reliable inferences are accessed compared to merely +giving a point estimate through MLE or MAP (as some machine learning models do under +the frequentist framework). This is achieved by incorporating into the inference process the +uncertainty in the posterior parameter estimate. Other benefits include posterior predictive +checks, which are conducted by checking for auto-consistency between generated data (y∗) +and observed data (y). +2.3 +Markov Chain Monte Carlo +Many of the integration problems central to Bayesian statistics, including those in +Equations 2.4 and 2.5, are analytically intractable. A class of sampling algorithms, known as +Markov chain Monte Carlo (MCMC), can be applied to approximate these (Andrieu et al. +2003). Suppose for some function of interest f(x), the objective is to obtain its integral, +with respect to a non-standard target distribution p(x) from which samples cannot be drawn +directly: +I(f) = +� +f(x) p(x) dx . +(2.6) +By constructing Markov chains that have p(x) as the invariant distribution, MCMC samplers, +while traversing the sample space X, are able to generate samples x(i) that mimic samples +12 + +drawn directly from the target distribution p(x). In other words, this mechanism makes it +possible to draw a set of samples {x(i)}N +i=1 from p(x). +Then, by the Monte Carlo principle, the integral I(f) can be approximated with a sum +IN(f): +IN(f) = 1 +N +N +� +i=1 +f(x(i)) +a.s. +−−−−→ +N−→∞ I(f) = +� +f(x) p(x) dx . +(2.7) +That is, the estimate IN(f) is unbiased and by the strong law of large numbers, it will +converge almost surely (a.s.) to I(f). That’s why MCMC is a powerful tool in Bayesian +analysis. In practice, the Metropolis-Hastings (MH) algorithm and Gibbs sampling have been +popular MCMC methods (Andrieu et al. 2003), but only when the parameter space is not +too high-dimensional (McElreath 2020). +Due to limited computing resources, it is impossible to run Markov chains infinitely long. +In other words, inference has to be made based on finitely many draws. One approach, +which is effectively leveraged in this research, is to run multiple chains in parallel and +monitor various statistics for diagnosing non-convergence. Besides the effective sample size +per transition of the Markov chain, the Gelman-Rubin statistic (Gelman and Rubin 1992), +denoted by ˆR, is used in this dissertation. The ˆR statistic quantifies whether the ensemble +of Markov chains initialized from diffuse points in parameter space finally converge to the +same equilibrium phase (Betancourt 2017b). When ˆR is sufficiently close to 1 (for example +ˆR < 1.05), convergence is declared to be achieved. As an example, Figure 2.2 presents how +four chains are started in different corners but approach stationarity and convergence after a +certain number of iterations. +For many of the problems in practice, including the models in this dissertation, the +parameter space is very high-dimensional and involves highly curving regions. The Metropolis- +Hastings algorithm and Gibbs sampling are far from efficient in these situations. Hamiltonian +Monte Carlo (HMC), originally proposed by Duane et al. (1987), really outshines the other +algorithms at this point and is the main sampling strategy adopted in this dissertation. +13 + +Figure 2.2: The evolution of four random walk Metropolis Markov chains (Carpenter 2020), +each started in a different location. The target density is a bivariate normal with unit variance +and correlation 0.9. After M = 5000 iterations, the four chains have mixed well and explored +most of the target density. +Specifically, No-U-Turn Sampler (NUTS) introduced by Hoffman and Gelman (2014), which +is an extension to HMC, is employed for sampling from posterior distributions. +2.4 +Machine Learning +Machine learning was defined by Mitchell (1997) as computers improving automatically +through experience. It can also be viewed as a function estimation problem (Vapnik 2000), +or as the process of extracting important patterns and trends from data (Hastie et al. 2009). +In terms of tasks, common types of learning consist of supervised, unsupervised, semi- +supervised, and reinforcement (Burkov 2019). Let xi ∈ X ⊆ Rd represent input, and yi ∈ Y +represent target, then the goals of the first two types are: +• Supervised learning aims to use the dataset, consisting of X = {xi}n +i=1 and y = {yi}n +i=1, +to produce a model that is able to predict an output (yj) given some new/unseen input +(xj), i.e., learning the underlying mapping f : X → Y. +• Unsupervised learning is used to find the hidden patterns in X; in this case there does +not exist any labels (y) or predefined targets. +14 + +M= 50 +M = 500 +M = 5000 +4 +0 +-4 +-4 +0 +4 +-4 +0 +4 +-4 +0 +4 +01Another variation of learning is online learning, in which case training data is fed to the +algorithm continuously or one example at a time (Abu-Mostafa et al. 2012). In other words, +streaming data is available that the algorithm has to process on the run. This is different +from batch learning, where data is provided beforehand and “frozen” during the learning +process. Online learning can be applied to the different tasks as discussed above (supervised +and others). +In terms of model characteristics, machine learning models can be categorized into +parametric and nonparametric models. Parametric models are characterized by a fixed +number of parameters, whereas nonparametric models have an infinite-dimensional parameter +space. For example, in the latter case the parameter space can be the set of continuous +functions in a regression setting (Orbanz and Teh 2010). In this dissertation, supervised and +unsupervised learning are leveraged while exploiting both parametric and nonparametric +models. +From Bayesian’s perspective, machine learning is essentially computing the posterior (de +Freitas 2013), which is then used for inference and prediction tasks. This is conducted +exactly through Equation 2.3a. In practice, machine learning conducted under Bayesian’s +framework follows a principled workflow (Figure 2.3), which is adapted for the modeling in +this dissertation. +2.5 +Analytics Toolset +For the past five to ten years, prosperity in contributions and progress in the open +source community has been witnessed. Ecosystems around Python, R, and Julia have been +prototyped, tested, and deployed in production environments in various industries. Powerful +probabilistic programming languages (PPL), for example Stan (Carpenter et al. 2017) and +PyMC3 (Salvatier et al. 2016), have become the workhorse for Bayesian machine learning. +The majority of this work is implemented in Python. Specifically, Bayesian learning +is performed by leveraging PyMC3. Some analytic visualizations are produced employing +ArviZ (Kumar et al. 2019). Geospatial operations are performed with the help of GeoPan- +15 + +1. Conceptual Analysis +4. Model Development +2. Define Observational Space +3. Construct Summary Statistics +5. Construct Summary Functions +6. Simulate Bayesian Ensemble +7. Prior Checks +8. Configure Algorithm +9. Fit Simulated Ensemble +10. Algorithmic Calibration +11. Inferential Calibration +12. Fit Observed Data +13. Diagnose Posterior Fit +14. Posterior Retrodictive Checks +15. Celebrate +Pre-M odel +Pre-Data +Post-M odel +Pre-Data +Post-M odel +Post-Data +Figure 2.3: The flowchart adapted from (Betancourt 2020) shows a principled Bayesian +workflow. +das (Jordahl et al. 2020). Satellite imagery is processed and analyzed in MATLAB, with +implementations mainly following Elvidge et al. (2013). +16 + +CHAPTER 3 +DATA PREPROCESSING AND EXPLORATORY DATA ANALYSIS +In this chapter, an overview of the flaring data is given. Some other variables which might +be correlated with the flaring statistics are also considered. Exploratory data analysis is +performed for choosing the subset of the variables as the focus in this dissertation. A state +level model is developed in the end which motivates the work in the next two chapters. +3.1 +Data Gathering +Four sources of data, L8 satellite images, VIIRS estimated flared volumes, NDIC monthly +production reports, and county/oilfield shapefiles for North Dakota were gathered for the +analysis used in this research. +3.1.1 +Landsat 8 Images +In total, 167 images (since 2013) were downloaded from Google Cloud using the criteria +below: +• From five Path/Row’s: 126/216, 126/217, 126/218, 127/216, and 127/217. +According to the Worldwide Reference System (WRS), the satellite imagery of any +portion of the world can be queried using Path and Row numbers. These five Path/Row’s +cover the majority of the areas in North Dakota that have production and flaring +activities. +• Nighttime images. +Only nocturnal Landsat 8 imagery are used for the purpose of flare detection. +• Cloud cover less than 10 %. +Images with low cloud cover percentages reveal more clearly land features including gas +flares, and thus are ideal for validating the developed methodologies. +17 + +• GeoTIFF Data Product. +Both the georeferencing information and the raw images of all the spectral bands are +preserved through the GeoTIFF format, which are necessary for the analysis. +3.1.2 +VIIRS Estimated Volumes +The VIIRS flare inventory and estimated volume dataset obtained from Mikhail N. Zhizhin +(personal communication) are used in this dissertation. This dataset includes monthly flare +detection records in North America from March 2012 to December 2018 (both inclusive) with +their associated: +• Timestamps giving the specific month +• Latitudes and longitudes in WGS 84 coordinates +• Flared volume estimations in bcm +3.1.3 +NDIC Monthly Production Reports +All the monthly production reports from May 2015 to April 2020 (both inclusive) which +have flaring information have been downloaded from NDIC. There is one Excel spreadsheet +per month; each row corresponds to a well (that was active in that month), and columns +are for various types of information, including flared gas volume (estimated and reported +by operator), oilfield, oil production, etc. A screenshot of the top ∼50 rows in one of the +spreadsheets is displayed in Figure 3.1. +3.1.4 +NDIC Shapefiles +The shapefiles for the counties and oilfields in North Dakota are downloaded from the +NDIC GIS Map Server. All the polygons are described in NAD 27 coordinates. The shapefiles +are for reverse geocoding the satellite detection locations to readable addresses, specifically +which county and oilfield is a flare located in. +18 + +Figure 3.1: A screenshot of the top ∼50 rows in the October 2018 production report. Each row corresponds to a well. There are +in total 17,135 rows in this spreadsheet, with the first row being the header. +19 + +c +E +H +M +0 +P +Q +FileNo Company +WellName +Quarter +Section Township Range County +FieldName +Pool +oil +Wtr +Days Runs +Gas +GasSold Flared Lat +Long +10/1/201833053038990000 +22021 EQUINORENERGY LP +BILL 14-23 2TFH +SWSW +11 +151 +101 MCK +ALEXANDER +BAKKEN +1044 +2361 +29 +1041 +1897 +1781 +47.90742645 +-103.5799043 +10/1/2018 33053048330000 +25091 EQUINOR ENERGY LP +BILL 14-23 3H +NWNE +44 +151 +101 MCK +ALEXANDER +BAKKEN +1977 +3837 +31 +1968 +4194 +4017 +53 +47.90455309 +-103.569154 +10/1/2018 33053050010000 +25645 EQUINOR ENERGY LP +BILL 14-23 4TFH +NWNE +151 +101 MCK +ALEXANDER +BAKKEN +1713 +7919 +31 +1705 +3251 +3086 +41 +47.90455315 +-103.5690316 +10/1/201833053048340000 +25092 EQUINOR ENERGY LP +BILL 14-23 5TFH +NWNE +11 +101 MCK +ALEXANDER +BAKKEN +2892 +1492 +1350 +804 +47.90455303 +-103.5692764 +25644 EQUINOR ENERGY LP +NWNE +101 MCK +ALEXANDER +BAKKEN +5150 +4960 +47.90455321 +-103.5689092 +22023 EQUINOR ENERGY LP +BILL 14-23 1H +sWSW +151 +101MCK +ALEXANDER +1593 +10/1/201833053039010000 +1599 +8115 +2062 +47.90748404 +-103.5798468 +HEINZ 18-19 XW #1H +SESW +151 +100MCK +ALEXANDER +4813 +8叫 +4769 +5991 +5290 +701 +47.9070795 +-103.5317558 +10/1/2018 33053042390000 +23319 EQUINOR ENERGY LP +PORTER 35-26 #1TFH +SESW +35 +151 +101MCK +ALEXANDER +956 +2394 +914 +1281 +1145 +12 +47.84918886 +-103.576799 +23320 EQUINOR ENERGY LP +PORTER 35-26 #2H +SESW +saag +151 +101 MCK +ALEXANDER +BAKKEN +1140 +1737 +1119 +2259 +2088 +47.84918891 +10/1/2018 33053042380000 +TIMBER CREEK 13-24 1TFH +NENW +151 +101MCK +ALEXANDER +1058 +2643 +920 +2217 +2088 +47.9051858 +-103.5513108 +23301 +EQUINOR ENERGY LP +TIMBER CREEK 13-24 2H +NENW +151 +101 MCK +ALEXANDER +BAKKEN +1617 +2803 +1839 +4403 +4358 +45 +47.90518511 +-103.5511884 +10/1/2018 33053043310000 +SESW +151 +101 MCK +ALEXANDER +BAKKEN +1039 +1120 +17 +1023 +1600 +1379 +0 +47.84914049 +-103.5533416 +10/1/201833053014360000 +9132 TAQA USA, INC +SKEDSVOLD +23-6 +SENW +24 +11 +101 MCK +ALEXANDER +DUPEROW +006 +1348 +47.886693 +-103.57491 +10/1/201833053015410000 +9617 TAQA USA, INC. +BOLKEN 24-12 +NWSW +101 MCK +ALEXANDER +MADISON +ooo +0 +0 +47.882622 +-103.558708 +8165 TAQA USA, INC. +SKEDSVOLD 1 +SENE +2221 +151 +101 MCK +ALEXANDER +MADISON +0 +0 +... +47.88620736 +-103.5647448 +10/1/2018 33053026140000 +15670 TAQA USA, INC. +HANSEN 2-23 +W2SE +151 +101 MCK +ALEXANDER +RED RIVER +0 +0 +47.88086054 +103.5688647 +10/1/201833053012110000 +8165 TAQA USA, INC. +SKEDSVOLD 1 +SENE +101 MCK +ALEXANDER +RED RIVER +0 +0 +0 +47.88620736 +103.5647448 +10/1/201833023008590000 +23323 HUNT OIL COMPANY +ALEXANDRIA 1-33-28HTF +LOT3 +101DIV +ALEXANDRIA +BAKKEN +671 +1184 +982 +102 +48.71889333 +-103.6779665 +10/1/2018 33023012220000 +28507 HUNT OIL COMPANY +ALEXANDRIA +161-100-21-16H-1 +SWSE +22 +100 DIV +ALEXANDRIA +BAKKEN +1689 +2541 +1771 +2449 +1575 +48.75011262 +-103.6683129 +28501 HUNT OIL COMPANY +ALEXANDRIA +161-100-22-15H-1 +SESW +161 +100 DIV +ALEXANDRIA +BAKKEN +1616 +2331 +31 +1719 +1435 +.616 +120 +48.75059075 +-103.6538032 +10/1/2018 33023010970000 +26256 HUNT OIL COMPANY +ALEXANDRIA +161-100-24-13H-1 +SESW +42278 +161 +100 DIV +ALEXANDRIA +BAKKEN +1665 +3195 +31 +1720 +2148 +1512 +145 +48.75017929 +-103.6098863 +ALEXANDRIA +SWSE +161 +100 DIV +ALEXANDRIA +BAKKEN +923 +1506 +27725 HUNT OIL COMPANY +161-100-32-29H-1 +1195 +699 +48.72106643 +-103.691812 +10/1/2018 33023010160000 +25096 HUNT OIL COMPANY +ALEXANDRIA +1-26-35H +NENW +161 +100 DIV +ALEXANDRIA +1577 +1242 +95 +48.74896506 +-103.6318854 +10/1/201833023011820000 +27921 HUNT OIL COMPANY +ALEXANDRIA +161-100-17-20H-1 +NENW +161 +100 DIV +ALEXANDRIA +BAKKEN +1607 +2569 +1742 +2280 +50 +1614 +48.7781167 +-103.6984434 +26 +10/1/2018 33023010470000 +25666 HUNT OIL COMPANY +ALEXANDRIA +161-100-18-19H-1 +NENW +161 +100 DIV +ALEXANDRIA +BAKKEN +1008 +1817 +31 +996 +1049 +573 +48.77706545 +-103.7183907 +ALEXANDRIA 161-100-23-14H-1 +NENW +287 +26259 HUNT OIL COMPANY +161 +100 DIV +ALEXANDRIA +BAKKEN +1551 +2768 +31 +1693 +1946 +1445 +28 +48.74884085 +-103.6323199 +10/1/2018 33023012210000 +28502 HUNT OIL COMPANY +ALEXANDRIA 161-100-27-34H-1 +SESW +161 +100 DIV +ALEXANDRIA +BAKKEN +1077 +1417 +1016 +1378 +574 +48.75059015 +-103.6542387 +10/1/201833023010480000 +25667 HUNT OIL COMPANY +ALEXANDRIA 161-100-7-6H-1 +NENW +161 +100 DIV +ALEXANDRIA +BAKKEN +29 +971 +1396 +31 +959 +1034 +539 +48.77735325 +-103.7183887 +10/1/2018 33023011830000 +27922 HUNT OIL COMPANY +ALEXANDRIA 161-100-8-5H-1 +NENW +161 +100 DIV +ALEXANDRIA +BAKKEN +1731 +2831 +31 +1969 +1728 +1065 +48.77811809 +-103.6988791 +26475 MUREX PETROLEUM CORPORATION GARY LEE 31-30H +SESW +18 +100 DIV +ALEXANDRIA +BAKKEN +1125 +1706 +28. +1202 +10/1/201833023011050000 +148 +48.72129419 +-103.7176511 +26752 PETRO-HUNT, L.L.C. +MARVIN 14-34HS +SESW +100 DIV +ALEXANDRIA +BAKKEN +2607 +48.808449 +103.6534915 +10/1/201 +130000 +26592 PETRO-HUNT, L.L.C. +PAUL 3-4HS +LOT3 +161 +100 DIV +ALEXANDRIA +BAKKEN +1227 +2303 +31 +1259 +1078 +861 +48.80668514 +-103.6753468 +21888 PETRO-HUNT, L.L.C. +TOMLINSON 3-1HS +LOT3 +161 +100 DIV +ALEXANDRIA +BAKKEN +1259 +2140 +31 +1254 +1400 +1180 +4 +48.80693638 +-103.610843 +25533 PETRO-HUNT, L.L.C. +TOMLINSON 3-2HS +LOT3 +161 +100 DIV +ALEXANDRIA +BAKKEN +1219 +10/1/201833023010360000 +865 +20 +48.80678597 +-103.6320128 +17850 EOG RESOURCES,INC +COTTONWOOD 2-35H +SESE +157 +92 MTL +ALGER +BAKKEN +48.37339884 +-102.5182998 +10/1/2018 33061012140000 +18562 EOG RESOURCES, INC. +COTTONWOOD 6-33H +SESE +157 +92 MTL +ALGER +0 +48.37324713 +-102.5648531 +10/1/2018 33061009710000 +17937 EOG RESOURCES, INC. +COTTONWOOD 5-34H +SWSE +157 +92 MTL +ALGER +415 +678 +381 +48 +48.37345232 +-102.5439178 +10/1/2018 33061010810000 +18214 EOG RESOURCES, INC. +JAMES HILL 1-31H +LOT4 +157 +91MTL +ALGER +BAKKEN +866 +885 +1166 +917 +0 +48.38625553 +-102.4928685 +40 +18024 EOG RESOURCES, INC. +ROSS +SWSE +897 +10/1/2018 33061009980000 +10-18H +156 +92 MTL +ALGER +BAKKEN +586 +412 +485 +633 +48.32796157 +-102.5531716 +18843 EOG RESOURCES, INC +ROSS +SESE +156 +92 MTL +ALGER +BAKKEN +1897 +2390 +1857 +3229 +10/1/201833061013070000 +18-10H +48.34293945 +-102.4835757 +ROSS 29-1716H +NWNE +156 +92 MTL +ALGER +BAKKEN +901 +690 +31 +809 +1198 +711 +175 +48.34081767 +-102.5341257 +10/1/2018 33061008300000 +17526 EOG RESOURCES, INC. +ROSS 4-05H +SESE +5990 +156 +92 MTL +ALGER +BAKKEN +846 +789 +31 +725 +1549 +1115 +123 +48.35699697 +-102.5273298 +子 +18444 EOG RESOURCES, INC. +ROSS 100-09H +SWSW +18 +92 MTL +ALGER +BAKKEN +771 +573 +638 +10/1/201833061011660000 +498 +48.3426108 +-102.5208672 +10/1/201833061038720000 +32137 EOG RESOURCES, INC. +ROSS 106-0915H +NWNE +92 MTL +ALGER +BAKKEN +2124 +5340 +2099 +5273 +485 +48.35482483 +-102.5101913 +10/1/2018 33061009750000 +17950 EOG RESOURCES, INC. +ROSS 1-09H +ESMS +920 +92 MTL +ALGER +BAKKEN +537 +467 +538 +228 +48.34259214 +102.510155 +47 +10/1/2018 +33061010130000 +18062 EOG RESOURCES, INC +ROSS 11-27H +SESE +92 MTL +ALGER +BAKKEN +126 +378 +612 +1124 +161 +48.29904089 +102.4836364 +03.2 +Satellite Image Processing +As discussed in Section 3.1.1, all the available L8 images have been downloaded. They +are processed in batch, following the workflow as outlined in Section 2.1. To compare and +contrast with VIIRS’ performance, specifically the nighttime combustion source detection +limits, all the flares detected from all of the L8 images are gathered and used to generate the +source area versus temperature scattergram shown in Figure 3.2. +Although it is expected that L8 would pick up smaller flares than VIIRS (which is +capable of detecting flares around the size of a whole cooktop area), the majority of the +detections as indicated on the scattergram are too small for natural gas flaring. To verify +if some hot pixels are clustered together and actually representing a single flare or flaring +site, HDBSCAN (Campello et al. 2013) with an implementation due to McInnes et al. +(2017) is executed on every L8 detection map to find out if large blobs of hot pixels are +present. HDBSCAN is a density-based clustering algorithm which keeps all the advantages +of the original DBSCAN (Ester et al. 1996), for example the capacity of finding clusters of +arbitrary shapes. It also outperforms DBSCAN by being able to build clusters of varying +density (Burkov 2019). Further, to get the most accurate results in this case, haversine metric +is chosen to handle the great-circle distances between the hot pixels; leaf clustering is used +instead of the default Excess of Mass method to produce more fine grained clusters. The +clustering results are illustrated in Figure 3.3. +To verify whether these clusters are really single flares or they are actually a large number +of neighboring wells (in which case each hot pixel still represents an individual flare), they +are tracked down by looking further into each detection map (KMZ file). It is found that +some large blobs of hot pixels are clustered and indeed represent single (huge) flares. One of +the examples is shown in Figure 3.4. This poses a challenge to situations where an accurate +estimate of the flare count is needed. +The reason for this processing artifact is that, for large flares, there is glow surrounding the +flare that was treated as many individual combustion sources. There are potential approaches +20 + +(a) VIIRS performance (Elvidge et al. 2019) +(b) L8 performance; figure provided by Mikhail N. Zhizhin (personal communication) +Figure 3.2: The nighttime combustion source detection limits of VIIRS (top) and L8 (bottom). +For natural gas flaring whose temperature is generally greater than 1500 K, L8 detected flares +show source areas (around 10−2 m2) orders of magnitude less than that of VIIRS (around +1 m2). +21 + +10° +-350 +300 +103 +-250 +Detectable Area, m? +102 +200 +Counts +·150 +10 +DNB +M07 +-100 +M08 +M10 +M11 +-50 +M12 +M13 +10 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +Temperature, KDNB +M07 +M08 +M10 +M11 +M12 +M13 +B05 +B06 +B07 +B10 +B111 +4 +5 +6 +7 +8 +9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 54 56 58 59 60 65 68 73 76 77 84120 +Number of detections within a cluster +0 +10000 +20000 +30000 +40000 +50000 +60000 +70000 +Count of clusters +75264 +10527 +647057984632 +306621561428982782621377298263232188148131120 91 72 70 61 48 45 47 35 24 23 19 22 17 18 13 +8 +9 +11 11 +7 +4 +6 +1 +3 +6 +2 +6 +3 +2 +2 +1 +4 +1 +2 +2 +3 +1 +1 +2 +1 +1 +1 +2 +Figure 3.3: A count plot showing the distribution of cluster sizes: clearly there are a certain +number of large clusters (as shown by the tail to the right). For example, there exists 2 +clusters each of which contains 120 hot pixels and there is one cluster with 84 hot pixels. +(a) Band 6 (SWIR) +(b) KMZ view +Figure 3.4: A large flare consisting of many hot pixels (detections), which is found by running +the nightfire algorithm on L8 images. Both the Band 6 (grayscale image) and the KMZ view +are shown and provided by Christopher D. Elvidge (personal communication). +to mitigate this to make the interpretation and estimation out of L8 more accurate. In this +work, the flares detected from VIIRS and the gas volumes estimated out of those are the +focus for analytics. +22 + +80 +883.3 +Reverse Geocoding +By reverse geocoding, the county information of every VIIRS flare that is in North Dakota +can be retrieved. For most of the flares, the oilfield information is also retrievable. Thereafter, +the flaring statistics from VIIRS and NDIC can be compared and contrasted at different +levels, for a certain point or period of time. +Shapefiles as discussed in Section 3.1.4 are used. With the help of GeoPandas, the +procedures for extracting counties and oilfields are the same: +1. Read the VIIRS records into a geospatial data object, with their original coordinates in +WGS 84. +2. Read the shapefile into a geospatial data object, with its original coordinates in NAD +27. +3. Transform all the geometries in the shapefile to WGS 84 coordinates. +4. Perform a spatial join of the two data objects to get the county or oilfield information +for each flare, if a specific county/oilfield’s polygon and the flare intersect, i.e., having +any boundary or interior point in common. +3.4 +Correlational Analysis +To study the correlations between oil/gas prices, flaring statistics, and production perfor- +mance, various time series are extracted for May 2015 to December 2018 (both inclusive). +The below list describes all the variables used with their associated labels: +VIIRS flared vol +monthly flared gas volume from VIIRS +NDIC flared vol +monthly flared gas volume from NDIC +WTI oil price +WTI crude oil price given by EIA (2020b) +Henry Hub gas price +Henry Hub natural gas price given by EIA (2020a) +23 + +NDIC oil prod +monthly oil production from NDIC +NDIC gas prod +monthly gas production from NDIC +VIIRS flare count +monthly flare detections count from VIIRS +NDIC flaring well count +monthly wells count which conduct flaring from NDIC +NDIC GOR +ratio of the NDIC gas production to the NDIC oil production +First, the monthly observations are extracted from each time series, and Spearman’s ρ +is employed to measure the statistical dependence between the variables. Spearman’s ρ is +a rank correlation, which quantifies the correlation between the rankings of two variables. +Compared to Pearson’s r, it assesses monotonic relationships which can be nonlinear and is +more robust to outliers, therefore is used in this section. The pairwise correlations between +the variables are presented in Figure 3.5. Since a correlation matrix is always symmetric with +unit diagonals, only the lower triangular part without the diagonal is plotted to minimize the +information redundancy. +It can be observed that most pairs show positive correlations. Financial factors (i.e., +the oil and gas prices) are not among any of the highly correlated pairs (e.g., above 0.80). +Nevertheless, it is indicated that the NDIC and VIIRS reportings have a positive correlation, +and oil production is positively correlated with flared gas volume. +In this analysis, due to the nature of the procedure (i.e., extract the monthly data and +then measure the rank correlations), all the information on the time scale is neglected. To +explore the correlations in the context of time series, the first differences (i.e., lag-1 differences) +are taken for each variable +y′ +t = yt − yt−1, +(3.1) +and then pairwise Spearman’s ρ is evaluated and visualized in Figure 3.6. In this case, there +aren’t many pairs of variables which are highly correlated, except the oil and gas production +are shown to be monotonically related on the lag-1 differences, which is unsurprising. In the +24 + +VIIRS flared vol +NDIC flared vol +WTI oil price +Henry Hub gas price +NDIC oil prod +NDIC gas prod +VIIRS flare count +NDIC flaring well count +NDIC GOR +VIIRS flared vol +NDIC flared vol +WTI oil price +Henry Hub gas price +NDIC oil prod +NDIC gas prod +VIIRS flare count +NDIC flaring well count +NDIC GOR +0.91 +0.59 +0.61 +0.31 +0.33 +0.48 +0.72 +0.82 +0.38 +-0.05 +0.68 +0.65 +0.58 +0.37 +0.54 +0.87 +0.84 +0.40 +0.05 +0.75 +0.40 +0.68 +0.84 +0.43 +0.18 +0.74 +0.54 +0.64 +0.52 +0.45 +0.62 +0.51 +0.20 +0.86 +0.17 +0.34 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Figure 3.5: A heat map showing the pairwise Spearman correlations between the original +time series’ monthly observations. The values are annotated in each cell, the corresponding +variables of which can be obtained by reading off the tick labels from the vertical and +horizontal axes. +remainder of this dissertation, the focus is put on flaring and production related statistics +instead of the financial factors. +25 + +VIIRS flared vol +NDIC flared vol +WTI oil price +Henry Hub gas price +NDIC oil prod +NDIC gas prod +VIIRS flare count +NDIC flaring well count +NDIC GOR +VIIRS flared vol +NDIC flared vol +WTI oil price +Henry Hub gas price +NDIC oil prod +NDIC gas prod +VIIRS flare count +NDIC flaring well count +NDIC GOR +0.61 +0.16 +0.26 +-0.07 +0.21 +0.09 +0.49 +0.59 +0.13 +-0.13 +0.50 +0.51 +0.09 +-0.20 +0.95 +0.69 +0.54 +0.05 +-0.09 +0.44 +0.45 +0.28 +0.66 +0.21 +0.16 +0.28 +0.24 +0.37 +-0.04 +-0.35 +-0.28 +-0.28 +-0.10 +0.16 +0.01 +-0.34 +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Figure 3.6: A heat map showing the pairwise Spearman correlations between the time series +after applying the first differences. The values are annotated in each cell, the corresponding +variables of which can be obtained by reading off the tick labels from the vertical and +horizontal axes. +3.5 +State Level Flaring Model +In this section, a regression model is built for the purpose of investigating the statistical +relationships between the NDIC and VIIRS reportings. Data from both sources are visualized +in Figure 3.7, which demonstrate a positive correlation. +Assuming a Gaussian observation model for the NDIC reporting with the location +parameter encoding VIIRS’ information, the model is specified through Expressions 3.2a– +26 + +2015-07 +2016-01 +2016-07 +2017-01 +2017-07 +2018-01 +2018-07 +2019-01 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Monthly bcm +VIIRS +NDIC +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +VIIRS Monthly Data (bcm) +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +NDIC Monthly Data (bcm) +Figure 3.7: Visualizations of both the NDIC and VIIRS reportings. Left figure shows the +time series. Right figure presents the scatterplot using the data points of each month. +3.2e: +α ∼ Half-Normal(0.2) +(3.2a) +β ∼ Gamma(2, 2) +(3.2b) +σ ∼ Half-Cauchy(0.1) +(3.2c) +µi = α + β × VIIRSi +(3.2d) +NDICi ∼ N(µi, σ) +(3.2e) +where α is the intercept and β is the slope, both of which are constrained to be non-negative +based on the nature of flaring volume; σ is the standard deviation in the Gaussian likelihood +function, which has to be non-negative as well; µi is the expected NDIC reporting of month +i, while NDICi and VIIRSi are the observed data (i.e., reported volumes) from NDIC and +VIIRS in month i, respectively. The notation used in defining this model communicates the +data generating process unambiguously and is adopted throughout this dissertation. Priors +and hyperpriors are on the top while the observation model is at the bottom. The prior +distributions for this model and all the others in this dissertation are chosen following the +principles below: +27 + +1. Prefer weakly informative priors, i.e., choose the priors based on the domain expertise +at hand before observing any data. They should be strong enough to reflect the domain +expertise and be weak enough to “let the data speak”, i.e., let the likelihood dominate +when there is a decent amount of data. For example, a prior of a gamma distribution +with mean Eβ = 2/2 = 1 is placed on β, reflecting the assumption that the satellite +interpretation workflow gives the same flared volume as the NDIC reporting, before +one observes any data. +2. Prefer priors with soft constraints as opposed to hard constraints, i.e., follow Cromwell’s +rule. For example, α, β and σ all have prior distributions with support on R>0 or +R≥0. Counterexamples include using a triangular distribution or a continuous uniform +distribution as the prior for such quantities, for which the author does not recommend. +3. Prefer maximum entropy distributions, i.e., make the most conservative assumptions +based on all the information at hand (obeying all the known constraints). For example, +the Gaussian and the binomial distributions are maximum entropy distributions and +used in this dissertation, the fact of which can be formally shown leveraging the +definition of Kullback–Leibler (KL) divergence. +Once the priors and likelihood are established, four Markov chains of Hamiltonian Monte +Carlo are run in parallel to sample from the posterior. The parameter estimates are reported +in Table 3.1, and the posterior distributions and trace plots are presented in Figure 3.8. The +four chains are plotted separately with different colors. The x-axis of the trace plot shows the +number of iterations. This layout is used consistently for the remainder of this dissertation. +Table 3.1: Parameter Estimates of State Level Flaring Model +Parameter +Variable +Point Estimate +90 % Credible Interval +α +Intercept +0.061 +(0.044, 0.079) +β +Slope +0.535 +(0.482, 0.590) +σ +Reporting variability +0.030 +(0.024, 0.035) +28 + +0.02 +0.04 +0.06 +0.08 +0.10 +0 +10 +20 +30 +Posterior Density +0 +500 +1000 +1500 +2000 +2500 +0.02 +0.04 +0.06 +0.08 +0.10 +Sample Value +0.45 +0.50 +0.55 +0.60 +0.65 +0 +5 +10 +Posterior Density +0 +500 +1000 +1500 +2000 +2500 +0.5 +0.6 +Sample Value +0.020 +0.025 +0.030 +0.035 +0.040 +0.045 +0.050 +0 +50 +100 +Posterior Density +0 +500 +1000 +1500 +2000 +2500 +0.02 +0.03 +0.04 +0.05 +Sample Value +Figure 3.8: Posterior distributions (left column) and trace plots (right column) for the state +level flaring model. Well mixing and convergence of the Markov chains have been achieved as +shown by the trace plots. +Utilizing the model and the trace, posterior predictive samples are generated to construct +the intervals (Figure 3.9). Point estimates and point predictions are easy to obtain for a +certain machine learning model, however it is the properly constructed intervals that will +provide insights into the uncertainty for decision making. The author would like to emphasize +the importance of quantifying uncertainties when using machine learning, no matter for +inference, prediction, or building intermediate models for integration into physics-based +models. This is unfortunately neglected or ignored in some of the applications/publications in +the petroleum engineering domain. The importance of properly quantifying the uncertainties +will also be stressed in the following chapters. +Whenever only one model specification is needed for making point predictions, it can be +recovered by the parameter estimates from Table 3.1: +NDICi = 0.061 + 0.535 × VIIRSi , +(3.3) +29 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +VIIRS Monthly Data (bcm) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +NDIC Monthly Data (bcm) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +VIIRS Monthly Data (bcm) +Figure 3.9: Intervals are constructed using posterior predictive samples. In both figures, the +line shows the “best” fit using point estimates (posterior means) of α and β. Shaded area in +the left figure presents the 90 % credible interval (CI) of the regression mean. Shaded area in +the right figure demonstrates the 90 % prediction interval for the future NDIC reporting, for +which most of the existing observations fall within. +where NDICi and VIIRSi are flared volumes in bcm of month i. The model also provides +clear interpretations for the NDIC reporting regression mean, on the whole state level: +1. The intercept indicates on average there is 90 % probability that 0.04 bcm to 0.08 bcm +reported volume per month will not be captured by the current VIIRS processing +workflow. The posterior mean is 0.061 bcm (≈ 2150 MMcf). +2. The slope indicates on average when satellite estimated volume increases by one unit, +under 90 % probability the NDIC reporting will increase by 0.48 unit to 0.59 unit. The +posterior mean is 0.535 unit. +This model, while serving as a decent calibration and estimation tool for NDIC reporting +on the state level, makes the assumption that the heterogeneity within the state (e.g., among +different counties) is negligible and all the monthly observations are conditionally independent +30 + +and identically distributed (i.i.d.). For the scenarios in which these assumptions do not hold, +other types of models can be built and are discussed in Chapter 4 and Chapter 5. +31 + +CHAPTER 4 +COUNTY LEVEL FLARING MODEL +“Multilevel regression deserves to be the default form of regression.” +— McElreath (2015) +4.1 +Learning the Heterogeneity +In this chapter, the author explores the heterogeneity in correlations between the state- +reported and satellite-detected flaring statistics, among different counties in North Dakota. +The motivations are threefold: +1. Provide more granular insights than merely investigating the whole state’s flaring +statistics. +2. Compare and contrast different counties’ reporting consistencies with the baseline (i.e., +the satellite detections). +3. Develop a dedicated model for each county for calibration and prediction purposes. +4.2 +Hierarchical Model +A common problem in learning from data is modeling individuals or units of a population. +For example, building models for different counties in a state, or for different well pads in +an oilfield. Usually from domain expertise, it is expected that the units would demonstrate +some differences, however they do not necessarily represent completely independent data +generating processes. In other words, the units are different in some ways, while being similar +in others. Unfortunately, the following two common modeling approaches are extreme and +not ideal: +1. Complete pooling +32 + +• This ignores heterogeneity and assumes that the observations from all the units are +generated/described by the exact same process. One set of parameters is learned +for the whole population. In this situation, the variance might be smaller, however +the bias could be huge. +2. No pooling +• This lets each unit learn its own set of parameters from its own data. +The +assumption is that the information from each unit tells one nothing about any +other unit. In this situation, the bias might be smaller, however the variance could +be huge. +In practice, neither of these approaches will be able to generalize well for insight extraction +or prediction tasks, due to the total generalization error being large. In fact, these two +extremes can be compromised by explicitly modeling the entire population of units. That +is, in order to investigate the correlations among the individual units, an explicit model is +introduced for the population. In the learning phase, the individual posteriors are used to fit +some population distribution, while the information of the population is then fed back to the +individuals. What happens in this case is that the individuals with diffuse likelihood functions +(e.g. with less data) are dragged more towards the population distribution, whereas the +individuals which are well informed by their data will have their posteriors mostly unchanged. +In this process, dynamic regularization is achieved, i.e., the total generalization error is much +smaller by partially pooling the data and balancing between the bias and variance. +In the context of county level model development, the question is now how might one model +the population. To motivate the choice of a particular class of models, some characteristics of +the counties have to be examined. In this work, the counties are considered to be exchangeable, +i.e., the joint probability p(θ1, θ2, . . . , θn) is invariant to permutation of the indices, where +θi, i = 1, 2, . . . , n is the parameters for the i-th county. That is, for any permutation π, +p(θ1, θ2, . . . , θn) = p(θπ1, θπ2, . . . , θπn) . +(4.1) +33 + +Furthermore, the list of counties can grow, i.e., although one might only look at a few counties +at this point, in the future new counties in terms of flaring activities might be considered. +If a population being modeled is exchangeable, and the population can grow arbitrarily +large, de Finetti’s theorem shows that the only distribution that respects exchangeability is a +hierarchical distribution: +p(θ1, θ2, . . . , θn) = +� � n +� +i=1 +p(θi | φ) +� +p(φ) dφ , +(4.2) +where φ is a population parameter (which can be generalized to multiple population parame- +ters) and p(φ) is a population prior. It asserts an important fact that if exchangeable data is +used for analytics, there must exist a population model (Jordan and Broderick 2010). This +provides guidance for the development of the county level flaring models in this chapter. +Equivalently, the individual and population parameters can be fitted jointly, achieving a +dynamic pooling of the data: +p(θ1, θ2, . . . , θn, φ) = +� n +� +i=1 +p(θi | φ) +� +p(φ) , +(4.3) +in which process not only the θ’s but also φ are learned. After adding the observations +component (D = {(xj, yj) | j = 1, . . . , m}) to it, the joint model becomes: +p +�� +yj, xj, θcounty[j], ψj +�m +j=1 , φ +� += +� m +� +j=1 +p(yj | xj, θcounty[j], ψj) p(θcounty[j] | φ) +� +p(φ) , +(4.4) +where θcounty[j] stands for the parameters for the j-th observation based on its county assign- +ment, and ψ are some other parameters in the likelihood function that are not necessarily +distributed according to a population model. Equation 4.4 characterizes a hierarchical model +that fits nicely into the Bayesian framework and is exploited for building the models in this +chapter. +As a fundamental approach to model heterogeneity, hierarchical models have been de- +pended upon routinely in various fields including ecological science (Bolker 2008), political +science (Gelman and Hill 2006), and biological science (McElreath 2015). The author believes +34 + +that they should be widely accepted and utilized in the petroleum engineering domain as well, +where the dataset is usually presented in hierarchies. For example, the shale gas wells in a +given basin were completed by different oilfield service companies. The information can then +be pooled among the service companies. A further discussion is given in Section 7.3. One +caveat, though, is that de Finetti’s theorem is based on the assumption that the population +(of units) is exchangeable and can grow arbitrarily large. Just like every other assumption +in machine learning, it should not be taken for granted and does not always hold. In the +context of county level flaring model development, one might argue that there are currently +53 counties in North Dakota and there might not be many new counties (as administrative +divisions) in any finite amount of time. In that regard, the author agrees with the claim of +Box et al. (2009) that, since assumptions “are never exactly true”, what shall be sought is the +useful models as opposed to the correct ones. That is the goal for applying the hierarchical +models in this chapter. +It is worth noting that the terminologies are not consistent when referring to these types of +models: some argue that hierarchical model and multilevel model are different names for the +same modeling technique (Bolker 2008; McElreath 2015), while others tried to differentiate +them (Carpenter 2019). In this dissertation, the model assumptions are communicated via the +mathematical structures instead of the terminologies, by writing out the full model definitions +whenever possible. +4.3 +Data Description +After performing the reverse geocoding as outlined in Section 3.3, there are twelve counties +found to have reported flaring activities from both VIIRS and NDIC. For each county’s +historical data from May 2015 to December 2018 (both inclusive), only the months that have +reported volumes from both sources are extracted. A scatterplot for each of the 12 counties +is presented in Figure 4.1, where the county abbreviations follow the convention from the +NDIC monthly production reports. Table 4.1 lists the full county names associated with each +abbreviation. +35 + +0.0 +0.1 +0.2 +0.3 +0.4 +NDIC +county = MCK +county = DUN +county = WIL +county = MTL +0.0 +0.1 +0.2 +0.3 +0.4 +NDIC +county = BOW +county = DIV +county = BRK +county = MCL +0.0 +0.1 +0.2 +0.3 +0.4 +VIIRS +0.0 +0.1 +0.2 +0.3 +0.4 +NDIC +county = BIL +0.0 +0.1 +0.2 +0.3 +0.4 +VIIRS +county = STK +0.0 +0.1 +0.2 +0.3 +0.4 +VIIRS +county = SLP +0.0 +0.1 +0.2 +0.3 +0.4 +VIIRS +county = GV +Figure 4.1: Scatterplots of NDIC and VIIRS reportings for different counties. Both the x- +and y-axis are shared among all the subplots. The x-axis is the monthly VIIRS reporting of +the flared volume in bcm, and the y-axis is for the NDIC reporting in the same unit. +It can be seen that the flaring magnitudes in terms of the flared volumes are quite diverse +for the different counties. To better visualize all of them, a zoomed-in view for each county is +shown in Figure 4.2. It becomes clear that most of the counties except SLP and GV have +more than ∼12 data points; however, only the four counties in the top row (i.e., MCK, DUN, +WIL and MTL) have the largest amount of data and indicate stronger positive correlations +between VIIRS and NDIC. +For the purpose of building county level models and investigating the heterogeneity among +the counties, the no pooling option discussed in the previous section will fail. Especially +with counties SLP (which has 3 observations) and GV (which has 2 observations), if a linear +36 + +Table 4.1: North Dakota County Abbreviations +Abbreviation +County +MCK +McKenzie County +DUN +Dunn County +WIL +Williams County +MTL +Mountrail County +BOW +Bowman County +DIV +Divide County +BRK +Burke County +MCL +McLean County +BIL +Billings County +STK +Stark County +SLP +Slope County +GV +Golden Valley County +model such as Equation 3.2d is fitted, the learned slope parameters βcounty will have point +estimates ˆβslp ≈ 0 and ˆβgv ≫ 0 with their associated samples. The interpretation of the +slope parameter (which was discussed right after Equation 3.3) implies that such inferences +are never possible. Some other counties, even with more data points (e.g., MCL), suffer +from the noise levels in their observations. Using their own dataset will frustrate accurate +inferences. Therefore, in order to build models robustly at a county level, the hierarchical +model discussed in the previous section is exploited. +4.4 +Model Specification +Motivated by the discussions in Section 4.2, partial pooling is performed by explicitly +modeling the entire population of counties. In this way, the counties such as MCL can +leverage the information from other counties to learn their own parameters. Counties with +“strong data” (i.e., very informative data which makes the likelihood dominate the structure +of the posterior), such as those in the top row of Figure 4.2, indicate a positive correlation +between VIIRS and NDIC. Therefore, a similar strategy as in Model 3.2 is adopted for the +counties, i.e., one set of slope and intercept is learned for each county. +37 + +0.1 +0.2 +0.3 +0.1 +0.2 +0.3 +NDIC +county = MCK +0.02 +0.04 +0.06 +0.08 +0.10 +0.02 +0.04 +0.06 +0.08 +0.10 +county = DUN +0.00 +0.05 +0.10 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +county = WIL +0.00 +0.05 +0.10 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +county = MTL +0.000 0.005 0.010 0.015 0.020 +0.000 +0.005 +0.010 +0.015 +0.020 +NDIC +county = BOW +0.000 +0.005 +0.010 +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +county = DIV +0.000 +0.002 +0.004 +0.000 +0.001 +0.002 +0.003 +0.004 +0.005 +county = BRK +0.0000 0.0005 0.0010 0.0015 +0.0000 +0.0005 +0.0010 +0.0015 +county = MCL +0.000 +0.001 +0.002 +0.003 +VIIRS +0.000 +0.001 +0.002 +0.003 +NDIC +county = BIL +0.0000 0.0005 0.0010 0.0015 +VIIRS +0.0000 +0.0005 +0.0010 +0.0015 +county = STK +0.0000 +0.0005 +0.0010 +VIIRS +0.00000 +0.00025 +0.00050 +0.00075 +0.00100 +0.00125 +county = SLP +0.0000 +0.0005 +0.0010 +0.0015 +VIIRS +0.00000 +0.00025 +0.00050 +0.00075 +0.00100 +0.00125 +0.00150 +county = GV +Figure 4.2: Scatterplots of NDIC and VIIRS reportings for different counties, without sharing +neither x- nor y-axis for all the subplots. Within each subplot, equal scaling and limits are +set for x- and y-axis. The axes’ meanings are the same as in Figure 4.1. +Since the slope and intercept are very interpretable, the meanings of which were discussed +right after Equation 3.3, partial pooling is also enabled across parameter types (i.e., intercepts +and slopes). In other words, knowing how much flared volume is missed from VIIRS (i.e., +the information carried by the intercept) might improve learning how VIIRS and NDIC will +covary (i.e., the information carried by the slope). Specifically, a population model with a +multivariate normal density is used for the different counties’ parameters. +The hierarchical model is specified through Expressions 4.5a–4.5j: +µα ∼ Half-Normal(0.1) +(4.5a) +µβ ∼ Gamma(2, 2) +(4.5b) +38 + +σα ∼ Half-Normal(0.1) +(4.5c) +σβ ∼ Half-Normal(0.1) +(4.5d) +σ ∼ Half-Normal(0.05) +(4.5e) +R ∼ LKJcorr(2) +(4.5f) +Σ = +� +σα +0 +0 +σβ +� +· R · +� +σα +0 +0 +σβ +� +(4.5g) +� +αcounty +βcounty +� +∼ MVNormal +� +� +� +µα +µβ +� +, Σ +� +� +(4.5h) +µj = αcounty[j] + βcounty[j] × VIIRSj +(4.5i) +NDICj ∼ N(µj, σ) +(4.5j) +where: +µα is the average intercept for all the counties; +µβ is the average slope for all the counties; +σα is the standard deviation among different counties’ intercepts; +σβ is the standard deviation among different counties’ slopes; +σ is the the standard deviation in NDIC reporting within the counties; +R is the correlation matrix distributed according to an LKJ distribution. It is 2-by-2 +in size and encodes the correlation between the intercepts and slopes; +Σ is the covariance matrix for the population model, which is constructed by multi- +plying the correlation matrix from both sides by a diagonal matrix of standard +deviations; +αcounty and βcounty are the intercept and slope for each county, whose prior distributions +are defined by a two-dimensional Gaussian population model; +county[j] (in the subscript) denotes the county index, i.e., county[j] ∈ {k ∈ N0 | +k ≤ 11}, such that αcounty[j] and βcounty[j] are the intercept and slope for the j-th +observation based on its county assignment; +VIIRSj is the VIIRS reported volume of the j-th observation; +39 + +µj denotes the underlying flared volume of the j-th observation; +NDICj is the NDIC reported volume of the j-th observation. +The LKJ distribution due to Lewandowski, Kurowicka, and Joe (2009) is a distribution +over positive-definite symmetric matrices with unit diagonals, i.e., correlation matrices. In +the model specification above, it directly influences the prior for the covariance matrix. +Before it was introduced and when HMC was not widely applicable, the usual choices for +modeling covariance matrices were Wishart or inverse-Wishart distributions, due to their +nice conjugacy properties. However, LKJ is better suited for modern Bayesian computational +settings (Betancourt 2015; Lambert 2018) and therefore employed in this work. +LKJ has a single parameter η, which can be interpreted as the shape parameter of a +symmetric beta distribution (Gelman et al. 2013). As η gets larger, the prior is more skeptical +of large correlations in the matrix, i.e., providing regularizing effects. The probability density +of LKJ with a few η values are displayed in Figure 4.3. In this work, LKJcorr(η = 2) is +chosen to define a weakly informative and regularizing prior. +−1.0 +−0.5 +0.0 +0.5 +1.0 +correlation +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Density +eta=1 +eta=2 +eta=4 +Figure 4.3: LKJcorr(η = eta) probability density. As η increases, larger correlations become +less plausible. +Model 4.5, while being expressive in the data generating process, is a centered parameter- +ization of the hierarchical structure (Papaspiliopoulos et al. 2007). In this parameterization, +40 + +the hierarchical parameters (such as βcounty) and the lower-level parameters in the prior (e.g., +µβ and σβ) are tightly coupled, and they are highly correlated in the posterior. Since this +model involves complex geometries and interactions in the posterior, HMC is leveraged +for sampling. When there is not a lot of data (which is the case for the current NDIC +and VIIRS reportings), this parameterization leads to very inefficient sampling and non- +convergences (Stan Development Team 2020). The noncentered parameterization is preferable +in these cases and therefore employed for building the county level models. +4.5 +Model Reparameterization +Reparameterization of hierarchical models can be applied to any distribution in the +location-scale family, for which the normal distribution is a good candidate. In the case of +reparameterizing a multivariate normal prior, suppose the prior for θ is a multivariate normal +with mean vector µ and covariance matrix Σ (such as Expression 4.5h), then a noncentered +parameterization is given by: +�θ ∼ MVNormal(0n, In) +(4.6a) +ϕ = µ + L · �θ +(4.6b) +where �θ has the same dimensions as θ and all of its elements i.i.d. according to N(0, 1), +L satisfies L · L⊤ = Σ, and ϕ recovers the exact same prior distribution for θ. +This +reparameterization leads to more efficient sampling by reducing the dependence between +µ, L, and �θ. One choice for L is the Cholesky factor of Σ, which provides implementation +convenience for the multivariate normal cases (Stan Development Team 2020) and is adopted +in this work. +The noncentered county level model is specified through Expressions 4.7a–4.7j, with the +reparameterized part (corresponding to Model 4.5) highlighted in blue: +µα ∼ Half-Normal(0.1) +(4.7a) +µβ ∼ Gamma(2, 2) +(4.7b) +41 + +σα ∼ Half-Normal(0.1) +(4.7c) +σβ ∼ Half-Normal(0.1) +(4.7d) +σ ∼ Half-Normal(0.05) +(4.7e) +L ∼ LKJCholeskyCov +� +η = 2, +� +σα +σβ +�⊺� +(4.7f) +� +zα +zβ +� +∼ MVNormal +� +� +� +0 +0 +� +, +� +1 +0 +0 +1 +�� +� +(4.7g) +� +αcounty +βcounty +� += +� +µα +µβ +� ++ L · +� +zα +zβ +� +(4.7h) +µj = αcounty[j] + βcounty[j] × VIIRSj +(4.7i) +NDICj ∼ N(µj, σ) +(4.7j) +where: +L is the Cholesky factor of the covariance matrix which has LKJ distributed correla- +tions; +zα and zβ are the standardized intercept and slope for each county. +The rest of the symbols have the same meaning as in Model 4.5. The noncentered model +imposes the exact same probabilistic structure as in Model 4.5, and is implemented for making +inference on each county’s parameters. +4.6 +Model Fitting +Four chains are sampled from the posterior distributions. The posterior distributions +and trace plots for the slopes and intercepts are presented in Figure 4.4 and Figure 4.5, +respectively. Well mixing and convergence have been achieved as shown by the trace plots. +To better compare and contrast the different counties’ parameters, the forest plots of +90 % highest density intervals (HDI) for the slopes and intercepts are given in Figure 4.6 +and Figure 4.7, respectively. In both figures, counties are ordered by the VIIRS reported +volumes, and those with the least amount of estimated volumes (such as SLP and GV) are at +the bottom. The thin lines present the 90 % HDI’s and the thicker line segments stand for +42 + +0.46 +0.48 +0.50 +0.52 +0.54 +0.56 +0.58 +0 +10 +20 +Posterior Density +county +MCK, slope +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.50 +0.55 +Sample Value +county +MCK, slope +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +0.0 +2.5 +5.0 +7.5 +Posterior Density +county +DUN, slope +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.3 +0.4 +0.5 +0.6 +Sample Value +county +DUN, slope +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +0 +5 +10 +Posterior Density +county +WIL, slope +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.5 +0.6 +0.7 +Sample Value +county +WIL, slope +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0 +5 +Posterior Density +county +MTL, slope +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.5 +0.6 +0.7 +0.8 +Sample Value +county +MTL, slope +0.0 +0.2 +0.4 +0.6 +0.8 +0 +2 +4 +Posterior Density +county +BOW, slope +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.0 +0.5 +Sample Value +county +BOW, slope +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +2 +4 +Posterior Density +county +DIV, slope +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.0 +0.5 +1.0 +Sample Value +county +DIV, slope +0.2 +0.4 +0.6 +0.8 +1.0 +0 +2 +4 +Posterior Density +county +BRK, slope +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.5 +1.0 +Sample Value +county +BRK, slope +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0 +2 +4 +Posterior Density +county +MCL, slope +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.0 +0.5 +1.0 +Sample Value +county +MCL, slope +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0 +2 +4 +Posterior Density +county +BIL, slope +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.0 +0.5 +1.0 +Sample Value +county +BIL, slope +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0 +2 +4 +Posterior Density +county +STK, slope +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.0 +0.5 +1.0 +Sample Value +county +STK, slope +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0 +2 +4 +Posterior Density +county +SLP, slope +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.0 +0.5 +1.0 +Sample Value +county +SLP, slope +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +2 +4 +Posterior Density +county +GV, slope +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.0 +0.5 +1.0 +Sample Value +county +GV, slope +Figure 4.4: Posterior distributions and trace plots of the slopes for each county. +43 + +0.010 +0.015 +0.020 +0.025 +0 +50 +100 +150 +Posterior Density +county +MCK, intercept +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.01 +0.02 +Sample Value +county +MCK, intercept +0.0000 +0.0025 +0.0050 +0.0075 +0.0100 +0.0125 +0.0150 +0.0175 +0 +50 +100 +150 +Posterior Density +county +DUN, intercept +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.00 +0.01 +Sample Value +county +DUN, intercept +0.004 +0.006 +0.008 +0.010 +0.012 +0.014 +0.016 +0 +100 +200 +Posterior Density +county +WIL, intercept +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.005 +0.010 +0.015 +Sample Value +county +WIL, intercept +0.004 +0.002 +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0 +100 +200 +Posterior Density +county +MTL, intercept +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.005 +0.000 +0.005 +0.010 +Sample Value +county +MTL, intercept +0.012 +0.014 +0.016 +0.018 +0 +100 +200 +300 +Posterior Density +county +BOW, intercept +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.0125 +0.0150 +0.0175 +Sample Value +county +BOW, intercept +0.001 +0.000 +0.001 +0.002 +0.003 +0.004 +0.005 +0.006 +0 +100 +200 +300 +Posterior Density +county +DIV, intercept +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.0000 +0.0025 +0.0050 +Sample Value +county +DIV, intercept +0.000 +0.002 +0.004 +0.006 +0 +200 +Posterior Density +county +BRK, intercept +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.0000 +0.0025 +0.0050 +Sample Value +county +BRK, intercept +0.003 +0.002 +0.001 +0.000 +0.001 +0.002 +0.003 +0.004 +0 +200 +Posterior Density +county +MCL, intercept +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.0025 +0.0000 +0.0025 +Sample Value +county +MCL, intercept +0.004 +0.002 +0.000 +0.002 +0.004 +0.006 +0 +100 +200 +Posterior Density +county +BIL, intercept +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.000 +0.005 +Sample Value +county +BIL, intercept +0.006 +0.004 +0.002 +0.000 +0.002 +0.004 +0.006 +0.008 +0 +100 +200 +Posterior Density +county +STK, intercept +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.005 +0.000 +0.005 +Sample Value +county +STK, intercept +0.010 +0.005 +0.000 +0.005 +0.010 +0.015 +0 +50 +100 +Posterior Density +county +SLP, intercept +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.01 +0.00 +0.01 +Sample Value +county +SLP, intercept +0.015 +0.010 +0.005 +0.000 +0.005 +0.010 +0.015 +0 +50 +100 +Posterior Density +county +GV, intercept +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.01 +0.00 +0.01 +Sample Value +county +GV, intercept +Figure 4.5: Posterior distributions and trace plots of the intercepts for each county. +44 + +the interquartile ranges (IQR). The points represent the posterior means. +0.4 +0.5 +0.6 +0.7 +county GV, slope +county SLP, slope +county STK, slope +county BIL, slope +county MCL, slope +county BRK, slope +county DIV, slope +county BOW, slope +county MTL, slope +county WIL, slope +county DUN, slope +county MCK, slope +90.0% HDI +Figure 4.6: A forest plot showing the uncertainties around each county’s slope estimate. The +counties at the bottom have insufficient or noisy datasets, therefore their estimates are largely +pulled towards the partially-pooled mean. +0.00 +0.01 +0.02 +county GV, intercept +county SLP, intercept +county STK, intercept +county BIL, intercept +county MCL, intercept +county BRK, intercept +county DIV, intercept +county BOW, intercept +county MTL, intercept +county WIL, intercept +county DUN, intercept +county MCK, intercept +90.0% HDI +Figure 4.7: A forest plot showing the uncertainties around each county’s intercept estimate. +The dotted line labels the zero intercept, for which some counties’ estimates are not significantly +different from. +In the case of the slopes (Figure 4.6), it can be seen the top four counties are quite +diverse. MTL has the largest point estimate in the entire population (ˆβmtl > 0.6) while +45 + +DUN has the smallest one (ˆβdun < 0.5). Furthermore, the HDI’s for DUN and MTL rarely +overlap, indicating that it is almost certain that MTL has a larger slope than DUN. The +counties with fewer observations (remaining eight counties) have greater uncertainties in their +parameter estimates, while all of their point estimates are pulled towards the partially-pooled +mean which is between 0.5 and 0.6. When there is not enough data for some counties, the +hierarchical model strives to reinforce information sharing among different counties, thus +providing more sensible results and also quantifying the uncertainties in such processes. From +domain expertise, these results make more physical sense than the no-pooling estimates +discussed in Section 4.3 (i.e., ˆβslp ≈ 0 and ˆβgv ≫ 0). +In the case of the intercepts (Figure 4.7), there is also heterogeneity among the counties. +In particular, by plotting a dotted line labeling the zero intercept, some counties are found to +likely have zero intercept (e.g., zero is covered by the IQR or HDI) while others have intercepts +that are significantly different from zero. It might not be surprising to get close-to-zero +intercepts and greater uncertainties for those counties with less data (such as SLP and GV), +however it is interesting to obtain the HDI for MTL that covers zero. Recall that the intercept +parameter can be interpreted as the NDIC reported volume which is not captured by VIIRS. +This finding for MTL, along with the fact that MTL has the largest slope point estimate +(where a larger slope denotes closer proximity to the satellite estimation), convinces the +author that MTL used to have persistent and stronger gas flares. They kept VIIRS from +missing the flaring events in general, and lead to the reported volumes from NDIC and +VIIRS being closer to each other. On the contrary, DUN’s smaller slope and larger intercept +characterize its flares as sporadic and weaker. One thing worth mentioning is that, with +the current interpretation of the intercept, it does not make much physical sense to have +negative intercepts. Although every county has positive point estimates for their intercepts, +some counties’ HDI’s show coverage over the negative values. This is a limitation of choosing +a 2D Gaussian population model for the intercepts and slopes. Since the 2D Gaussian is +supported on R2, in the context of some counties having “weak data”, negative values make +46 + +an appearance in their HDI’s. +The discussions above naturally lead to the question of whether the slopes and intercepts +are correlated. It turns out that, by partially pooling the different types of parameters, a +probable negative correlation between the slopes and intercepts is revealed (Figure 4.8). The +correlation is learned from the heterogeneity in flare characteristics among the counties: +• Persistent flares yield smaller intercepts and larger slopes. +• Sporadic flares yield larger intercepts and smaller slopes. +In other words, intercepts and slopes covary in the entire population of counties. By pooling +information across parameter types, what the model learns in the intercept can improve +learning about slopes, and vice versa. With this “experience” or “knowledge”, the hierarchical +model will be able to quickly update its expectation for any new counties’ parameters even +with just a few observations in the beginning. It should be noted that there is also some +probability mass for the positive correlation values, i.e., the negative correlation is not very +strong. This could be due to that some counties do not have a lot of data at this time. The +posterior will be updated as more data is brought in. +Finally, the parameter estimates are reported in Table 4.2, from which the parametric +model for each county can be recovered, and then deployed in calibration and prediction +usage scenarios. +4.7 +Model Extensibility +Looking back at the hierarchical model and the reparameterization strategy from the +previous sections, there are four potential deployment scenarios that are worth discussing. +They demonstrate the extensibility and flexibility of the chosen approach in the context of +flaring data analytics: +1. New counties are present in terms of the reported flaring statistics from both VIIRS +and NDIC. +47 + +1.0 +0.5 +0.0 +0.5 +1.0 +Correlation +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Density +prior +posterior +Figure 4.8: Correlation between the intercepts and slopes. Blue: Posterior distribution of +the correlation, the mode of which is below zero. Dashed: Prior distribution, the LKJcorr(2) +density. +At this time, there are 12 counties that have reported flaring statistics from both VIIRS +and NDIC. If flaring data becomes available for some other counties in the future, the +hierarchical model allows the population to be immediately expanded to accommodate +the new counties. This can be seen from the conditional structure in Equation 4.3: by +taking a model for n + 1 counties +p(θ1, θ2, . . . , θn, θn+1, φ) = +�n+1 +� +i=1 +p(θi | φ) +� +p(φ) , +(4.8) +then pulling out the term for the (n + 1)-th county from the right-hand side (RHS) +p(θ1, θ2, . . . , θn, θn+1, φ) = p(θn+1 | φ) +� n +� +i=1 +p(θi | φ) +� +p(φ) , +(4.9) +it can be recognized that the remaining part on the RHS is the hierarchical model for +n counties +p(θ1, θ2, . . . , θn, θn+1, φ) = p(θn+1 | φ) p(θ1, θ2, . . . , θn, φ) . +(4.10) +48 + +Table 4.2: Parameter Estimates of County Level Flaring Model +Parameter +Variable +County +Point Estimate +90 % CI +αcounty +Intercept +MCK +0.019 +(0.015, 0.023) +DUN +0.008 +(0.004, 0.013) +WIL +0.010 +(0.007, 0.013) +MTL +0.002 +(−0.001, 0.006) +BOW +0.015 +(0.013, 0.017) +DIV +0.003 +(0.001, 0.005) +BRK +0.003 +(0.001, 0.004) +MCL +0.000 +(−0.001, 0.002) +BIL +0.001 +(−0.001, 0.003) +STK +0.001 +(−0.003, 0.004) +SLP +0.001 +(−0.005, 0.007) +GV +0.002 +(−0.005, 0.009) +βcounty +Slope +MCK +0.519 +(0.493, 0.542) +DUN +0.464 +(0.385, 0.547) +WIL +0.549 +(0.495, 0.605) +MTL +0.623 +(0.553, 0.693) +BOW +0.516 +(0.370, 0.677) +DIV +0.554 +(0.395, 0.719) +BRK +0.556 +(0.389, 0.715) +MCL +0.563 +(0.391, 0.730) +BIL +0.560 +(0.395, 0.727) +STK +0.562 +(0.393, 0.729) +SLP +0.561 +(0.406, 0.752) +GV +0.560 +(0.398, 0.731) +This indicates the newly introduced counties will only depend on the population +parameters φ, i.e., how the new counties interact with the existing ones (from the initial +dataset) is not explicitly specified but being mediated through φ. This mechanism +allows the population (of counties) to be expanded arbitrarily. In practice, without any +modification, Model 4.7 can be re-fitted with the new dataset as a whole. +2. More data are available for those counties which used to have very few observations. +In the event of more data becoming available for those counties with wide HDI’s such +as SLP and GV, the posteriors will be updated according to that information. Their +49 + +HDI’s would become narrower and narrower as more and more data are available, and +since the hierarchical model pools information among the counties, these counties will +contribute to updating the population model’s and other counties’ parameters. Similar +to Item 1 above, Model 4.7 does not need modifying and can be re-fitted with the new +data. +3. Sample sizes among counties become more unbalanced. +In general, when there is a lot of data for each county, the centered parameterization +(Model 4.5) is more efficient. When the sample size is not large, which is the case for +the current VIIRS and NDIC reportings, the noncentered parameterization (Model 4.7) +is better. However, the parameterization for hierarchical models is not a monolithic +tactic. If the reported flaring data becomes very unbalanced across counties, e.g., some +counties have a huge amount of data whereas others have very little data, then each +county can be parameterized differently. More specifically, +• For the counties that have strong data such that their likelihood functions dominate, +centered parameterization can be applied through Expressions 4.5f–4.5h. +• For the counties that have weak data such that their prior models dominate, +noncentered parameterization can be applied through Expressions 4.7f–4.7h. +All in all, this is still one hierarchical model which defines the exact same probabilistic +structure as Model 4.5 or Model 4.7, but avoids inefficiencies and non-convergences in +the sampling from posteriors. +4. Oilfield level heterogeneity needs to be examined. +Under the assumptions that the oilfields in North Dakota are exchangeable and the +population of oilfields (which conduct flaring) can grow, the hierarchical model developed +in this chapter can be directly applied to investigate the heterogeneity in different +oilfields’ parameters. Following the reverse geocoding as discussed in Section 3.3, there +50 + +are 258 oilfields that have both NDIC and VIIRS reportings for the same study period +as in this chapter. Some oilfields have very few observations and can benefit from the +hierarchical model through pooling information among the entire population of oilfields. +Furthermore, due to the number of oilfields being relatively large, the population +model could be learned with more ease (because more information is available for the +population). In the case of the county level model developed in this chapter, since +there are only 12 individuals (counties) in the population, some uncertainties about the +population are inevitably present and reflected through the posteriors. +The models developed in this chapter, while capturing the heterogeneity among the +different counties in North Dakota, rely on the assumption that all the monthly observations +within a certain county are conditionally i.i.d. For situations where the temporal structure +has to be taken into consideration, other types of models can be built and are discussed in +the next chapter. +51 + +CHAPTER 5 +FLARING TIME SERIES ANALYTICS +“Were neural networks over-hyped, or have we underestimated +the power of smoothing methods? +I think both these propositions are true.” +— MacKay (2003) +5.1 +Learning the Flaring Pattern and Behavior +In this chapter, the author develops a generic framework for revealing flaring patterns +and behaviors. The main challenges are fourfold: +1. Observed data are noisy. +• Companies estimate the flaring volumes and conduct self-reporting. Satellites +could miss some events. However, having knowledge about the underlying process +is vital in lots of situations including when the state and local governments need +to make key decisions based on the data. In the meantime, understanding the +underlying process helps with anomaly detection by differentiating between true +anomalies in reporting and ordinary noise or stochasticity. +2. A probabilistic approach is desirable to be adopted. +• A set of most probable functions (characterizing the underlying process) are +preferable over one single best fit function. +3. The observations of a certain entity are time series. +• The temporal structure is intrinsic to the dataset and thus must be harnessed. +4. The framework should be generic enough for automated insights extraction. +52 + +• There are more than 200 operators and 500 oilfields operating in North Dakota. +Choosing a specific parametric form of model (e.g., ARIMA or LSTM) for each +entity and then fitting the model to the data is not only time consuming, but also +prevents easy integration into automation pipelines (for extracting insights for +example). +It is striking that the elegant properties of Gaussian process make it a natural choice to +tackle all of these challenges and is therefore employed in this chapter. +5.2 +Gaussian Process +A Gaussian process (GP) can be viewed as a distribution over infinite-dimensional +Hilbert space of functions. It is formally defined as “a collection of random variables, any +finite number of which have a joint Gaussian distribution” (Rasmussen and Williams 2006). +Gaussian processes are extremely powerful nonparametric learning techniques, which provide +a composite of flexibility and interpretability. +They are well suited to problems which +necessitate principled handling of uncertainty and interpretation, in the presence of noisy +and dynamic datasets. Such scenarios include smoothing (Deisenroth et al. 2012) and time +series modeling (Roberts et al. 2013). They are also well established in different fields under +various names, for example kriging in geostatistics and Kalman filters both correspond to +Gaussian processes (MacKay 1998). +In this work, the motivation is to develop a generic framework for recognizing the underly- +ing unknown processes f(x) which reflect flaring strategies and behaviors. Thus inference is +conducted directly in the function space employing GP as a prior. A Gaussian process is com- +pletely specified by its mean function m(x) and covariance function k(x, x′) (Bandyopadhyay +2018), which are defined as: +m(x) = E[f(x)] , +(5.1) +k(x, x′) = E[(f(x) − m(x))(f(x′) − m(x′))] , +(5.2) +53 + +and the function distributed as a Gaussian process is denoted by +f(x) ∼ GP +� +m(x), k(x, x′) +� +. +(5.3) +5.2.1 +Mean Function +In this work, the mean functions are always chosen to be zero, since there is no prior +knowledge on the mean of the latent processes. In the meantime, for GPs with a zero +mean function, the mean of the posterior process is not confined to be zero (Rasmussen and +Williams 2006). All the latent functions modeled with a GP prior in this dissertation follow +f(x) ∼ GP +� +0, k(x, x′) +� +, +(5.4) +where k is some covariance function. +5.2.2 +Covariance Function +Covariance function, also known as kernel, is the crucial ingredient in a GP, as it encodes +one’s assumptions about how the function should behave by defining similarity. The fun- +damental assumption is that data points with inputs x which are close would have similar +target values y. This assumption is usually very reasonable in areas including time series +modeling, and it is theoretically backed by Tobler’s first law of geography. The covariance +functions used in this dissertation include: +1. The Mat´ern class of covariance functions, which is given by: +kν(r) = 21−ν +Γ(ν) +� +√ +2ν r +ℓ +�ν +Kν +� +√ +2ν r +ℓ +� +, +(5.5) +where Γ(·) is the gamma function, Kν is a modified Bessel function of the second kind +of order ν, r =∥x − x′∥, and ℓ is the lengthscale controlling the smoothness from one +perspective: large ℓ characterizes functions which change slowly and can be reliably +extrapolated further away. +The Mat´ern covariance functions can be written as a product of an exponential and a +polynomial of order p, when ν is half-integer: ν = p+1/2, p ∈ N0. The hyperparameter +54 + +ν controls the smoothness from another perspective: when ν = 1/2, the Mat´ern kernel +becomes the exponential kernel (continuous but not differentiable); as ν → ∞, it +becomes the exponentiated quadratic kernel (infinitely differentiable). Rasmussen and +Williams (2006) argued that the most interesting cases for machine learning would be +ν = 3/2 and ν = 5/2. +For gas flaring time series, as operators might change flaring strategy at any given time +due to policy changes, gas processing facility deployment, gas price fluctuation, etc., the +latent process might not be as smooth as infinitely differentiable. Instead the Mat´ern +kernel is harnessed which is capable of inducing non-smooth function realizations to +handle those discontinuities. Specifically the Mat´ern kernel with ν = 5/2 is chosen for +this dissertation with the input space X ⊆ R1: +kmat´ern52(x, x′; ℓ) := +� +1 + +� +5(x − x′)2 +ℓ ++ 5(x − x′)2 +3ℓ2 +� +exp +� +− +� +5(x − x′)2 +ℓ +� +, +(5.6) +where x vary over the time domain. +2. The standard periodic kernel due to MacKay (1998): +kperiodic(x, x′; T, ℓ) := exp +� +−sin2(π|x − x′| 1 +T ) +2ℓ2 +� +, +(5.7) +where T denotes the period. This kernel is used for modeling seasonal behaviors. +3. The white noise kernel, which is given by: +kWhiteNoise(x, x′; δ) := δ2In, +(5.8) +where δ2 is the variance of the noise. In this dissertation, the usage of the white noise +kernel is for stabilizing the computation of the covariance matrix. Adding a small value +of diagonal shift will try to guarantee the resulting covariance matrix is always positive +semi-definite. +55 + +A nice property is that the sum and product of the established kernels are still valid +kernels. This fact is also exploited in the model building process in this work. +5.2.3 +Inference and Model Reparameterization +In practice, one always works with a dataset of finite size. In such situations, a multivariate +normal prior distribution is placed on the vector of function values f, +f ∼ MVNormal(mx, Kxx) , +(5.9) +where the vector mx and the matrix Kxx are the mean function and covariance function +evaluated over the inputs x. +A key question which has significant impact on the inference is how to learn the hyperpa- +rameters from data. A natural (and popular) approach is to conduct maximum likelihood +estimation, i.e., generating point estimates leveraging the data. However, as Betancourt +(2017a) showed with experiment results, both regularized and unregularized maximum +marginal likelihood have limited performance in terms of fitting robustly and recovering +the true data generating process. Technically, given a particular kernel with particular +hyperparameters, a GP does not support an entire Hilbert space but only a slice through +that space; changing the hyperparameters by an infinitesimal amount yields a different slice +which has no overlap with the original one. Therefore in this dissertation, a full Bayesian +approach is taken for the GP inference, i.e., the entire Hilbert space of functions is considered +by taking into account all of the possible hyperparameters for a specific kernel. +For the class of problems which have Gaussian observation models, GP has nice closed-form +posterior results. However, for the situations which do not have Gaussian observation models, +for examples the ones in this dissertation which employ Student-t or Poisson likelihood, there +does not exist analytical solutions. HMC as discussed in Section 2.3 is used to sample from +the posteriors. +Specifically, the noncentered parameterization of the latent multivariate Gaussian is +exploited. The reparameterized model is +56 + +�f ∼ MVNormal(0n, In) +(5.10a) +L = Cholesky(Kxx) +(5.10b) +f = mx + L · �f +(5.10c) +which defines the same distribution as Expression 5.9 but induces a nicer posterior geometry +for HMC to explore and sample from (Betancourt 2017a). +Once the learning on hyperparameters is done, posterior predictive distribution of the +latent function values which are not part of the original dataset is obtained by +f∗ | f ∼ MVNormal +� +m∗ + K⊤ +x∗K−1 +xx(f − mx), K∗∗ − K⊤ +x∗K−1 +xxKx∗ +� +, +(5.11) +where m∗ is the mean function evaluated at the new inputs, K∗∗ is the covariance between +the new inputs, and Kx∗ is the covariance between the original inputs and the new inputs. +5.3 +Suite of Models for Pattern Recognition +This section presents models built from various angles, with the goal of providing a +coherent framework for learning the flaring pattern and behavior in a principled manner. +Each model is tested on real flaring data from North Dakota. Whenever more granular +analytics capabilities are demonstrated through investigations at oilfield level or operator +level, the data from a major producing field, the Blue Buttes Oilfield (Alexeyev et al. 2017), +and one operator, denoted by ‘Operator A’ are used. +5.3.1 +Modeling Proportion of Gas Flared +The proportion of gas production that is flared is an indicator of flaring intensity and energy +efficiency. It is interesting to investigate whether the proportion has changed over a period of +time for certain operators and oilfields. The model is specified through Expressions 5.12a– +5.12i: +ℓ ∼ Gamma(2, 1) +(5.12a) +η ∼ Half-Cauchy(5) +(5.12b) +57 + +ν ∼ Gamma(2, 0.1) +(5.12c) +ˆσ2 ∼ Half-Cauchy(5) +(5.12d) +k = η2 × kmat´ern52(x, x′; ℓ) +(5.12e) +f ∼ GP(0, k) +(5.12f) +πi = logit−1� +f(xi) +� +(5.12g) +µi = πi × Gi +(5.12h) +Fi ∼ Student-t(ν, µi, 1/ˆσ2) +(5.12i) +where: +ℓ is the lengthscale for the Mat´ern kernel; +η is the marginal deviation parameter controlling how strongly the latent functions +vary in the output space; +ν is the degrees of freedom for the Student-t likelihood; +ˆσ2 controls the inverse scaling parameter of the Student-t likelihood (analogous to +the precision of a Gaussian distribution); +k is the covariance function for the GP; +f denotes the latent process, which is distributed according to the GP; +πi is the underlying flaring gas proportion of month i. Since proportion is bounded +between 0 and 1, the inverse-logit function is applied to the latent process; +Gi is the total gas production of month i; +µi denotes the underlying flared volume of month i; +Fi is the reported flared volume, which is modeled using a Student-t observation +model. +The reasoning behind choosing a Student-t observation model is to make the model +specification be able to generalize to as many entities as possible and be robust to (potentially +many) outliers and noisy data points. This is due to the fact that at this time, operators have +to estimate the flared volume by their own procedures and conduct reporting, in which case +inaccuracies are introduced unintentionally or intentionally. The heavier tail of Student’s +58 + +t-distribution is a natural decision in modeling to deal with those phenomena. This line +of thought, i.e., design models that are generic and robust, is indeed reflected in choosing +the half-Cauchy priors (which are heavy-tailed and very weakly informative) and GP as a +nonparametric regression technique. +To demonstrate this model’s capability on real data, both the Blue Buttes Oilfield and +Operator A are tested. The production and flared volumes coming from NDIC are used. For +the oilfield, the posterior distributions and trace plots of the hyperparameters are presented +in Figure 5.1. The posterior predictive samples for the underlying process of gas flaring +proportions (πi) are demonstrated in Figure 5.2, which depict the trend very clearly. The +colored bands have the below coverage for the posterior samples: +• The darkest colored band (in the center at a certain x location) represents the 49th +percentile to 51st percentile; +• The lightest colored band (characterized by the widest interval at a certain x location) +represents the 1st percentile to 99th percentile. +Additionally, 30 random samples are drawn from the GP posterior and plotted on the +same figure, showing as thin lines. The latent functions do not go through all the observed +data points, in which case the model would have been overfitted; instead they present the +possible functions which are most compatible with the data as well as the assumptions +inherent in the model. On one hand, the insights are already obtained, i.e., the underlying +process is inferred. On the other hand, this serves as an anomaly detection tool. For example, +the state government might be interested to look into that observed data in the second half +of 2019 which deviated quite a lot from the “true” process, e.g. to audit the reporting for +that month or to investigate what had happened that led to a sudden huge drop in flaring in +just one month. +With the exact same model specification, the model is also run with the operator’s data. +The posterior distributions and trace plots of the hyperparameters are presented in Figure 5.3. +59 + +4 +6 +8 +10 +12 +14 +16 +0.0 +0.1 +0.2 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +5 +10 +15 +Sample Value +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.5 +1.0 +1.5 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +1 +2 +3 +Sample Value +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +0 +100 +200 +300 +Posterior Density +_sq +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.005 +0.010 +Sample Value +_sq +0 +20 +40 +60 +80 +100 +120 +0.00 +0.01 +0.02 +0.03 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0 +50 +100 +Sample Value +Figure 5.1: Posterior distributions and trace plots for the Blue Buttes Oilfield gas flaring +proportion model. Well mixing and convergence have been achieved. +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +2016 +2017 +2018 +2019 +2020 +Report Date +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Proportion of Gas Flared in Total Gas Produced +Observed Proportion of Gas Flared +Figure 5.2: Posterior predictive samples showing the gas flaring proportion variations at the +Blue Buttes Oilfield. Blue points are the observed data while red lines present the posterior +samples. +60 + +The posterior predictive samples for the underlying process of gas flaring proportions (πi) are +demonstrated in Figure 5.4. It can be seen this operator’s flaring proportion time series is +more jagged than the Blue Buttes Oilfield (which is operated by more than five companies). A +operator can change flaring strategies more swiftly which can be captured as well. Nevertheless +the long-term trend is also available. Comparing Figure 5.1 and Figure 5.3, it can be seen the +posterior distributions are very different. However the priors for them were specified in the +exact same way. This showcases the power of Bayesian approach. Taking ℓ as an example, +a Gamma(2, 1) prior is placed on it. However, after conditioning on the data, the operator +model reports smaller lengthscale values on average (indicating jagged processes), whereas +the oilfield model reports larger lengthscale values (suggesting smoother processes). +2 +4 +6 +8 +10 +12 +14 +0.0 +0.1 +0.2 +Posterior Density +0 +500 +1000 +1500 +2000 +5 +10 +15 +Sample Value +0.2 +0.4 +0.6 +0.8 +1.0 +0 +1 +2 +3 +4 +Posterior Density +0 +500 +1000 +1500 +2000 +0.2 +0.4 +0.6 +0.8 +1.0 +Sample Value +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +1 +2 +3 +4 +Posterior Density +_sq +0 +500 +1000 +1500 +2000 +0.00 +0.25 +0.50 +0.75 +1.00 +Sample Value +_sq +0 +20 +40 +60 +80 +100 +0.00 +0.01 +0.02 +0.03 +Posterior Density +0 +500 +1000 +1500 +2000 +0 +25 +50 +75 +100 +Sample Value +Figure 5.3: Posterior distributions and trace plots for the Operator A gas flaring proportion +model. Well mixing and convergence have been achieved. Notice the differences between +these inference results and those in Figure 5.1, both of which are based on exactly the same +priors and likelihood, demonstrating the model specification’s wide applicability. +61 + +0.2 +0.3 +0.4 +0.5 +0.6 +2016 +2017 +2018 +2019 +2020 +Report Date +0.2 +0.3 +0.4 +0.5 +0.6 +Proportion of Gas Flared in Total Gas Produced +Observed Proportion of Gas Flared +Figure 5.4: Posterior predictive samples showing the gas flaring proportion variations of +Operator A. Blue points are the observed data while red lines present the posterior samples. +Order 24665, which is established by the North Dakota Industrial Commission, defines +the gas capture percentage pcap as +pcap = Gsold + Gused + Gproc +Gprod +, +(5.13) +where: +Gsold is the monthly gas sold; +Gused is the monthly gas used on lease; +Gproc is the monthly gas processed; +Gprod is the monthly gas produced. +Since North Dakota bans the venting of natural gas (U.S. Department of Energy 2019b), +it is obvious the model developed in this section provides a powerful tool for NDIC to evaluate +compliance with the gas capture goals: at a given month i, pcap = 1 − πi. Furthermore, when +looking at the model specification, there is nothing special that encodes the data sources and +location information. A user of this model is free to use satellite estimation as the observed +data or apply it to the Permian Basin, and conduct inference on the flaring proportion. This +is a benefit from using nonparametric and interpretable models as opposed to black box +62 + +models (such as the neural networks, in which case the learned weights and bias inside the +network provide little or no domain insights). The author hopes this section provides a +comprehensive view in terms of how and why to use GP, with real data. Models built and +presented in later sections follow a similar flow. +5.3.2 +Modeling Proportion of Wells Flaring +The proportion of wells that conduct flaring in a month can reflect a company’s flaring +strategy and is an indicator of flaring magnitude. It is interesting to investigate how this +indicator varies for a certain entity in a certain time period. The model is specified through +Expressions 5.14a–5.14f: +ℓ ∼ Gamma(2, 1) +(5.14a) +η ∼ Half-Cauchy(5) +(5.14b) +k = η2 × kmat´ern52(x, x′; ℓ) +(5.14c) +f ∼ GP(0, k) +(5.14d) +pi = logit−1� +f(xi) +� +(5.14e) +Wi ∼ Binomial(Ni, pi) +(5.14f) +where pi is the unobserved “true” proportion of wells that conduct flaring in month i, Ni +is the total number of active wells in month i, and Wi is the observed (i.e., estimated and +reported by company) number of wells that conduct flaring in month i. The rest of the +symbols have the same meaning as in Model 5.12. +To demonstrate this model’s capability on actual data, both the Blue Buttes Oilfield +and Operator A are tested. For the oilfield, the posterior distributions and trace plots +of the hyperparameters are presented in Figure 5.5. The posterior predictive samples for +the underlying process of well flaring proportion (pi) are demonstrated in Figure 5.6. The +visualization strategy (different colors represent different percentiles, etc.) is the same as in +Section 5.3.1. +63 + +2 +4 +6 +8 +10 +12 +14 +0.0 +0.1 +0.2 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +5 +10 +15 +Sample Value +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0 +1 +2 +3 +4 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.5 +1.0 +Sample Value +Figure 5.5: Posterior distributions and trace plots for the Blue Buttes Oilfield well flaring +proportion model. Well mixing and convergence have been achieved. +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +2016 +2017 +2018 +2019 +2020 +Report Date +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +Proportion of Wells that Conducted Flaring +Observed Proportion of Wells Flaring +Figure 5.6: Posterior predictive samples showing the well flaring proportion variations at the +Blue Buttes Oilfield. Blue points are the observed data while red lines present the posterior +samples. +With the exact same model specification, this model is also tested with the operator’s data. +The posterior distributions and trace plots of the hyperparameters are presented in Figure 5.7. +The posterior predictive samples for the underlying process of well flaring proportion (pi) +are demonstrated in Figure 5.8. Comparing the two sets of figures from the oilfield and the +operator, it can be seen: +64 + +1. With the same prior placed on the lengthscale ℓ, the oilfield model learns from the data +and gives a posterior mode around 5.5, whereas the operator model gives a posterior +mode around 10.0. This is also reflected in the posterior samples time series plot: the +oilfield experienced some well flaring proportion changes in relative shorter time periods, +whereas the operator underwent changes on a longer time span. +2. The oilfield’s posterior samples time series show narrower percentile bands while the +operator’s show wider percentile bands. This is due to the fact that the operator chosen +here had smaller number of wells than the oilfield. Since the binomial observation model +is used for each month’s flaring well count, this naturally represents and quantifies the +uncertainties (i.e., binary data contains less information especially when the sample +size is small), as well as aligns with the expectation that when there is more data, there +should be less uncertainties; when there is less data, there should be more uncertainties. +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +22.5 +0.00 +0.05 +0.10 +0.15 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +5 +10 +15 +20 +Sample Value +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +0 +1 +2 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.5 +1.0 +1.5 +2.0 +Sample Value +Figure 5.7: Posterior distributions and trace plots for the Operator A well flaring proportion +model. Well mixing and convergence have been achieved. Notice the differences between +these inference results and those in Figure 5.5, both of which are based on exactly the same +priors and likelihood, demonstrating the model specification’s wide applicability. +This really showcases how and why to encode domain expertise in flaring data analytics +while exploiting machine learning models, which is also the reason to choose the Bayesian +approach. One could fit a black box model either with target values Wi ∈ R, or without +any probabilistic view (e.g., to optimize for the best deterministic function mapping in the +65 + +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +2016 +2017 +2018 +2019 +2020 +Report Date +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Proportion of Wells that Conducted Flaring +Observed Proportion of Wells Flaring +Figure 5.8: Posterior predictive samples showing the well flaring proportion variations of +Operator A. Blue points are the observed data while red lines present the posterior samples. +hypothesis space). But either of those would be fundamentally flawed. Domain expertise +indicates the well count has to be a non-positive integer, i.e., Wi ∈ N0. Furthermore, neither +the NDIC reporting nor the satellite estimation is ever produced in a noise-free environment, +and therefore probabilistic modeling is a must. Compared to frequentist machine learning, +Bayesian learning is entirely probabilistic and gives one the capability and freedom to encode +his/her domain expertise. +5.3.3 +Modeling Flare Detection Count +Satellite detected flare count provides an unbiased indicator of flaring intensity. How this +indicator varies in a certain time period for a certain entity is valuable information to obtain. +The model is specified through Expressions 5.15a–5.15f. Essentially the latent process is +modeled as a Gaussian Cox process (Adams et al. 2009), where the Poisson process has +varying intensity across time domain and a GP prior is placed on this intensity. +ℓ ∼ Gamma(2, 1) +(5.15a) +η ∼ Half-Cauchy(5) +(5.15b) +k = η2 × kmat´ern52(x, x′; ℓ) +(5.15c) +66 + +f ∼ GP(0, k) +(5.15d) +λi = exp +� +f(xi) +� +(5.15e) +Ci ∼ Poisson(λi) +(5.15f) +where λi is the unobserved flaring intensity (“true” count) in month i and Ci is the reported +VIIRS detection count in month i. Since λi is bounded to be positive, the natural exponential +function is applied to the latent process. The rest of the symbols have the same meaning as +in Model 5.12. +For the task of flaring pattern recognition, the author believes this approach (leveraging +a Gaussian Cox process) is a nicer surrogate than a popular change point model presented +in (Davidson-Pilon 2015; Salvatier et al. 2016; Stan Development Team 2020), which is +specified by: +e ∼ Exponential(re) +(5.16a) +l ∼ Exponential(rl) +(5.16b) +s ∼ Uniform(1, T) +(5.16c) +Ci ∼ Poisson(i < s ? e : l) +(5.16d) +where e and l are the early and late rates respectively, re and rl controls the priors for the +early and late rates, s is the change point, T is the total time period, and the rate in the +Poisson likelihood is decided through a ternary conditional operator (?:). The reason is +that, although this model could be generalized to more than one change point, its usage +is restricted by the assumption that any period between two adjacent change points has a +constant rate. This limitation becomes obvious when analyzing the actual flaring data in the +discussions below, and is a major disadvantage of the change point model. +The Gaussian Cox process model is tested with the Blue Buttes Oilfield’s data. Since +only VIIRS data is used, the whole time series is analyzed beginning in 2012. The posterior +distributions and trace plots of the hyperparameters are presented in Figure 5.9. +The +posterior predictive samples for the underlying process of flare count (Ci) are demonstrated +67 + +in Figure 5.10. The visualization strategy (different colors represent different percentiles, etc.) +is the same as in Section 5.3.1. From the time series plot, it can be seen the observations +from 2014 to 2017 can possibly be described by a change point model (with late 2015 being a +potential change point), but the steady growth before and after that time span will frustrate +accurate inference with such a model. +10 +15 +20 +25 +30 +0.00 +0.05 +0.10 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +10 +15 +20 +25 +30 +Sample Value +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.5 +1.0 +1.5 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +1 +2 +3 +Sample Value +Figure 5.9: Posterior distributions and trace plots for the Blue Buttes Oilfield flare count +model. Well mixing and convergence have been achieved. +5 +10 +15 +20 +25 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +Report Date +5 +10 +15 +20 +25 +Detection Rate +Observed Detection Count +Figure 5.10: Posterior predictive samples showing the flare count variations at the Blue Buttes +Oilfield. Blue points are the observed data while red lines present the posterior samples. +68 + +This model’s inference results serve as a type of confirmation, if not evidence, in terms of +whether or not an entity achieves the goal/target in reducing the number of wells flaring, +when the detection count is used as a surrogate for the number of wells flaring. In practice, +reducing the number of wells flaring is exactly the second goal of the regulatory policy +introduced by the North Dakota Industrial Commission in 2014. If the state government is +interested in this order’s effectiveness from a macroscopic standpoint, the model can also be +used to conduct inferences with the state level data. In this case, the posterior distributions +and trace plots of the hyperparameters are presented in Figure 5.11. The posterior predictive +samples for the underlying process of flare count (Ci) are demonstrated in Figure 5.12. +10 +12 +14 +16 +18 +20 +22 +24 +0.00 +0.05 +0.10 +0.15 +0.20 +Posterior Density +0 +500 +1000 +1500 +2000 +10 +15 +20 +Sample Value +2 +3 +4 +5 +6 +7 +0.0 +0.2 +0.4 +0.6 +Posterior Density +0 +500 +1000 +1500 +2000 +2 +4 +6 +Sample Value +Figure 5.11: Posterior distributions and trace plots for the North Dakota flare count model. +Well mixing and convergence have been achieved. Notice the differences between these +inference results and those in Figure 5.9, both of which are based on exactly the same priors +and likelihood, demonstrating the model specification’s wide applicability. +The percentile bands in this case are quite narrow, which indicate greater confidence +in the inferences about the data generating process given the model assumptions. By not +(over)fitting to each and every observation, interesting patterns are discovered, for example +in every year there is one and only one peak that happened around June. It is worth +pointing out that there is no model that can tell the modeler if his/her assumptions are good, +only domain expertise might. This model employing a Poisson observation model could be +considered “rigid” due to the fact that a Poisson likelihood has only one parameter λ (to +69 + +200 +300 +400 +500 +600 +2013 +2014 +2015 +2016 +2017 +2018 +2019 +Report Date +200 +300 +400 +500 +600 +Detection Rate +Observed Detection Count +Figure 5.12: Posterior predictive samples showing the flare count variations in North Dakota. +Blue points are the observed data while red lines present the posterior samples. +control both the mean and variance) and, furthermore, when λ is large as in this scenario, +a Poisson distribution is well approximated by a normal distribution. Whenever the state +government believes that overdispersion might exist, other observation models such as the +negative binomial distribution could be considered. In such cases, only Expression 5.15f needs +to be changed to the negative binomial likelihood, with a prior added for the overdispersion +parameter. The specific parameterization is given by Equation 6.4 in Section 6.3. This +really showcases both the flexibility and interpretability of taking a Bayesian approach for +high-stakes decision making areas including flaring data analytics. +5.3.4 +Modeling Proportion of Oil Flared +As crude oil (as opposed to natural gas) is the main commodity at this time, the amount +of gas in a barrel of oil equivalent (BOE) that is flared provides an indicator of production +efficiency due to flaring. In this work, the normalized quantity, proportion of oil production +being flared, is used such that the model specification is generic for large and small entities. +The model is specified through Expressions 5.17a–5.17j: +ℓ ∼ Gamma(2, 1) +(5.17a) +70 + +η ∼ Half-Cauchy(5) +(5.17b) +ν ∼ Gamma(2, 0.1) +(5.17c) +ˆσ2 ∼ Half-Cauchy(5) +(5.17d) +k = η2 × kmat´ern52(x, x′; ℓ) +(5.17e) +f ∼ GP(0, k) +(5.17f) +πi = logit−1� +f(xi) +� +(5.17g) +µi = πi × Oi +(5.17h) +c = 6 Mcf +1 BOE +(5.17i) +Fi/c =: Ei ∼ Student-t(ν, µi, 1/ˆσ2) +(5.17j) +where: +πi is the underlying flaring BOE proportion of month i; +Oi is the total oil production of month i; +µi denotes the “true” flared BOE of month i; +c denotes the conversion factor that 6 Mcf equals 1 BOE, given by the United States +Geological Survey (2000); +Ei is the reported flared BOE, which is modeled using a Student-t observation model. +The rest of the symbols have the same meaning as in Model 5.12. To test this model’s +performance on real data, both the Blue Buttes Oilfield and Operator A are used. For the +oilfield, the posterior distributions and trace plots of the hyperparameters are presented in +Figure 5.13. The posterior predictive samples for the underlying process of BOE flaring +proportion (πi) are demonstrated in Figure 5.14. The visualization strategy (different colors +represent different percentiles, etc.) is the same as in Section 5.3.1. +With the exact same model specification, this model is also tested with the operator’s +data. The posterior distributions and trace plots of the hyperparameters are presented in +Figure 5.15. The posterior predictive samples for the underlying process of BOE flaring +proportion (πi) are demonstrated in Figure 5.16. +Comparing the two sets of figures from the oilfield and the operator, it can be observed: +71 + +6 +8 +10 +12 +14 +16 +18 +20 +0.00 +0.05 +0.10 +0.15 +0.20 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +5 +10 +15 +20 +Sample Value +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +0.00 +0.25 +0.50 +0.75 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +1 +2 +3 +4 +5 +Sample Value +0.00005 +0.00010 +0.00015 +0.00020 +0.00025 +0.00030 +0.00035 +0 +5000 +10000 +Posterior Density +_sq +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.0001 +0.0002 +0.0003 +Sample Value +_sq +0 +20 +40 +60 +80 +100 +120 +0.00 +0.01 +0.02 +0.03 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0 +50 +100 +Sample Value +Figure 5.13: Posterior distributions and trace plots for the Blue Buttes Oilfield BOE flaring +proportion model. Well mixing and convergence have been achieved. +0.02 +0.04 +0.06 +0.08 +0.10 +2016 +2017 +2018 +2019 +2020 +Report Date +0.02 +0.04 +0.06 +0.08 +0.10 +Proportion of BOE Flared in Total Oil Produced +Observed Proportion of BOE Flared +Figure 5.14: Posterior predictive samples showing the BOE flaring proportion variations +at the Blue Buttes Oilfield. Blue points are the observed data while red lines present the +posterior samples. +72 + +10 +15 +20 +25 +0.00 +0.05 +0.10 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +10 +15 +20 +25 +Sample Value +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +0.0 +0.5 +1.0 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +1 +2 +3 +4 +Sample Value +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0 +50 +100 +Posterior Density +_sq +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0.01 +0.02 +0.03 +Sample Value +_sq +0 +20 +40 +60 +80 +100 +120 +140 +0.00 +0.01 +0.02 +0.03 +Posterior Density +0 +250 +500 +750 +1000 +1250 +1500 +1750 +0 +50 +100 +Sample Value +Figure 5.15: Posterior distributions and trace plots of the BOE flaring proportion model +for Operator A. Well mixing and convergence have been achieved. Notice the differences +between these inference results and those in Figure 5.13, both of which are based on exactly +the same priors and likelihood, demonstrating the model specification’s wide applicability. +1. With the same prior placed on the lengthscale ℓ, which has a mean of 2 (months), both +models have updated the posterior to move away from this mean, reflecting a long +range variation. The oilfield has a posterior mode about 1 year while the operator has +a mode around 15 months. The operator has much larger reporting variability, shown +by the parameter ˆσ2. +2. With a Student-t likelihood, both models demonstrate robustness to outliers and +overfitting. This can be seen from the oilfield’s late 2019 observations and the operator’s +early 2016 observations. For the posterior function samples, shown as the thin lines, +some of them are indeed pulled towards those “outliers”. However, the percentile plots +73 + +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +2016 +2017 +2018 +2019 +2020 +Report Date +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +Proportion of BOE Flared in Total Oil Produced +Observed Proportion of BOE Flared +Figure 5.16: Posterior predictive samples showing the BOE flaring proportion variations of +Operator A. Blue points are the observed data while red lines present the posterior samples. +(shown as the colored bands) are not impacted and those really can be interpreted as +the trend which is most compatible with the data and the assumptions. This built-in +Occam’s razor of the Bayesian approach when choosing appropriate priors is very +impressive. In many of the frequentist machine learning methods, if the regularization +strategy is not implemented well especially when the sample size is not huge enough +for the asymptotic properties to kick in, outliers become “influential observations” that +will have a huge undesirable effect on the inference results. +5.3.5 +Modeling Scale Factor between VIIRS and NDIC +Both NDIC and VIIRS reporting give (estimated) flared gas volume. The scale factor +between the two sources provides insights into whether NDIC reporting is consistent: +1. for different entities (e.g., among a group of operators), and +2. for one entity when looking at a certain time period. +This is based on the fact that the satellite detection processing algorithm is unbiased and +consistent. Item 2 is particularly interesting in terms of time series analytics. The model is +specified through Expressions 5.18a–5.18n: +74 + +ℓmat ∼ Gamma(8, 2) +(5.18a) +ηmat ∼ Half-Cauchy(5) +(5.18b) +T ∼ N(12, 1) +(5.18c) +ℓper ∼ Gamma(4, 3) +(5.18d) +ηper ∼ Half-Cauchy(5) +(5.18e) +ν ∼ Gamma(2, 0.1) +(5.18f) +ˆσ2 ∼ Half-Cauchy(5) +(5.18g) +kmat = η2 +mat × kmat´ern52(x, x′; ℓmat) +(5.18h) +kper = η2 +per × kperiodic(x, x′; T, ℓper) +(5.18i) +kwn = kWhiteNoise(x, x′; δ = 1e−6) +(5.18j) +f ∼ GP(0, kmat + kper + kwn) +(5.18k) +βi = exp +� +f(xi) +� +(5.18l) +µi = βi × VIIRSi +(5.18m) +NDICi ∼ Student-t(ν, µi, 1/ˆσ2) +(5.18n) +where: +ℓmat is the lengthscale for the Mat´ern kernel; +ηmat is the marginal deviation for the Mat´ern kernel; +T is the period for the periodic kernel; +ℓper is the lengthscale for the periodic kernel; +ηper is the marginal deviation for the periodic kernel; +kmat is the Mat´ern kernel (component); +kper is the periodic kernel (component); +kwn is the white noise kernel (component); +f denotes the latent process, which is distributed according to a GP whose covariance +function is the sum of 3 kernels; +βi is the underlying scale factor between VIIRS and NDIC of month i. Since this scale +factor is bounded to be positive, the natural exponential function is applied to +75 + +the latent process; +VIIRSi is the VIIRS reported volume of month i; +µi denotes the underlying flared volume of month i; +NDICi is the NDIC reported volume of month i, which is modeled using a Student-t +observation model. +The rest of the symbols have the same meaning as in Model 5.12. The reason for adding +a periodic kernel is to investigate if there are any seasonal patterns. Maintaining a proper +Bayesian workflow lets the data speak for itself, i.e., whether there exists seasonal behaviors +or not, as shown by the two case studies in this section. +The model is first fitted with the state level data to investigate the macroscopic reporting +consistency. The posterior distributions and trace plots of the hyperparameters are presented +in Figure 5.17. The posterior predictive samples for the underlying process of the scale factor +variations (βi) are demonstrated in Figure 5.18. The visualization strategy (different colors +represent different percentiles, etc.) is the same as in Section 5.3.1. From the posterior time +series plot, it can be seen in general the volumes from NDIC reporting is smaller than that of +VIIRS reporting, except for the times when the total flaring magnitude was small (indicated +by the smaller points). More importantly, within each and every year from 2015 to 2018, +there is a decreasing trend in the values of the scale factor (βi) around midyear. Each year’s +latent process from Q2 to Q3 can be viewed as a “seesaw”, with July being the middle pivot +point and the months after July always going down. Note that within each year, the NDIC +reporting of flared volumes might increase steadily or a lot (which was actually happening +from the time series plot in Figure 3.7), however this scale factor declining trends indicate the +satellites observed much greater flaring activities than what was reported by the companies! +This finding suggests that the NDIC reporting is very likely not consistent throughout the +year, and the state government should be concerned that some companies might underreport +their flared volumes especially in the second half of the year. +76 + +2 +4 +6 +8 +10 +12 +0.0 +0.1 +0.2 +0.3 +Posterior Density +_mat +0 +500 +1000 +1500 +2000 +2.5 +5.0 +7.5 +10.0 +Sample Value +_mat +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +2 +4 +6 +8 +Posterior Density +_mat +0 +500 +1000 +1500 +2000 +0.0 +0.2 +0.4 +Sample Value +_mat +10 +11 +12 +13 +14 +0.0 +0.5 +1.0 +Posterior Density +T +0 +500 +1000 +1500 +2000 +10 +11 +12 +13 +14 +Sample Value +T +1 +2 +3 +4 +5 +0.0 +0.2 +0.4 +0.6 +0.8 +Posterior Density +_per +0 +500 +1000 +1500 +2000 +2 +4 +Sample Value +_per +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.5 +1.0 +1.5 +Posterior Density +_per +0 +500 +1000 +1500 +2000 +0 +1 +2 +3 +Sample Value +_per +0.0000 +0.0005 +0.0010 +0.0015 +0.0020 +0.0025 +0.0030 +0 +500 +1000 +1500 +Posterior Density +_sq +0 +500 +1000 +1500 +2000 +0.000 +0.001 +0.002 +0.003 +Sample Value +_sq +0 +20 +40 +60 +80 +100 +0.00 +0.01 +0.02 +0.03 +0.04 +Posterior Density +0 +500 +1000 +1500 +2000 +0 +25 +50 +75 +100 +Sample Value +Figure 5.17: Posterior distributions and trace plots for the North Dakota VIIRS-NDIC scale +factor model. Well mixing and convergence have been achieved. +77 + +0.6 +0.8 +1.0 +1.2 +2015-07 +2016-01 +2016-07 +2017-01 +2017-07 +2018-01 +2018-07 +2019-01 +Report Date +0.6 +0.8 +1.0 +1.2 +Scale Factor Applied to Satellite Estimation +VIIRS_bcm +0.0 +0.2 +0.4 +0.6 +0.8 +Figure 5.18: Posterior predictive samples showing the scale factor variations of North Dakota. +Blue points are the observed data while red lines present the posterior samples. Larger points +indicate greater flaring magnitude as observed from VIIRS. +A interesting question arises: is this seasonal behavior universal across all the entities? +The answer is unfortunately no, which indicates some operators likely reported their flared +volume in an inconsistent manner throughout the entire year. In fact, if the Blue Buttes +Oilfield data is used to fit the model, rather consistent behavior is observed. In this case, the +posterior distributions and trace plots of the hyperparameters are presented in Figure 5.19. +The posterior predictive samples for the underlying process of the scale factor variations (βi) +are demonstrated in Figure 5.20. With the exact same model specification incorporating +the periodic kernel, no apparent seasonal behaviors are discerned by the inference process. +There are much uncertainties around the time of early 2016, where the point sizes indicate +the overall flaring magnitudes were small as observed from VIIRS, and the NDIC reported +volumes were actually larger than that of VIIRS. This could be due to the truncation effects +instead of the reporting inconsistencies, i.e., when the flares are sporadic and weaker, they are +not easily captured by the satellites, resulting in a truncated sample for the VIIRS processing +workflow. By applying this model and workflow to the other major producing fields, it will +likely pick up the ones who have the “seesaw” behaviors in their reporting. +78 + +2 +4 +6 +8 +10 +12 +0.0 +0.1 +0.2 +0.3 +Posterior Density +_mat +0 +1000 +2000 +3000 +4000 +2.5 +5.0 +7.5 +10.0 +12.5 +Sample Value +_mat +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0 +1 +2 +Posterior Density +_mat +0 +1000 +2000 +3000 +4000 +0 +1 +2 +Sample Value +_mat +9 +10 +11 +12 +13 +14 +15 +16 +0.0 +0.2 +0.4 +Posterior Density +T +0 +1000 +2000 +3000 +4000 +10 +12 +14 +16 +Sample Value +T +1 +2 +3 +4 +5 +6 +0.0 +0.2 +0.4 +0.6 +Posterior Density +_per +0 +1000 +2000 +3000 +4000 +2 +4 +6 +Sample Value +_per +0 +2 +4 +6 +8 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +Posterior Density +_per +0 +1000 +2000 +3000 +4000 +0.0 +2.5 +5.0 +7.5 +10.0 +Sample Value +_per +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1e +5 +0 +100000 +200000 +300000 +Posterior Density +_sq +0 +1000 +2000 +3000 +4000 +0.0 +0.5 +1.0 +Sample Value +1e +5 +_sq +0 +20 +40 +60 +80 +100 +0.00 +0.01 +0.02 +0.03 +Posterior Density +0 +1000 +2000 +3000 +4000 +0 +25 +50 +75 +100 +Sample Value +Figure 5.19: Posterior distributions and trace plots for the Blue Buttes Oilfield VIIRS-NDIC +scale factor model. Well mixing and convergence have been achieved. Notice the differences +between these inference results and those in Figure 5.17, both of which are based on exactly +the same priors and likelihood, demonstrating the model specification’s wide applicability. +79 + +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2015-07 +2016-01 +2016-07 +2017-01 +2017-07 +2018-01 +2018-07 +2019-01 +Report Date +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Scale Factor Applied to Satellite Estimation +VIIRS_bcm +0.000 +0.025 +0.050 +0.075 +Figure 5.20: Posterior predictive samples showing the scale factor variations in the Blue +Buttes Oilfield. Blue points are the observed data while red lines present the posterior +samples. Larger points indicate greater flaring magnitude as observed from VIIRS. +5.3.6 +Predicting NDIC Flared Volume +GP is not only fully capable of making predictions once the model hyperparameters are +learned, but it can provide rigorously constructed intervals quantifying uncertainties as well +through Expression 5.11, for which many of the frequentist machine learning methods fail to +do. The author chooses to present one particular prediction case study, that is to predict +NDIC reported volume based on the projected scale factor between VIIRS and NDIC. This +will be a particular interesting deployment scenario once fast satellite detection/estimation +is available, which takes less time than waiting on company reports followed by compiling +everything into an analytics-ready format. +The predictions are generated in the form of posterior predictive samples. Along with +the historical observations, the predictions of the scale factor for the next six months are +presented in Figure 5.21. The very wide percentile bands in the forecasting indicate that the +seasonal behaviors will likely take effect again, however with great uncertainties. If point +predictions (i.e., without the prediction intervals) are needed, one can always use the posterior +mean, mode, etc. to construct that “best” function; however this showcases why predicting +80 + +the future is generally very difficult and uncertainties should always be properly characterized. +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +2015-07 +2016-01 +2016-07 +2017-01 +2017-07 +2018-01 +2018-07 +2019-01 +2019-07 +Report Date +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +Scale Factor Applied to Satellite Estimation +VIIRS_bcm +0.0 +0.2 +0.4 +0.6 +0.8 +Figure 5.21: Posterior predictive samples showing predictions of the scale factor for the next +six months. Blue points are the observed data while red lines present the posterior samples. +Larger points indicate greater flaring magnitude as observed from VIIRS. +5.3.7 +A Look Back at the Prior Choices +Looking back at the suite of models developed, the set of priors for the latent functions +have been the same (except the scale factor model where a periodic kernel is added). However +the posteriors are all updated (i.e., “learned”) based on each dataset and modeling goal. This +means the below set of priors +ℓ ∼ Gamma(2, 1) +(5.19a) +η ∼ Half-Cauchy(5) +(5.19b) +k = η2 × kmat´ern52(x, x′; ℓ) +(5.19c) +f ∼ GP(0, k) +(5.19d) +serves as a generic framework and can be recommended for flaring time series analytics +in general, in a GP context. Notice this prior choice gives latent function values in the +unconstrained space, i.e., f(x) ∈ R. However, in many situations, the domain expertise +81 + +indicates the quantities of interest live in constrained space, such as: +• R>0 for Poisson rate parameter when modeling count data, and +• [0, 1] for binomial success probability when modeling flaring well proportion. +To better reflect the domain expertise, the link functions can be leveraged. For the above +scenarios, the log link function and the logit link function can be applied, respectively. +Although this prior configuration is the result of several design iterations and tested with +real data, there is no reason to think that it is optimal for every entity. Indeed, the model +for scale factor between VIIRS and NDIC has bespoke components in its priors. The Stan +Development Team (2020) also gave some general prior choice recommendations for GP. +The whole suite of models demonstrate full capability of harnessing the temporal structure +in flaring time series at different levels for different entities. This provides huge potential for +extracting insights from noisy monthly data streams. For the situations where cross-sectional +data analytics is desirable, for example when the latest monthly data is available and the +state government needs insights from merely that month (before appending it to the whole +historical data for a longitudinal study), other types of models can be built. Such is discussed +in the next chapter. +82 + +CHAPTER 6 +UNSUPERVISED LEARNING FROM MULTIPLE PERSPECTIVES +“Estimation of densities is a universal problem of +statistics (knowing the densities one can solve +various problems).” +— Vapnik (2000) +6.1 +Learning the Distribution +In this chapter, the author studies how to describe the flaring related quantities’ distribu- +tion among the oilfields in North Dakota in a cross-sectional setting. That is, data collected +for one point or a period of time (such as a certain month or quarter) is analyzed. In this +setting, the data used for learning is unlabeled: +U = {x1, x2, . . . , xN} , +(6.1) +where xi, i = 1, 2, . . . , N, are the observations for the i-th oilfield. Thus unsupervised learning +is naturally applied. The model to be learned is in the form of a conditional probability +distribution Pθ(x | z) where z is some latent structure and θ represents the parameters. +This has many application scenarios in practice. When the latest month’s or quarter’s +data is available, the government of North Dakota might need distributional insights of +the population (of oilfields), preferably beyond some forms of the order statistics (such +as the five-number summary). This cross-sectional study is especially valuable and worth +conducting when a direct comparison with previous months/quarters (which can be either +the immediately previous one, or the same month/quarter in previous years) is desirable, +or deeper understanding of the population is needed, such as looking for potential clusters +among the entities. +83 + +6.2 +Probability Model Estimation +The task of learning distributions is a probability model estimation problem in unsu- +pervised settings (Li 2019). It sometimes takes the form of density estimation, which is +considered by some statisticians as the most fundamental topic in probabilistic machine +learning (Yu 2017). A basic and common technique, the histogram, can be easily misused +which leads to biased understanding of the dataset (Figure 6.1). +15 +10 +5 +0 +5 +10 +15 +x +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +p(x) +15 +10 +5 +0 +5 +10 +15 +x +0.00 +0.05 +0.10 +0.15 +0.20 +p(x) +Figure 6.1: Effective usage of histograms can be surprisingly subtle. With the exact same +dataset adapted from (VanderPlas et al. 2012), the two histograms with different bin sizes +demonstrate different multimodal features. Accepting some default configuration from some +software package yields only one view of the distribution. +In general, assuming that the data is generated by a probability model, the structure +and parameters of that model are learned from the data. The type of the structure, i.e., +the set of possible probability models is usually given (assumed), while the specifics of the +structure and the parameters have to be learned. The goal is to find the model structure and +the parameters which are most likely to have generated the data. +The probability model can be a mixture model or a graphical model. In this dissertation, +the mixture model is considered, where the assumption is that data comes from a mixture +of distributions. Mathematically, mixture models describe a distribution p(x) by a convex +combination of K base distributions: +84 + +p(x) = +K +� +k=1 +πkpk(x) +(6.2a) +K +� +k=1 +πk = 1, πk ≥ 0, +(6.2b) +where pk are the components in the mixture and πk are the mixture weights. Mixture models +can be interpreted as the overall population being a combination of distinct subpopulations. +Mixture models can be generalized to the continuous cases as well. For example, both the +negative binomial distribution and Student’s t-distribution can be thought of a mixture of +some continuous distributions (Martin 2018). +In the model representation Pθ(x | z), x stands for the observations which can be discrete +or continuous quantities; z represents the latent structure which is a discrete random variable. +The model is parameterized by θ. When the model is assumed to be a mixture type, z +represents the different components. The knowledge of the model structure and parameters are +learned from the data U = {x1, x2, . . . , xN}, where in this work xi ∈ X ⊆ R1, i = 1, 2, . . . , N, +is the observation for the i-th oilfield. +6.3 +Modeling VIIRS Detection Count +In Section 5.3.3, methods are developed for analyzing the time series of VIIRS detection +count for any given oilfield. This section tackles the problem of how to extract insights from +any given month’s flare detection count in North Dakota’s oilfields. Specifically, by learning +from each oilfield’s detection count, the population of the oilfields is summarized, through +which the state government can gain distributional insights. +Following the general form in Section 6.2, this problem becomes a special case that +the latent structure z does not exist, i.e., satisfying Pθ(x | z) = Pθ(x), where x represents +the detection count. It is when estimating conditional probability distributions becomes +estimating probability distributions, therefore, only estimating the parameters of Pθ(x) is +enough. Density estimation in classical statistics, for instance the Gaussian parameters +85 + +estimation, is an example of such scenarios. +Since the count data is modeled, the author compares the four observation models below +with many randomly chosen months’ data: +1. Poisson likelihood +2. Negative binomial likelihood +3. Zero-inflated Poisson (ZIP) likelihood +4. Zero-inflated negative binomial (ZINB) likelihood +Items 3 and 4 above are experimented with because many of the oilfields in North Dakota +did not have detection records from VIIRS for a given month. Therefore, zero-inflated models +are tried as well. Through the posterior predictive checks, it is found that the negative +binomial observation model fits data in the most compatible manner, which is employed in +this work. +The model is specified through Expressions 6.3a–6.3c: +µ ∼ Gamma(2, 1) +(6.3a) +φ ∼ Exponential(1) +(6.3b) +Ci ∼ NegBinomial(µ, φ) +(6.3c) +where Ci denotes the detection count for the i-th oilfield. The probability mass function of +the negative binomial likelihood is parameterized by a location parameter µ ∈ R>0, and an +overdispersion parameter φ ∈ R>0, in the following way: +P(X = n | µ, φ) = Γ(φ + n) +n! Γ(φ) +� +µ +µ + φ +�n � +φ +µ + φ +�φ +for n ∈ N0 , +(6.4) +where Γ(·) is the gamma function. Through this parameterization, the expectation and +variance of a random variable X ∼ P are: +E[X] = µ +and +V[X] = µ + µ2 +φ . +(6.5) +86 + +As the negative binomial distribution describes a Poisson random variable whose rate +parameter is gamma distributed, and due to the fact that Poisson(µ) has variance µ, the +learned parameters provide nice interpretations for the state government: +• µ indicates a mean intensity from the detection count’s perspective, just like the +interpretation of a Poisson’s rate parameter. The larger the value of µ, the more flare +detections are present on average at an oilfield level. +• φ indicates the heterogeneity among the oilfields in North Dakota. Specifically, µ2/φ is +the additional variance above that of a Poisson with rate µ. The smaller the value of φ, +the more oilfields with extreme detection counts (away from µ) are present. +To demonstrate this model’s compatibility with the observations, the data from October +2018 is used. There are 506 oilfields in total. The distribution of the detection count for all +the oilfields is illustrated in Figure 6.2. +0 +5 +10 +15 +20 +25 +Number of Detections in an Oilfield +0 +50 +100 +150 +200 +250 +300 +350 +Count +Figure 6.2: A histogram for the distribution of the oilfield detection counts from October +2018. There are lots of zeros (more than 350) and a few oilfields have relatively high detection +counts (e.g., greater than or equal to 20). +87 + +After fitting Model 6.3, the posterior distributions and trace plots of the hyperparameters +are presented in Figure 6.3. The parameter estimation results are reported in Table 6.1. +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +0 +1 +2 +3 +Posterior Density +0 +500 +1000 +1500 +2000 +2500 +1.0 +1.5 +Sample Value +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +0.22 +0.24 +0.26 +0 +5 +10 +15 +20 +Posterior Density +0 +500 +1000 +1500 +2000 +2500 +0.10 +0.15 +0.20 +0.25 +Sample Value +Figure 6.3: Posterior distributions and trace plots for the oilfield detection counts distribution, +fitted with the data from October 2018. Well mixing and convergence have been achieved. +Table 6.1: Parameter Estimates of Oilfield Detection Count Distribution +Parameter +Variable +Point Estimate +90 % CI +µ +Intensity +1.005 +(0.814, 1.200) +φ +Heterogeneity +0.168 +(0.135, 0.202) +The point estimate for the intensity parameter µ is relatively small (ˆµ ≈ 1), which possibly +results from the model being overwhelmed by the large number of zero counts. However, +by inspecting the histogram from Figure 6.2, the tail of the distribution definitely extends +far beyond ˆµ. Therefore, posterior predictive checks are performed to scrutinize Model 6.3’s +compatibility with the observations. +These types of checks substantially harness the information from the samples drawn from +the posterior distributions. By combining the uncertainty about the parameters, as described +by the posterior, with the uncertainty about the outcomes, as described by the likelihood, the +generative model is employed to simulate the implied observations. Subsequently, posterior +predictive plots are generated to display the model-based predictions along with the raw data. +Such a plot for the detection count distribution model is given in Figure 6.4. +88 + +0 +5 +10 +15 +20 +25 +Number of Detections in an Oilfield +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +Density +Posterior Predictive Simulation +Observed Data +Figure 6.4: Histograms for the distribution of the oilfield detection counts from October +2018. +Blue: original data observed from VIIRS. Gray: posterior predictive simulation +results obtained from Model 6.3. Each set of the simulation results is plotted using gray +with transparency via alpha blending (setting α = 0.15), such that the darker gray on the +histograms indicates the simulated data which is more aligned with the model’s expectation. +In Figure 6.4, the histograms for the original VIIRS observations, as well as all of the +posterior predictive simulations are displayed. Each set of the parameter values (of µ and φ) +are used in simulating one synthetic snapshot of the oilfields in North Dakota for October +2018, and there are in total 12,000 snapshots (constructed by the samples from the four +Markov chains, each of which was setup for 3000 sampling iterations). Every histogram is +visualized through an unfilled line chart, i.e., rendering the “step” histogram. +Through Figure 6.4, it appears that the model is very compatible with the observations +from October 2018, in that there is no obvious and consistent discrepancy between the +observed and simulated data. To delve into the tail behaviors, i.e., beyond the zero count, a +zoomed-in view is depicted in Figure 6.5. A few discrepancies are observed from this view, +for example, when the count Ci = 11 and Ci = 12. One thing to note is that, with such a low +mean (ˆµ ≈ 1), even with a relatively large overdispersion (ˆφ ≈ 0.2), the model would still be +surprised by the high detection count, e.g., when Ci ≥ 20. +89 + +0 +5 +10 +15 +20 +25 +Number of Detections in an Oilfield +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +Density +Posterior Predictive Simulation +Observed Data +Figure 6.5: Histograms for the distribution of the oilfield detection counts from October +2018, with the y-axis clipped to better present those counts which are greater than zero. The +legend with the associated color scheme is the same as in Figure 6.4. +The thorough performance of Model 6.3 that is characterized by a negative binomial +likelihood, and the complicatedness of the real data manifest themselves through the posterior +predictive checks. As discussed earlier in Section 6.3, the negative binomial likelihood was +compared with three other likelihoods (Poisson, ZIP and ZINB) on many randomly chosen +months, and found to outperform them in terms of the compatibility with the data in general. +In fact, there are some months’ data that are distributed in a “cleaner” way, i.e., almost +perfectly described by Model 6.3. The author chooses not to cherry-pick those data, in the +hope of not misleading the readers about the performance of the developed model. +Nevertheless, the simplicity, interpretability, and effectiveness of Model 6.3 proves itself in +the mission of modeling detection count distribution. In practice, the state government can +benefit from this model in the two use cases below: +1. When the latest month’s data becomes available, Model 6.3 can be fitted to obtain an +estimate for µ and φ. These parameter estimates along with the credible intervals can +be compared with those from the earlier times. In the case of the discussions above, the +90 + +learned parameters can be compared either with August/September from 2018, or with +October from 2016/2017. From the comparison, it provides insights into whether there +are more detection counts on average (characterized by a larger µ), or if more oilfields +with an atypical number of detections are spotted (characterized by a smaller φ). +2. After the model is fitted, it is recommended to perform the posterior predictive checks +as demonstrated in Figure 6.4 and Figure 6.5, to identify any issues of the fits. The list +of the oilfields which have large deviations from the simulated data, especially those on +the far tail (e.g., when Ci ≥ 20), are worth tracking. That is, to investigate whether +the “anomalies” from each month are random samples from the population or do not +change from month to month. This provides further understanding of how the oilfields +population behave, from the perspective of the detection count. +A distributional summary of the detection counts exhibits only one facet of the flaring +landscape, while the flared volumes distribution provides another crucial one, which is +discussed next. +6.4 +Modeling Flared Volume +In this section, the VIIRS estimated flared volumes for different oilfields are studied +from a distributional point of view. The dataset from a three-month period is analyzed +for demonstration purposes. Specifically, following the reverse geocoding as discussed in +Section 3.3, all the oilfields’ cumulative flared volumes during Q4 2018 are computed and +compiled for analysis. +There are in total 152 oilfields that have VIIRS reported volumes in this time span. The +data is highly skewed (Figure 6.6). Therefore, for each oilfield, the order of magnitude of +the flared volume (in bcm) is computed for the analysis, instead of working with the original +absolute volumes. +From an applied perspective, taking the log of a measure converts the measure into +magnitudes (McElreath 2015), which is applied to each oilfield’s flared volume: +91 + +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +Flared Volume (in bcm) of Different Oilfields in Q4 2018 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Count +Figure 6.6: Histogram for the distribution of the oilfield flared volumes from Q4 2018. Most +of the oilfields have relatively small flared volumes (e.g., less than 0.01 bcm), while a few +oilfields have volumes that are greater than 0.1 bcm. +Li = log(Fi), +(6.6) +where Fi is the original flared volume in bcm, and Li is the flared volume magnitude, both of +which are for the i-th oilfield. In this dissertation, base e is always used for the logarithm (i.e., +natural logarithm). A univariate distribution of the magnitudes is visualized in Figure 6.7. +Among the three approaches used to visualize the distribution, only the rug plot does +not lead to subtleties due to the hyperparameters used. However, as a 1D scatter plot, +its representation ability is naturally limited. The histogram suffers from the problem as +illustrated in Figure 6.1. The curve is generated by kernel density estimation (KDE). For a +given dataset as defined in Equation 6.1, KDE represents the underlying distribution as: +ˆp(x) = +1 +Nh +N +� +i=1 +K +�x − xi +h +� +, +(6.7) +where K(·) is a kernel function and h is a bandwidth parameter. To generate Figure 6.7, the +Gaussian kernel is used, which is given by: +92 + +12 +10 +8 +6 +4 +2 +0 +log_bcm of Different Oilfields in Q4 2018 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +Density +Figure 6.7: Distribution of the oilfield flared volume magnitudes from Q4 2018. The rug +plot marks the value for each oilfield. The histogram is generated with nine bins. The curve +displays a Gaussian kernel density estimate. +K(z) = +1 +√ +2π exp +� +−z2 +2 +� +, +(6.8) +and h is chosen based on Scott’s rule. +Since the bandwidth plays a similar role as the bin size in histograms, KDE can also +lead to the same issue as in histograms. Nevertheless, all three (the rug plot, histogram and +KDE) agree that a single Gaussian approximation of the density which generates this data +would be a poor approximation. Therefore, Gaussian mixture model (GMM) is employed +to represent the data, i.e. the base distributions in Model 6.2 are chosen to be Gaussians. +GMM provides more expressive modeling capabilities and also possibilities for clustering. +6.4.1 +Model Specification +As discussed earlier, since the flared volume is a continuous quantity, density estimation is +applicable and tackled with GMM. At first, the data generating process is considered, which +paves the way for potential clustering applications. That is, each data point Li (defined in +93 + +Equation 6.6) is assumed to be generated by exactly one mixture component. The number +of components, K, is unknown, and up to seven components are tried to fit the dataset +visualized in Figure 6.7. A relatively small number of components are experimented, because +as the number of components increases, it becomes more difficult to interpret the modeling +results. The model is specified through Expressions 6.9a–6.9i, ∀K ∈ {2, . . . , 7}: +α = (α1, . . . , αK) = 6 · 1K +(6.9a) +p ∼ Dirichlet(α) +(6.9b) +zi ∼ Categorical(p) +(6.9c) +l1 = min{L1, . . . , Ln} +(6.9d) +l2 = max{L1, . . . , Ln} +(6.9e) +�µk = l1 + (k − 1) +�l2 − l1 +K − 1 +� +, +k = 1, . . . , K +(6.9f) +µk ∼ N(�µk, 2), +k = 1, . . . , K +(6.9g) +σk ∼ Half-Normal(2), +k = 1, . . . , K +(6.9h) +Li | (zi = j) ∼ N(µj, σj) +j ∈ {1, . . . , K} +(6.9i) +where: +α is the vector of concentration parameters for the Dirichlet distribution, which is a +multivariate generalization of the beta distribution; +p is the simplex of probabilities for the mixture components, which is assigned +a Dirichlet prior. This prior with each value inside α being 6, is a weakly +informative prior, expecting any pk inside p could be bigger or smaller than +the others. Ten random draws from Dirichlet([6, 6, 6, 6, 6, 6, 6]) are displayed in +Figure 6.8; +zi is the probable mixture component that the i-th oilfield belongs to; +l1 and l2 are the lower and upper bound for {Li}n +i=1, respectively; +�µk is used in “initializing” the location of the k-th mixture component, and {�µk}K +k=1 +essentially represent the K evenly spaced points between [l1, l2]; +94 + +µk is the mean for the k-th Gaussian component; +σk is the standard deviation for the k-th Gaussian component; +Li is the flared volume magnitude of the i-th oilfield, which is generated by the +mixture component zi. +1 +2 +3 +4 +5 +6 +7 +Index +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Probability +Figure 6.8: Ten random draws from a Dirichlet prior with α = (6, 6, 6, 6, 6, 6, 6). One draw is +highlighted to show that this prior is weak, in that it does not force all the probabilities (in +any single draw) to be equal. +Model 6.9, while unambiguously expressing the assumed generative process, relies on +sampling the discrete latent variables zn, which is controlled by a categorical mixing dis- +tribution. This reliance causes slow mixing and ineffective exploration of the posterior +distribution. An equivalent parameterization which addresses these problems is to marginal- +ize out the z parameter. The marginalized model is specified through Expressions 6.10a–6.10h, +∀K ∈ {2, . . . , 7}: +α = (α1, . . . , αK) = 6 · 1K +(6.10a) +w ∼ Dirichlet(α) +(6.10b) +l1 = min{L1, . . . , Ln} +(6.10c) +l2 = max{L1, . . . , Ln} +(6.10d) +�µk = l1 + (k − 1) +�l2 − l1 +K − 1 +� +, +k = 1, . . . , K +(6.10e) +95 + +µk ∼ N(�µk, 2), +k = 1, . . . , K +(6.10f) +σk ∼ Half-Normal(2), +k = 1, . . . , K +(6.10g) +Li ∼ +K +� +j=1 +wj N(µj, σj) +(6.10h) +where w are the mixture weights (i.e., mixing proportions), and the rest of the symbols have +the same meaning as in Model 6.9. The likelihood function, defined in Expression 6.10h, +corresponds with the density of a mixture model expressed in its general form (Equation 6.2a). +Model 6.10 is implemented and fitted six times (∀K ∈ {2, . . . , 7}) to compare the inference +results with different number of components specified. For each K, rapid mixing and fast +convergence of the Markov chains are obtained. The modeling results are displayed in +Figure 6.9, where the KDE (same as in Figure 6.7) and the Gaussian components inferred +are plotted along with the posterior samples. +It can be observed that, when using a mixture of Gaussians, the multimodal features +can be represented in a relative effortlessly way, and all the mean fits are quite close to +the one obtained with KDE. As the number of components increases, for example when +K = 6 or K = 7, the mean density estimation using GMM resembles KDE more closely, but +the samples from the posterior show more stochasticity, which is an indicator of potential +overfitting. This naturally leads to the question of how to decide the number of components +for this dataset. +6.4.2 +Model Comparison +Choosing the best K is a model comparison problem, for which there does not exist a +silver bullet. In this dissertation, the author chooses to take the information criteria approach, +specifically leveraging the widely applicable information criterion (WAIC) introduced by +Watanabe (2010). Information criteria provide a theoretical estimate of the relative out-of- +sample KL divergence (McElreath 2020), and thus a lower value is better. Following Martin +(2018) and McElreath (2020), WAIC is computed by: +96 + +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Density +K = 2 +K = 3 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Density +K = 4 +K = 5 +−12 +−10 +−8 +−6 +−4 +−2 +0 +log bcm of Different Oilfields in Q4 2018 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Density +K = 6 +−12 +−10 +−8 +−6 +−4 +−2 +0 +log bcm of Different Oilfields in Q4 2018 +K = 7 +Figure 6.9: GMM inference results with different K’s. The thick blue line denotes the +posterior mean fit of the underlying density. The light blue lines show 50 random samples +from the posterior. The dashed lines represent the posterior mean Gaussian components. +The red curve shows the fit using KDE. +WAIC(y, Θ) = −2 × lppd(y, Θ) + 2pwaic +(6.11a) += −2 +n +� +i=1 +log +� +1 +S +S +� +j=1 +p(yi | Θj) +� ++ 2 +n +� +i=1 +VΘ[log p(yi | Θj)] , +(6.11b) +where: +y denotes the observations and yi is the i-th observation; +Θ is the posterior distribution and Θj is the j-th set of sampled parameter values; +97 + +S is the number of posterior samples; +lppd(·) calculates the log pointwise predictive density; +pwaic is the penalty term given by summing up the variance in the log-likelihood over +the S posterior samples, for each observation i. +Fundamentally, model comparison is performed by leveraging Occam’s razor, i.e., parsi- +monious models are preferred in light of predictive performance. The models are compared +based on their WAIC values, which are summarized using Figure 6.10. +630 +635 +640 +645 +650 +655 +WAIC +2 +3 +4 +5 +6 +7 +Number of Components +Figure 6.10: WAIC values with different K’s. The open points denote the WAIC values. The +long horizontal line segments represent the standard error for each WAIC. Standard error of +the difference in WAIC (between each model and the top-ranked one) is shown by the lighter +line segment with the triangle on it. +It can be seen that the model with two Gaussian components are the best (smallest +WAIC), however, there are considerable overlaps among all of the models when the estimated +standard error is taken into consideration. Considering the fact that K = 2 gives the simplest +model, also that there are only 152 observations (oilfields) in this dataset, the GMM with +two components would be the best choice. +98 + +6.4.3 +Clustering +When looking at the developed model from a latent variable perspective (Model 6.9), it +becomes obvious that the mixture model serves as a natural candidate for solving clustering +tasks, in that every observation (Li) can be drawn from one of the K data generating +processes, each with its own set of parameters, N(Li | µk, σk). Since a probabilistic model +is built, for the purpose of clustering, a reasonable choice is to assign a data point to the +mixture component (i.e., cluster) with the highest posterior probabilities (which are also +interpreted as the responsibilities). In the case of the 2-component GMM trained from the +previous sections, for a particular observation x, the probability that it belongs to cluster +one (z = 1) can be computed using Bayes’ theorem (Equation 2.3a): +p(z = 1 | x) = +p(z = 1) N(x | µ1, σ1) +p(z = 1) N(x | µ1, σ1) + p(z = 2) N(x | µ2, σ2) , +(6.12) +where every part in the formula can be obtained from the posterior samples (e.g., using the +posterior means). +Clustering, as an unsupervised approach, can be used to reveal the hidden groups in +the observations. In the case of the oilfield flaring magnitudes data in this chapter, the two +clusters can be directly mapped to concepts such as major and minor flaring fields. However, +it is usually the deeper insights into what caused these clusters that the state government +is mostly interested in, for the sake of decision- and policy-making for example. If the +oilfields belonging to the major flaring cluster seem to be a volatile membership when more +months/quarters data are analyzed, the variations in flared volumes are possibly tied more +closely to company strategies and movements. On the other hand, if there exists a group of +oilfields that are found to join the major flaring cluster on a regular basis, this could provide +a perspective in regards to where to construct the next natural gas processing plants, i.e., +the locations/capacities of the new gas plants should be optimized based on those oilfields’ +situations. +99 + +In this chapter, the dataset compiled for unsupervised learning is univariate, i.e., xi ∈ +X ⊆ R1. GMM are also suitable for the density estimation and clustering tasks when the +data goes beyond 1D. As an example, for the same oilfields studied for Q4 2018, if their oil +production volumes are extracted from NDIC, a scatterplot of gas flaring versus oil production +magnitudes is shown in Figure 6.11. It is very possible that the density of the underlying +distribution can be modeled by a bivariate normal distribution or a 2D GMM. In such +cases, the mixture components become multivariate normal distributions, and the component +covariance matrices can be constructed with the help of the LKJ distribution (which is +employed in Models 4.5 and 4.7). The developed density model can be used, for example, +in anomaly detections, looking for any oilfields which have a tendency to creep toward the +upper left corner (characterized by very little oil production and a huge flaring magnitude). +Similar to all the inferences presented throughout this dissertation, one advantage of doing +such is that the decision making can be based on some consistent metrics (such as probability +scores), instead of some criteria based on human eyeballing or improvising. +102 +103 +104 +105 +106 +107 +NDIC Reported Oil Production (bbl) +10−4 +10−3 +10−2 +10−1 +VIIRS Reported Flared Volume (bcm) +Figure 6.11: A scatterplot of oil production and flared gas volumes for different oilfields +in Q4 2018. Both the x- and y-axis are in log scale, showing the relationship between the +magnitudes. +100 + +This concludes the statistical modeling journey of this dissertation. In the next chapter, +discussions are presented on one extension scenario and one bigger picture viewpoint, from +applying Bayesian learning to flaring data. +101 + +CHAPTER 7 +DISCUSSION +This chapter discusses the possibility of operator level monitoring and analytics, potential +result inconsistencies, and relates the endeavors of learning from flaring data to the larger +process of applying machine learning in the petroleum engineering domain. +7.1 +Operator Level Monitoring and Analytics +Up till this point, the satellite-detected flaring statistics have been applied to the state, +county, and oilfield levels. This is made possible by the reverse geocoding discussed in +Section 3.3. An ideal application scenario is operator level monitoring and analytics by +leveraging the information from the satellite detections. +Unfortunately, assigning flares to corresponding companies is not a straightforward +operation. One possible solution is to make use of the shapefiles of the leases, which are +not provided by NDIC. Some data vendors have such files in their database. However, after +spending some effort investigating the lease shapefiles from one vendor, the author believes it +is possible to create more problems than solving the existing ones, when bringing in such +information. In particular, some reasons include: +• Multiple companies exist on a single lease. +• The company names from the lease shapefiles do not always correspond with those on +the NDIC monthly production reports. +• Some leases in the vendor’s database miss start date or end date data. +• It takes time for the vendor to compile and digitize such information, which makes the +available lease shapefiles not up to date. +102 + +Nevertheless, for such an important use case, the author managed to develop a nearest- +neighbor-based approach which partly solves the problem (Algorithm 7.1). The essence of +this approach is to cautiously assign the closest well’s operator to each satellite-detected +flare. The closest wells are found based on the corresponding time window. For example, for +the flares detected in January 2016, only the active wells reported on the NDIC production +report from the same month are looked up. The function FindClosestOperator() returns +the closest operator (OPj) for each VIIRS detection, as well as the calculated distance (dj) +between each pair (of flare and well). The distance is calculated based on the haversine metric, +i.e., the great-circle distance, thus the Earth radius (RE) is needed. The function is essentially +performing the k-nearest-neighbors (k-NN) search for k = 1. When the sample is as large as +in this case, i.e., there are usually a few hundred VIIRS detections and more than 15,000 +wells for each month, linear scanning each well’s location for each VIIRS detection is too +slow. Therefore, in this work, the function internally depends on a ball tree implementation +from scikit-learn (Pedregosa et al. 2011) for speedup on the k-NN search. +Once the 2-tuple, (OPj, dj), is obtained for each VIIRS detection, some logics are imple- +mented to decide whether to drop or keep the operator assignment. The idea is straightforward: +the assignment is immediately kept or discarded, when dj is very small or very large, respec- +tively. If dj is mid-range, i.e., dsecure ≤ dj ≤ dcutoff, the assignment will be in effect, only if +the flare and the operator are found to be located on the same township/range/section. The +township/range/section shapefiles, as part of the input for Algorithm 7.1, are available from +the NDIC GIS Map Server. The reverse geocoding follows the exact same procedure as in +Section 3.3. After the processing is completed, a small portion of the VIIRS detections are +not used for operator level analytics, because either they are too far away from the reported +well locations, or the townships/ranges/sections fail to match. It should be noted that, the +pseudocode for Algorithm 7.1 is written in a way that illustrates the precise details in the data +processing logics. For the implementation in this work, some of the for-loops are replaced by +the vectorized operations for enhanced performance. +103 + +Algorithm 7.1: Nearest-Neighbor-Based Flare Owner Assignment +Input: both VIIRS and NDIC reportings in WGS 84 coordinates, the +township/range/section shapefiles for North Dakota, dsecure, dcutoff, RE +Output: operators being assigned to most VIIRS detections +1 n ← number of months +2 for i ← 1 to n do +3 +VIIRSi ← the i-th month’s observations from VIIRS +4 +NDICi ← the i-th month’s reportings from NDIC +5 +(OP, d) ← FindClosestOperator(VIIRSi, NDICi, RE) +6 +m ← number of records in OP or d +7 +for j ← 1 to m do +8 +OPj ← the closest operator found on the j-th record +9 +dj ← the distance between the flare and the closest well, for the j-th record +10 +if dj > dcutoff then +11 +drop OPj +12 +else if dj < dsecure then +13 +keep OPj +14 +else +15 +if township/range/section agree then +16 +keep OPj +17 +else +18 +drop OPj +19 +end +20 +end +21 +end +22 end +The developed approach is tested with real flaring data from North Dakota. For the +demonstrated cases in this section, the values below are chosen for Algorithm 7.1: +dsecure = 300 m +(7.1a) +dcutoff = 800 m +(7.1b) +RE = 6371 km +(7.1c) +Some operators are found to show positive correlations between the NDIC and VIIRS +reported volumes. Examples of two operators, denoted by Operator B and Operator C, are +shown in Figure 7.1. The axes’ meanings are the same as in the right panel of Figure 3.7. The +104 + +legend shows the results of fitting Equation 3.2d by ordinary least squares (OLS). R2 +adj stands +for the adjusted R2. Although the differences in ˆβoperator indicate that there is heterogeneity +among the different companies, these operators show some consistency in terms of their own +reporting and have good matches with the VIIRS data up to a scale factor (as the intercepts +are very close to zero). +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +VIIRS Monthly Data (bcm) +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +NDIC Monthly Data (bcm) +NDICi = 0.000 + 0.378 × VIIRSi +R2 +Adj = 0.959 +45-degree line +(a) Operator B +0.00 +0.01 +0.02 +0.03 +0.04 +VIIRS Monthly Data (bcm) +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +0.040 +NDIC Monthly Data (bcm) +NDICi = 0.001 + 0.658 × VIIRSi +R2 +Adj = 0.861 +45-degree line +(b) Operator C +Figure 7.1: Examples of good fits between the NDIC and VIIRS reported volumes, at the +operator level. +However, some operators (e.g., Operator D and Operator E) show discrepancies between +their reportings and the satellite-detected flaring statistics, which are manifested through the +poor fits (Figure 7.2). Certainly, a poor fit with the linear model does not indicate much on +its own. Nonetheless, there exists a pattern in both scatterplots that, some points seem to +be “pushed down” towards the x-axis. If the time series of these two operators are drawn, +it shows that this behavior is due to company-reported volumes leveling off for a certain +period of time (Figure 7.3). The VIIRS curves in the time series imply that there were flaring +intensity variations for those times. This workflow, driven by Algorithm 7.1, is capable of +raising a flag when it comes across datasets like these, and can serve as a powerful monitoring +105 + +and analytics tool, however, strong cautions need to be applied. +0.00 +0.02 +0.04 +0.06 +0.08 +VIIRS Monthly Data (bcm) +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +NDIC Monthly Data (bcm) +NDICi = 0.002 + 0.207 × VIIRSi +R2 +Adj = 0.180 +45-degree line +(a) Operator D +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +VIIRS Monthly Data (bcm) +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +NDIC Monthly Data (bcm) +NDICi = 0.000 + 0.171 × VIIRSi +R2 +Adj = 0.256 +45-degree line +(b) Operator E +Figure 7.2: Examples of poor fits between the NDIC and VIIRS reported volumes, at the +operator level. +The introduced approach, although it looks promising, is by no means a one-stop solution +and has the potential for being misapplied. First, there is the possibility of misassigning the +satellite-detected flares to the operators. Whenever the concern is raised, further investigations +can be conducted by looking into the detection maps as well as the satellite imagery of +the operators’ production sites. In addition, this method is more effective for the relatively +large producing/flaring operators, because when a company conducts very little flaring, the +truncation effects discussed for the peak in Figure 5.20 are magnified. +7.2 +Warnings Regarding Inconsistencies +Given the resolution of the satellite imagery, assigning specific flaring volumes to a given +operator is fraught with challenges. Although the VIIRS processing workflow is capable of +picking up flares with areas around 1 m2 (Figure 3.2(a)), the pixel footprint is much larger +(Table 2.1). Since the latitude and longitude of the pixel center is stored for each individual +VIIRS observation (Elvidge et al. 2015), when multiple operators have sub-pixel combustion +106 + +The company-reported volumes +leveled off at small values. +(a) Operator D +The company-reported volumes +leveled off at small values. +The company +reported zero flaring. +(b) Operator E +Figure 7.3: Time series of the two example operators whose reporting did not quite align +with the VIIRS detected trends/patterns. The points or periods in time for which the +company-reported data were significantly different from the satellite detections are annotated. +sources, it makes flare owner assignment extremely challenging. In such situations, conclusions +reached by merely benchmarking company reporting against VIIRS reporting would likely +be inaccurate. In fact, in the realm of NDIC reporting, warnings must be issued regarding +any inconsistencies in those results, with considerations from three aspects. First, the report +from the U.S. Department of Energy (2019a) presents data supporting that North Dakota +107 + +VIIRS +0.08 +NDIC +0.07 +0.06 +bcm +0.05 +Monthly +0.04 +0.03 +0.02 +0.01 +0.00 +2015-07 +2016-01 +2016-07 +2017-01 +2017-07 +2018-01 +2018-07 +2019-01VIIRS +NDIC +0.008 +bcm +0.006 +Monthly +0.004 +0.002 +0.000 +2015-07 +2016-01 +2016-07 +2017-01 +2017-07 +2018-01 +2018-07 +2019-01shows closer agreement between the NOAA estimations and state reportings (of flared gas +volumes), when compared with Texas and New Mexico. Second, flaring is preferred over +venting because methane (the main component of natural gas) is more potent than carbon +dioxide which is the main product of flaring (EIA 2019b). Since North Dakota bans venting, +the massive flaring magnitude indicates that the direct release of gas into the atmosphere is +minimized. Third, estimation of flaring volumes is inherently a difficult task. When it is not +practicable to meter the flared gas, the Canadian Association of Petroleum Producers (2002) +gives guidelines on available volume estimation methods. Every category of methods, no +matter using rules of thumb, or experimentally determined correlations, or process simulators, +has its own limitations and accuracy issues. Considering the fact that the VIIRS volumes +used in this work were largely calibrated using the Cedigaz reported data (Section 2.1), which +has its own error bars (Elvidge et al. 2015), the difference between company reporting and +VIIRS reporting is inconclusive and unsurprising, especially when the standard error of the +difference is larger than the difference itself. +By inspecting a more comprehensive profile of time series, both Operator D and Operator E +from the previous section are self-consistent in their reportings to the NDIC. Their time +series are displayed in Figure 7.4 and Figure 7.5, respectively. The variables and associated +labels (shown in the legends) follow the same definitions from Section 3.4. The units for +all the variables are given in Table 7.1. Clearly, the reported flared volumes show good +correspondence with the gas production and GOR profiles. Some rapid variations in their +flared volumes match the fluctuations in the gas prices, i.e., when the gas price drops, the +operators tend to flare more, whereas when the gas price reaches peak, there is little flaring. +In summary, to nail down the decisions and conclusions with regard to operator reporting +quality, better resolution satellite data and a more comprehensive review of the time series +profiles are required. +108 + +0.5 +1.0 +×106 +NDIC flared vol +20 +40 +60 +WTI oil price +2 +3 +4 +Henry Hub gas price +2 +3 +×106 +NDIC oil prod +2 +4 +6 +×106 +NDIC gas prod +200 +400 +NDIC flaring well count +2016 +2017 +2018 +2019 +2020 +1.5 +2.0 +NDIC GOR +Figure 7.4: A more comprehensive time series plot for Operator D. The increase in the +reported flared volume in early 2019 corresponds to the gas price declining in the same period. +109 + +0 +50000 +NDIC flared vol +20 +40 +60 +WTI oil price +2 +3 +4 +Henry Hub gas price +200000 +300000 +NDIC oil prod +0 +50000 +100000 +NDIC gas prod +0 +100 +NDIC flaring well count +2016 +2017 +2018 +2019 +2020 +0.0 +0.2 +0.4 +NDIC GOR +Figure 7.5: A more comprehensive time series plot for Operator E. The sudden drop in the +reported flared volume in late 2018 corresponds to the halted gas production. +110 + +Table 7.1: Units for Operator Time Series in Figures 7.4 and 7.5 +Variable +Unit +NDIC flared vol +Mcf +WTI oil price +$/bbl +Henry Hub gas price +$/MMBtu +NDIC oil prod +bbl +NDIC gas prod +Mcf +NDIC flaring well count +1 +NDIC GOR +Mcf/bbl +7.3 +Caveats in Petroleum Data Analytics +As a petroleum engineer, the author is thrilled to witness the oil and gas industry and +academia are embracing data-driven mindsets and solutions, while being part of it through +writing this dissertation. However, there are certainly areas that could be continuously +improved, and this section provides a discussion on one of those. That is, extending a +cautious welcome to some black box models. +The pervasive influence of some black box models in the recent years can be seen by +performing a rough search on OnePetro (Table 7.2). One thing to note is that, from an +algorithmic point of view, these methods are rather “glass boxes” as opposed to “black boxes”, +i.e., everything under the hood in terms of implementation is well understood. For example, +backpropagation, which is the core of neural network training, is based on the chain rule. +However, for a given task, the learned parameters inside the network provide little or no +insights for the problem domain. Therefore, it is considered a black box. +The wide adoption of such models is largely due to the availability of the open source +libraries, for example in the Python ecosystem, construction and training of neural networks +become much simpler thanks to TensorFlow and PyTorch, and gradient boosting models can +be built within a few lines of code with the help of XGBoost, LightGBM, or CatBoost. In +other words, with the mathematical details of those statistical routines abstracted away, for +a practitioner, implementing those models is almost as easy as pushing a Learning button on +111 + +Table 7.2: Publication Count Rise on OnePetro +Exact Phrase +Searched +Year Method +Introduced +Publication Count +2010–2014 +2015–2019 +neural network +1958† +843 +2044 +gradient boosting +2001‡ +1 +110 +random forest +2001§ +9 +245 +† Based on (Rosenblatt 1958) +‡ Based on (Friedman 2001) +§ Based on (Breiman 2001) +a GUI. +Unfortunately, easiness in the implementation does not imply appropriateness for the +problem. In particular, those black box models face the challenges below: +1. How to incorporate domain expertise. +A lot of the black box models in the frequentist framework make the assumption that +the observations are conditionally i.i.d. The hope is that by feeding a huge number of +i.i.d. samples to a universal approximator, such as a neural network, some function for +prediction can be optimized with a certain accuracy. For some applications, the domain +expertise is often encoded in the feature selection process. For example, to train a +model to predict oil production, the analyst might choose some completion parameters +other than the API well number or well name, as input features. +However, in the author’s opinion, this way of incorporating domain expertise is still +a shallow one, which is far from what the oil and gas industry have accumulated in +many decades. For example, the phenomena of well interference through fracture hits +leave the assumption of some neighboring wells being i.i.d. in an unfavorable position. +Another example would be, when looking at a populations of wells from one basin +that are completed by N oilfield service companies, domain expertise might indicate +that, each company deserves its own model while each company is not completely +112 + +independent from others in terms of the completion technologies, etc. In this situation, +the hierarchical model employed in Chapter 4 might be a better choice, in which case a +lot of the prior knowledge about the different service companies can be incorporated +into the population model. +2. How to interpret the results. +As discussed earlier, the black box models suffer from the interpretability issues. Using +the shale gas wells example from Item 1 above, if a black box model is trained, it is +impossible (at this point) to attribute the failure in capturing the well interference effects +to a certain part of the neural network, or to a certain portion of the decision trees (in +the case of gradient boosted trees or random forest). Rudin (2019) asserted that people +should “stop explaining black box machine learning models” and use interpretable +models for high-stakes decisions. In the petroleum industry, there are a number of +high-stakes decision scenarios, such as real-time well integrity anomaly detection and +production forecasting in a high well cost context. Blindly applying black box models +to those scenarios might involve serious losses. In terms of providing interpretability, +the Bayesian approach employed throughout this dissertation is much more effective. +Each and every assumption is expressed in the generative model through either the +priors or the likelihood. +3. How to quantify the uncertainties, especially in the context of risk management and +decision making. +Along the lines of Item 2 above, error bars are vital, especially in high-stakes prediction +applications. In the case of predicting oil production using a trained data-driven model, +point prediction results such as 1000 bbl/day are not really insightful. In fact, if the +95 % prediction interval (PI) is 1000 ± 50 bbl/day, that point prediction becomes more +informative. However, if the 95 % PI is 1000 ± 1500 bbl/day, that same point prediction +is unhelpful or misleading. What shall be reported instead is either the considered +113 + +model yields much uncertainty in this given task, or there is possibility that the entity +will not produce anything at all. +It should be noted that, the ‘95’ in the CI/PI is not a “magic number”. A state govern- +ment or an oil company might want to make decisions based on 73 % or 99.6 % confidence, +or any other arbitrary choices. What really matters is the necessity of a principled +way to quantify the uncertainties in machine learning-based estimations/predictions, +such that any intervals can be computed. As presented throughout this dissertation, +the Bayesian approach provides full capacity and flexibility is this regard. In fact, for +parameter estimates, the author chooses to give 90 % CI instead of the “conventional” +95 %, to emphasize that this should be a domain’s consideration rather than a statistical +one. +A lot of the black box models in the frequentist framework, however, fall short of this +requirement. Maximum likelihood estimation (MLE), which is fundamentally relied +upon by some frequentist learning methods, enjoys really nice properties and is capable +of quantifying uncertainties, but only when a massive amount of data is at hand such +that the asymptotic properties could take effect. Unfortunately, that is not the case in +many scenarios for the petroleum engineering domain, which is discussed next. +4. How to mitigate overfitting when the data is not “big”. +Two aspects are worth discussing here. For one thing, the big data is not everywhere. +Indeed, the author believes that the claim of Gelman (2015) that, “sample sizes are +never large”, applies to a lot of problems in the petroleum industry. The reason is that, +if the data were large, the analyst would already be on to the next problem for which +more data is needed. For example, a sample of 500 producing wells in the Bakken +Formation could make some general study possible. When the analyst has access to a +dataset of more than 15,000 wells, some granular insights are desirable. Especially, if +partial pooling is needed among the different service companies/operators, different +114 + +members of the formation, or different completion technologies, data for some units of +the population could be very small (which happens for the analysis in Chapter 4). +On the other hand, the sample size should be inspected in the light of model complexity. +The number of parameters provides one measure of such. For example, consider a +hypothetical classification problem, whose goal is to determine if a given well will deliver +good or average or poor production performance. Ten completion parameters (features) +are available to train the multilayer perceptron illustrated in Figure 7.6. +x1 +x2 +x10 +Input layer: +ten completion +parameters +h(1) +1 +h(1) +2 +h(1) +3 +h(1) +20 +Hidden +layer 1 +h(2) +1 +h(2) +2 +h(2) +10 +Hidden +layer 2 +ˆy1 +ˆy2 +ˆy3 +Output layer: +good/average/poor +production +performance +... +... +... +Figure 7.6: A neural network designed for the hypothetical well performance classification +problem. The input layer has 10 neurons for the completion parameters. The first and second +hidden layer has 20 and 10 neurons, respectively. The output layer has three neurons for +multiclass classification. +In this (small) neural network, the number of parameters np is given by: +np = 11 × 20 + 21 × 10 + 11 × 3 = 463, +(7.2) +when considering a single bias node for every layer except the last one. To train this +model, a dataset of 500 wells would definitely be a small sample. There is still possibility +to train such a model with a small sample, however, great efforts in regularization +have to be made, in the hope that the neural network will learn something that can be +generalized, instead of merely memorizing the observed samples (i.e., overfitting). +115 + +By utilizing the regularizing priors, the Bayesian approach’s built-in Occam’s razor +greatly mitigate the risk of overfitting. In particular, Bayesian nonparametric models, +such as the Gaussian processes employed in Chapter 5, are very attractive in a sense +that the sizes of models are allowed to grow with the size of data (Orbanz and Teh +2010). This makes the developed model flexible while being robust to overfitting. +Although the Bayesian learning models (such as the ones developed in this work) have +outstanding merits and deserve wider utilization in petroleum data analytics, they are not +cure-alls. Recently researchers have started to stress the necessity of bespoke statistical +models (Andorra 2020; McElreath 2020). The argument is that, off-the-shelf models, no +matter neural networks or generalized linear models, interrupt the incorporation of domain +expertise. This is especially relevant in the field of petroleum engineering. For instance, +when conducting data-driven analysis for hydraulic fracturing performance, it makes sense +to bring in the fracture propagation models to the machine learning workflow. That way, +statistical models are motivated by the physically informed models. The Bayesian framework, +as employed throughout this dissertation, readily embraces this strategy, in that the domain +knowledge, which is represented by differential equations for example, can be inserted into +the generative model. One advantage is that a lot of the parameters will have direct scientific +meanings, and more informative priors can be placed based on scientific constraints, field +experience, etc. The final outcome should be better inferences and predictions. +116 + +CHAPTER 8 +CONCLUSIONS AND RECOMMENDATIONS +In this dissertation, the effectiveness of a full Bayesian approach has been observed in +learning models from natural gas flaring data. The author hopes this work contributes to the +understanding of the options and considerations when applying data-driven approaches to +gas flaring. In closing, this chapter presents the major conclusions and recommendations for +future work. +8.1 +Conclusions +The major conclusions are: +1. Bayesian learning implemented using Hamiltonian Monte Carlo can be effectively +applied to real problems in gas flaring analytics, in both supervised and unsupervised +settings. The advantages of the Bayesian approach indicate it deserves wider usage in +the petroleum engineering domain in general; these advantages are listed below: +(a) Petrotechnical domain expertise can be incorporated in a principled way. +(b) Model interpretability is drastically improved, facilitating communications with +petroleum engineers. +(c) Quantification of uncertainty leads to more robust decision making, which is +important for oil exploration and production companies. +(d) The built-in Occam’s razor makes the model less prone to overfitting, in the +context of noisy field measurements. +2. The development of a suite of models (Table 8.1), with both parametric and nonpara- +metric techniques, provides guidance on how insights can be extracted from various +117 + +angles. The presented models are designed and tested to be able to generalize to +different entities at various levels. +3. To investigate the heterogeneity among the different entities (such as counties or +oilfields), partial pooling is recommended, because some entities have very little data. +4. Gaussian processes demonstrate very attractive traits in revealing the patterns and +trends from flaring time series. A set of priors with the Mat´ern 5/2 kernel works very +well across different modeling goals, observation models, and data sources. +5. From a distributional point of view, the negative binomial and Gaussian mixture models +are good representations of the oilfield flare counts and flared volumes, respectively. +The learned parameters and structures are very interpretable. Hidden clusters are found +by fitting Gaussian mixture models. +6. A nearest-neighbor-based approach for operator level monitoring and analytics is +introduced. Its performance is tested on real data and defendable results are obtained. +However, better resolution satellite data is needed for the scenario of multiple operators’ +wells being very close to each other. +7. All the dissertation objectives (Section 1.2) have been achieved. In particular, the flared +volumes missed from VIIRS for the state and each county are estimated via fitting the +intercept parameter and reported in Table 3.1 and Table 4.2. The nighttime combustion +source detection limits of Landsat 8, without being corrected for artifacts due to +glow, are determined and reported in Figure 3.2(b). Correlations between financial +factors, production performance, and flared volumes at a state level are computed using +Spearman’s ρ and reported in Figure 3.5 and Figure 3.6 for the original data and lag-1 +differences, respectively. Most pairs of the variables do not show strong correlations on +the lag-1 differences. Robust Gaussian process modeling serves as a generic framework +for addressing the rest of the objectives, including demonstrating operator approaches, +118 + +evaluating if the goals of the North Dakota regulatory policy (Order 24665) have been +achieved, and predicting NDIC flared volumes. +Table 8.1: Models Developed in this Dissertation +Numbering +Target of Modeling +Page +Model 3.2 +Associations between VIIRS and NDIC at a state level +27 +Model 4.5 +Associations between VIIRS and NDIC at a county level (centered) +38 +Model 4.7 +Associations between VIIRS and NDIC at a county level (noncentered) +41 +Model 5.12 +Proportion of gas production being flared as time series +57 +Model 5.14 +Proportion of wells that conduct flaring as time series +63 +Model 5.15 +VIIRS detection count as time series +66 +Model 5.17 +Proportion of oil being flared as time series +70 +Model 5.18 +Scale factor between VIIRS and NDIC as time series +74 +Model 6.3 +VIIRS detection count distribution for oilfields +86 +Model 6.9 +VIIRS volume distribution for oilfields (latent discrete parameterization) +94 +Model 6.10 +VIIRS volume distribution for oilfields (marginalized) +95 +8.2 +Future Work +A number of areas for future research include: +1. L8 processing workflow. +The studies of Section 3.2 indicate that the inclusion of L8 information (using the +existing VIIRS workflow) faces the challenges of the processing artifacts due to glow. It +would be interesting to tailor the processing algorithm for L8, which opens the door for +data fusion of VIIRS and L8, providing much better resolution interpretations. +2. Fast detection of flares on a monthly basis. +The development of a rapid flare detection and volume estimation method (based on +satellite imagery) will lead to continuous monthly data streams. Since NDIC needs +about two months’ turnaround time to compile and digitize the company reports, many +of the machine learning workflows proposed in this dissertation will be able to provide +predictive insights with rapid detection data. +119 + +3. Hierarchical Gaussian processes. +The models in Chapter 5 are learned from each entity’s own data. It would be interesting +to see how far the scheme of partial pooling (Chapter 4) can be taken. Can pooling +across different entities via hierarchical Gaussian processes improve the inferences? +4. Spatial-temporal analysis. +One step further from Item 3 above, the efficacy of spatial-temporal models (which +allow for pooling information across time and space) are worth investigating. Are +neighboring entities exhibiting close resemblance in flaring behaviors? +5. Unify everything under Bayesian nonparametrics. +The model comparison for GMMs in Chapter 6 depends on specifying the potential +numbers of clusters a priori. In fact, Dirichlet process, as an infinite-dimensional gener- +alization of the Dirichlet distribution, is nonparametric and allows for automatically +choosing the number of necessary clusters. Considering the effectiveness of GP (Chap- +ter 5), it would be interesting to see how far the nonparametric models can be taken in +flaring data analytics. 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SPE- +199707-MS. https://doi.org/10.2118/199707-MS. +128 + diff --git a/p9E2T4oBgHgl3EQf0ggt/content/tmp_files/load_file.txt b/p9E2T4oBgHgl3EQf0ggt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..505e9256ef9ffc49e6cfd127ed93dcdf4b39a6d2 --- /dev/null +++ b/p9E2T4oBgHgl3EQf0ggt/content/tmp_files/load_file.txt @@ -0,0 +1,10596 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf,len=10595 +page_content='APPLICATION OF MACHINE LEARNING TO GAS FLARING by Rong Lu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04141v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='LG] 11 Jan 2023 © Copyright by Rong Lu, 2020 All Rights Reserved A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Petroleum Engineering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Golden, Colorado Date Signed: Rong Lu Signed: Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Jennifer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Miskimins Thesis Advisor Golden, Colorado Date Signed: Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Jennifer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Miskimins Professor and Head Department of Petroleum Engineering ii ABSTRACT Currently in the petroleum industry, operators often flare the produced gas instead of commodifying it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The flaring magnitudes are large in some states, which constitute problems with energy waste and CO2 emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In North Dakota, operators are required to estimate and report the volume flared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The questions are, how good is the quality of this reporting, and what insights can be drawn from it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Apart from the company-reported statistics, which are available from the North Dakota Industrial Commission (NDIC), flared volumes can be estimated via satellite remote sensing, serving as an unbiased benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since interpretation of the Landsat 8 imagery is hindered by artifacts due to glow, the estimated volumes based on the Visible Infrared Imaging Radiometer Suite (VIIRS) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Reverse geocoding is performed for comparing and contrasting the NDIC and VIIRS data at different levels, such as county and oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' With all the data gathered and preprocessed, Bayesian learning implemented by Markov chain Monte Carlo methods is performed to address three problems: county level model development, flaring time series analytics, and distribution estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' First, there is heterogeneity among the different counties, in the associations between the NDIC and VIIRS volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In light of such, models are developed for each county by exploiting hierarchical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Second, the flaring time series, albeit noisy, contains information regarding trends and patterns, which provide some insights into operator approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Gaussian processes are found to be effective in many different pattern recognition scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Third, distributional insights are obtained through unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The negative binomial and Gaussian mixture models are found to effectively describe the oilfield flare count and flared volume distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Finally, a nearest-neighbor-based approach for operator level monitoring and analytics is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' iii TABLE OF CONTENTS ABSTRACT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' iii LIST OF FIGURES .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' xii LIST OF SYMBOLS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' xiii LIST OF ABBREVIATIONS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5 CHAPTER 2 LITERATURE REVIEW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 26 CHAPTER 4 COUNTY LEVEL FLARING MODEL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 70 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 Modeling Scale Factor between VIIRS and NDIC .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 74 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 Predicting NDIC Flared Volume .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 80 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 A Look Back at the Prior Choices .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 81 CHAPTER 6 UNSUPERVISED LEARNING FROM MULTIPLE PERSPECTIVES .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 83 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Learning the Distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 119 vi LIST OF FIGURES Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Top 30 countries ranked by flared gas volume in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 The time series show the trend of gas flaring for the top two states in the United States (EIA 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 This Google Earth imagery shows gas flaring being conducted on a well location in North Dakota (Google Earth 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 Part of the original poster (Earth Observation Group at Payne Institute 2019) which uses one year accumulation of VIIRS low light imaging data to showcase human activities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', gas flaring, fishing, and city lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Landsat 8’s spatial resolution (NASA 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 8 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 The evolution of four random walk Metropolis Markov chains (Carpenter 2020), each started in a different location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 14 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 The flowchart adapted from (Betancourt 2020) shows a principled Bayesian workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 16 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 A screenshot of the top ∼50 rows in the October 2018 production report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Each row corresponds to a well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' There are in total 17,135 rows in this spreadsheet, with the first row being the header.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 19 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 The nighttime combustion source detection limits of VIIRS (top) and L8 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For natural gas flaring whose temperature is generally greater than 1500 K, L8 detected flares show source areas (around 10−2 m2) orders of magnitude less than that of VIIRS (around 1 m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 21 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 A count plot showing the distribution of cluster sizes: clearly there are a certain number of large clusters (as shown by the tail to the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 22 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 A large flare consisting of many hot pixels (detections), which is found by running the nightfire algorithm on L8 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Both the Band 6 (grayscale image) and the KMZ view are shown and provided by Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Elvidge (personal communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 22 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 A heat map showing the pairwise Spearman correlations between the original time series’ monthly observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 25 vii Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 A heat map showing the pairwise Spearman correlations between the time series after applying the first differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 26 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 Visualizations of both the NDIC and VIIRS reportings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 27 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 Posterior distributions (left column) and trace plots (right column) for the state level flaring model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 29 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9 Intervals are constructed using posterior predictive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 30 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Scatterplots of NDIC and VIIRS reportings for different counties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 36 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Scatterplots of NDIC and VIIRS reportings for different counties, without sharing neither x- nor y-axis for all the subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 38 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 LKJcorr(η = eta) probability density.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 40 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 Posterior distributions and trace plots of the slopes for each county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 43 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 Posterior distributions and trace plots of the intercepts for each county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 44 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 A forest plot showing the uncertainties around each county’s slope estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 45 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 A forest plot showing the uncertainties around each county’s intercept estimate.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 45 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 Correlation between the intercepts and slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 48 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Posterior distributions and trace plots for the Blue Buttes Oilfield gas flaring proportion model.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Posterior predictive samples showing the gas flaring proportion variations at the Blue Buttes Oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 60 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Posterior distributions and trace plots for the Operator A gas flaring proportion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 Posterior predictive samples showing the gas flaring proportion variations of Operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 62 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 Posterior distributions and trace plots for the Blue Buttes Oilfield well flaring proportion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 64 viii Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 Posterior predictive samples showing the well flaring proportion variations at the Blue Buttes Oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 64 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 Posterior distributions and trace plots for the Operator A well flaring proportion model.' metadata={'source': 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Oilfield flare count model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 70 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='13 Posterior distributions and trace plots for the Blue Buttes Oilfield BOE flaring proportion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 72 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14 Posterior predictive samples showing the BOE flaring proportion variations at the Blue Buttes Oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 72 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 Posterior distributions and trace plots of the BOE flaring proportion model for Operator A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 73 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='16 Posterior predictive samples showing the BOE flaring proportion variations of Operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 74 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17 Posterior distributions and trace plots for the North Dakota VIIRS-NDIC scale factor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 77 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18 Posterior predictive samples showing the scale factor variations of North Dakota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 78 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='19 Posterior distributions and trace plots for the Blue Buttes Oilfield VIIRS-NDIC scale factor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 79 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 Posterior predictive samples showing the scale factor variations in the Blue Buttes Oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 80 ix Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='21 Posterior predictive samples showing predictions of the scale factor for the next six months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 81 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Effective usage of histograms can be surprisingly subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 84 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 A histogram for the distribution of the oilfield detection counts from October 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 87 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Posterior distributions and trace plots for the oilfield detection counts distribution, fitted with the data from October 2018.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 88 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 Histograms for the distribution of the oilfield detection counts from October 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 89 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 Histograms for the distribution of the oilfield detection counts from October 2018, with the y-axis clipped to better present those counts which are greater than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 90 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 Histogram for the distribution of the oilfield flared volumes from Q4 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 92 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 Distribution of the oilfield flared volume magnitudes from Q4 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 93 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 Ten random draws from a Dirichlet prior with α = (6, 6, 6, 6, 6, 6, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 95 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9 GMM inference results with different K’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 97 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 WAIC values with different K’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 98 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='11 A scatterplot of oil production and flared gas volumes for different oilfields in Q4 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 100 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Examples of good fits between the NDIC and VIIRS reported volumes, at the operator level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 105 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Examples of poor fits between the NDIC and VIIRS reported volumes, at the operator level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 106 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Time series of the two example operators whose reporting did not quite align with the VIIRS detected trends/patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The points or periods in time for which the company-reported data were significantly different from the satellite detections are annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 107 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 A more comprehensive time series plot for Operator D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 109 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 A more comprehensive time series plot for Operator E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 110 x Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 A neural network designed for the hypothetical well performance classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 115 xi LIST OF TABLES Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Resolutions of Landsat 8 and VIIRS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 7 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Parameter Estimates of State Level Flaring Model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 28 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 North Dakota County Abbreviations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 37 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Parameter Estimates of County Level Flaring Model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 49 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Parameter Estimates of Oilfield Detection Count Distribution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 88 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Units for Operator Time Series in Figures 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 111 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Publication Count Rise on OnePetro .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 112 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Models Developed in this Dissertation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 119 xii LIST OF SYMBOLS Vectors and matrices are in bold type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A subscript asterisk, such as in y∗, indicates reference to a test set quantity or a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' General Nomenclature ℓ2 norm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ∥·∥ Data set: D = {(xi, yi) | i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , n} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' D Natural numbers with zero .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' N0 Pi (italic) representing a variable .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' π Pi (upright) denoting the transcendental constant (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14159 · · · ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' π Prediction for a test input x∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' y∗ Proportional to;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', p(x | y) ∝ p(x, y) means that p(x | y) is equal to p(x, y) times a factor which is independent of x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ∝ Real numbers .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' R Universal quantifier: for all x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' p(· | ·) Expectation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' expectation of g(x) when x ∼ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' E or Ep[g(x)] Probability density function .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' p(·) xiii Probability mass function .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' P(·) Random variable X is distributed according to p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' X ∼ p Variance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' variance of g(x) when x ∼ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' V or Vp[g(x)] Probability Distributions Binomial distribution with parameters n, p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Binomial(n, p) Categorical distribution with parameter p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Categorical(p) Continuous uniform distribution with parameters a, b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Uniform(a, b) Dirichlet distribution with parameter α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Dirichlet(α) Distribution over Cholesky decomposed covariance matrices with parameters η, σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' LKJCholeskyCov(η, σ) Exponential distribution with parameter λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Exponential(λ) Gamma distribution with parameters α, β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Gamma(α, β) Half-Cauchy distribution with parameter γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Half-Cauchy(γ) Half-Normal distribution with parameter σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Half-Normal(σ) LKJ distribution with parameter η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' LKJcorr(η) Multivariate Gaussian distribution with parameters µ, Σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' MVNormal(µ, Σ) Negative binomial distribution with parameters µ, φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' NegBinomial(µ, φ) Poisson distribution with parameter λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Poisson(λ) Student’s t-distribution with parameters ν, µ, σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' N(µ, σ) Gaussian Processes Covariance function evaluated at x and x′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' m(x) Vector of latent function values, f = (f(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , f(xn))⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' f Vectors and Matrices Cholesky decomposition: L is a lower triangular matrix such that L · L⊤ = K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Cholesky(K) Identity matrix of size n × n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In Transpose of matrix L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' log(·) Units barrels of oil equivalent .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ZIP Zero-inflated negative binomial .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ZINB xviii ACKNOWLEDGMENTS In the very first place, I want to express my deepest appreciation to my advisor, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Jennifer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Miskimins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Miskimins has been my MS/PhD advisor and mentor since 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since I returned to Mines to start my PhD in 2017, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Miskimins has been providing me with the best guidance, the greatest support, and the most opportunities that I could imagine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' During the first semester, I worked as a lab assistant in the High Bay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' in a later semester, I worked as a teaching assistant in her well stimulation course;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ever since I started to become interested in machine learning, she has provided me with a huge number of opportunities to connect with different groups of people, for brainstorming and pursuing my research interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To a certain extent, I feel like I finally become a “qualified” FAST student member, thanks to all of these precious experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' What I have achieved, including this dissertation, would have never been possible without the guidance and support from Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Miskimins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Her world-class technical expertise, attitudes toward work/life, and art of managing different teams at various levels are what I hope I can learn from in my career and personal life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' I am deeply grateful to my dissertation committee members: Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Soutir Bandyopadhyay, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Alfred W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Eustes III, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Yilin Fan, and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Jim Crompton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' My competency in my research field, as well as the shape of this dissertation are built with the help of those fruitful discussions and insightful comments from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' I am indebted to my mentors, colleagues, and friends from the Payne Institute for Public Policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Especially, I want to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Mikhail N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Zhizhin, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Elvidge, and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Morgan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Bazilian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It is such an eye-opening experience for me to work with these world-class experts in remote sensing and satellite imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' I would particularly like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Zhizhin for his help, insights, and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' I am really grateful to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Bandyopadhyay and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Luis Tenorio from the AMS Department, for their fantastic teaching, knowing me personally and motivating me to work hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Looking xix back at what I have learned in machine learning which makes this dissertation possible, taking their classes are definitely the most important resources for myself (excuse me for not being a probabilist at this moment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' By taking their statistical methods classes, I started to appreciate what really is machine learning, and falling in love with mathematics, more specifically, probability theory and statistical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The TA experience at Mines makes me a better PhD student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' What I have learned, tech- nical or non-technical, made their way into this dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' I want to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Crompton, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Eustes, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Mark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Miller, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Linda A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Battalora, and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Miskimins for providing me with those valuable TA opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' I am grateful to all of my students for their support and feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' I want to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Yu-Shu Wu, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Xiaolong Yin, and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Yilin Fan for their care, support, and encouragement throughout my PhD study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' I would like to thank Denise Winn-Bower, Rachel McDonald, and Joe Chen for their help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' I really appreciate the feedback from the FAST member companies’ representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A lot of the discussions and the reflections following those were incorporated into this dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Especially, I want to thank Ty Woodworth for his time and help, in the process of collecting plunger lift data for me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' I got very warm welcomes every time I visited their Windsor office in Northern Colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Ty kindly introduced me to the team he led, and I got the great opportunities to ask questions and discuss with many field experts in different areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Those discussions helped me tremendously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Special thanks go to the open source community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the process of conducting this research and typesetting this dissertation, I benefited a lot from the ecosystems around Linux/GNU, TEX/LATEX, and Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Especially, I want to thank the people behind PyMC3, a probabilistic programming language that this dissertation is heavily dependent upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Last but not least, I would like to thank my family and friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Thank you to my beloved wife Xiaodan, for all her love, support, and delicious dishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' I also want to thank my parents and parents-in-law for their support, encouragement, and understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' xx I dedicate this work to my mother, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Lingying Ni, and my father, Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Honggang Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 谁言寸草心，报得三春晖。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' xxi CHAPTER 1 INTRODUCTION Currently in the petroleum industry, for wells which produce both crude oil and natural gas, operators often choose to flare the produced gas instead of commodifying it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The rationales behind such decisions are multifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Variations in natural gas price can be an important factor, especially when the processing and transportation cost is higher than the value of gas (Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The amount of gas being flared each year on a national level is huge, and an increasing trend can be observed for the top flaring countries (Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Source: NOAA, Colorado School of Mines, GGFR The new ranking – top 30 flaring countries (2014 – 2018) Ranked by 2018 flare volume Million m3 gas/year flared Public Disclosure Authorized Public Disclosure Authorized Public Disclosure Authorized Public Disclosure Authorized Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: Top 30 countries ranked by flared gas volume in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' United States ranks No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4 and has a large increase from 2017 to 2018 (World Bank 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Due to the boom of unconventional resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', shale gas reservoirs) development in the recent decade, the United States has been among the top flaring countries in terms 1 24,000 22,000 20,000 18,000 16,000 14,000 12,000 10,000 8,000 6,000 4,000 2,000 0 2014 ■2015 ■2016 2017 ■2018of total volume flared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is backed by the data from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Energy Information Administration (EIA) (2019) showing North Dakota, which is underlain by the Bakken Formation, and Texas, which houses the Permian Basin and the Eagle Ford Shale, are the top two flaring states since 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The two states’ annual flaring volume time series are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Some flaring sites can be clearly identified from Google Earth’s imagery (Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 1995 2000 2005 2010 2015 Date 0 20000 40000 60000 80000 100000 120000 Gas Vented and Flared (MMcf) North Dakota Texas Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2: The time series show the trend of gas flaring for the top two states in the United States (EIA 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Texas regained the lead in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Natural gas flaring constitutes a problem of energy waste and CO2 emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In recent years, various organizations and government agencies have advocated reducing or eliminating routine gas flaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, the North Dakota Industrial Commission (NDIC) introduced a gas flaring regulatory policy (Order 24665) in 2014, with goals of reducing flaring in different aspects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', volume of gas flared).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The World Bank launched the “Zero Routine Flaring by 2030” initiative in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To monitor and benchmark flaring activity’s magnitude, a precise and accurate method to obtain quantitative flaring information is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, in certain situations, this information is only available through self-reporting mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3: This Google Earth imagery shows gas flaring being conducted on a well location in North Dakota (Google Earth 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Inaccuracies might be introduced either intentionally or unintentionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Satellite remote sensing is one unbiased approach for solving this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It can help detect active flares especially during nighttime and can be used to calibrate the estimation for flared gas volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For this work, two different types of sensors are considered, including the Landsat 8 (L8)’s Operational Land Imager (OLI) and Thermal Infrared Sensor (TIRS), as well as the Visible Infrared Imaging Radiometer Suite (VIIRS) that is on the Suomi National Polar-orbiting Partnership (NPP) and NOAA-20 satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the remainder of this dissertation, they are referred to as L8 and VIIRS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' An example of detecting flaring with VIIRS low light imaging data is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Research Goal This research is undertaken to achieve the following goals: Evaluate the methodology for estimating flared gas volume leveraging satellite imagery;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' and, 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4: Part of the original poster (Earth Observation Group at Payne Institute 2019) which uses one year accumulation of VIIRS low light imaging data to showcase human activities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', gas flaring, fishing, and city lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' As annotated, North Dakota’s flaring activities are very visible from space at night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Find insights into operators’ gas flaring behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Dissertation Objectives To achieve the goals outlined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1, more specific objectives are listed below: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Compare and contrast the flaring data from VIIRS and NDIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Compare the VIIRS flared volumes to the NDIC, using the NDIC as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Evaluate the effectiveness of using Landsat 8 nighttime images to improve flare detection and volume estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Determine the detection limits of Landsat 8 and compare it with VIIRS’ capabili- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Investigate operator approaches for gas flaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4 Shale Oil The boom of shale oil production is clearly depicted by the number of gas flares ected-inNorthDakota• Determine the correlation between gas price / oil price / oil production and flared gas volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Evaluate if the North Dakota regulatory policy (Order 24665) achieved its goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Develop a model that can predict flared gas volume at a state level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Find any hidden structure/clusters from all the producing entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Outline and Contributions The main contribution of this dissertation is demonstrating that Bayesian learning implemented by Markov chain Monte Carlo methods is very effective in flaring data analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A series of parametric and nonparametric machine learning models are developed for various analytics goals and granularities, providing direct guidance for future modeling endeavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To demonstrate the effectiveness and robustness, they are all tested with real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The superiority of this approach is based on the fact that the inference stage is entirely probabilistic, in that the parametric uncertainties arising from probable models as well as the stochastic uncertainties arising from noisy observations are all properly characterized and quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It makes the extracted insights robust and interpretable for decision- and policy-making by, for example, a state government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In Chapter 2, a literature review is given for the state of the art in satellite imagery processing, Bayesian inference, Markov chain Monte Carlo methods, and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In Chapter 3, the data gathering processes are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Results from some exploratory data analysis are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In Chapter 4, county level models are built to study the correlations between VIIRS and NDIC, and to explore the heterogeneity among the counties in North Dakota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In Chapter 5, flaring time series analytics is presented for the purposes of revealing trends and patterns at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In Chapter 6, unsupervised learning is applied on flaring data to characterize the latent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5 In Chapter 7, a method of operator level monitoring and analytics is introduced, and some discussions about applying Bayesian learning are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In Chapter 8, major conclusions drawn are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Recommendations based on this work are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A number of future research areas are outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 6 CHAPTER 2 LITERATURE REVIEW In the 1990s, the World Bank started gathering nighttime satellite images, from which big cities and oilfields were both bright and needed to be sorted using extra information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The situation changed in 2012 when infrared data became available from VIIRS (Rassenfoss and Zborowski 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One of the data products, VIIRS Nightfire (VNF) specializes in natural gas flaring observation and is even able to distinguish between biomass burning and gas flaring (Elvidge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' VNF’s development was based upon VIIRS imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To improve the performance of flare detection and gas volume estimation, other sources of information, such as L8 imagery, can be leveraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 presents a comparison of L8 and VIIRS spatial and temporal resolutions (NASA 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Wikipedia 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 illustrates L8’s spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In addition, L8 collects data in 11 different spectral bands of the electromagnetic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' VIIRS has 22 bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Both L8 and VIIRS are in near-polar orbits of the earth and can reveal rich features in the landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Therefore, L8 should be able to identify smaller gas flares compared to VIIRS’ capability, although its longer satellite revisit time poses a challenge to identify less persistent flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' More details on the processing steps of VNF are discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1, the essence of which will be applied to L8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: Resolutions of Landsat 8 and VIIRS Resolution Type Spatial [m] Temporal [d] Landsat 8 15 to 100† 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0‡ VIIRS 375 to 750† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 † Depends on the band of the electro- magnetic spectrum ‡ For daytime mode 7 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: Landsat 8’s spatial resolution (NASA 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Each Landsat pixel (30 by 30 meter area) is roughly the size of a baseball diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Nowadays, one resource which is more than abundant is data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For a certain discipline or research field, new sources of data bring in new dimensions of information, such as satellite images are now playing a role in gas flaring analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' How to analyze data effectively and intelligently to gain insights is a central problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the petroleum engineering domain, for example, data driven approaches have been proposed to analyze stimulation treatments (Kaza- kov and Miskimins 2011) and predict screenouts (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Machine learning is a powerful tool for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It is at the core of artificial intelligence and data science, and lies at the intersection of statistics and computer science (Jordan and Mitchell 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Frameworks in computational learning theory, such as the PAC learning proposed by Valiant (1984), help provide a theoretical backbone for some learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One subset of machine learning, deep learning (DL), had its debut in 2006 when Hinton and Salakhutdinov introduced Deep Belief Networks (DBN), but it did not gain wide acceptance until 2012 when AlexNet showed the breakthrough performance on classification accuracy in the ImageNet competition (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=" AlexNet is a DL-based model (more 8 Landsat 8's Spatial Resolution 30 m 15 m 100m Vis-NIR-SWIR = 30 m Panchromatic=15m Thermal IR = 100 m (Resampledto30m)specifically a convolutional neural network) and achieved an error rate of 15." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 %, which is more than 10 % lower than the runner-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' DL dominated the competition thereafter, and DL-based models finally surpassed human performance on the classification data set in 2015 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Although neural network-based models have gained much success in recent years, it should be noted that no one type of model can always be the best candidate for all problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This has been formally shown by Wolpert (1996), and is usually referred to as the “no free lunch” (NFL) theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' More recently, Olson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (2017) empirically assessed 13 classification algorithms on 165 different problem sets, and the results aligned with the theorem: even the union of the top five best performing algorithms cannot dominate all of the problem sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the following sections, a detailed review is given for the aspects below, which serve as the foundation and inspiration for this work: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Satellite image processing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Bayesian inference 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Markov chain Monte Carlo 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Machine learning 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Analytics toolset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Satellite Image Processing Satellite images are utilized to estimate flared gas volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The fire detection algorithm based on Planck curve fitting and physical laws, known as VIIRS Nightfire (VNF) due to Elvidge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (2013), serves as a starting point for analyzing L8 images in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The method consists of several major steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Detection of hot pixels During nighttime, the sensors mainly record instrument noise which approximately follows a Gaussian distribution, except for the few pixels that contain an infrared 9 emitter such as a gas flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Therefore hot pixels can be identified by setting a cutoff on the tail of the distribution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', those pixels with digital numbers exceeding the mean plus four standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Noise filtering Hot pixels that are detected in only one spectral band are treated as noise and filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Atmospheric correction Losses in radiance due to scattering and absorption effects can be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' MOD- TRAN ® 5 (Berk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2006), parameterized with atmospheric water vapor and tem- perature profiles, is used to derive the correction coefficients for each spectral band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Planck curve fitting Planck curves are modeled for gas flares, which appear as gray bodies because they are sub-pixel sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Therefore the output of the fitting is an estimate of the temperature and an emission scaling factor (the emissivity term in the Planck function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The latter is used subsequently to estimate the source area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Calculation of source area The source area S is calculated using S = εA , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) where ε is the emission scaling factor and A is the size of the pixel footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Calculation of radiant heat The radiant heat is calculated using the Stefan–Boltzmann law: RH = σT 4S , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2) 10 where RH is the radiant heat in MW, σ is the Stefan–Boltzmann constant, T is the temperature in K, and S is the source area in m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Once RH is obtained, previous work by Elvidge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (2015) developed a calibration for estimating flared gas volume, utilizing nation-level flaring reporting provided by Cedigaz (2015) and state-level reporting from Texas and North Dakota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The developed calibration can then be applied to each individual flaring site worldwide for estimation of flared gas volume, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Bayesian Inference Bayesian inference leverages conditional probability theory to establish a formal procedure for learning from data (Betancourt 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Bayesian models provide full joint probability distributions p(D, θ) over observable data D and unobservable model parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The essence of Bayesian analysis is to obtain the posterior distribution p(θ | D), which characterizes the conditional probability of parameters θ given some data D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It can be derived through Bayes’ theorem: p(θ | D) = p(D | θ) p(θ) p(D) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3a) = p(D | θ) p(θ) � p(D | θ′) p(θ′) dθ′ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3b) ∝ p(D | θ) p(θ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3c) where p(D | θ) is the likelihood (also referred to as the observation model) which denotes how likely the data is given a certain set of parameters, and p(θ) is the prior which models the probability of the parameters before observing any data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The prior encodes domain expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Once some observations are given, it is updated into a posterior which quantifies how consistent the model configurations are with both the domain knowledge and the observed data (Betancourt 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' After the posterior is obtained, most if not all inferential questions can then be answered with posterior expectation values of certain functions (Betancourt 11 2019): Ep[g(θ)] = � g(θ) p(θ | D) dθ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4) where g(θ) is the function encoding some inferential question (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', where in the model configuration space the posterior concentrates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Predictions can be made in the form of a posterior predictive distribution: p(y∗ | x∗, D) = � p(y∗ | θ, x∗) p(θ | D) dθ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5) where y∗ is the predictions based on the training set D for a test input x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Essentially this is integrating the prediction p(y∗ | θ, x∗) over the posterior distribution of parameters (Ras- mussen and Williams 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Note that by giving the final results in terms of a probability distribution, richer information and more reliable inferences are accessed compared to merely giving a point estimate through MLE or MAP (as some machine learning models do under the frequentist framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is achieved by incorporating into the inference process the uncertainty in the posterior parameter estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Other benefits include posterior predictive checks, which are conducted by checking for auto-consistency between generated data (y∗) and observed data (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Markov Chain Monte Carlo Many of the integration problems central to Bayesian statistics, including those in Equations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5, are analytically intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A class of sampling algorithms, known as Markov chain Monte Carlo (MCMC), can be applied to approximate these (Andrieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Suppose for some function of interest f(x), the objective is to obtain its integral, with respect to a non-standard target distribution p(x) from which samples cannot be drawn directly: I(f) = � f(x) p(x) dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6) By constructing Markov chains that have p(x) as the invariant distribution, MCMC samplers, while traversing the sample space X, are able to generate samples x(i) that mimic samples 12 drawn directly from the target distribution p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In other words, this mechanism makes it possible to draw a set of samples {x(i)}N i=1 from p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Then, by the Monte Carlo principle, the integral I(f) can be approximated with a sum IN(f): IN(f) = 1 N N � i=1 f(x(i)) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' −−−−→ N−→∞ I(f) = � f(x) p(x) dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7) That is, the estimate IN(f) is unbiased and by the strong law of large numbers, it will converge almost surely (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=') to I(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' That’s why MCMC is a powerful tool in Bayesian analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In practice, the Metropolis-Hastings (MH) algorithm and Gibbs sampling have been popular MCMC methods (Andrieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2003), but only when the parameter space is not too high-dimensional (McElreath 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Due to limited computing resources, it is impossible to run Markov chains infinitely long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In other words, inference has to be made based on finitely many draws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One approach, which is effectively leveraged in this research, is to run multiple chains in parallel and monitor various statistics for diagnosing non-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Besides the effective sample size per transition of the Markov chain, the Gelman-Rubin statistic (Gelman and Rubin 1992), denoted by ˆR, is used in this dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The ˆR statistic quantifies whether the ensemble of Markov chains initialized from diffuse points in parameter space finally converge to the same equilibrium phase (Betancourt 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' When ˆR is sufficiently close to 1 (for example ˆR < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05), convergence is declared to be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' As an example, Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 presents how four chains are started in different corners but approach stationarity and convergence after a certain number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For many of the problems in practice, including the models in this dissertation, the parameter space is very high-dimensional and involves highly curving regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The Metropolis- Hastings algorithm and Gibbs sampling are far from efficient in these situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Hamiltonian Monte Carlo (HMC), originally proposed by Duane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (1987), really outshines the other algorithms at this point and is the main sampling strategy adopted in this dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 13 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2: The evolution of four random walk Metropolis Markov chains (Carpenter 2020), each started in a different location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The target density is a bivariate normal with unit variance and correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' After M = 5000 iterations, the four chains have mixed well and explored most of the target density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Specifically, No-U-Turn Sampler (NUTS) introduced by Hoffman and Gelman (2014), which is an extension to HMC, is employed for sampling from posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 Machine Learning Machine learning was defined by Mitchell (1997) as computers improving automatically through experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It can also be viewed as a function estimation problem (Vapnik 2000), or as the process of extracting important patterns and trends from data (Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In terms of tasks, common types of learning consist of supervised, unsupervised, semi- supervised, and reinforcement (Burkov 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Let xi ∈ X ⊆ Rd represent input, and yi ∈ Y represent target, then the goals of the first two types are: Supervised learning aims to use the dataset, consisting of X = {xi}n i=1 and y = {yi}n i=1, to produce a model that is able to predict an output (yj) given some new/unseen input (xj), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', learning the underlying mapping f : X → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Unsupervised learning is used to find the hidden patterns in X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' in this case there does not exist any labels (y) or predefined targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 14 M= 50 M = 500 M = 5000 4 0 4 4 0 4 4 0 4 4 0 4 01Another variation of learning is online learning, in which case training data is fed to the algorithm continuously or one example at a time (Abu-Mostafa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In other words, streaming data is available that the algorithm has to process on the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is different from batch learning, where data is provided beforehand and “frozen” during the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Online learning can be applied to the different tasks as discussed above (supervised and others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In terms of model characteristics, machine learning models can be categorized into parametric and nonparametric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Parametric models are characterized by a fixed number of parameters, whereas nonparametric models have an infinite-dimensional parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, in the latter case the parameter space can be the set of continuous functions in a regression setting (Orbanz and Teh 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this dissertation, supervised and unsupervised learning are leveraged while exploiting both parametric and nonparametric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' From Bayesian’s perspective, machine learning is essentially computing the posterior (de Freitas 2013), which is then used for inference and prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is conducted exactly through Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In practice, machine learning conducted under Bayesian’s framework follows a principled workflow (Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3), which is adapted for the modeling in this dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 Analytics Toolset For the past five to ten years, prosperity in contributions and progress in the open source community has been witnessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Ecosystems around Python, R, and Julia have been prototyped, tested, and deployed in production environments in various industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Powerful probabilistic programming languages (PPL), for example Stan (Carpenter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2017) and PyMC3 (Salvatier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2016), have become the workhorse for Bayesian machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The majority of this work is implemented in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Specifically, Bayesian learning is performed by leveraging PyMC3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Some analytic visualizations are produced employing ArviZ (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Geospatial operations are performed with the help of GeoPan- 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Conceptual Analysis 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Model Development 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Define Observational Space 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Construct Summary Statistics 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Construct Summary Functions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Simulate Bayesian Ensemble 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Prior Checks 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Configure Algorithm 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Fit Simulated Ensemble 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Algorithmic Calibration 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Inferential Calibration 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Fit Observed Data 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Diagnose Posterior Fit 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Posterior Retrodictive Checks 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Celebrate Pre-M odel Pre-Data Post-M odel Pre-Data Post-M odel Post-Data Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3: The flowchart adapted from (Betancourt 2020) shows a principled Bayesian workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' das (Jordahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Satellite imagery is processed and analyzed in MATLAB, with implementations mainly following Elvidge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 16 CHAPTER 3 DATA PREPROCESSING AND EXPLORATORY DATA ANALYSIS In this chapter, an overview of the flaring data is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Some other variables which might be correlated with the flaring statistics are also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Exploratory data analysis is performed for choosing the subset of the variables as the focus in this dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A state level model is developed in the end which motivates the work in the next two chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Data Gathering Four sources of data, L8 satellite images, VIIRS estimated flared volumes, NDIC monthly production reports, and county/oilfield shapefiles for North Dakota were gathered for the analysis used in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Landsat 8 Images In total, 167 images (since 2013) were downloaded from Google Cloud using the criteria below: From five Path/Row’s: 126/216, 126/217, 126/218, 127/216, and 127/217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' According to the Worldwide Reference System (WRS), the satellite imagery of any portion of the world can be queried using Path and Row numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' These five Path/Row’s cover the majority of the areas in North Dakota that have production and flaring activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Nighttime images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Only nocturnal Landsat 8 imagery are used for the purpose of flare detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Cloud cover less than 10 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Images with low cloud cover percentages reveal more clearly land features including gas flares, and thus are ideal for validating the developed methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 17 GeoTIFF Data Product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Both the georeferencing information and the raw images of all the spectral bands are preserved through the GeoTIFF format, which are necessary for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 VIIRS Estimated Volumes The VIIRS flare inventory and estimated volume dataset obtained from Mikhail N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Zhizhin (personal communication) are used in this dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This dataset includes monthly flare detection records in North America from March 2012 to December 2018 (both inclusive) with their associated: Timestamps giving the specific month Latitudes and longitudes in WGS 84 coordinates Flared volume estimations in bcm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 NDIC Monthly Production Reports All the monthly production reports from May 2015 to April 2020 (both inclusive) which have flaring information have been downloaded from NDIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' There is one Excel spreadsheet per month;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' each row corresponds to a well (that was active in that month), and columns are for various types of information, including flared gas volume (estimated and reported by operator), oilfield, oil production, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A screenshot of the top ∼50 rows in one of the spreadsheets is displayed in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 NDIC Shapefiles The shapefiles for the counties and oilfields in North Dakota are downloaded from the NDIC GIS Map Server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' All the polygons are described in NAD 27 coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The shapefiles are for reverse geocoding the satellite detection locations to readable addresses, specifically which county and oilfield is a flare located in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 18 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: A screenshot of the top ∼50 rows in the October 2018 production report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Each row corresponds to a well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' There are in total 17,135 rows in this spreadsheet, with the first row being the header.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 19 c E H M 0 P Q FileNo Company WellName Quarter Section Township Range County FieldName Pool oil Wtr Days Runs Gas GasSold Flared Lat Long 10/1/201833053038990000 22021 EQUINORENERGY LP BILL 14-23 2TFH SWSW 11 151 101 MCK ALEXANDER BAKKEN 1044 2361 29 1041 1897 1781 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='90742645 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5799043 10/1/2018 33053048330000 25091 EQUINOR ENERGY LP BILL 14-23 3H NWNE 44 151 101 MCK ALEXANDER BAKKEN 1977 3837 31 1968 4194 4017 53 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='90455309 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='569154 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' MARVIN 14-34HS SESW 100 DIV ALEXANDRIA BAKKEN 2607 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='808449 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6534915 10/1/201 130000 26592 PETRO-HUNT, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5273298 子 18444 EOG RESOURCES, INC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ROSS 100-09H SWSW 18 92 MTL ALGER BAKKEN 771 573 638 10/1/201833061011660000 498 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3426108 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5208672 10/1/201833061038720000 32137 EOG RESOURCES, INC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ROSS 106-0915H NWNE 92 MTL ALGER BAKKEN 2124 5340 2099 5273 485 48.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='29904089 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4836364 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Satellite Image Processing As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1, all the available L8 images have been downloaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' They are processed in batch, following the workflow as outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To compare and contrast with VIIRS’ performance, specifically the nighttime combustion source detection limits, all the flares detected from all of the L8 images are gathered and used to generate the source area versus temperature scattergram shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Although it is expected that L8 would pick up smaller flares than VIIRS (which is capable of detecting flares around the size of a whole cooktop area), the majority of the detections as indicated on the scattergram are too small for natural gas flaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To verify if some hot pixels are clustered together and actually representing a single flare or flaring site, HDBSCAN (Campello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2013) with an implementation due to McInnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (2017) is executed on every L8 detection map to find out if large blobs of hot pixels are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' HDBSCAN is a density-based clustering algorithm which keeps all the advantages of the original DBSCAN (Ester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 1996), for example the capacity of finding clusters of arbitrary shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It also outperforms DBSCAN by being able to build clusters of varying density (Burkov 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Further, to get the most accurate results in this case, haversine metric is chosen to handle the great-circle distances between the hot pixels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' leaf clustering is used instead of the default Excess of Mass method to produce more fine grained clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The clustering results are illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To verify whether these clusters are really single flares or they are actually a large number of neighboring wells (in which case each hot pixel still represents an individual flare), they are tracked down by looking further into each detection map (KMZ file).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It is found that some large blobs of hot pixels are clustered and indeed represent single (huge) flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One of the examples is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This poses a challenge to situations where an accurate estimate of the flare count is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The reason for this processing artifact is that, for large flares, there is glow surrounding the flare that was treated as many individual combustion sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' There are potential approaches 20 (a) VIIRS performance (Elvidge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2019) (b) L8 performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' figure provided by Mikhail N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Zhizhin (personal communication) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2: The nighttime combustion source detection limits of VIIRS (top) and L8 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For natural gas flaring whose temperature is generally greater than 1500 K, L8 detected flares show source areas (around 10−2 m2) orders of magnitude less than that of VIIRS (around 1 m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 21 10° 350 300 103 250 Detectable Area, m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 102 200 Counts 150 10 DNB M07 100 M08 M10 M11 50 M12 M13 10 250 500 750 1000 1250 1500 1750 2000 Temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' KDNB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='M07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='M08 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='M10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='M11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='M12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='M13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='B05 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='B06 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='B07 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 54 56 58 59 60 65 68 73 76 77 84120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='Number of detections within a cluster ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='30000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='40000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='50000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='60000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='70000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='Count of clusters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='75264 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10527 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='647057984632 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='306621561428982782621377298263232188148131120 91 72 70 61 48 45 47 35 24 23 19 22 17 18 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='11 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3: A count plot showing the distribution of cluster sizes: clearly there are a certain number of large clusters (as shown by the tail to the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, there exists 2 clusters each of which contains 120 hot pixels and there is one cluster with 84 hot pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (a) Band 6 (SWIR) (b) KMZ view Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4: A large flare consisting of many hot pixels (detections), which is found by running the nightfire algorithm on L8 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Both the Band 6 (grayscale image) and the KMZ view are shown and provided by Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Elvidge (personal communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' to mitigate this to make the interpretation and estimation out of L8 more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this work, the flares detected from VIIRS and the gas volumes estimated out of those are the focus for analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 22 80 883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Reverse Geocoding By reverse geocoding, the county information of every VIIRS flare that is in North Dakota can be retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For most of the flares, the oilfield information is also retrievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Thereafter, the flaring statistics from VIIRS and NDIC can be compared and contrasted at different levels, for a certain point or period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Shapefiles as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' With the help of GeoPandas, the procedures for extracting counties and oilfields are the same: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Read the VIIRS records into a geospatial data object, with their original coordinates in WGS 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Read the shapefile into a geospatial data object, with its original coordinates in NAD 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Transform all the geometries in the shapefile to WGS 84 coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Perform a spatial join of the two data objects to get the county or oilfield information for each flare, if a specific county/oilfield’s polygon and the flare intersect, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', having any boundary or interior point in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 Correlational Analysis To study the correlations between oil/gas prices, flaring statistics, and production perfor- mance, various time series are extracted for May 2015 to December 2018 (both inclusive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='The below list describes all the variables used with their associated labels: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='VIIRS flared vol ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='monthly flared gas volume from VIIRS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='NDIC flared vol ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='monthly flared gas volume from NDIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='WTI oil price ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='WTI crude oil price given by EIA (2020b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='Henry Hub gas price ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='Henry Hub natural gas price given by EIA (2020a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='NDIC oil prod ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='monthly oil production from NDIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='NDIC gas prod ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='monthly gas production from NDIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='VIIRS flare count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='monthly flare detections count from VIIRS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='NDIC flaring well count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='monthly wells count which conduct flaring from NDIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='NDIC GOR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='ratio of the NDIC gas production to the NDIC oil production ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' the monthly observations are extracted from each time series,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' and Spearman’s ρ is employed to measure the statistical dependence between the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Spearman’s ρ is a rank correlation, which quantifies the correlation between the rankings of two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Compared to Pearson’s r, it assesses monotonic relationships which can be nonlinear and is more robust to outliers, therefore is used in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The pairwise correlations between the variables are presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since a correlation matrix is always symmetric with unit diagonals, only the lower triangular part without the diagonal is plotted to minimize the information redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It can be observed that most pairs show positive correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Financial factors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the oil and gas prices) are not among any of the highly correlated pairs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Nevertheless, it is indicated that the NDIC and VIIRS reportings have a positive correlation, and oil production is positively correlated with flared gas volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this analysis, due to the nature of the procedure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', extract the monthly data and then measure the rank correlations), all the information on the time scale is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To explore the correlations in the context of time series, the first differences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', lag-1 differences) are taken for each variable y′ t = yt − yt−1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) and then pairwise Spearman’s ρ is evaluated and visualized in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this case, there aren’t many pairs of variables which are highly correlated, except the oil and gas production are shown to be monotonically related on the lag-1 differences, which is unsurprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the 24 VIIRS flared vol NDIC flared vol WTI oil price Henry Hub gas price NDIC oil prod NDIC gas prod VIIRS flare count NDIC flaring well count NDIC GOR VIIRS flared vol NDIC flared vol WTI oil price Henry Hub gas price NDIC oil prod NDIC gas prod VIIRS flare count NDIC flaring well count NDIC GOR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5: A heat map showing the pairwise Spearman correlations between the original time series’ monthly observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The values are annotated in each cell, the corresponding variables of which can be obtained by reading off the tick labels from the vertical and horizontal axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' remainder of this dissertation, the focus is put on flaring and production related statistics instead of the financial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 25 VIIRS flared vol NDIC flared vol WTI oil price Henry Hub gas price NDIC oil prod NDIC gas prod VIIRS flare count NDIC flaring well count NDIC GOR VIIRS flared vol NDIC flared vol WTI oil price Henry Hub gas price NDIC oil prod NDIC gas prod VIIRS flare count NDIC flaring well count NDIC GOR 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6: A heat map showing the pairwise Spearman correlations between the time series after applying the first differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The values are annotated in each cell, the corresponding variables of which can be obtained by reading off the tick labels from the vertical and horizontal axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 State Level Flaring Model In this section, a regression model is built for the purpose of investigating the statistical relationships between the NDIC and VIIRS reportings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Data from both sources are visualized in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7, which demonstrate a positive correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Assuming a Gaussian observation model for the NDIC reporting with the location parameter encoding VIIRS’ information, the model is specified through Expressions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2a– 26 2015-07 2016-01 2016-07 2017-01 2017-07 2018-01 2018-07 2019-01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 Monthly bcm VIIRS NDIC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 VIIRS Monthly Data (bcm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 NDIC Monthly Data (bcm) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7: Visualizations of both the NDIC and VIIRS reportings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Left figure shows the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Right figure presents the scatterplot using the data points of each month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2e: α ∼ Half-Normal(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2a) β ∼ Gamma(2, 2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2b) σ ∼ Half-Cauchy(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2c) µi = α + β × VIIRSi (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2d) NDICi ∼ N(µi, σ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2e) where α is the intercept and β is the slope, both of which are constrained to be non-negative based on the nature of flaring volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' σ is the standard deviation in the Gaussian likelihood function, which has to be non-negative as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' µi is the expected NDIC reporting of month i, while NDICi and VIIRSi are the observed data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', reported volumes) from NDIC and VIIRS in month i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The notation used in defining this model communicates the data generating process unambiguously and is adopted throughout this dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Priors and hyperpriors are on the top while the observation model is at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The prior distributions for this model and all the others in this dissertation are chosen following the principles below: 27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Prefer weakly informative priors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', choose the priors based on the domain expertise at hand before observing any data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' They should be strong enough to reflect the domain expertise and be weak enough to “let the data speak”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', let the likelihood dominate when there is a decent amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, a prior of a gamma distribution with mean Eβ = 2/2 = 1 is placed on β, reflecting the assumption that the satellite interpretation workflow gives the same flared volume as the NDIC reporting, before one observes any data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Prefer priors with soft constraints as opposed to hard constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', follow Cromwell’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, α, β and σ all have prior distributions with support on R>0 or R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Counterexamples include using a triangular distribution or a continuous uniform distribution as the prior for such quantities, for which the author does not recommend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Prefer maximum entropy distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', make the most conservative assumptions based on all the information at hand (obeying all the known constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, the Gaussian and the binomial distributions are maximum entropy distributions and used in this dissertation, the fact of which can be formally shown leveraging the definition of Kullback–Leibler (KL) divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Once the priors and likelihood are established, four Markov chains of Hamiltonian Monte Carlo are run in parallel to sample from the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The parameter estimates are reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1, and the posterior distributions and trace plots are presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The four chains are plotted separately with different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The x-axis of the trace plot shows the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This layout is used consistently for the remainder of this dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: Parameter Estimates of State Level Flaring Model Parameter Variable Point Estimate 90 % Credible Interval α Intercept 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='061 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='044, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='079) β Slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='535 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='482, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='590) σ Reporting variability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='030 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='024, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='035) 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0 10 20 30 Posterior Density 0 500 1000 1500 2000 2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 Sample Value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='65 0 5 10 Posterior Density 0 500 1000 1500 2000 2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 Sample Value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='050 0 50 100 Posterior Density 0 500 1000 1500 2000 2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 Sample Value Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8: Posterior distributions (left column) and trace plots (right column) for the state level flaring model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence of the Markov chains have been achieved as shown by the trace plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Utilizing the model and the trace, posterior predictive samples are generated to construct the intervals (Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Point estimates and point predictions are easy to obtain for a certain machine learning model, however it is the properly constructed intervals that will provide insights into the uncertainty for decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The author would like to emphasize the importance of quantifying uncertainties when using machine learning, no matter for inference, prediction, or building intermediate models for integration into physics-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is unfortunately neglected or ignored in some of the applications/publications in the petroleum engineering domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The importance of properly quantifying the uncertainties will also be stressed in the following chapters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Whenever only one model specification is needed for making point predictions, it can be recovered by the parameter estimates from Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: NDICi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='061 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='535 × VIIRSi , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3) 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 VIIRS Monthly Data (bcm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 NDIC Monthly Data (bcm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 VIIRS Monthly Data (bcm) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9: Intervals are constructed using posterior predictive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In both figures, the line shows the “best” fit using point estimates (posterior means) of α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Shaded area in the left figure presents the 90 % credible interval (CI) of the regression mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Shaded area in the right figure demonstrates the 90 % prediction interval for the future NDIC reporting, for which most of the existing observations fall within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' where NDICi and VIIRSi are flared volumes in bcm of month i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The model also provides clear interpretations for the NDIC reporting regression mean, on the whole state level: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The intercept indicates on average there is 90 % probability that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 bcm to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='08 bcm reported volume per month will not be captured by the current VIIRS processing workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior mean is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='061 bcm (≈ 2150 MMcf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The slope indicates on average when satellite estimated volume increases by one unit, under 90 % probability the NDIC reporting will increase by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='48 unit to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='59 unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior mean is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='535 unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This model, while serving as a decent calibration and estimation tool for NDIC reporting on the state level, makes the assumption that the heterogeneity within the state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', among different counties) is negligible and all the monthly observations are conditionally independent 30 and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the scenarios in which these assumptions do not hold, other types of models can be built and are discussed in Chapter 4 and Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 31 CHAPTER 4 COUNTY LEVEL FLARING MODEL “Multilevel regression deserves to be the default form of regression.” — McElreath (2015) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Learning the Heterogeneity In this chapter, the author explores the heterogeneity in correlations between the state- reported and satellite-detected flaring statistics, among different counties in North Dakota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The motivations are threefold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Provide more granular insights than merely investigating the whole state’s flaring statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Compare and contrast different counties’ reporting consistencies with the baseline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the satellite detections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Develop a dedicated model for each county for calibration and prediction purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Hierarchical Model A common problem in learning from data is modeling individuals or units of a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, building models for different counties in a state, or for different well pads in an oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Usually from domain expertise, it is expected that the units would demonstrate some differences, however they do not necessarily represent completely independent data generating processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In other words, the units are different in some ways, while being similar in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Unfortunately, the following two common modeling approaches are extreme and not ideal: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Complete pooling 32 This ignores heterogeneity and assumes that the observations from all the units are generated/described by the exact same process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One set of parameters is learned for the whole population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this situation, the variance might be smaller, however the bias could be huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' No pooling This lets each unit learn its own set of parameters from its own data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The assumption is that the information from each unit tells one nothing about any other unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this situation, the bias might be smaller, however the variance could be huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In practice, neither of these approaches will be able to generalize well for insight extraction or prediction tasks, due to the total generalization error being large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In fact, these two extremes can be compromised by explicitly modeling the entire population of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' That is, in order to investigate the correlations among the individual units, an explicit model is introduced for the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the learning phase, the individual posteriors are used to fit some population distribution, while the information of the population is then fed back to the individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' What happens in this case is that the individuals with diffuse likelihood functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' with less data) are dragged more towards the population distribution, whereas the individuals which are well informed by their data will have their posteriors mostly unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this process, dynamic regularization is achieved, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the total generalization error is much smaller by partially pooling the data and balancing between the bias and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the context of county level model development, the question is now how might one model the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To motivate the choice of a particular class of models, some characteristics of the counties have to be examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this work, the counties are considered to be exchangeable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the joint probability p(θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , θn) is invariant to permutation of the indices, where θi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , n is the parameters for the i-th county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' That is, for any permutation π, p(θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , θn) = p(θπ1, θπ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , θπn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) 33 Furthermore, the list of counties can grow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', although one might only look at a few counties at this point, in the future new counties in terms of flaring activities might be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' If a population being modeled is exchangeable, and the population can grow arbitrarily large, de Finetti’s theorem shows that the only distribution that respects exchangeability is a hierarchical distribution: p(θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , θn) = � � n � i=1 p(θi | φ) � p(φ) dφ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2) where φ is a population parameter (which can be generalized to multiple population parame- ters) and p(φ) is a population prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It asserts an important fact that if exchangeable data is used for analytics, there must exist a population model (Jordan and Broderick 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This provides guidance for the development of the county level flaring models in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Equivalently, the individual and population parameters can be fitted jointly, achieving a dynamic pooling of the data: p(θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , θn, φ) = � n � i=1 p(θi | φ) � p(φ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3) in which process not only the θ’s but also φ are learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' After adding the observations component (D = {(xj, yj) | j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , m}) to it, the joint model becomes: p �� yj, xj, θcounty[j], ψj �m j=1 , φ � = � m � j=1 p(yj | xj, θcounty[j], ψj) p(θcounty[j] | φ) � p(φ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4) where θcounty[j] stands for the parameters for the j-th observation based on its county assign- ment, and ψ are some other parameters in the likelihood function that are not necessarily distributed according to a population model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 characterizes a hierarchical model that fits nicely into the Bayesian framework and is exploited for building the models in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' As a fundamental approach to model heterogeneity, hierarchical models have been de- pended upon routinely in various fields including ecological science (Bolker 2008), political science (Gelman and Hill 2006), and biological science (McElreath 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The author believes 34 that they should be widely accepted and utilized in the petroleum engineering domain as well, where the dataset is usually presented in hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, the shale gas wells in a given basin were completed by different oilfield service companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The information can then be pooled among the service companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A further discussion is given in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One caveat, though, is that de Finetti’s theorem is based on the assumption that the population (of units) is exchangeable and can grow arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Just like every other assumption in machine learning, it should not be taken for granted and does not always hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the context of county level flaring model development, one might argue that there are currently 53 counties in North Dakota and there might not be many new counties (as administrative divisions) in any finite amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In that regard, the author agrees with the claim of Box et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (2009) that, since assumptions “are never exactly true”, what shall be sought is the useful models as opposed to the correct ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' That is the goal for applying the hierarchical models in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It is worth noting that the terminologies are not consistent when referring to these types of models: some argue that hierarchical model and multilevel model are different names for the same modeling technique (Bolker 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' McElreath 2015), while others tried to differentiate them (Carpenter 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this dissertation, the model assumptions are communicated via the mathematical structures instead of the terminologies, by writing out the full model definitions whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Data Description After performing the reverse geocoding as outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3, there are twelve counties found to have reported flaring activities from both VIIRS and NDIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For each county’s historical data from May 2015 to December 2018 (both inclusive), only the months that have reported volumes from both sources are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A scatterplot for each of the 12 counties is presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1, where the county abbreviations follow the convention from the NDIC monthly production reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 lists the full county names associated with each abbreviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 NDIC county = MCK county = DUN county = WIL county = MTL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 NDIC county = BOW county = DIV county = BRK county = MCL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 VIIRS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 NDIC county = BIL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 VIIRS county = STK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 VIIRS county = SLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 VIIRS county = GV Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: Scatterplots of NDIC and VIIRS reportings for different counties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Both the x- and y-axis are shared among all the subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The x-axis is the monthly VIIRS reporting of the flared volume in bcm, and the y-axis is for the NDIC reporting in the same unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It can be seen that the flaring magnitudes in terms of the flared volumes are quite diverse for the different counties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To better visualize all of them, a zoomed-in view for each county is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It becomes clear that most of the counties except SLP and GV have more than ∼12 data points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' however, only the four counties in the top row (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', MCK, DUN, WIL and MTL) have the largest amount of data and indicate stronger positive correlations between VIIRS and NDIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the purpose of building county level models and investigating the heterogeneity among the counties, the no pooling option discussed in the previous section will fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Especially with counties SLP (which has 3 observations) and GV (which has 2 observations), if a linear 36 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: North Dakota County Abbreviations Abbreviation County MCK McKenzie County DUN Dunn County WIL Williams County MTL Mountrail County BOW Bowman County DIV Divide County BRK Burke County MCL McLean County BIL Billings County STK Stark County SLP Slope County GV Golden Valley County model such as Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2d is fitted, the learned slope parameters βcounty will have point estimates ˆβslp ≈ 0 and ˆβgv ≫ 0 with their associated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The interpretation of the slope parameter (which was discussed right after Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3) implies that such inferences are never possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Some other counties, even with more data points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', MCL), suffer from the noise levels in their observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Using their own dataset will frustrate accurate inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Therefore, in order to build models robustly at a county level, the hierarchical model discussed in the previous section is exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 Model Specification Motivated by the discussions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2, partial pooling is performed by explicitly modeling the entire population of counties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this way, the counties such as MCL can leverage the information from other counties to learn their own parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Counties with “strong data” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', very informative data which makes the likelihood dominate the structure of the posterior), such as those in the top row of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2, indicate a positive correlation between VIIRS and NDIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Therefore, a similar strategy as in Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 is adopted for the counties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', one set of slope and intercept is learned for each county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 NDIC county = MCK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 county = DUN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='125 county = WIL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='125 county = MTL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='020 NDIC county = BOW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0125 county = DIV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005 county = BRK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0015 county = MCL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='003 VIIRS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='003 NDIC county = BIL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0015 VIIRS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0015 county = STK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0010 VIIRS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00125 county = SLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0015 VIIRS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00150 county = GV Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2: Scatterplots of NDIC and VIIRS reportings for different counties, without sharing neither x- nor y-axis for all the subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Within each subplot, equal scaling and limits are set for x- and y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The axes’ meanings are the same as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since the slope and intercept are very interpretable, the meanings of which were discussed right after Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3, partial pooling is also enabled across parameter types (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', intercepts and slopes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In other words, knowing how much flared volume is missed from VIIRS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the information carried by the intercept) might improve learning how VIIRS and NDIC will covary (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the information carried by the slope).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Specifically, a population model with a multivariate normal density is used for the different counties’ parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The hierarchical model is specified through Expressions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5a–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5j: µα ∼ Half-Normal(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5a) µβ ∼ Gamma(2, 2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5b) 38 σα ∼ Half-Normal(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5c) σβ ∼ Half-Normal(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5d) σ ∼ Half-Normal(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5e) R ∼ LKJcorr(2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5f) Σ = � σα 0 0 σβ � R · � σα 0 0 σβ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5g) � αcounty βcounty � ∼ MVNormal � � � µα µβ � , Σ � � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5h) µj = αcounty[j] + βcounty[j] × VIIRSj (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5i) NDICj ∼ N(µj, σ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5j) where: µα is the average intercept for all the counties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' µβ is the average slope for all the counties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' σα is the standard deviation among different counties’ intercepts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' σβ is the standard deviation among different counties’ slopes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' σ is the the standard deviation in NDIC reporting within the counties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' R is the correlation matrix distributed according to an LKJ distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It is 2-by-2 in size and encodes the correlation between the intercepts and slopes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Σ is the covariance matrix for the population model, which is constructed by multi- plying the correlation matrix from both sides by a diagonal matrix of standard deviations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' αcounty and βcounty are the intercept and slope for each county, whose prior distributions are defined by a two-dimensional Gaussian population model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' county[j] (in the subscript) denotes the county index, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', county[j] ∈ {k ∈ N0 | k ≤ 11}, such that αcounty[j] and βcounty[j] are the intercept and slope for the j-th observation based on its county assignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' VIIRSj is the VIIRS reported volume of the j-th observation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 39 µj denotes the underlying flared volume of the j-th observation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' NDICj is the NDIC reported volume of the j-th observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The LKJ distribution due to Lewandowski, Kurowicka, and Joe (2009) is a distribution over positive-definite symmetric matrices with unit diagonals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', correlation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the model specification above, it directly influences the prior for the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Before it was introduced and when HMC was not widely applicable, the usual choices for modeling covariance matrices were Wishart or inverse-Wishart distributions, due to their nice conjugacy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, LKJ is better suited for modern Bayesian computational settings (Betancourt 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Lambert 2018) and therefore employed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' LKJ has a single parameter η, which can be interpreted as the shape parameter of a symmetric beta distribution (Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' As η gets larger, the prior is more skeptical of large correlations in the matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', providing regularizing effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The probability density of LKJ with a few η values are displayed in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this work, LKJcorr(η = 2) is chosen to define a weakly informative and regularizing prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 Density eta=1 eta=2 eta=4 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3: LKJcorr(η = eta) probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' As η increases, larger correlations become less plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5, while being expressive in the data generating process, is a centered parameter- ization of the hierarchical structure (Papaspiliopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this parameterization, 40 the hierarchical parameters (such as βcounty) and the lower-level parameters in the prior (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', µβ and σβ) are tightly coupled, and they are highly correlated in the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since this model involves complex geometries and interactions in the posterior, HMC is leveraged for sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' When there is not a lot of data (which is the case for the current NDIC and VIIRS reportings), this parameterization leads to very inefficient sampling and non- convergences (Stan Development Team 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The noncentered parameterization is preferable in these cases and therefore employed for building the county level models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 Model Reparameterization Reparameterization of hierarchical models can be applied to any distribution in the location-scale family, for which the normal distribution is a good candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the case of reparameterizing a multivariate normal prior, suppose the prior for θ is a multivariate normal with mean vector µ and covariance matrix Σ (such as Expression 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5h), then a noncentered parameterization is given by: �θ ∼ MVNormal(0n, In) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6a) ϕ = µ + L · �θ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6b) where �θ has the same dimensions as θ and all of its elements i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' according to N(0, 1), L satisfies L · L⊤ = Σ, and ϕ recovers the exact same prior distribution for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This reparameterization leads to more efficient sampling by reducing the dependence between µ, L, and �θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One choice for L is the Cholesky factor of Σ, which provides implementation convenience for the multivariate normal cases (Stan Development Team 2020) and is adopted in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The noncentered county level model is specified through Expressions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7a–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7j, with the reparameterized part (corresponding to Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5) highlighted in blue: µα ∼ Half-Normal(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7a) µβ ∼ Gamma(2, 2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7b) 41 σα ∼ Half-Normal(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7c) σβ ∼ Half-Normal(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7d) σ ∼ Half-Normal(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7e) L ∼ LKJCholeskyCov � η = 2, � σα σβ �⊺� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7f) � zα zβ � ∼ MVNormal � � � 0 0 � , � 1 0 0 1 �� � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7g) � αcounty βcounty � = � µα µβ � + L · � zα zβ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7h) µj = αcounty[j] + βcounty[j] × VIIRSj (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7i) NDICj ∼ N(µj, σ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7j) where: L is the Cholesky factor of the covariance matrix which has LKJ distributed correla- tions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' zα and zβ are the standardized intercept and slope for each county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The rest of the symbols have the same meaning as in Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The noncentered model imposes the exact same probabilistic structure as in Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5, and is implemented for making inference on each county’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 Model Fitting Four chains are sampled from the posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior distributions and trace plots for the slopes and intercepts are presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 and Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence have been achieved as shown by the trace plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To better compare and contrast the different counties’ parameters, the forest plots of 90 % highest density intervals (HDI) for the slopes and intercepts are given in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 and Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In both figures, counties are ordered by the VIIRS reported volumes, and those with the least amount of estimated volumes (such as SLP and GV) are at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The thin lines present the 90 % HDI’s and the thicker line segments stand for 42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='58 0 10 20 Posterior Density county MCK, slope 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='55 Sample Value county MCK, slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 Posterior Density county DUN, slope 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 Sample Value county DUN, slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='70 0 5 10 Posterior Density county WIL, slope 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 Sample Value county WIL, slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='80 0 5 Posterior Density county MTL, slope 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 Sample Value county MTL, slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': 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+page_content='0 Sample Value county DIV, slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0 2 4 Posterior Density county BRK, slope 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 Sample Value county BRK, slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0 2 4 Posterior Density county MCL, slope 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 Sample Value county MCL, slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0 2 4 Posterior Density county BIL, slope 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} 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+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0 2 4 Posterior Density county STK, slope 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 Sample Value county STK, slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0 2 4 Posterior Density county SLP, slope 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} 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+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0 2 4 Posterior Density county GV, slope 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 Sample Value county GV, slope Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4: Posterior distributions and trace plots of the slopes for each county.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 0 100 200 Posterior Density county MTL, intercept 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='015 0 50 100 Posterior Density county SLP, intercept 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 Sample Value county SLP, intercept 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='015 0 50 100 Posterior Density county GV, intercept 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 Sample Value county GV, intercept Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5: Posterior distributions and trace plots of the intercepts for each county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 44 the interquartile ranges (IQR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The points represent the posterior means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 county GV, slope county SLP, slope county STK, slope county BIL, slope county MCL, slope county BRK, slope county DIV, slope county BOW, slope county MTL, slope county WIL, slope county DUN, slope county MCK, slope 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0% HDI Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6: A forest plot showing the uncertainties around each county’s slope estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The counties at the bottom have insufficient or noisy datasets, therefore their estimates are largely pulled towards the partially-pooled mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 county GV, intercept county SLP, intercept county STK, intercept county BIL, intercept county MCL, intercept county BRK, intercept county DIV, intercept county BOW, intercept county MTL, intercept county WIL, intercept county DUN, intercept county MCK, intercept 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0% HDI Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7: A forest plot showing the uncertainties around each county’s intercept estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The dotted line labels the zero intercept, for which some counties’ estimates are not significantly different from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the case of the slopes (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6), it can be seen the top four counties are quite diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' MTL has the largest point estimate in the entire population (ˆβmtl > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6) while 45 DUN has the smallest one (ˆβdun < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Furthermore, the HDI’s for DUN and MTL rarely overlap, indicating that it is almost certain that MTL has a larger slope than DUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The counties with fewer observations (remaining eight counties) have greater uncertainties in their parameter estimates, while all of their point estimates are pulled towards the partially-pooled mean which is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' When there is not enough data for some counties, the hierarchical model strives to reinforce information sharing among different counties, thus providing more sensible results and also quantifying the uncertainties in such processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' From domain expertise, these results make more physical sense than the no-pooling estimates discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', ˆβslp ≈ 0 and ˆβgv ≫ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the case of the intercepts (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7), there is also heterogeneity among the counties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In particular, by plotting a dotted line labeling the zero intercept, some counties are found to likely have zero intercept (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', zero is covered by the IQR or HDI) while others have intercepts that are significantly different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It might not be surprising to get close-to-zero intercepts and greater uncertainties for those counties with less data (such as SLP and GV), however it is interesting to obtain the HDI for MTL that covers zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Recall that the intercept parameter can be interpreted as the NDIC reported volume which is not captured by VIIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This finding for MTL, along with the fact that MTL has the largest slope point estimate (where a larger slope denotes closer proximity to the satellite estimation), convinces the author that MTL used to have persistent and stronger gas flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' They kept VIIRS from missing the flaring events in general, and lead to the reported volumes from NDIC and VIIRS being closer to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' On the contrary, DUN’s smaller slope and larger intercept characterize its flares as sporadic and weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One thing worth mentioning is that, with the current interpretation of the intercept, it does not make much physical sense to have negative intercepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Although every county has positive point estimates for their intercepts, some counties’ HDI’s show coverage over the negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is a limitation of choosing a 2D Gaussian population model for the intercepts and slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since the 2D Gaussian is supported on R2, in the context of some counties having “weak data”, negative values make 46 an appearance in their HDI’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The discussions above naturally lead to the question of whether the slopes and intercepts are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It turns out that, by partially pooling the different types of parameters, a probable negative correlation between the slopes and intercepts is revealed (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The correlation is learned from the heterogeneity in flare characteristics among the counties: Persistent flares yield smaller intercepts and larger slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Sporadic flares yield larger intercepts and smaller slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In other words, intercepts and slopes covary in the entire population of counties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' By pooling information across parameter types, what the model learns in the intercept can improve learning about slopes, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' With this “experience” or “knowledge”, the hierarchical model will be able to quickly update its expectation for any new counties’ parameters even with just a few observations in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It should be noted that there is also some probability mass for the positive correlation values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the negative correlation is not very strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This could be due to that some counties do not have a lot of data at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior will be updated as more data is brought in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Finally, the parameter estimates are reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2, from which the parametric model for each county can be recovered, and then deployed in calibration and prediction usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 Model Extensibility Looking back at the hierarchical model and the reparameterization strategy from the previous sections, there are four potential deployment scenarios that are worth discussing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' They demonstrate the extensibility and flexibility of the chosen approach in the context of flaring data analytics: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' New counties are present in terms of the reported flaring statistics from both VIIRS and NDIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 Correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 Density prior posterior Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8: Correlation between the intercepts and slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue: Posterior distribution of the correlation, the mode of which is below zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Dashed: Prior distribution, the LKJcorr(2) density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' At this time, there are 12 counties that have reported flaring statistics from both VIIRS and NDIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' If flaring data becomes available for some other counties in the future, the hierarchical model allows the population to be immediately expanded to accommodate the new counties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This can be seen from the conditional structure in Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3: by taking a model for n + 1 counties p(θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , θn, θn+1, φ) = �n+1 � i=1 p(θi | φ) � p(φ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8) then pulling out the term for the (n + 1)-th county from the right-hand side (RHS) p(θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , θn, θn+1, φ) = p(θn+1 | φ) � n � i=1 p(θi | φ) � p(φ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9) it can be recognized that the remaining part on the RHS is the hierarchical model for n counties p(θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , θn, θn+1, φ) = p(θn+1 | φ) p(θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , θn, φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10) 48 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2: Parameter Estimates of County Level Flaring Model Parameter Variable County Point Estimate 90 % CI αcounty Intercept MCK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='019 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='015, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='023) DUN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='008 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='004, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='013) WIL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='007, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='013) MTL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='006) BOW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='015 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='013, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='017) DIV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005) BRK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='003 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='004) MCL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002) BIL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='003) STK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='004) SLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='007) GV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='009) βcounty Slope MCK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='519 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='493, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='542) DUN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='464 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='385, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='547) WIL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='549 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='495, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='605) MTL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='623 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='553, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='693) BOW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='516 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='370, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='677) DIV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='554 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='395, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='719) BRK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='556 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='389, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='715) MCL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='563 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='391, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='730) BIL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='560 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='395, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='727) STK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='562 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='393, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='729) SLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='561 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='406, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='752) GV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='560 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='398, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='731) This indicates the newly introduced counties will only depend on the population parameters φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', how the new counties interact with the existing ones (from the initial dataset) is not explicitly specified but being mediated through φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This mechanism allows the population (of counties) to be expanded arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In practice, without any modification, Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 can be re-fitted with the new dataset as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' More data are available for those counties which used to have very few observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the event of more data becoming available for those counties with wide HDI’s such as SLP and GV, the posteriors will be updated according to that information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Their 49 HDI’s would become narrower and narrower as more and more data are available, and since the hierarchical model pools information among the counties, these counties will contribute to updating the population model’s and other counties’ parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Similar to Item 1 above, Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 does not need modifying and can be re-fitted with the new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Sample sizes among counties become more unbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In general, when there is a lot of data for each county, the centered parameterization (Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5) is more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' When the sample size is not large, which is the case for the current VIIRS and NDIC reportings, the noncentered parameterization (Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7) is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, the parameterization for hierarchical models is not a monolithic tactic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' If the reported flaring data becomes very unbalanced across counties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', some counties have a huge amount of data whereas others have very little data, then each county can be parameterized differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' More specifically, For the counties that have strong data such that their likelihood functions dominate, centered parameterization can be applied through Expressions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5f–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the counties that have weak data such that their prior models dominate, noncentered parameterization can be applied through Expressions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7f–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' All in all, this is still one hierarchical model which defines the exact same probabilistic structure as Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 or Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7, but avoids inefficiencies and non-convergences in the sampling from posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Oilfield level heterogeneity needs to be examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Under the assumptions that the oilfields in North Dakota are exchangeable and the population of oilfields (which conduct flaring) can grow, the hierarchical model developed in this chapter can be directly applied to investigate the heterogeneity in different oilfields’ parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Following the reverse geocoding as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3, there 50 are 258 oilfields that have both NDIC and VIIRS reportings for the same study period as in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Some oilfields have very few observations and can benefit from the hierarchical model through pooling information among the entire population of oilfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Furthermore, due to the number of oilfields being relatively large, the population model could be learned with more ease (because more information is available for the population).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the case of the county level model developed in this chapter, since there are only 12 individuals (counties) in the population, some uncertainties about the population are inevitably present and reflected through the posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The models developed in this chapter, while capturing the heterogeneity among the different counties in North Dakota, rely on the assumption that all the monthly observations within a certain county are conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For situations where the temporal structure has to be taken into consideration, other types of models can be built and are discussed in the next chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 51 CHAPTER 5 FLARING TIME SERIES ANALYTICS “Were neural networks over-hyped, or have we underestimated the power of smoothing methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' I think both these propositions are true.” — MacKay (2003) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Learning the Flaring Pattern and Behavior In this chapter, the author develops a generic framework for revealing flaring patterns and behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The main challenges are fourfold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Observed data are noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Companies estimate the flaring volumes and conduct self-reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Satellites could miss some events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, having knowledge about the underlying process is vital in lots of situations including when the state and local governments need to make key decisions based on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the meantime, understanding the underlying process helps with anomaly detection by differentiating between true anomalies in reporting and ordinary noise or stochasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A probabilistic approach is desirable to be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A set of most probable functions (characterizing the underlying process) are preferable over one single best fit function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The observations of a certain entity are time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The temporal structure is intrinsic to the dataset and thus must be harnessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The framework should be generic enough for automated insights extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 52 There are more than 200 operators and 500 oilfields operating in North Dakota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Choosing a specific parametric form of model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', ARIMA or LSTM) for each entity and then fitting the model to the data is not only time consuming, but also prevents easy integration into automation pipelines (for extracting insights for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It is striking that the elegant properties of Gaussian process make it a natural choice to tackle all of these challenges and is therefore employed in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Gaussian Process A Gaussian process (GP) can be viewed as a distribution over infinite-dimensional Hilbert space of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It is formally defined as “a collection of random variables, any finite number of which have a joint Gaussian distribution” (Rasmussen and Williams 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Gaussian processes are extremely powerful nonparametric learning techniques, which provide a composite of flexibility and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' They are well suited to problems which necessitate principled handling of uncertainty and interpretation, in the presence of noisy and dynamic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Such scenarios include smoothing (Deisenroth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2012) and time series modeling (Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' They are also well established in different fields under various names, for example kriging in geostatistics and Kalman filters both correspond to Gaussian processes (MacKay 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this work, the motivation is to develop a generic framework for recognizing the underly- ing unknown processes f(x) which reflect flaring strategies and behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Thus inference is conducted directly in the function space employing GP as a prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A Gaussian process is com- pletely specified by its mean function m(x) and covariance function k(x, x′) (Bandyopadhyay 2018), which are defined as: m(x) = E[f(x)] , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) k(x, x′) = E[(f(x) − m(x))(f(x′) − m(x′))] , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2) 53 and the function distributed as a Gaussian process is denoted by f(x) ∼ GP � m(x), k(x, x′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Mean Function In this work, the mean functions are always chosen to be zero, since there is no prior knowledge on the mean of the latent processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the meantime, for GPs with a zero mean function, the mean of the posterior process is not confined to be zero (Rasmussen and Williams 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' All the latent functions modeled with a GP prior in this dissertation follow f(x) ∼ GP � 0, k(x, x′) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4) where k is some covariance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Covariance Function Covariance function, also known as kernel, is the crucial ingredient in a GP, as it encodes one’s assumptions about how the function should behave by defining similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The fun- damental assumption is that data points with inputs x which are close would have similar target values y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This assumption is usually very reasonable in areas including time series modeling, and it is theoretically backed by Tobler’s first law of geography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The covariance functions used in this dissertation include: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The Mat´ern class of covariance functions, which is given by: kν(r) = 21−ν Γ(ν) � √ 2ν r ℓ �ν Kν � √ 2ν r ℓ � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5) where Γ(·) is the gamma function, Kν is a modified Bessel function of the second kind of order ν, r =∥x − x′∥, and ℓ is the lengthscale controlling the smoothness from one perspective: large ℓ characterizes functions which change slowly and can be reliably extrapolated further away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The Mat´ern covariance functions can be written as a product of an exponential and a polynomial of order p, when ν is half-integer: ν = p+1/2, p ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The hyperparameter 54 ν controls the smoothness from another perspective: when ν = 1/2, the Mat´ern kernel becomes the exponential kernel (continuous but not differentiable);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' as ν → ∞, it becomes the exponentiated quadratic kernel (infinitely differentiable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Rasmussen and Williams (2006) argued that the most interesting cases for machine learning would be ν = 3/2 and ν = 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For gas flaring time series, as operators might change flaring strategy at any given time due to policy changes, gas processing facility deployment, gas price fluctuation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the latent process might not be as smooth as infinitely differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Instead the Mat´ern kernel is harnessed which is capable of inducing non-smooth function realizations to handle those discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Specifically the Mat´ern kernel with ν = 5/2 is chosen for this dissertation with the input space X ⊆ R1: kmat´ern52(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ℓ) := � 1 + � 5(x − x′)2 ℓ + 5(x − x′)2 3ℓ2 � exp � − � 5(x − x′)2 ℓ � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6) where x vary over the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The standard periodic kernel due to MacKay (1998): kperiodic(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' T, ℓ) := exp � −sin2(π|x − x′| 1 T ) 2ℓ2 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7) where T denotes the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This kernel is used for modeling seasonal behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The white noise kernel, which is given by: kWhiteNoise(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' δ) := δ2In, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8) where δ2 is the variance of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this dissertation, the usage of the white noise kernel is for stabilizing the computation of the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Adding a small value of diagonal shift will try to guarantee the resulting covariance matrix is always positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 55 A nice property is that the sum and product of the established kernels are still valid kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This fact is also exploited in the model building process in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Inference and Model Reparameterization In practice, one always works with a dataset of finite size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In such situations, a multivariate normal prior distribution is placed on the vector of function values f, f ∼ MVNormal(mx, Kxx) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9) where the vector mx and the matrix Kxx are the mean function and covariance function evaluated over the inputs x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A key question which has significant impact on the inference is how to learn the hyperpa- rameters from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A natural (and popular) approach is to conduct maximum likelihood estimation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', generating point estimates leveraging the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, as Betancourt (2017a) showed with experiment results, both regularized and unregularized maximum marginal likelihood have limited performance in terms of fitting robustly and recovering the true data generating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Technically, given a particular kernel with particular hyperparameters, a GP does not support an entire Hilbert space but only a slice through that space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' changing the hyperparameters by an infinitesimal amount yields a different slice which has no overlap with the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Therefore in this dissertation, a full Bayesian approach is taken for the GP inference, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the entire Hilbert space of functions is considered by taking into account all of the possible hyperparameters for a specific kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the class of problems which have Gaussian observation models, GP has nice closed-form posterior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, for the situations which do not have Gaussian observation models, for examples the ones in this dissertation which employ Student-t or Poisson likelihood, there does not exist analytical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' HMC as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 is used to sample from the posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Specifically, the noncentered parameterization of the latent multivariate Gaussian is exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The reparameterized model is 56 �f ∼ MVNormal(0n, In) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10a) L = Cholesky(Kxx) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10b) f = mx + L · �f (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10c) which defines the same distribution as Expression 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9 but induces a nicer posterior geometry for HMC to explore and sample from (Betancourt 2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Once the learning on hyperparameters is done, posterior predictive distribution of the latent function values which are not part of the original dataset is obtained by f∗ | f ∼ MVNormal � m∗ + K⊤ x∗K−1 xx(f − mx), K∗∗ − K⊤ x∗K−1 xxKx∗ � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='11) where m∗ is the mean function evaluated at the new inputs, K∗∗ is the covariance between the new inputs, and Kx∗ is the covariance between the original inputs and the new inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Suite of Models for Pattern Recognition This section presents models built from various angles, with the goal of providing a coherent framework for learning the flaring pattern and behavior in a principled manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Each model is tested on real flaring data from North Dakota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Whenever more granular analytics capabilities are demonstrated through investigations at oilfield level or operator level, the data from a major producing field, the Blue Buttes Oilfield (Alexeyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2017), and one operator, denoted by ‘Operator A’ are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Modeling Proportion of Gas Flared The proportion of gas production that is flared is an indicator of flaring intensity and energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It is interesting to investigate whether the proportion has changed over a period of time for certain operators and oilfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The model is specified through Expressions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12a– 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12i: ℓ ∼ Gamma(2, 1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12a) η ∼ Half-Cauchy(5) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12b) 57 ν ∼ Gamma(2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12c) ˆσ2 ∼ Half-Cauchy(5) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12d) k = η2 × kmat´ern52(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ℓ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12e) f ∼ GP(0, k) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12f) πi = logit−1� f(xi) � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12g) µi = πi × Gi (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12h) Fi ∼ Student-t(ν, µi, 1/ˆσ2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12i) where: ℓ is the lengthscale for the Mat´ern kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' η is the marginal deviation parameter controlling how strongly the latent functions vary in the output space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ν is the degrees of freedom for the Student-t likelihood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ˆσ2 controls the inverse scaling parameter of the Student-t likelihood (analogous to the precision of a Gaussian distribution);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' k is the covariance function for the GP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' f denotes the latent process, which is distributed according to the GP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' πi is the underlying flaring gas proportion of month i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since proportion is bounded between 0 and 1, the inverse-logit function is applied to the latent process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Gi is the total gas production of month i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' µi denotes the underlying flared volume of month i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Fi is the reported flared volume, which is modeled using a Student-t observation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The reasoning behind choosing a Student-t observation model is to make the model specification be able to generalize to as many entities as possible and be robust to (potentially many) outliers and noisy data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is due to the fact that at this time, operators have to estimate the flared volume by their own procedures and conduct reporting, in which case inaccuracies are introduced unintentionally or intentionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The heavier tail of Student’s 58 t-distribution is a natural decision in modeling to deal with those phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This line of thought, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', design models that are generic and robust, is indeed reflected in choosing the half-Cauchy priors (which are heavy-tailed and very weakly informative) and GP as a nonparametric regression technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To demonstrate this model’s capability on real data, both the Blue Buttes Oilfield and Operator A are tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The production and flared volumes coming from NDIC are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the oilfield, the posterior distributions and trace plots of the hyperparameters are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior predictive samples for the underlying process of gas flaring proportions (πi) are demonstrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2, which depict the trend very clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The colored bands have the below coverage for the posterior samples: The darkest colored band (in the center at a certain x location) represents the 49th percentile to 51st percentile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The lightest colored band (characterized by the widest interval at a certain x location) represents the 1st percentile to 99th percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Additionally, 30 random samples are drawn from the GP posterior and plotted on the same figure, showing as thin lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The latent functions do not go through all the observed data points, in which case the model would have been overfitted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' instead they present the possible functions which are most compatible with the data as well as the assumptions inherent in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' On one hand, the insights are already obtained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the underlying process is inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' On the other hand, this serves as an anomaly detection tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, the state government might be interested to look into that observed data in the second half of 2019 which deviated quite a lot from the “true” process, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' to audit the reporting for that month or to investigate what had happened that led to a sudden huge drop in flaring in just one month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' With the exact same model specification, the model is also run with the operator’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior distributions and trace plots of the hyperparameters are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 59 4 6 8 10 12 14 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Posterior Density 0 250 500 750 1000 1250 1500 1750 5 10 15 Sample Value 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='03 Posterior Density 0 250 500 750 1000 1250 1500 1750 0 50 100 Sample Value Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: Posterior distributions and trace plots for the Blue Buttes Oilfield gas flaring proportion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='40 2016 2017 2018 2019 2020 Report Date 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='40 Proportion of Gas Flared in Total Gas Produced Observed Proportion of Gas Flared Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2: Posterior predictive samples showing the gas flaring proportion variations at the Blue Buttes Oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue points are the observed data while red lines present the posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 60 The posterior predictive samples for the underlying process of gas flaring proportions (πi) are demonstrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It can be seen this operator’s flaring proportion time series is more jagged than the Blue Buttes Oilfield (which is operated by more than five companies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A operator can change flaring strategies more swiftly which can be captured as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Nevertheless the long-term trend is also available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Comparing Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 and Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3, it can be seen the posterior distributions are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However the priors for them were specified in the exact same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This showcases the power of Bayesian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Taking ℓ as an example, a Gamma(2, 1) prior is placed on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, after conditioning on the data, the operator model reports smaller lengthscale values on average (indicating jagged processes), whereas the oilfield model reports larger lengthscale values (suggesting smoother processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2 4 6 8 10 12 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 Sample Value _sq 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='03 Posterior Density 0 500 1000 1500 2000 0 25 50 75 100 Sample Value Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3: Posterior distributions and trace plots for the Operator A gas flaring proportion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Notice the differences between these inference results and those in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1, both of which are based on exactly the same priors and likelihood, demonstrating the model specification’s wide applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 2016 2017 2018 2019 2020 Report Date 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 Proportion of Gas Flared in Total Gas Produced Observed Proportion of Gas Flared Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4: Posterior predictive samples showing the gas flaring proportion variations of Operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue points are the observed data while red lines present the posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Order 24665, which is established by the North Dakota Industrial Commission, defines the gas capture percentage pcap as pcap = Gsold + Gused + Gproc Gprod , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='13) where: Gsold is the monthly gas sold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Gused is the monthly gas used on lease;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Gproc is the monthly gas processed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Gprod is the monthly gas produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since North Dakota bans the venting of natural gas (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Department of Energy 2019b), it is obvious the model developed in this section provides a powerful tool for NDIC to evaluate compliance with the gas capture goals: at a given month i, pcap = 1 − πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Furthermore, when looking at the model specification, there is nothing special that encodes the data sources and location information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A user of this model is free to use satellite estimation as the observed data or apply it to the Permian Basin, and conduct inference on the flaring proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is a benefit from using nonparametric and interpretable models as opposed to black box 62 models (such as the neural networks, in which case the learned weights and bias inside the network provide little or no domain insights).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The author hopes this section provides a comprehensive view in terms of how and why to use GP, with real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Models built and presented in later sections follow a similar flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Modeling Proportion of Wells Flaring The proportion of wells that conduct flaring in a month can reflect a company’s flaring strategy and is an indicator of flaring magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It is interesting to investigate how this indicator varies for a certain entity in a certain time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The model is specified through Expressions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14a–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14f: ℓ ∼ Gamma(2, 1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14a) η ∼ Half-Cauchy(5) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14b) k = η2 × kmat´ern52(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ℓ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14c) f ∼ GP(0, k) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14d) pi = logit−1� f(xi) � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14e) Wi ∼ Binomial(Ni, pi) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14f) where pi is the unobserved “true” proportion of wells that conduct flaring in month i, Ni is the total number of active wells in month i, and Wi is the observed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', estimated and reported by company) number of wells that conduct flaring in month i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The rest of the symbols have the same meaning as in Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To demonstrate this model’s capability on actual data, both the Blue Buttes Oilfield and Operator A are tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the oilfield, the posterior distributions and trace plots of the hyperparameters are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior predictive samples for the underlying process of well flaring proportion (pi) are demonstrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The visualization strategy (different colors represent different percentiles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=') is the same as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 63 2 4 6 8 10 12 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Posterior Density 0 250 500 750 1000 1250 1500 1750 5 10 15 Sample Value 0.' metadata={'source': 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1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 Sample Value Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5: Posterior distributions and trace plots for the Blue Buttes Oilfield well flaring proportion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='75 2016 2017 2018 2019 2020 Report Date 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='75 Proportion of Wells that Conducted Flaring Observed Proportion of Wells Flaring Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6: Posterior predictive samples showing the well flaring proportion variations at the Blue Buttes Oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue points are the observed data while red lines present the posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' With the exact same model specification, this model is also tested with the operator’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior distributions and trace plots of the hyperparameters are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior predictive samples for the underlying process of well flaring proportion (pi) are demonstrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Comparing the two sets of figures from the oilfield and the operator, it can be seen: 64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' With the same prior placed on the lengthscale ℓ, the oilfield model learns from the data and gives a posterior mode around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5, whereas the operator model gives a posterior mode around 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is also reflected in the posterior samples time series plot: the oilfield experienced some well flaring proportion changes in relative shorter time periods, whereas the operator underwent changes on a longer time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The oilfield’s posterior samples time series show narrower percentile bands while the operator’s show wider percentile bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is due to the fact that the operator chosen here had smaller number of wells than the oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since the binomial observation model is used for each month’s flaring well count, this naturally represents and quantifies the uncertainties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', binary data contains less information especially when the sample size is small), as well as aligns with the expectation that when there is more data, there should be less uncertainties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' when there is less data, there should be more uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0 1 2 Posterior Density 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 Sample Value Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7: Posterior distributions and trace plots for the Operator A well flaring proportion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Notice the differences between these inference results and those in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5, both of which are based on exactly the same priors and likelihood, demonstrating the model specification’s wide applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This really showcases how and why to encode domain expertise in flaring data analytics while exploiting machine learning models, which is also the reason to choose the Bayesian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One could fit a black box model either with target values Wi ∈ R, or without any probabilistic view (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', to optimize for the best deterministic function mapping in the 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='40 2016 2017 2018 2019 2020 Report Date 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='40 Proportion of Wells that Conducted Flaring Observed Proportion of Wells Flaring Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8: Posterior predictive samples showing the well flaring proportion variations of Operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue points are the observed data while red lines present the posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' hypothesis space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' But either of those would be fundamentally flawed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Domain expertise indicates the well count has to be a non-positive integer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', Wi ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Furthermore, neither the NDIC reporting nor the satellite estimation is ever produced in a noise-free environment, and therefore probabilistic modeling is a must.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Compared to frequentist machine learning, Bayesian learning is entirely probabilistic and gives one the capability and freedom to encode his/her domain expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Modeling Flare Detection Count Satellite detected flare count provides an unbiased indicator of flaring intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' How this indicator varies in a certain time period for a certain entity is valuable information to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The model is specified through Expressions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15a–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Essentially the latent process is modeled as a Gaussian Cox process (Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2009), where the Poisson process has varying intensity across time domain and a GP prior is placed on this intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ℓ ∼ Gamma(2, 1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15a) η ∼ Half-Cauchy(5) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15b) k = η2 × kmat´ern52(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ℓ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15c) 66 f ∼ GP(0, k) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15d) λi = exp � f(xi) � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15e) Ci ∼ Poisson(λi) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15f) where λi is the unobserved flaring intensity (“true” count) in month i and Ci is the reported VIIRS detection count in month i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since λi is bounded to be positive, the natural exponential function is applied to the latent process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The rest of the symbols have the same meaning as in Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the task of flaring pattern recognition, the author believes this approach (leveraging a Gaussian Cox process) is a nicer surrogate than a popular change point model presented in (Davidson-Pilon 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Salvatier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Stan Development Team 2020), which is specified by: e ∼ Exponential(re) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='16a) l ∼ Exponential(rl) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='16b) s ∼ Uniform(1, T) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='16c) Ci ∼ Poisson(i < s ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' e : l) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='16d) where e and l are the early and late rates respectively, re and rl controls the priors for the early and late rates, s is the change point, T is the total time period, and the rate in the Poisson likelihood is decided through a ternary conditional operator (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=':).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The reason is that, although this model could be generalized to more than one change point, its usage is restricted by the assumption that any period between two adjacent change points has a constant rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This limitation becomes obvious when analyzing the actual flaring data in the discussions below, and is a major disadvantage of the change point model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The Gaussian Cox process model is tested with the Blue Buttes Oilfield’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since only VIIRS data is used, the whole time series is analyzed beginning in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior distributions and trace plots of the hyperparameters are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior predictive samples for the underlying process of flare count (Ci) are demonstrated 67 in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The visualization strategy (different colors represent different percentiles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=') is the same as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' From the time series plot, it can be seen the observations from 2014 to 2017 can possibly be described by a change point model (with late 2015 being a potential change point), but the steady growth before and after that time span will frustrate accurate inference with such a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 10 15 20 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 Posterior Density 0 250 500 750 1000 1250 1500 1750 10 15 20 25 30 Sample Value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 Posterior Density 0 250 500 750 1000 1250 1500 1750 1 2 3 Sample Value Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9: Posterior distributions and trace plots for the Blue Buttes Oilfield flare count model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5 10 15 20 25 2013 2014 2015 2016 2017 2018 2019 Report Date 5 10 15 20 25 Detection Rate Observed Detection Count Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10: Posterior predictive samples showing the flare count variations at the Blue Buttes Oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue points are the observed data while red lines present the posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 68 This model’s inference results serve as a type of confirmation, if not evidence, in terms of whether or not an entity achieves the goal/target in reducing the number of wells flaring, when the detection count is used as a surrogate for the number of wells flaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In practice, reducing the number of wells flaring is exactly the second goal of the regulatory policy introduced by the North Dakota Industrial Commission in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' If the state government is interested in this order’s effectiveness from a macroscopic standpoint, the model can also be used to conduct inferences with the state level data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this case, the posterior distributions and trace plots of the hyperparameters are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior predictive samples for the underlying process of flare count (Ci) are demonstrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 10 12 14 16 18 20 22 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 Posterior Density 0 500 1000 1500 2000 10 15 20 Sample Value 2 3 4 5 6 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 Posterior Density 0 500 1000 1500 2000 2 4 6 Sample Value Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='11: Posterior distributions and trace plots for the North Dakota flare count model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Notice the differences between these inference results and those in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9, both of which are based on exactly the same priors and likelihood, demonstrating the model specification’s wide applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The percentile bands in this case are quite narrow, which indicate greater confidence in the inferences about the data generating process given the model assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' By not (over)fitting to each and every observation, interesting patterns are discovered, for example in every year there is one and only one peak that happened around June.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It is worth pointing out that there is no model that can tell the modeler if his/her assumptions are good, only domain expertise might.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This model employing a Poisson observation model could be considered “rigid” due to the fact that a Poisson likelihood has only one parameter λ (to 69 200 300 400 500 600 2013 2014 2015 2016 2017 2018 2019 Report Date 200 300 400 500 600 Detection Rate Observed Detection Count Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12: Posterior predictive samples showing the flare count variations in North Dakota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue points are the observed data while red lines present the posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' control both the mean and variance) and, furthermore, when λ is large as in this scenario, a Poisson distribution is well approximated by a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Whenever the state government believes that overdispersion might exist, other observation models such as the negative binomial distribution could be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In such cases, only Expression 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15f needs to be changed to the negative binomial likelihood, with a prior added for the overdispersion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The specific parameterization is given by Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This really showcases both the flexibility and interpretability of taking a Bayesian approach for high-stakes decision making areas including flaring data analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 Modeling Proportion of Oil Flared As crude oil (as opposed to natural gas) is the main commodity at this time, the amount of gas in a barrel of oil equivalent (BOE) that is flared provides an indicator of production efficiency due to flaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this work, the normalized quantity, proportion of oil production being flared, is used such that the model specification is generic for large and small entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The model is specified through Expressions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17a–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17j: ℓ ∼ Gamma(2, 1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17a) 70 η ∼ Half-Cauchy(5) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17b) ν ∼ Gamma(2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17c) ˆσ2 ∼ Half-Cauchy(5) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17d) k = η2 × kmat´ern52(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ℓ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17e) f ∼ GP(0, k) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17f) πi = logit−1� f(xi) � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17g) µi = πi × Oi (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17h) c = 6 Mcf 1 BOE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17i) Fi/c =: Ei ∼ Student-t(ν, µi, 1/ˆσ2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17j) where: πi is the underlying flaring BOE proportion of month i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Oi is the total oil production of month i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' µi denotes the “true” flared BOE of month i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' c denotes the conversion factor that 6 Mcf equals 1 BOE, given by the United States Geological Survey (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Ei is the reported flared BOE, which is modeled using a Student-t observation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The rest of the symbols have the same meaning as in Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To test this model’s performance on real data, both the Blue Buttes Oilfield and Operator A are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the oilfield, the posterior distributions and trace plots of the hyperparameters are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior predictive samples for the underlying process of BOE flaring proportion (πi) are demonstrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The visualization strategy (different colors represent different percentiles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=') is the same as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' With the exact same model specification, this model is also tested with the operator’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior distributions and trace plots of the hyperparameters are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior predictive samples for the underlying process of BOE flaring proportion (πi) are demonstrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Comparing the two sets of figures from the oilfield and the operator, it can be observed: 71 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 Posterior Density 0 250 500 750 1000 1250 1500 1750 5 10 15 20 Sample Value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': 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Posterior Density 0 250 500 750 1000 1250 1500 1750 0 50 100 Sample Value Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='13: Posterior distributions and trace plots for the Blue Buttes Oilfield BOE flaring proportion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 Proportion of BOE Flared in Total Oil Produced Observed Proportion of BOE Flared Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14: Posterior predictive samples showing the BOE flaring proportion variations at the Blue Buttes Oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue points are the observed data while red lines present the posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 72 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': 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+page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='035 0 50 100 Posterior Density _sq 0 250 500 750 1000 1250 1500 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='03 Sample Value _sq 0 20 40 60 80 100 120 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='03 Posterior Density 0 250 500 750 1000 1250 1500 1750 0 50 100 Sample Value Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15: Posterior distributions and trace plots of the BOE flaring proportion model for Operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Notice the differences between these inference results and those in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='13, both of which are based on exactly the same priors and likelihood, demonstrating the model specification’s wide applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' With the same prior placed on the lengthscale ℓ, which has a mean of 2 (months), both models have updated the posterior to move away from this mean, reflecting a long range variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The oilfield has a posterior mode about 1 year while the operator has a mode around 15 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The operator has much larger reporting variability, shown by the parameter ˆσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' With a Student-t likelihood, both models demonstrate robustness to outliers and overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This can be seen from the oilfield’s late 2019 observations and the operator’s early 2016 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the posterior function samples, shown as the thin lines, some of them are indeed pulled towards those “outliers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, the percentile plots 73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='09 2016 2017 2018 2019 2020 Report Date 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='09 Proportion of BOE Flared in Total Oil Produced Observed Proportion of BOE Flared Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='16: Posterior predictive samples showing the BOE flaring proportion variations of Operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue points are the observed data while red lines present the posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (shown as the colored bands) are not impacted and those really can be interpreted as the trend which is most compatible with the data and the assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This built-in Occam’s razor of the Bayesian approach when choosing appropriate priors is very impressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In many of the frequentist machine learning methods, if the regularization strategy is not implemented well especially when the sample size is not huge enough for the asymptotic properties to kick in, outliers become “influential observations” that will have a huge undesirable effect on the inference results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 Modeling Scale Factor between VIIRS and NDIC Both NDIC and VIIRS reporting give (estimated) flared gas volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The scale factor between the two sources provides insights into whether NDIC reporting is consistent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' for different entities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', among a group of operators), and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' for one entity when looking at a certain time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is based on the fact that the satellite detection processing algorithm is unbiased and consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Item 2 is particularly interesting in terms of time series analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The model is specified through Expressions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18a–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18n: 74 ℓmat ∼ Gamma(8, 2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18a) ηmat ∼ Half-Cauchy(5) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18b) T ∼ N(12, 1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18c) ℓper ∼ Gamma(4, 3) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18d) ηper ∼ Half-Cauchy(5) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18e) ν ∼ Gamma(2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18f) ˆσ2 ∼ Half-Cauchy(5) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18g) kmat = η2 mat × kmat´ern52(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ℓmat) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18h) kper = η2 per × kperiodic(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' T, ℓper) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18i) kwn = kWhiteNoise(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' δ = 1e−6) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18j) f ∼ GP(0, kmat + kper + kwn) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18k) βi = exp � f(xi) � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18l) µi = βi × VIIRSi (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18m) NDICi ∼ Student-t(ν, µi, 1/ˆσ2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18n) where: ℓmat is the lengthscale for the Mat´ern kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ηmat is the marginal deviation for the Mat´ern kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' T is the period for the periodic kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ℓper is the lengthscale for the periodic kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ηper is the marginal deviation for the periodic kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' kmat is the Mat´ern kernel (component);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' kper is the periodic kernel (component);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' kwn is the white noise kernel (component);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' f denotes the latent process, which is distributed according to a GP whose covariance function is the sum of 3 kernels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' βi is the underlying scale factor between VIIRS and NDIC of month i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since this scale factor is bounded to be positive, the natural exponential function is applied to 75 the latent process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' VIIRSi is the VIIRS reported volume of month i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' µi denotes the underlying flared volume of month i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' NDICi is the NDIC reported volume of month i, which is modeled using a Student-t observation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The rest of the symbols have the same meaning as in Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The reason for adding a periodic kernel is to investigate if there are any seasonal patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Maintaining a proper Bayesian workflow lets the data speak for itself, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', whether there exists seasonal behaviors or not, as shown by the two case studies in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The model is first fitted with the state level data to investigate the macroscopic reporting consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior distributions and trace plots of the hyperparameters are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior predictive samples for the underlying process of the scale factor variations (βi) are demonstrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The visualization strategy (different colors represent different percentiles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=') is the same as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' From the posterior time series plot, it can be seen in general the volumes from NDIC reporting is smaller than that of VIIRS reporting, except for the times when the total flaring magnitude was small (indicated by the smaller points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' More importantly, within each and every year from 2015 to 2018, there is a decreasing trend in the values of the scale factor (βi) around midyear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Each year’s latent process from Q2 to Q3 can be viewed as a “seesaw”, with July being the middle pivot point and the months after July always going down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Note that within each year, the NDIC reporting of flared volumes might increase steadily or a lot (which was actually happening from the time series plot in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7), however this scale factor declining trends indicate the satellites observed much greater flaring activities than what was reported by the companies!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This finding suggests that the NDIC reporting is very likely not consistent throughout the year, and the state government should be concerned that some companies might underreport their flared volumes especially in the second half of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 76 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': 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_per 0 500 1000 1500 2000 0 1 2 3 Sample Value _per 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0030 0 500 1000 1500 Posterior Density _sq 0 500 1000 1500 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='003 Sample Value _sq 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 Posterior Density 0 500 1000 1500 2000 0 25 50 75 100 Sample Value Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17: Posterior distributions and trace plots for the North Dakota VIIRS-NDIC scale factor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 2015-07 2016-01 2016-07 2017-01 2017-07 2018-01 2018-07 2019-01 Report Date 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Scale Factor Applied to Satellite Estimation VIIRS_bcm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18: Posterior predictive samples showing the scale factor variations of North Dakota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue points are the observed data while red lines present the posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Larger points indicate greater flaring magnitude as observed from VIIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A interesting question arises: is this seasonal behavior universal across all the entities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The answer is unfortunately no, which indicates some operators likely reported their flared volume in an inconsistent manner throughout the entire year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In fact, if the Blue Buttes Oilfield data is used to fit the model, rather consistent behavior is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this case, the posterior distributions and trace plots of the hyperparameters are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The posterior predictive samples for the underlying process of the scale factor variations (βi) are demonstrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' With the exact same model specification incorporating the periodic kernel, no apparent seasonal behaviors are discerned by the inference process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' There are much uncertainties around the time of early 2016, where the point sizes indicate the overall flaring magnitudes were small as observed from VIIRS, and the NDIC reported volumes were actually larger than that of VIIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This could be due to the truncation effects instead of the reporting inconsistencies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', when the flares are sporadic and weaker, they are not easily captured by the satellites, resulting in a truncated sample for the VIIRS processing workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' By applying this model and workflow to the other major producing fields, it will likely pick up the ones who have the “seesaw” behaviors in their reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 78 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Posterior Density _mat 0 1000 2000 3000 4000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} 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+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0 1 2 Posterior Density _mat 0 1000 2000 3000 4000 0 1 2 Sample Value _mat 9 10 11 12 13 14 15 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 Posterior Density T 0 1000 2000 3000 4000 10 12 14 16 Sample Value T 1 2 3 4 5 6 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 Sample Value _per 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 1e 5 0 100000 200000 300000 Posterior Density _sq 0 1000 2000 3000 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 Sample Value 1e 5 _sq 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='03 Posterior Density 0 1000 2000 3000 4000 0 25 50 75 100 Sample Value Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='19: Posterior distributions and trace plots for the Blue Buttes Oilfield VIIRS-NDIC scale factor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Notice the differences between these inference results and those in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17, both of which are based on exactly the same priors and likelihood, demonstrating the model specification’s wide applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 1.' metadata={'source': 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+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 Scale Factor Applied to Satellite Estimation VIIRS_bcm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='075 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20: Posterior predictive samples showing the scale factor variations in the Blue Buttes Oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue points are the observed data while red lines present the posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Larger points indicate greater flaring magnitude as observed from VIIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 Predicting NDIC Flared Volume GP is not only fully capable of making predictions once the model hyperparameters are learned, but it can provide rigorously constructed intervals quantifying uncertainties as well through Expression 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='11, for which many of the frequentist machine learning methods fail to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The author chooses to present one particular prediction case study, that is to predict NDIC reported volume based on the projected scale factor between VIIRS and NDIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This will be a particular interesting deployment scenario once fast satellite detection/estimation is available, which takes less time than waiting on company reports followed by compiling everything into an analytics-ready format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The predictions are generated in the form of posterior predictive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Along with the historical observations, the predictions of the scale factor for the next six months are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The very wide percentile bands in the forecasting indicate that the seasonal behaviors will likely take effect again, however with great uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' If point predictions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', without the prediction intervals) are needed, one can always use the posterior mean, mode, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' to construct that “best” function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' however this showcases why predicting 80 the future is generally very difficult and uncertainties should always be properly characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 2015-07 2016-01 2016-07 2017-01 2017-07 2018-01 2018-07 2019-01 2019-07 Report Date 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Scale Factor Applied to Satellite Estimation VIIRS_bcm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='21: Posterior predictive samples showing predictions of the scale factor for the next six months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue points are the observed data while red lines present the posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Larger points indicate greater flaring magnitude as observed from VIIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 A Look Back at the Prior Choices Looking back at the suite of models developed, the set of priors for the latent functions have been the same (except the scale factor model where a periodic kernel is added).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However the posteriors are all updated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', “learned”) based on each dataset and modeling goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This means the below set of priors ℓ ∼ Gamma(2, 1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='19a) η ∼ Half-Cauchy(5) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='19b) k = η2 × kmat´ern52(x, x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' ℓ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='19c) f ∼ GP(0, k) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='19d) serves as a generic framework and can be recommended for flaring time series analytics in general, in a GP context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Notice this prior choice gives latent function values in the unconstrained space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', f(x) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, in many situations, the domain expertise 81 indicates the quantities of interest live in constrained space, such as: R>0 for Poisson rate parameter when modeling count data, and [0, 1] for binomial success probability when modeling flaring well proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To better reflect the domain expertise, the link functions can be leveraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the above scenarios, the log link function and the logit link function can be applied, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Although this prior configuration is the result of several design iterations and tested with real data, there is no reason to think that it is optimal for every entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Indeed, the model for scale factor between VIIRS and NDIC has bespoke components in its priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The Stan Development Team (2020) also gave some general prior choice recommendations for GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The whole suite of models demonstrate full capability of harnessing the temporal structure in flaring time series at different levels for different entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This provides huge potential for extracting insights from noisy monthly data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the situations where cross-sectional data analytics is desirable, for example when the latest monthly data is available and the state government needs insights from merely that month (before appending it to the whole historical data for a longitudinal study), other types of models can be built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Such is discussed in the next chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 82 CHAPTER 6 UNSUPERVISED LEARNING FROM MULTIPLE PERSPECTIVES “Estimation of densities is a universal problem of statistics (knowing the densities one can solve various problems).” — Vapnik (2000) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Learning the Distribution In this chapter, the author studies how to describe the flaring related quantities’ distribu- tion among the oilfields in North Dakota in a cross-sectional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' That is, data collected for one point or a period of time (such as a certain month or quarter) is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this setting, the data used for learning is unlabeled: U = {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , xN} , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1) where xi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , N, are the observations for the i-th oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Thus unsupervised learning is naturally applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The model to be learned is in the form of a conditional probability distribution Pθ(x | z) where z is some latent structure and θ represents the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This has many application scenarios in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' When the latest month’s or quarter’s data is available, the government of North Dakota might need distributional insights of the population (of oilfields), preferably beyond some forms of the order statistics (such as the five-number summary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This cross-sectional study is especially valuable and worth conducting when a direct comparison with previous months/quarters (which can be either the immediately previous one, or the same month/quarter in previous years) is desirable, or deeper understanding of the population is needed, such as looking for potential clusters among the entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 83 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Probability Model Estimation The task of learning distributions is a probability model estimation problem in unsu- pervised settings (Li 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It sometimes takes the form of density estimation, which is considered by some statisticians as the most fundamental topic in probabilistic machine learning (Yu 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A basic and common technique, the histogram, can be easily misused which leads to biased understanding of the dataset (Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 15 10 5 0 5 10 15 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14 p(x) 15 10 5 0 5 10 15 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 p(x) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: Effective usage of histograms can be surprisingly subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' With the exact same dataset adapted from (VanderPlas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2012), the two histograms with different bin sizes demonstrate different multimodal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Accepting some default configuration from some software package yields only one view of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In general, assuming that the data is generated by a probability model, the structure and parameters of that model are learned from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The type of the structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the set of possible probability models is usually given (assumed), while the specifics of the structure and the parameters have to be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The goal is to find the model structure and the parameters which are most likely to have generated the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The probability model can be a mixture model or a graphical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this dissertation, the mixture model is considered, where the assumption is that data comes from a mixture of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Mathematically, mixture models describe a distribution p(x) by a convex combination of K base distributions: 84 p(x) = K � k=1 πkpk(x) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2a) K � k=1 πk = 1, πk ≥ 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2b) where pk are the components in the mixture and πk are the mixture weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Mixture models can be interpreted as the overall population being a combination of distinct subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Mixture models can be generalized to the continuous cases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, both the negative binomial distribution and Student’s t-distribution can be thought of a mixture of some continuous distributions (Martin 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the model representation Pθ(x | z), x stands for the observations which can be discrete or continuous quantities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' z represents the latent structure which is a discrete random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The model is parameterized by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' When the model is assumed to be a mixture type, z represents the different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The knowledge of the model structure and parameters are learned from the data U = {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , xN}, where in this work xi ∈ X ⊆ R1, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , N, is the observation for the i-th oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Modeling VIIRS Detection Count In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3, methods are developed for analyzing the time series of VIIRS detection count for any given oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This section tackles the problem of how to extract insights from any given month’s flare detection count in North Dakota’s oilfields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Specifically, by learning from each oilfield’s detection count, the population of the oilfields is summarized, through which the state government can gain distributional insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Following the general form in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2, this problem becomes a special case that the latent structure z does not exist, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', satisfying Pθ(x | z) = Pθ(x), where x represents the detection count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It is when estimating conditional probability distributions becomes estimating probability distributions, therefore, only estimating the parameters of Pθ(x) is enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Density estimation in classical statistics, for instance the Gaussian parameters 85 estimation, is an example of such scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since the count data is modeled, the author compares the four observation models below with many randomly chosen months’ data: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Poisson likelihood 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Negative binomial likelihood 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Zero-inflated Poisson (ZIP) likelihood 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Zero-inflated negative binomial (ZINB) likelihood Items 3 and 4 above are experimented with because many of the oilfields in North Dakota did not have detection records from VIIRS for a given month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Therefore, zero-inflated models are tried as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Through the posterior predictive checks, it is found that the negative binomial observation model fits data in the most compatible manner, which is employed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The model is specified through Expressions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3a–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3c: µ ∼ Gamma(2, 1) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3a) φ ∼ Exponential(1) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3b) Ci ∼ NegBinomial(µ, φ) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3c) where Ci denotes the detection count for the i-th oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The probability mass function of the negative binomial likelihood is parameterized by a location parameter µ ∈ R>0, and an overdispersion parameter φ ∈ R>0, in the following way: P(X = n | µ, φ) = Γ(φ + n) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Γ(φ) � µ µ + φ �n � φ µ + φ �φ for n ∈ N0 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4) where Γ(·) is the gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Through this parameterization, the expectation and variance of a random variable X ∼ P are: E[X] = µ and V[X] = µ + µ2 φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5) 86 As the negative binomial distribution describes a Poisson random variable whose rate parameter is gamma distributed, and due to the fact that Poisson(µ) has variance µ, the learned parameters provide nice interpretations for the state government: µ indicates a mean intensity from the detection count’s perspective, just like the interpretation of a Poisson’s rate parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The larger the value of µ, the more flare detections are present on average at an oilfield level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' φ indicates the heterogeneity among the oilfields in North Dakota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Specifically, µ2/φ is the additional variance above that of a Poisson with rate µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The smaller the value of φ, the more oilfields with extreme detection counts (away from µ) are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To demonstrate this model’s compatibility with the observations, the data from October 2018 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' There are 506 oilfields in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The distribution of the detection count for all the oilfields is illustrated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 0 5 10 15 20 25 Number of Detections in an Oilfield 0 50 100 150 200 250 300 350 Count Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2: A histogram for the distribution of the oilfield detection counts from October 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' There are lots of zeros (more than 350) and a few oilfields have relatively high detection counts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', greater than or equal to 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 87 After fitting Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3, the posterior distributions and trace plots of the hyperparameters are presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The parameter estimation results are reported in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8 0 1 2 3 Posterior Density 0 500 1000 1500 2000 2500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 Sample Value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='26 0 5 10 15 20 Posterior Density 0 500 1000 1500 2000 2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='25 Sample Value Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3: Posterior distributions and trace plots for the oilfield detection counts distribution, fitted with the data from October 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Well mixing and convergence have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: Parameter Estimates of Oilfield Detection Count Distribution Parameter Variable Point Estimate 90 % CI µ Intensity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='814, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='200) φ Heterogeneity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='168 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='135, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='202) The point estimate for the intensity parameter µ is relatively small (ˆµ ≈ 1), which possibly results from the model being overwhelmed by the large number of zero counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, by inspecting the histogram from Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2, the tail of the distribution definitely extends far beyond ˆµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Therefore, posterior predictive checks are performed to scrutinize Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3’s compatibility with the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' These types of checks substantially harness the information from the samples drawn from the posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' By combining the uncertainty about the parameters, as described by the posterior, with the uncertainty about the outcomes, as described by the likelihood, the generative model is employed to simulate the implied observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Subsequently, posterior predictive plots are generated to display the model-based predictions along with the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Such a plot for the detection count distribution model is given in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 88 0 5 10 15 20 25 Number of Detections in an Oilfield 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 Density Posterior Predictive Simulation Observed Data Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4: Histograms for the distribution of the oilfield detection counts from October 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blue: original data observed from VIIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Gray: posterior predictive simulation results obtained from Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Each set of the simulation results is plotted using gray with transparency via alpha blending (setting α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15), such that the darker gray on the histograms indicates the simulated data which is more aligned with the model’s expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4, the histograms for the original VIIRS observations, as well as all of the posterior predictive simulations are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Each set of the parameter values (of µ and φ) are used in simulating one synthetic snapshot of the oilfields in North Dakota for October 2018, and there are in total 12,000 snapshots (constructed by the samples from the four Markov chains, each of which was setup for 3000 sampling iterations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Every histogram is visualized through an unfilled line chart, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', rendering the “step” histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Through Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4, it appears that the model is very compatible with the observations from October 2018, in that there is no obvious and consistent discrepancy between the observed and simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To delve into the tail behaviors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', beyond the zero count, a zoomed-in view is depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A few discrepancies are observed from this view, for example, when the count Ci = 11 and Ci = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One thing to note is that, with such a low mean (ˆµ ≈ 1), even with a relatively large overdispersion (ˆφ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2), the model would still be surprised by the high detection count, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', when Ci ≥ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 89 0 5 10 15 20 25 Number of Detections in an Oilfield 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12 Density Posterior Predictive Simulation Observed Data Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5: Histograms for the distribution of the oilfield detection counts from October 2018, with the y-axis clipped to better present those counts which are greater than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The legend with the associated color scheme is the same as in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The thorough performance of Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 that is characterized by a negative binomial likelihood, and the complicatedness of the real data manifest themselves through the posterior predictive checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' As discussed earlier in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3, the negative binomial likelihood was compared with three other likelihoods (Poisson, ZIP and ZINB) on many randomly chosen months, and found to outperform them in terms of the compatibility with the data in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In fact, there are some months’ data that are distributed in a “cleaner” way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', almost perfectly described by Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The author chooses not to cherry-pick those data, in the hope of not misleading the readers about the performance of the developed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Nevertheless, the simplicity, interpretability, and effectiveness of Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 proves itself in the mission of modeling detection count distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In practice, the state government can benefit from this model in the two use cases below: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' When the latest month’s data becomes available, Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 can be fitted to obtain an estimate for µ and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' These parameter estimates along with the credible intervals can be compared with those from the earlier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the case of the discussions above, the 90 learned parameters can be compared either with August/September from 2018, or with October from 2016/2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' From the comparison, it provides insights into whether there are more detection counts on average (characterized by a larger µ), or if more oilfields with an atypical number of detections are spotted (characterized by a smaller φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' After the model is fitted, it is recommended to perform the posterior predictive checks as demonstrated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 and Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5, to identify any issues of the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The list of the oilfields which have large deviations from the simulated data, especially those on the far tail (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', when Ci ≥ 20), are worth tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' That is, to investigate whether the “anomalies” from each month are random samples from the population or do not change from month to month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This provides further understanding of how the oilfields population behave, from the perspective of the detection count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A distributional summary of the detection counts exhibits only one facet of the flaring landscape, while the flared volumes distribution provides another crucial one, which is discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 Modeling Flared Volume In this section, the VIIRS estimated flared volumes for different oilfields are studied from a distributional point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The dataset from a three-month period is analyzed for demonstration purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Specifically, following the reverse geocoding as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3, all the oilfields’ cumulative flared volumes during Q4 2018 are computed and compiled for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' There are in total 152 oilfields that have VIIRS reported volumes in this time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The data is highly skewed (Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Therefore, for each oilfield, the order of magnitude of the flared volume (in bcm) is computed for the analysis, instead of working with the original absolute volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' From an applied perspective, taking the log of a measure converts the measure into magnitudes (McElreath 2015), which is applied to each oilfield’s flared volume: 91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='150 Flared Volume (in bcm) of Different Oilfields in Q4 2018 0 10 20 30 40 50 60 70 80 Count Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6: Histogram for the distribution of the oilfield flared volumes from Q4 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Most of the oilfields have relatively small flared volumes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 bcm), while a few oilfields have volumes that are greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 bcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Li = log(Fi), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6) where Fi is the original flared volume in bcm, and Li is the flared volume magnitude, both of which are for the i-th oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this dissertation, base e is always used for the logarithm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', natural logarithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A univariate distribution of the magnitudes is visualized in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Among the three approaches used to visualize the distribution, only the rug plot does not lead to subtleties due to the hyperparameters used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, as a 1D scatter plot, its representation ability is naturally limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The histogram suffers from the problem as illustrated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The curve is generated by kernel density estimation (KDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For a given dataset as defined in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1, KDE represents the underlying distribution as: ˆp(x) = 1 Nh N � i=1 K �x − xi h � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7) where K(·) is a kernel function and h is a bandwidth parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To generate Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7, the Gaussian kernel is used, which is given by: 92 12 10 8 6 4 2 0 log_bcm of Different Oilfields in Q4 2018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='200 Density Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7: Distribution of the oilfield flared volume magnitudes from Q4 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The rug plot marks the value for each oilfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The histogram is generated with nine bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The curve displays a Gaussian kernel density estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' K(z) = 1 √ 2π exp � −z2 2 � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8) and h is chosen based on Scott’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since the bandwidth plays a similar role as the bin size in histograms, KDE can also lead to the same issue as in histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Nevertheless, all three (the rug plot, histogram and KDE) agree that a single Gaussian approximation of the density which generates this data would be a poor approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Therefore, Gaussian mixture model (GMM) is employed to represent the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' the base distributions in Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 are chosen to be Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' GMM provides more expressive modeling capabilities and also possibilities for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Model Specification As discussed earlier, since the flared volume is a continuous quantity, density estimation is applicable and tackled with GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' At first, the data generating process is considered, which paves the way for potential clustering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' That is, each data point Li (defined in 93 Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6) is assumed to be generated by exactly one mixture component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The number of components, K, is unknown, and up to seven components are tried to fit the dataset visualized in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A relatively small number of components are experimented, because as the number of components increases, it becomes more difficult to interpret the modeling results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The model is specified through Expressions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9a–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9i, ∀K ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , 7}: α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , αK) = 6 · 1K (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9a) p ∼ Dirichlet(α) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9b) zi ∼ Categorical(p) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9c) l1 = min{L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , Ln} (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9d) l2 = max{L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , Ln} (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9e) �µk = l1 + (k − 1) �l2 − l1 K − 1 � , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , K (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9f) µk ∼ N(�µk, 2), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , K (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9g) σk ∼ Half-Normal(2), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , K (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9h) Li | (zi = j) ∼ N(µj, σj) j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , K} (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9i) where: α is the vector of concentration parameters for the Dirichlet distribution, which is a multivariate generalization of the beta distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' p is the simplex of probabilities for the mixture components, which is assigned a Dirichlet prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This prior with each value inside α being 6, is a weakly informative prior, expecting any pk inside p could be bigger or smaller than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Ten random draws from Dirichlet([6, 6, 6, 6, 6, 6, 6]) are displayed in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' zi is the probable mixture component that the i-th oilfield belongs to;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' l1 and l2 are the lower and upper bound for {Li}n i=1, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' �µk is used in “initializing” the location of the k-th mixture component, and {�µk}K k=1 essentially represent the K evenly spaced points between [l1, l2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 94 µk is the mean for the k-th Gaussian component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' σk is the standard deviation for the k-th Gaussian component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Li is the flared volume magnitude of the i-th oilfield, which is generated by the mixture component zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 1 2 3 4 5 6 7 Index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='30 Probability Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='8: Ten random draws from a Dirichlet prior with α = (6, 6, 6, 6, 6, 6, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One draw is highlighted to show that this prior is weak, in that it does not force all the probabilities (in any single draw) to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9, while unambiguously expressing the assumed generative process, relies on sampling the discrete latent variables zn, which is controlled by a categorical mixing dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This reliance causes slow mixing and ineffective exploration of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' An equivalent parameterization which addresses these problems is to marginal- ize out the z parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The marginalized model is specified through Expressions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10a–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10h, ∀K ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , 7}: α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , αK) = 6 · 1K (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10a) w ∼ Dirichlet(α) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10b) l1 = min{L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , Ln} (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10c) l2 = max{L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , Ln} (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10d) �µk = l1 + (k − 1) �l2 − l1 K − 1 � , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , K (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10e) 95 µk ∼ N(�µk, 2), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , K (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10f) σk ∼ Half-Normal(2), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , K (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10g) Li ∼ K � j=1 wj N(µj, σj) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10h) where w are the mixture weights (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', mixing proportions), and the rest of the symbols have the same meaning as in Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The likelihood function, defined in Expression 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10h, corresponds with the density of a mixture model expressed in its general form (Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 is implemented and fitted six times (∀K ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' , 7}) to compare the inference results with different number of components specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For each K, rapid mixing and fast convergence of the Markov chains are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The modeling results are displayed in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9, where the KDE (same as in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7) and the Gaussian components inferred are plotted along with the posterior samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It can be observed that, when using a mixture of Gaussians, the multimodal features can be represented in a relative effortlessly way, and all the mean fits are quite close to the one obtained with KDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' As the number of components increases, for example when K = 6 or K = 7, the mean density estimation using GMM resembles KDE more closely, but the samples from the posterior show more stochasticity, which is an indicator of potential overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This naturally leads to the question of how to decide the number of components for this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Model Comparison Choosing the best K is a model comparison problem, for which there does not exist a silver bullet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this dissertation, the author chooses to take the information criteria approach, specifically leveraging the widely applicable information criterion (WAIC) introduced by Watanabe (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Information criteria provide a theoretical estimate of the relative out-of- sample KL divergence (McElreath 2020), and thus a lower value is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Following Martin (2018) and McElreath (2020), WAIC is computed by: 96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='25 Density K = 2 K = 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='25 Density K = 4 K = 5 −12 −10 −8 −6 −4 −2 0 log bcm of Different Oilfields in Q4 2018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='25 Density K = 6 −12 −10 −8 −6 −4 −2 0 log bcm of Different Oilfields in Q4 2018 K = 7 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9: GMM inference results with different K’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The thick blue line denotes the posterior mean fit of the underlying density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The light blue lines show 50 random samples from the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The dashed lines represent the posterior mean Gaussian components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The red curve shows the fit using KDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' WAIC(y, Θ) = −2 × lppd(y, Θ) + 2pwaic (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='11a) = −2 n � i=1 log � 1 S S � j=1 p(yi | Θj) � + 2 n � i=1 VΘ[log p(yi | Θj)] , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='11b) where: y denotes the observations and yi is the i-th observation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Θ is the posterior distribution and Θj is the j-th set of sampled parameter values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 97 S is the number of posterior samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' lppd(·) calculates the log pointwise predictive density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' pwaic is the penalty term given by summing up the variance in the log-likelihood over the S posterior samples, for each observation i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Fundamentally, model comparison is performed by leveraging Occam’s razor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', parsi- monious models are preferred in light of predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The models are compared based on their WAIC values, which are summarized using Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 630 635 640 645 650 655 WAIC 2 3 4 5 6 7 Number of Components Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10: WAIC values with different K’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The open points denote the WAIC values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The long horizontal line segments represent the standard error for each WAIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Standard error of the difference in WAIC (between each model and the top-ranked one) is shown by the lighter line segment with the triangle on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It can be seen that the model with two Gaussian components are the best (smallest WAIC), however, there are considerable overlaps among all of the models when the estimated standard error is taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Considering the fact that K = 2 gives the simplest model, also that there are only 152 observations (oilfields) in this dataset, the GMM with two components would be the best choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 98 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Clustering When looking at the developed model from a latent variable perspective (Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9), it becomes obvious that the mixture model serves as a natural candidate for solving clustering tasks, in that every observation (Li) can be drawn from one of the K data generating processes, each with its own set of parameters, N(Li | µk, σk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since a probabilistic model is built, for the purpose of clustering, a reasonable choice is to assign a data point to the mixture component (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', cluster) with the highest posterior probabilities (which are also interpreted as the responsibilities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the case of the 2-component GMM trained from the previous sections, for a particular observation x, the probability that it belongs to cluster one (z = 1) can be computed using Bayes’ theorem (Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3a): p(z = 1 | x) = p(z = 1) N(x | µ1, σ1) p(z = 1) N(x | µ1, σ1) + p(z = 2) N(x | µ2, σ2) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12) where every part in the formula can be obtained from the posterior samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', using the posterior means).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Clustering, as an unsupervised approach, can be used to reveal the hidden groups in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the case of the oilfield flaring magnitudes data in this chapter, the two clusters can be directly mapped to concepts such as major and minor flaring fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, it is usually the deeper insights into what caused these clusters that the state government is mostly interested in, for the sake of decision- and policy-making for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' If the oilfields belonging to the major flaring cluster seem to be a volatile membership when more months/quarters data are analyzed, the variations in flared volumes are possibly tied more closely to company strategies and movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' On the other hand, if there exists a group of oilfields that are found to join the major flaring cluster on a regular basis, this could provide a perspective in regards to where to construct the next natural gas processing plants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the locations/capacities of the new gas plants should be optimized based on those oilfields’ situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 99 In this chapter, the dataset compiled for unsupervised learning is univariate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', xi ∈ X ⊆ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' GMM are also suitable for the density estimation and clustering tasks when the data goes beyond 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' As an example, for the same oilfields studied for Q4 2018, if their oil production volumes are extracted from NDIC, a scatterplot of gas flaring versus oil production magnitudes is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It is very possible that the density of the underlying distribution can be modeled by a bivariate normal distribution or a 2D GMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In such cases, the mixture components become multivariate normal distributions, and the component covariance matrices can be constructed with the help of the LKJ distribution (which is employed in Models 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The developed density model can be used, for example, in anomaly detections, looking for any oilfields which have a tendency to creep toward the upper left corner (characterized by very little oil production and a huge flaring magnitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Similar to all the inferences presented throughout this dissertation, one advantage of doing such is that the decision making can be based on some consistent metrics (such as probability scores), instead of some criteria based on human eyeballing or improvising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 102 103 104 105 106 107 NDIC Reported Oil Production (bbl) 10−4 10−3 10−2 10−1 VIIRS Reported Flared Volume (bcm) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='11: A scatterplot of oil production and flared gas volumes for different oilfields in Q4 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Both the x- and y-axis are in log scale, showing the relationship between the magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 100 This concludes the statistical modeling journey of this dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the next chapter, discussions are presented on one extension scenario and one bigger picture viewpoint, from applying Bayesian learning to flaring data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 101 CHAPTER 7 DISCUSSION This chapter discusses the possibility of operator level monitoring and analytics, potential result inconsistencies, and relates the endeavors of learning from flaring data to the larger process of applying machine learning in the petroleum engineering domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Operator Level Monitoring and Analytics Up till this point, the satellite-detected flaring statistics have been applied to the state, county, and oilfield levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is made possible by the reverse geocoding discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' An ideal application scenario is operator level monitoring and analytics by leveraging the information from the satellite detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Unfortunately, assigning flares to corresponding companies is not a straightforward operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One possible solution is to make use of the shapefiles of the leases, which are not provided by NDIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Some data vendors have such files in their database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, after spending some effort investigating the lease shapefiles from one vendor, the author believes it is possible to create more problems than solving the existing ones, when bringing in such information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In particular, some reasons include: Multiple companies exist on a single lease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The company names from the lease shapefiles do not always correspond with those on the NDIC monthly production reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Some leases in the vendor’s database miss start date or end date data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It takes time for the vendor to compile and digitize such information, which makes the available lease shapefiles not up to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 102 Nevertheless, for such an important use case, the author managed to develop a nearest- neighbor-based approach which partly solves the problem (Algorithm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The essence of this approach is to cautiously assign the closest well’s operator to each satellite-detected flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The closest wells are found based on the corresponding time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, for the flares detected in January 2016, only the active wells reported on the NDIC production report from the same month are looked up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The function FindClosestOperator() returns the closest operator (OPj) for each VIIRS detection, as well as the calculated distance (dj) between each pair (of flare and well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The distance is calculated based on the haversine metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', the great-circle distance, thus the Earth radius (RE) is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The function is essentially performing the k-nearest-neighbors (k-NN) search for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' When the sample is as large as in this case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', there are usually a few hundred VIIRS detections and more than 15,000 wells for each month, linear scanning each well’s location for each VIIRS detection is too slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Therefore, in this work, the function internally depends on a ball tree implementation from scikit-learn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2011) for speedup on the k-NN search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Once the 2-tuple, (OPj, dj), is obtained for each VIIRS detection, some logics are imple- mented to decide whether to drop or keep the operator assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The idea is straightforward: the assignment is immediately kept or discarded, when dj is very small or very large, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' If dj is mid-range, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', dsecure ≤ dj ≤ dcutoff, the assignment will be in effect, only if the flare and the operator are found to be located on the same township/range/section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The township/range/section shapefiles, as part of the input for Algorithm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1, are available from the NDIC GIS Map Server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The reverse geocoding follows the exact same procedure as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' After the processing is completed, a small portion of the VIIRS detections are not used for operator level analytics, because either they are too far away from the reported well locations, or the townships/ranges/sections fail to match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It should be noted that, the pseudocode for Algorithm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 is written in a way that illustrates the precise details in the data processing logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the implementation in this work, some of the for-loops are replaced by the vectorized operations for enhanced performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 103 Algorithm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: Nearest-Neighbor-Based Flare Owner Assignment Input: both VIIRS and NDIC reportings in WGS 84 coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' the township/range/section shapefiles for North Dakota,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' dsecure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' dcutoff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' RE Output: operators being assigned to most VIIRS detections 1 n ← number of months 2 for i ← 1 to n do 3 VIIRSi ← the i-th month’s observations from VIIRS 4 NDICi ← the i-th month’s reportings from NDIC 5 (OP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' d) ← FindClosestOperator(VIIRSi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' NDICi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' RE) 6 m ← number of records in OP or d 7 for j ← 1 to m do 8 OPj ← the closest operator found on the j-th record 9 dj ← the distance between the flare and the closest well,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' for the j-th record 10 if dj > dcutoff then 11 drop OPj 12 else if dj < dsecure then 13 keep OPj 14 else 15 if township/range/section agree then 16 keep OPj 17 else 18 drop OPj 19 end 20 end 21 end 22 end The developed approach is tested with real flaring data from North Dakota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For the demonstrated cases in this section, the values below are chosen for Algorithm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: dsecure = 300 m (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1a) dcutoff = 800 m (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1b) RE = 6371 km (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1c) Some operators are found to show positive correlations between the NDIC and VIIRS reported volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Examples of two operators, denoted by Operator B and Operator C, are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The axes’ meanings are the same as in the right panel of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The 104 legend shows the results of fitting Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2d by ordinary least squares (OLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' R2 adj stands for the adjusted R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Although the differences in ˆβoperator indicate that there is heterogeneity among the different companies, these operators show some consistency in terms of their own reporting and have good matches with the VIIRS data up to a scale factor (as the intercepts are very close to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 VIIRS Monthly Data (bcm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='012 NDIC Monthly Data (bcm) NDICi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='378 × VIIRSi R2 Adj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='959 45-degree line (a) Operator B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 VIIRS Monthly Data (bcm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='040 NDIC Monthly Data (bcm) NDICi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='001 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='658 × VIIRSi R2 Adj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='861 45-degree line (b) Operator C Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: Examples of good fits between the NDIC and VIIRS reported volumes, at the operator level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, some operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', Operator D and Operator E) show discrepancies between their reportings and the satellite-detected flaring statistics, which are manifested through the poor fits (Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Certainly, a poor fit with the linear model does not indicate much on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Nonetheless, there exists a pattern in both scatterplots that, some points seem to be “pushed down” towards the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' If the time series of these two operators are drawn, it shows that this behavior is due to company-reported volumes leveling off for a certain period of time (Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The VIIRS curves in the time series imply that there were flaring intensity variations for those times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This workflow, driven by Algorithm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1, is capable of raising a flag when it comes across datasets like these, and can serve as a powerful monitoring 105 and analytics tool, however, strong cautions need to be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='08 VIIRS Monthly Data (bcm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='08 NDIC Monthly Data (bcm) NDICi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='207 × VIIRSi R2 Adj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='180 45-degree line (a) Operator D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 VIIRS Monthly Data (bcm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='010 NDIC Monthly Data (bcm) NDICi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='171 × VIIRSi R2 Adj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='256 45-degree line (b) Operator E Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2: Examples of poor fits between the NDIC and VIIRS reported volumes, at the operator level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The introduced approach, although it looks promising, is by no means a one-stop solution and has the potential for being misapplied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' First, there is the possibility of misassigning the satellite-detected flares to the operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Whenever the concern is raised, further investigations can be conducted by looking into the detection maps as well as the satellite imagery of the operators’ production sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In addition, this method is more effective for the relatively large producing/flaring operators, because when a company conducts very little flaring, the truncation effects discussed for the peak in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='20 are magnified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Warnings Regarding Inconsistencies Given the resolution of the satellite imagery, assigning specific flaring volumes to a given operator is fraught with challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Although the VIIRS processing workflow is capable of picking up flares with areas around 1 m2 (Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2(a)), the pixel footprint is much larger (Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since the latitude and longitude of the pixel center is stored for each individual VIIRS observation (Elvidge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2015), when multiple operators have sub-pixel combustion 106 The company-reported volumes leveled off at small values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (a) Operator D The company-reported volumes leveled off at small values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The company reported zero flaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (b) Operator E Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3: Time series of the two example operators whose reporting did not quite align with the VIIRS detected trends/patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The points or periods in time for which the company-reported data were significantly different from the satellite detections are annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' sources, it makes flare owner assignment extremely challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In such situations, conclusions reached by merely benchmarking company reporting against VIIRS reporting would likely be inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In fact, in the realm of NDIC reporting, warnings must be issued regarding any inconsistencies in those results, with considerations from three aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' First, the report from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Department of Energy (2019a) presents data supporting that North Dakota 107 VIIRS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='08 NDIC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='06 bcm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='05 Monthly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='00 2015-07 2016-01 2016-07 2017-01 2017-07 2018-01 2018-07 2019-01VIIRS NDIC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='008 bcm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='006 Monthly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='000 2015-07 2016-01 2016-07 2017-01 2017-07 2018-01 2018-07 2019-01shows closer agreement between the NOAA estimations and state reportings (of flared gas volumes), when compared with Texas and New Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Second, flaring is preferred over venting because methane (the main component of natural gas) is more potent than carbon dioxide which is the main product of flaring (EIA 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since North Dakota bans venting, the massive flaring magnitude indicates that the direct release of gas into the atmosphere is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Third, estimation of flaring volumes is inherently a difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' When it is not practicable to meter the flared gas, the Canadian Association of Petroleum Producers (2002) gives guidelines on available volume estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Every category of methods, no matter using rules of thumb, or experimentally determined correlations, or process simulators, has its own limitations and accuracy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Considering the fact that the VIIRS volumes used in this work were largely calibrated using the Cedigaz reported data (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1), which has its own error bars (Elvidge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2015), the difference between company reporting and VIIRS reporting is inconclusive and unsurprising, especially when the standard error of the difference is larger than the difference itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' By inspecting a more comprehensive profile of time series, both Operator D and Operator E from the previous section are self-consistent in their reportings to the NDIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Their time series are displayed in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 and Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The variables and associated labels (shown in the legends) follow the same definitions from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The units for all the variables are given in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Clearly, the reported flared volumes show good correspondence with the gas production and GOR profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Some rapid variations in their flared volumes match the fluctuations in the gas prices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', when the gas price drops, the operators tend to flare more, whereas when the gas price reaches peak, there is little flaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In summary, to nail down the decisions and conclusions with regard to operator reporting quality, better resolution satellite data and a more comprehensive review of the time series profiles are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 ×106 NDIC flared vol 20 40 60 WTI oil price 2 3 4 Henry Hub gas price 2 3 ×106 NDIC oil prod 2 4 6 ×106 NDIC gas prod 200 400 NDIC flaring well count 2016 2017 2018 2019 2020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 NDIC GOR Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4: A more comprehensive time series plot for Operator D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The increase in the reported flared volume in early 2019 corresponds to the gas price declining in the same period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 109 0 50000 NDIC flared vol 20 40 60 WTI oil price 2 3 4 Henry Hub gas price 200000 300000 NDIC oil prod 0 50000 100000 NDIC gas prod 0 100 NDIC flaring well count 2016 2017 2018 2019 2020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 NDIC GOR Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5: A more comprehensive time series plot for Operator E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The sudden drop in the reported flared volume in late 2018 corresponds to the halted gas production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 110 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: Units for Operator Time Series in Figures 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='4 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 Variable Unit NDIC flared vol Mcf WTI oil price $/bbl Henry Hub gas price $/MMBtu NDIC oil prod bbl NDIC gas prod Mcf NDIC flaring well count 1 NDIC GOR Mcf/bbl 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 Caveats in Petroleum Data Analytics As a petroleum engineer, the author is thrilled to witness the oil and gas industry and academia are embracing data-driven mindsets and solutions, while being part of it through writing this dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, there are certainly areas that could be continuously improved, and this section provides a discussion on one of those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' That is, extending a cautious welcome to some black box models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The pervasive influence of some black box models in the recent years can be seen by performing a rough search on OnePetro (Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One thing to note is that, from an algorithmic point of view, these methods are rather “glass boxes” as opposed to “black boxes”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', everything under the hood in terms of implementation is well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, backpropagation, which is the core of neural network training, is based on the chain rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, for a given task, the learned parameters inside the network provide little or no insights for the problem domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Therefore, it is considered a black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The wide adoption of such models is largely due to the availability of the open source libraries, for example in the Python ecosystem, construction and training of neural networks become much simpler thanks to TensorFlow and PyTorch, and gradient boosting models can be built within a few lines of code with the help of XGBoost, LightGBM, or CatBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In other words, with the mathematical details of those statistical routines abstracted away, for a practitioner, implementing those models is almost as easy as pushing a Learning button on 111 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2: Publication Count Rise on OnePetro Exact Phrase Searched Year Method Introduced Publication Count 2010–2014 2015–2019 neural network 1958† 843 2044 gradient boosting 2001‡ 1 110 random forest 2001§ 9 245 † Based on (Rosenblatt 1958) ‡ Based on (Friedman 2001) § Based on (Breiman 2001) a GUI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Unfortunately, easiness in the implementation does not imply appropriateness for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In particular, those black box models face the challenges below: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' How to incorporate domain expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A lot of the black box models in the frequentist framework make the assumption that the observations are conditionally i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The hope is that by feeding a huge number of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' samples to a universal approximator, such as a neural network, some function for prediction can be optimized with a certain accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For some applications, the domain expertise is often encoded in the feature selection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, to train a model to predict oil production, the analyst might choose some completion parameters other than the API well number or well name, as input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, in the author’s opinion, this way of incorporating domain expertise is still a shallow one, which is far from what the oil and gas industry have accumulated in many decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, the phenomena of well interference through fracture hits leave the assumption of some neighboring wells being i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' in an unfavorable position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Another example would be, when looking at a populations of wells from one basin that are completed by N oilfield service companies, domain expertise might indicate that, each company deserves its own model while each company is not completely 112 independent from others in terms of the completion technologies, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this situation, the hierarchical model employed in Chapter 4 might be a better choice, in which case a lot of the prior knowledge about the different service companies can be incorporated into the population model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' How to interpret the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' As discussed earlier, the black box models suffer from the interpretability issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Using the shale gas wells example from Item 1 above, if a black box model is trained, it is impossible (at this point) to attribute the failure in capturing the well interference effects to a certain part of the neural network, or to a certain portion of the decision trees (in the case of gradient boosted trees or random forest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Rudin (2019) asserted that people should “stop explaining black box machine learning models” and use interpretable models for high-stakes decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the petroleum industry, there are a number of high-stakes decision scenarios, such as real-time well integrity anomaly detection and production forecasting in a high well cost context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Blindly applying black box models to those scenarios might involve serious losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In terms of providing interpretability, the Bayesian approach employed throughout this dissertation is much more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Each and every assumption is expressed in the generative model through either the priors or the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' How to quantify the uncertainties, especially in the context of risk management and decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Along the lines of Item 2 above, error bars are vital, especially in high-stakes prediction applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In the case of predicting oil production using a trained data-driven model, point prediction results such as 1000 bbl/day are not really insightful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In fact, if the 95 % prediction interval (PI) is 1000 ± 50 bbl/day, that point prediction becomes more informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, if the 95 % PI is 1000 ± 1500 bbl/day, that same point prediction is unhelpful or misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' What shall be reported instead is either the considered 113 model yields much uncertainty in this given task, or there is possibility that the entity will not produce anything at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It should be noted that, the ‘95’ in the CI/PI is not a “magic number”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A state govern- ment or an oil company might want to make decisions based on 73 % or 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 % confidence, or any other arbitrary choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' What really matters is the necessity of a principled way to quantify the uncertainties in machine learning-based estimations/predictions, such that any intervals can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' As presented throughout this dissertation, the Bayesian approach provides full capacity and flexibility is this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In fact, for parameter estimates, the author chooses to give 90 % CI instead of the “conventional” 95 %, to emphasize that this should be a domain’s consideration rather than a statistical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A lot of the black box models in the frequentist framework, however, fall short of this requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Maximum likelihood estimation (MLE), which is fundamentally relied upon by some frequentist learning methods, enjoys really nice properties and is capable of quantifying uncertainties, but only when a massive amount of data is at hand such that the asymptotic properties could take effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Unfortunately, that is not the case in many scenarios for the petroleum engineering domain, which is discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' How to mitigate overfitting when the data is not “big”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Two aspects are worth discussing here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For one thing, the big data is not everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Indeed, the author believes that the claim of Gelman (2015) that, “sample sizes are never large”, applies to a lot of problems in the petroleum industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The reason is that, if the data were large, the analyst would already be on to the next problem for which more data is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, a sample of 500 producing wells in the Bakken Formation could make some general study possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' When the analyst has access to a dataset of more than 15,000 wells, some granular insights are desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Especially, if partial pooling is needed among the different service companies/operators, different 114 members of the formation, or different completion technologies, data for some units of the population could be very small (which happens for the analysis in Chapter 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' On the other hand, the sample size should be inspected in the light of model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The number of parameters provides one measure of such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For example, consider a hypothetical classification problem, whose goal is to determine if a given well will deliver good or average or poor production performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Ten completion parameters (features) are available to train the multilayer perceptron illustrated in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' x1 x2 x10 Input layer: ten completion parameters h(1) 1 h(1) 2 h(1) 3 h(1) 20 Hidden layer 1 h(2) 1 h(2) 2 h(2) 10 Hidden layer 2 ˆy1 ˆy2 ˆy3 Output layer: good/average/poor production performance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6: A neural network designed for the hypothetical well performance classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The input layer has 10 neurons for the completion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The first and second hidden layer has 20 and 10 neurons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The output layer has three neurons for multiclass classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In this (small) neural network, the number of parameters np is given by: np = 11 × 20 + 21 × 10 + 11 × 3 = 463, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2) when considering a single bias node for every layer except the last one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To train this model, a dataset of 500 wells would definitely be a small sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' There is still possibility to train such a model with a small sample, however, great efforts in regularization have to be made, in the hope that the neural network will learn something that can be generalized, instead of merely memorizing the observed samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', overfitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 115 By utilizing the regularizing priors, the Bayesian approach’s built-in Occam’s razor greatly mitigate the risk of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In particular, Bayesian nonparametric models, such as the Gaussian processes employed in Chapter 5, are very attractive in a sense that the sizes of models are allowed to grow with the size of data (Orbanz and Teh 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This makes the developed model flexible while being robust to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Although the Bayesian learning models (such as the ones developed in this work) have outstanding merits and deserve wider utilization in petroleum data analytics, they are not cure-alls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Recently researchers have started to stress the necessity of bespoke statistical models (Andorra 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' McElreath 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The argument is that, off-the-shelf models, no matter neural networks or generalized linear models, interrupt the incorporation of domain expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' This is especially relevant in the field of petroleum engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' For instance, when conducting data-driven analysis for hydraulic fracturing performance, it makes sense to bring in the fracture propagation models to the machine learning workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' That way, statistical models are motivated by the physically informed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The Bayesian framework, as employed throughout this dissertation, readily embraces this strategy, in that the domain knowledge, which is represented by differential equations for example, can be inserted into the generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One advantage is that a lot of the parameters will have direct scientific meanings, and more informative priors can be placed based on scientific constraints, field experience, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The final outcome should be better inferences and predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 116 CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS In this dissertation, the effectiveness of a full Bayesian approach has been observed in learning models from natural gas flaring data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The author hopes this work contributes to the understanding of the options and considerations when applying data-driven approaches to gas flaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In closing, this chapter presents the major conclusions and recommendations for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 Conclusions The major conclusions are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Bayesian learning implemented using Hamiltonian Monte Carlo can be effectively applied to real problems in gas flaring analytics, in both supervised and unsupervised settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The advantages of the Bayesian approach indicate it deserves wider usage in the petroleum engineering domain in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' these advantages are listed below: (a) Petrotechnical domain expertise can be incorporated in a principled way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (b) Model interpretability is drastically improved, facilitating communications with petroleum engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (c) Quantification of uncertainty leads to more robust decision making, which is important for oil exploration and production companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' (d) The built-in Occam’s razor makes the model less prone to overfitting, in the context of noisy field measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The development of a suite of models (Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1), with both parametric and nonpara- metric techniques, provides guidance on how insights can be extracted from various 117 angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The presented models are designed and tested to be able to generalize to different entities at various levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' To investigate the heterogeneity among the different entities (such as counties or oilfields), partial pooling is recommended, because some entities have very little data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Gaussian processes demonstrate very attractive traits in revealing the patterns and trends from flaring time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A set of priors with the Mat´ern 5/2 kernel works very well across different modeling goals, observation models, and data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' From a distributional point of view, the negative binomial and Gaussian mixture models are good representations of the oilfield flare counts and flared volumes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The learned parameters and structures are very interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Hidden clusters are found by fitting Gaussian mixture models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' A nearest-neighbor-based approach for operator level monitoring and analytics is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Its performance is tested on real data and defendable results are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' However, better resolution satellite data is needed for the scenario of multiple operators’ wells being very close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' All the dissertation objectives (Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2) have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In particular, the flared volumes missed from VIIRS for the state and each county are estimated via fitting the intercept parameter and reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The nighttime combustion source detection limits of Landsat 8, without being corrected for artifacts due to glow, are determined and reported in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Correlations between financial factors, production performance, and flared volumes at a state level are computed using Spearman’s ρ and reported in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='6 for the original data and lag-1 differences, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Most pairs of the variables do not show strong correlations on the lag-1 differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Robust Gaussian process modeling serves as a generic framework for addressing the rest of the objectives, including demonstrating operator approaches, 118 evaluating if the goals of the North Dakota regulatory policy (Order 24665) have been achieved, and predicting NDIC flared volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='1: Models Developed in this Dissertation Numbering Target of Modeling Page Model 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Associations between VIIRS and NDIC at a state level 27 Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='5 Associations between VIIRS and NDIC at a county level (centered) 38 Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='7 Associations between VIIRS and NDIC at a county level (noncentered) 41 Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='12 Proportion of gas production being flared as time series 57 Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='14 Proportion of wells that conduct flaring as time series 63 Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='15 VIIRS detection count as time series 66 Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='17 Proportion of oil being flared as time series 70 Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='18 Scale factor between VIIRS and NDIC as time series 74 Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='3 VIIRS detection count distribution for oilfields 86 Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='9 VIIRS volume distribution for oilfields (latent discrete parameterization) 94 Model 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='10 VIIRS volume distribution for oilfields (marginalized) 95 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 Future Work A number of areas for future research include: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' L8 processing workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The studies of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2 indicate that the inclusion of L8 information (using the existing VIIRS workflow) faces the challenges of the processing artifacts due to glow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It would be interesting to tailor the processing algorithm for L8, which opens the door for data fusion of VIIRS and L8, providing much better resolution interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Fast detection of flares on a monthly basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The development of a rapid flare detection and volume estimation method (based on satellite imagery) will lead to continuous monthly data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Since NDIC needs about two months’ turnaround time to compile and digitize the company reports, many of the machine learning workflows proposed in this dissertation will be able to provide predictive insights with rapid detection data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 119 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Hierarchical Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The models in Chapter 5 are learned from each entity’s own data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' It would be interesting to see how far the scheme of partial pooling (Chapter 4) can be taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Can pooling across different entities via hierarchical Gaussian processes improve the inferences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Spatial-temporal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' One step further from Item 3 above, the efficacy of spatial-temporal models (which allow for pooling information across time and space) are worth investigating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Are neighboring entities exhibiting close resemblance in flaring behaviors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Unify everything under Bayesian nonparametrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' The model comparison for GMMs in Chapter 6 depends on specifying the potential numbers of clusters a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' In fact, Dirichlet process, as an infinite-dimensional gener- alization of the Dirichlet distribution, is nonparametric and allows for automatically choosing the number of necessary clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Considering the effectiveness of GP (Chap- ter 5), it would be interesting to see how far the nonparametric models can be taken in flaring data analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Can all of the gas flaring 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=', Baugh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Extending Nighttime Combustion Source Detection Limits with Short Wavelength VIIRS Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' Remote Sensing 11 (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' https://doi.' metadata={'source': 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Texas, USA, 4-6 February.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' SPE- 199707-MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content='2118/199707-MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} +page_content=' 128' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9E2T4oBgHgl3EQf0ggt/content/2301.04141v1.pdf'} diff --git a/p9FLT4oBgHgl3EQfiS8Z/content/tmp_files/2301.12106v1.pdf.txt b/p9FLT4oBgHgl3EQfiS8Z/content/tmp_files/2301.12106v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e28ef70a5c7f7cd867c9ca6660bccb95baa403c5 --- /dev/null +++ b/p9FLT4oBgHgl3EQfiS8Z/content/tmp_files/2301.12106v1.pdf.txt @@ -0,0 +1,1983 @@ +Covariate-assisted bounds on causal effects with instrumental +variables +Alexander W. Levis1, Matteo Bonvini1, Zhenghao Zeng1∗, +Luke Keele2, Edward H. Kennedy1 +1Department of Statistics & Data Science, +Carnegie Mellon University +2Department of Surgery, +University of Pennsylvania +{alevis, mbonvini, zhenghaz} @ andrew.cmu.edu; +luke.keele@gmail.com; edward@stat.cmu.edu +Abstract +When an exposure of interest is confounded by unmeasured factors, an instrumental +variable (IV) can be used to identify and estimate certain causal contrasts. Identification of +the marginal average treatment effect (ATE) from IVs typically relies on strong untestable +structural assumptions. When one is unwilling to assert such structural assumptions, IVs +can nonetheless be used to construct bounds on the ATE. Famously, Balke and Pearl +(1997) employed linear programming techniques to prove tight bounds on the ATE for a +binary outcome, in a randomized trial with noncompliance and no covariate information. +We demonstrate how these bounds remain useful in observational settings with baseline +confounders of the IV, as well as randomized trials with measured baseline covariates. +The resulting lower and upper bounds on the ATE are non-smooth functionals, and thus +standard nonparametric efficiency theory is not immediately applicable. To remedy this, +we propose (1) estimators of smooth approximations of these bounds, and (2) under a +novel margin condition, influence function-based estimators of the ATE bounds that can +attain parametric convergence rates when the nuisance functions are modeled flexibly. We +propose extensions to continuous outcomes, and finally, illustrate the proposed estimators +in a randomized experiment studying the effects of influenza vaccination encouragement +on flu-related hospital visits. +Keywords: causal inference, partial identification, instrumental variables, influence functions +∗MB and ZZ contributed equally. +arXiv:2301.12106v1 [stat.ME] 28 Jan 2023 + +1 +Introduction +A primary goal in many scientific endeavors is to determine whether an intervention has a +causal effect on an outcome. However, simple comparisons between the outcomes of treated +and control groups are often complicated by confounding: differences in outcomes between +those who are and are not treated due to pre-treatment differences rather than the effect +of the treatment itself. One solution to selection bias of this form is to allocate the treat- +ment via randomization, rendering pre-treatment distributions equal, on average. In many +circumstances, however, randomization is infeasible or unethical. When this is the case, one +alternative is to identify an instrumental variable (IV). For a variable to be an IV, it must +meet the three following conditions: (a) the IV must be associated with the exposure; (b) the +IV must be randomly or as-if randomly assigned; and (c) the IV cannot have a direct effect on +the outcome (Angrist et al., 1996a). Under these conditions, as well as some further structural +assumptions, an IV can provide a consistent estimate of a causal effect even in the presence +of unobserved confounding between the exposure and the outcome. See Baiocchi et al. (2014) +and Imbens (2014) for general reviews of IV methods. +In most analyses, investigators focus on point identification–asserting sufficient structure +so that a single parameter describing a causal effect can be expressed as a function of the +observed data distribution. In fact, assumptions (a)–(c), on their own, are insufficient to point +identify an average causal effect. Point identification requires investigators to assume either +some form of homogeneity (e.g., lack of effect modification by unmeasured confounders) or +an assumption known as monotonicity (Robins, 1994; Hernán and Robins, 2006a; Tan, 2010; +Angrist et al., 1996b). Critically, several of these structural IV assumptions are untestable +and may be controversial in many applications. One alternative approach is to relax key IV +assumptions using partial identification. Under the partial identification approach, analysts +seek to estimate bounds on the parameter of interest, which can typically be done with weaker +assumptions (Manski, 1990a, 1995). There is a large body of work on partial identification +in IV designs with foundational work done by, e.g., Balke and Pearl (1997) and Manski and +Pepper (2009, 2000). See Swanson et al. (2018) for a general overview of partial identification +approaches to an IV analysis. +Extant IV partial identification methods, notably, do not allow for the use of auxiliary +information based on baseline covariates. Conditioning on covariates offers the potential to +render the bounds more informative by narrowing the bounds. In other settings, baseline +covariates must be included to control for confounding and satisfy the instrumental variable +assumptions. In this paper, we extend the Balke-Pearl bounds to include baseline covariate +information to remedy these issues. Our proposed estimators are based on semiparametric +efficiency theory and use influence functions, which allow for flexible and efficient estimation +via a variety of machine learning based methods. The use of influence functions also allows us +to derive simple closed-form variance estimators that are consistent. In addition, we outline +how to employ sample splitting to ensure honest inference. +The remainder of the paper is organized as follows. In Section 2, we define notation, review +key IV assumptions, and outline our target causal estimand: the average treatment effect. In +Section 3, we review extant methods for bounding the average treatment effect (ATE) based +on an IV. We also provide a simple illustration to demonstrate the benefits of incorporating +covariate information, and present a general result to guide which covariates to include in +the ensuing analyses. Importantly, the nonparametric bounds outlined in Section 3 are non- +smooth functionals of the observed data distribution, so standard semiparametric efficiency +theory does not immediately apply. In view of this challenging setting, Sections 4, 5 and +2 + +6 detail the three main methodological contributions of the paper: (i) valid but conservative +bounds on the ATE based on estimators of smooth functional approximations to the bounds (in +Section 4); (ii) efficient estimators of the true non-smooth bound functionals under a margin +condition (in Section 5); and (iii) extensions of the proposed methods to the case of a general +bounded outcome, discrete or continuous (in Section 6). Finally, in Section 7, we apply our +proposed methods to data from a randomized trial assessing the effects of the influenza vaccine +on flu-related outcomes. Proofs of all results can be found in the Supplementary Materials. +2 +Background: Notation, Assumption, and Estimand +2.1 +Notation and Assumptions +Consider the standard instrumental variable setup, in which the observed data are n iid copies +of O = (X, Z, A, Y ) ∼ P, where X ∈ X ⊆ Rd is a vector of covariates, and Z, A ∈ {0, 1} +are binary instrument and exposure variables, respectively. Our focus is mostly on binary +outcomes, Y ∈ {0, 1}; we will discuss extensions for non-binary outcomes in Section 6. We let +A(z) and Y (z) denote the counterfactual exposure and outcome values, had the instrument +been set to Z = z, for z ∈ {0, 1}. Similarly, we also define Y (a) and Y (z, a) to be the potential +outcomes under an intervention that sets A = a, and an intervention that sets both Z = z +and A = a, respectively. For z ∈ {0, 1}, let λz(X) = P[Z = z | X]. Next, we review the +set of assumptions that we take as given in the analysis. First, we make the two following +assumptions: +Assumption 1 (Consistency). A(Z) = A, and Y (Z, A) = Y (A) = Y , almost surely. +Assumption 2 (Positivity). For some ϵ > 0, λ1(X) ∈ [ϵ, 1 − ϵ], almost surely. +In words, Assumption 1 asserts that interventions on Z and A are well-defined, and that +there is no interference between subjects, so that a unit’s potential treatment and outcome +can be linked to their observed variables; Assumption 2 asserts that either instrument value +is possible, over all strata determined by X. These two assumptions are not unique to the IV +framework, and are commonly invoked in many causal inference settings. Next, we make the +following “core” IV assumptions: +Assumption 3 (Unconfoundedness). Z ⊥⊥ (A(z), Y (z)) | X. +Assumption 4 (Exclusion Restriction). Y (z, a) ≡ Y (a), for z, a ∈ {0, 1}. +Assumption 3 asserts that the effect of Z on A and Y is unconfounded, given measured +covariates X. Note that in certain special cases, Assumption 3 holds unconditionally, and +we can identify the IV effect without conditioning on X. For example, when Z is marginally +randomized (e.g., in a trial), Assumption 3 holds by design. Alternatively, in certain natural +experiments, analysts may assert that Assumption 3 holds unconditionally, e.g. Angrist and +Evans (1998). Hernán and Robins (2006b) refer to such IVs as causal IVs, and we adopt this +terminology. Assumption 4 asserts that the effect of Z on Y acts entirely through its effect on +A, i.e., Z has no direct effect on Y . We note that while in most IV settings a notion of non-zero +association between Z and A (often referred to as a “relevance” assumption) is asserted, this +is not formally required for partial identification of the average treatment effect. +3 + +2.2 +Estimands and Additional Structural Assumptions +In what follows, we focus on the average treatment effect (ATE) as the target causal estimand: +E (Y (a = 1) − Y (a = 0)) . +Critically, under the four assumptions introduced in the previous section, the ATE is not point +identified. Analysts typically take one of two approaches for point identification. The first +approach invokes some type of homogeneity assumptions and places various restrictions on +how the effects of A and Z vary from unit to unit in the study population. See Hernan and +Robins (2019) and Wang and Tchetgen Tchetgen (2018) for prominent examples. See Levis +et al. (2023) for a detailed review. However, homogeneity assumptions are often implausible or +difficult to verify in specific applications. The second approach invokes an assumption known +as monotonicity, which has the following form: A(z = 1) ≥ A(z = 0), i.e., if A(z = 0) = 1 +then A(z = 1) = 1 (Imbens and Angrist, 1994). Under monotonicity, the target estimand is +no longer the ATE, but instead is the local average treatment effect (LATE): +E(Y (a = 1) − Y (a = 0) | A(z = 1) > A(z = 0)) +Here, investigators must be content with a more local estimand that may not generalize to the +ATE. In this paper, we only assume that Assumptions 1–4 hold, and aim to construct tight +bounds on the ATE parameter. +3 +Partial Identification for IVs and the Role of Covariate Infor- +mation +Next, we provide an overview of one partial identification approach for the IV framework. +We then provide a brief illustration to demonstrate how covariate information can alter the +bounds on the ATE. We also present theoretical results to elucidate when we may expect the +inclusion of covariates to tighten the bounds on the ATE. +3.1 +Review: Balke-Pearl Bounds +In their seminal work, Alexander Balke and Judea Pearl leveraged symbolic linear program- +ming to develop sharp nonparametric bounds on the ATE for a binary outcome (Balke and +Pearl, 1994; Balke, 1995; Balke and Pearl, 1997). +Notably, their bounds only invoke As- +sumptions 1–4, and are provably tight under these assumptions. In both applications and +simulations they demonstrated substantial narrowing of the bounds on the ATE compared to +partial identification results in Robins (1989) and Manski (1990b). +The results in Balke and Pearl (1997) focused exclusively on causal IVs such that X = ∅. +As such, their methods do not make use of any measured covariates. Nonetheless, their results +hold just as well for bounding the conditional average treatment effect (CATE), E[Y (a = +1) − Y (a = 0) | X = x], for any x ∈ X, in two important settings where Assumptions 1–4 +may hold: (i) an experiment with Z marginally randomized and baseline covariates X are +measured, and (ii) an observational setting where X are baseline confounders required for +Z to be a valid IV. In case (i), we will demonstrate that incorporating baseline covariate +information X can both provide tighter theoretical bounds, and improve statistical precision. +In case (ii), conditioning on X is required for the IV assumptions to hold, and thus for the +bounds to be valid. +4 + +To begin, we review the main theoretical result from Balke and Pearl (1997). For each +y, a, z ∈ {0, 1}, define πya.z(X) = P(Y = y, A = a | X, Z = z). Moreover, define the two +8-vectors θℓ(X) = (θℓ,1(X), . . . , θℓ,8(X)), θu(X) = (θu,1(X), . . . , θu,8(X)) ∈ [−1, 1]8, where +omitting inputs, +θℓ,1 = π11.1 + π00.0 − 1 +θℓ,2 = π11.0 + π00.1 − 1 +θℓ,3 = −π01.1 − π10.1 +θℓ,4 = −π01.0 − π10.0 +θℓ,5 = π11.0 − π11.1 − π10.1 − π01.0 − π10.0 +θℓ,6 = π11.1 − π11.0 − π10.0 − π01.1 − π10.1 +θℓ,7 = π00.1 − π01.1 − π10.1 − π01.0 − π00.0 +θℓ,8 = π00.0 − π01.0 − π10.0 − π01.1 − π00.1 +(1) +and +θu,1 = 1 − π01.1 − π10.0 +θu,2 = 1 − π01.0 − π10.1 +θu,3 = π11.1 + π00.1 +θu,4 = π11.0 + π00.0 +θu,5 = −π01.0 + π01.1 + π00.1 + π11.0 + π00.0 +θu,6 = −π01.1 + π11.1 + π00.1 + π01.0 + π00.0 +θu,7 = −π10.1 + π11.1 + π00.1 + π11.0 + π10.0 +θu,8 = −π10.0 + π11.0 + π00.0 + π11.1 + π10.1. +(2) +Finally, define γℓ(X) = max1≤j≤8 θℓ,j(X) and γu(X) = min1≤j≤8 θu,j(X). Balke and Pearl +proved (e.g., see the main result in Balke and Pearl (1997), p. 1173) that the CATE is bounded +between γℓ and γu. We summarize their result in the following theorem. +Theorem 1 (Balke and Pearl (1997)). Under Assumptions 1–4, for (Z, A, Y ) ∈ {0, 1}3, the +CATE can be bounded as +γℓ(X) ≤ E(Y (a = 1) − Y (a = 0) | X) ≤ γu(X). +Moreover, these bounds are tight in the nonparametric model. +By marginalizing the bounds in Theorem 1, one obtains tight bounds on the ATE. By defining +L(P) = EP(γℓ(X)) and U(P) = EP(γu(X)), it follows that +L(P) ≤ E(Y (a = 1) − Y (a = 0)) ≤ U(P). +(3) +The primary goal of this paper is to construct efficient estimators of L(P) and U(P). +Statistically, this is a challenging task, since these are means of a pointwise maximum and +minimum, each of which is a non-smooth function. We will outline two strategies to deal +with this difficulty: (1) targeting a smooth approximation to the Balke-Pearl bounds (see +Section 4), or (2) invoking a margin condition (see Section 5). +Next, we demonstrate the benefit of using covariate-assisted bounds. In Section 3.2, we give +a simple illustration in the randomized experimental setting, showing that covariate-assisted +bounds can be substantially narrower. In Section 3.3, we provide a general result to guide the +choice of adjustment set X when more than one is possible. +5 + +3.2 +Motivating Illustration +Consider a hypothetical randomized experiment with arm assignment Z ∼ Bernoulli(0.5), in- +dependent of baseline covariates X1 ∼ Bernoulli(0.7), X2 ∼ Unif(−1, 1), with X1 ⊥⊥ X2. In +this hypothetical example, X1 represents a behavioral or demographic factor, and X2 repre- +sents an underlying risk score (i.e., low values represent good health and low risk, high values +represent poor health and high risk). In this experiment, we suppose that there is a degree of +noncompliance, which is completely determined by the covariates X1 and X2. Specifically, we +set: +A(z = 0)A(z = 1) = 1(X2 > 0.5) +{1 − A(z = 0)}{1 − A(z = 1)} = 1(X2 ≤ −0.5) +{1 − A(z = 0)}A(z = 1) = X11(X2 ∈ (−0.5, 0.5]) +A(z = 0){1 − A(z = 1)} = (1 − X1)1(X2 ∈ (−0.5, 0.5]) +In words, we can define four principal strata: “always takers” are those with highest risk, +“never takers” are those with lowest risk, “compliers” are those with intermediate risk and +behavioral factor X1 = 1, and “defiers” are those with intermediate risk and behavioral factor +X1 = 0 (Angrist et al., 1996a; Frangakis and Rubin, 2002). Moreover, we suppose the potential +outcomes Y (a) are completely determined by the compliance classes U = (A(z = 0), A(z = 1)), +and set Y (a) | U ∼ Bernoulli(pa(U)), with +pa = +� +� +� +� +� +� +� +� +� +� +� +0.10 + 0.85a, +if A(z = 0)A(z = 1) = 1, +0.85 + 0.10a, +if {1 − A(z = 0)}{1 − A(z = 1)} = 1, +0.35 + 0.60a, +if {1 − A(z = 0)}A(z = 1) = 1, +0.65 + 0.30a, +if A(z = 0){1 − A(z = 1)} = 1. +. +Figure 1 shows the covariate-agnostic and covariate-adjusted Balke-Pearl bounds resulting +from Theorem 1. In Figure 1, the true bounds are plotted in blue and the estimated bounds +are plotted in green. The covariate-agnostic bounds, while simple to compute as the maximum +and minimum of 8 (true or empirical) probabilities, are fairly wide and cover the null treatment +effect of zero. The covariate-adjusted bounds, on the other hand, are substantially narrower +and do not cover zero. +Note that in this example, the covariates X1, X2 happen to comprise all confounders of +the relationship between A and Y , so that direct adjustment (e.g., with the g-formula) would +identify the ATE. A practitioner without this knowledge, but with the foresight to measure X1 +and X2, could still use these covariates to construct more informative bounds. In what follows, +we show that such narrowing of the bounds is a general phenomenon, and occurs even in the +presence of residual confounding given X. Indeed, one will want to adjust for X whenever +these covariates are predictive of A and Y . +3.3 +Width of the Bounds +In the illustration above, both choices of adjustment sets (no covariates or (X1, X2)) result in +valid bounds since Assumptions 1–4 are satisfied with either choice. +In our example, the +covariate-assisted Balke-Pearl bounds improve substantially upon naïve covariate-agnostic +bounds. In fact, such additional adjustment in general cannot do worse, and can only im- +prove upon unadjusted bounds. We formalize this idea in the following result. +6 + +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +Average Treatment Effect +Covariate−Agnostic +Covariate−Adjusted +Figure 1: Balke-Pearl bounds for the ATE, with and without covariate adjustment. The red +dashed line represents the true ATE. Blue bars represent the true theoretical bounds, and +green bars are point estimates from a single sample of 10,000 subjects. The solid black line +indicates the reference value of zero average effect. +Proposition 1. Suppose X renders Z a valid instrument according to Assumptions 1–4. +Suppose that Assumptions 1–4 also hold after augmenting X with G, and Z ⊥⊥ G | X, i.e., G +doesn’t predict Z, except possibly through association with X. Then Balke-Pearl bounds based +on (X, G) are at least as tight as those based on X, and may be strictly tighter. +Note that this last result applies to both causal and non-causal IVs. According to Proposi- +tion 1, we would want to include any pure predictors of (A, Y ) that are not direct confounders +of the effect of Z on A and Y , when constructing Balke-Pearl bounds. This is somewhat +analogous to Lemma 4 in Rotnitzky and Smucler (2020), which says that influence function- +based estimation of the ATE from an observational study is improved by adding pure outcome +predictors—so-called “precision variables”—to an already sufficient set of confounders. Note, +however, that in that setting improvement corresponded to lower variance, whereas here we +are concerned both with the width of the theoretical bounds, as well as the variance of our +estimators. Specializing Proposition 1 to a causal IV (e.g., Z marginally randomized) yields +the following corollary: +Corollary 1. Suppose Z ⊥⊥ X and Assumptions 1–4 hold unconditionally as well as given +X. Then Balke-Pearl bounds based on X are at least as tight as unadjusted bounds. +7 + +Corollary 1 explains what we observed in the motivating example of Section 3.2. That is, +a causal IV guarantees that inclusion of baseline covariates X will result in bounds that are +no worse than unadjusted bounds. The degree to which such inclusion narrows the bounds is +of great interest, and will help in deciding which covariates to measure in practice—a more +precise characterization of this improvement we leave for future research. For the remainder +of the paper, we focus on constructing valid and efficient estimators of the covariate-assisted +Balke-Pearl bounds. +4 +Targeting Smooth Approximations +Briefly, we now review the statistical framework we use to derive our estimators and to evaluate +the theoretical properties of our methods. One key goal is to develop methods that are flexible +and resistant to bias from model misspecification. +Here, we reduce the potential for model misspecification by using flexible nonparametric +machine learning (ML) tools. More specifically, we construct bias-corrected estimators using +influence functions – a central element of semiparametric theory. As an example, we review +estimation of the average treatment effect (ATE) E[Y (a = 1) − Y (a = 0)] from this perspec- +tive. Under no unmeasured confounding and other assumptions, E[Y (a = 1)] (i.e., the mean +outcome if every unit in the population is treated) can be written as an averaged regression +function: ψ = E{E(Y | A = 1, X)}. Estimating E(Y | A = 1, X) is a standard regression +problem and flexible ML methods are preferred to more restrictive estimation methods such +as linear regression to avoid model misspecification. However, if ψ is estimated as the average +of predictions from a ML model, rather than from a linear model, the estimate will generally +inherit first-order smoothing bias from the nonparametric estimates. Instead, one can find +a function of the data, which we can denote generically as ϕ, such that estimating ψ as the +average value of (an estimated version of) ϕ will correct this first-order bias. The optimal +choice of this function is referred to as the influence function of ψ, and in this case is based on +two regression models, E(Y | A = 1, X) and P(A = 1 | X). Thus, the analyst fits two models: +a model for the outcome regressed against the treatment and all confounders (the outcome +model); and a model regressing the treatment against all confounders (the propensity model). +These models are combined to estimate the effect of interest. This approach is “doubly robust”, +since it is consistent if either the propensity model or the outcome model is correctly specified +(Scharfstein et al., 1999), and also leads to “doubly reduced” second-order bias. Next, the +estimation process is combined with sample-splitting or cross-fitting, to prevent over-fitting +by separating estimation of the components of the influence function from estimation of its +mean (Robins et al., 2008; Zheng and van der Laan, 2010; Chernozhukov et al., 2018). +4.1 +Approximation Based on the LSE Function +The covariate-adjusted bound functionals L(P) = EP(γℓ(X)) and U(P) = EP(γu(X)) are non- +smooth, since the pointwise maximum, γℓ(X) = max1≤j≤8 θℓ,j(X), and pointwise minimum, +γu(X) = min1≤j≤8 θu,j(X), are not differentiable everywhere. In the language of semipara- +metric theory, L(P) and U(P) are—without further conditions (see Section 5)—non-pathwise +differentiable functionals. Thus, they do not have influence functions to facilitate robust and +efficient estimation (Bickel et al., 1993; Kennedy, 2022). In response, our first proposal is to +instead target approximations to the bounds, Lg(P) and Uh(P), where g and h are sufficiently +smooth approximate pointwise maximum and minimum functions, respectively. +8 + +While other approximations are possible, we will focus on the log-sum-exp (LSE) function +as an approximate maximum g. Namely, for any t > 0, define gt : Rk → R via +gt(v) = 1 +t log +� +� +k +� +j=1 +etvj +� +� , for v ∈ Rk. +(4) +The LSE is a convex and smooth function (Boyd et al., 2004), with gradient and Hessian given +by +∇gt = +z +1T z , +∇2gt = +t +(1T z)2 +�� +1T z +� +diag(z) − zzT � +, +where z = (etv1, . . . , etvk). Importantly, for our purposes, the following inequality shows that +LSE approximates the pointwise maximum function: +max {v1, . . . , vk} < gt(v) ≤ max {v1, . . . , vk} + log k +t +. +Increasing the tuning parameter t yields smaller approximation error. On the other hand, the +Hessian matrix of gt also depends critically on t; as we will see in the following discussions, as +t increases, the operator norm +��∇2gt +�� increases, inducing larger estimation error. +The LSE function is commonly employed across a range of disciplines, including statisti- +cal mechanics (Aldous, 2005) and machine learning (Calafiore et al., 2020). Using the LSE +function to approximate the maximum function is also common in the statistics literature. +For instance, one way to prove Sudakov-Fernique’s inequality on Gaussian comparison is to +apply the functional form of Slepian’s inequality (which requires the function considered to be +twice-differentiable) to the LSE function and let t → ∞ (Vershynin, 2018; Wainwright, 2019). +As another example, in proving a high-dimensional Gaussian comparison version of the central +limit theorem, Chernozhukov et al. (2012) used Slepian’s Gaussian interpolation together with +Stein’s leave-one-out expansions. A typical tool to evaluate Stein’s leave-one-out expansion +is a Taylor’s expansion, which requires differentiability of the function. Hence, Chernozhukov +et al. (2012) also approximated the maximum function with the LSE function, and selected +the tuning parameter t to limit approximation error while controlling the derivatives of the +LSE function. +In the context of our problem, we replace the maximum function in E [max1≤j≤8 θℓ,j(X)] +with the LSE function and focus on estimating the smooth functional E[gt(θℓ(X))] where +θℓ(X) = (θℓ,1(X), . . . , θℓ,8(X)) ∈ R8. By the approximation property, E[gt(θℓ(X))] satisfies +E +� +max +1≤j≤8 θℓ,j(X) +� +≤ E[gt(θℓ(X))] ≤ E +� +max +1≤j≤8 θℓ,j(X) +� ++ log 8 +t +. +We can similarly define a smooth approximation for pointwise minimum function as ht = g−t +and estimate the smooth functional E[ht(θu(X))] for the upper bound E [min1≤j≤8 θu,j(X)] +where θu(X) = (θu,1(X), . . . , θu,8(X)) ∈ R8. The smooth approximation for the minimum +function satisfies +E +� +min +1≤j≤8 θu,j(X) +� +− log 8 +t +≤ E[ht(θu(X))] ≤ E +� +min +1≤j≤8 θu,j(X) +� +. +In the remaining part of this section, we develop efficiency theory for these smooth functional +approximations of the Balke-Pearl bounds, and propose robust and efficient estimators. +9 + +4.2 +Efficiency Theory +Next, we present the efficient influence function for functionals of the form Lg(P) and Uh(P). +In semiparametric efficiency theory (Bickel et al., 1993; Tsiatis, 2006; van der Vaart, 2000; +Kennedy, 2016), a fundamental goal is to characterize the (efficient) influence function. Math- +ematically, an influence function is the derivative in a von Mises expansion of the target +statistical functional (analogous to the usual derivative of a function in Taylor expansion). In +robust statistics, it coincides with the Gateaux derivative of the functional in the direction of +a point-mass contamination distribution. The influence function serves a number of purposes. +First, the variance of the efficient influence function is equal to the efficiency bound of the +target statistical functional, which serves as a lower bound of the variance for regular estima- +tors. It characterizes the inherent estimation difficulty of the target functional and provides +a benchmark to compare against when we construct estimators. Moreover, it enables us to +correct for first-order bias in the plug-in estimator and motivates the robust estimator, which +has a general second-order bias property even if nonparametric and flexible machine learning +methods with relatively slow rates are used for estimating the nuisance functions. +We summarize the nonparametric influence functions for Lg(P) and Uh(P) in the following +theorem. +Theorem 2. For each y, a, z ∈ {0, 1}, define +ψya.z(O; P) = 1(Z = z) +λz(X) +{1(Y = y, A = a) − πya.z(X)} . +The nonparametric influence function of Lg(P) is +˙Lg(O; P) = g (θℓ(X)) − Lg(P) + +8 +� +j=1 +∂g (θℓ(X)) +∂θℓ,j(X) Lj(O; P), +where Lj(O; P) is obtained by replacing πya.z(X) with ψya.z(O; P) in θℓ,j(X) and omitting +constant 1 whenever it appears. Similarly, the nonparametric influence function of Uh(P) is +˙Uh(O; P) = h (θu(X)) − Uh(P) + +8 +� +j=1 +∂h (θu(X)) +∂θu,j(X) Uj(O; P), +where Uj(O; P) is obtained by replacing πya.z(X) with ψya.z(O; P) in θu,j(X) and omitting +constant 1 whenever it appears. +Theorem 2 implies that there are two terms contributing to the influence function (and hence +estimation) of the smooth functionals Lg(P) and Uh(P). The first term, g (θℓ(X)) − Lg(P), +comes from estimating EP[g(θℓ(X))] when the functions θℓ,j(X)’s are known and do not need +to be estimated, while the second term, �8 +j=1 +∂g(θℓ(X)) +∂θℓ,j(X) Lj(O; P), comes from estimating these +unknown functions. +After characterizing the influence functions for Lg(P) and Uh(P), we can use them to +correct for the first-order bias in the von Mises expansion and arrive at robust estimators, +which allows us to perform estimation and inference efficiently. +10 + +4.3 +Estimation & Inference +Next, we propose and analyze a robust estimator for the smooth lower bound functional: +Lg(P). Similar results hold for Uh(P) using the same arguments. Assume we train models +for the nuisance functions λz(X), πya.z(X) based on a separate independent sample Dn. The +robust estimator is defined as: +�Lg = Lg(�P) + Pn +� +˙Lg(O; �P) +� += Pn +� +�g +� +�θℓ(X) +� ++ +8 +� +j=1 +∂g +� +�θℓ(X) +� +∂�θℓ,j(X) +Lj(O; �P) +� +� +The following theorem characterizes the conditional bias (given the training data) of the robust +estimator �Lg and establishes its asymptotic normality under additional conditions. +Theorem 3. Suppose g : R8 �→ R is twice continuously differentiable, such that ∥∇g(θ)∥∞ ≤ +C1 and ∥∇2g(θ)∥ ≤ C2 over θ. The nuisance functions �πya.z, �λz are estimated from a separate +independent sample. Moreover, there exists positive constant ϵ such that +P +� +ϵ ≤ �λ1(X) ≤ 1 − ϵ +� += 1. +Then the conditional bias of the robust estimator �Lg can be bounded as +���E[ �Lg | Dn] − Lg(P) +��� +≲ +� +max +y,a,z∈{0,1} ∥�πya.z − πya.z∥ +� � +C1 +����λ1 − λ1 +��� + C2 +max +y,a,z∈{0,1} ∥�πya.z − πya.z∥ +� +. +Let f(O) = ˙Lg(O; P) + Lg(P) be the non-centered influence function. If we further assume +��� �f − f +��� = oP(1) and the nuisance estimators satisfy the following convergence rate +����λ1 − λ1 +��� +� +max +y,a,z∈{0,1} ∥�πya.z − πya.z∥ +� += oP(n−1/2), +max +y,a,z∈{0,1} ∥�πya.z − πya.z∥2 = oP(n−1/2). +Then we have +�Lg − Lg(P) = Pn +� +˙Lg(O; P) +� ++ oP(n−1/2), +implying the robust estimator is √n-consistent and achieves the nonparametric efficiency +bound. +Sample splitting allows us to avoid complicated empirical process conditions (Chernozhukov +et al., 2016; Zheng and van der Laan, 2010; Kennedy et al., 2020) and derive bounds on the +conditional bias of estimator in terms of the convergence rate of the nuisance functions. Note +that in Theorem 3 we do not require the individual nuisance functions to converge at √n- +rate. +The conditions are on the convergence rates squared. +This is a key advantage of a +robust estimator: after we correct for the first-order bias, the remaining bias only involves +second-order terms and is “doubly small.” +As implied by Theorem 3, if the nuisance esti- +mators satisfy +����λ1 − λ1 +��� = OP +� +n−1/4� +and maxy,a,z∈{0,1} ∥�πya.z − πya.z∥ = oP +� +n−1/4� +, the +robust estimator will be √n-consistent and achieves the efficiency bound, which allows us +to perform inference efficiently. Causal IVs present a special case of interest. In this case, +Z ⊥⊥ X holds by randomization of treatment assignment Z and we know λz(X) is equal +to a constant λz by design. +Here, the requirement for the convergence rate is reduced to +maxy,a,z∈{0,1} ∥�πya.z − πya.z∥ = oP +� +n−1/4� +. +11 + +4.4 +Wald-type Confidence Interval for the ATE +Combining Theorem 1 with the smooth approximation we have +Lgt(P) − log 8 +t +≤ E[Y (a = 1) − Y (a = 0)] ≤ Uht(P) + log 8 +t +. +Thus the following +� +�Lgt − log 8 +t +− z1−α/2 +√n +�Vt, �Uht + log 8 +t ++ z1−α/2 +√n +� +Wt +� +is an asymptotically valid (though potentially conservative) 100(1 − α)% Wald-type confi- +dence interval for the ATE, where zβ is the β-th quantile of the standard normal distribution +and �V 2 +t , � +W 2 +t are plug-in estimators of the nonparametric efficiency bounds VarP +� +˙Lgt(O; P) +� +and VarP +� +˙Uht(O; P) +� +. The choice of the tuning parameter t requires balancing the smooth +approximation error log 8/t and the conditional bias term E[ �Lg | Dn] − Lg(P). Note that +∥∇2gt(θ)∥ ≤ t and Theorem 3 implies the conditional bias increases as we select a larger t. +Hence a smaller t is preferred to reduce the conditional bias. Yet reducing the approximation +error log 8/t requires a larger t. How to choose t in a data-driven fashion to arrive at the +shortest confidence interval remains an open problem left for future investigation. +5 +Bounds Using a Margin Condition +While the approach in Section 4 yields valid and generally conservative bounds on the ATE, +the estimates of the smoothed bounds do not converge to the true Balke-Pearl bounds L(P) +and U(P) at √n-rates, even with the tuning parameters optimized for a given sample size n +and known nuisance estimation error. Indeed, we should not expect it to be possible in the +nonparametric model to estimate the covariate-adjusted bounds at parametric rates, owing +to the non-pathwise differentiability of these functionals. Next, we outline a second approach +that exploits an additional assumption known as a margin condition in order to achieve faster +rates. +5.1 +An infeasible estimator +To motivate the proposed estimator, we first consider an infeasible estimator of the bounds. +Let dℓ(X) ∈ arg max1≤j≤8 θℓ,j(X) and du(X) ∈ arg min1≤j≤8 θu,j(X). The bounds can thus +be written as +L(P) = +8 +� +j=1 +E [1{dℓ(X) = j}θℓ,j(X)] +and +U(P) = +8 +� +j=1 +E [1{du(X) = j}θu,j(X)] . +Under the assumption that X �→ dℓ(X) and X �→ du(X) are known functions, it can be +shown that the uncentered influence functions of L(P) and U(P) are +ϕℓ(O; P, dℓ) = +8 +� +j=1 +1{dℓ(X) = j} {Lj(O; P) + θℓ,j(X)} +ϕu(O; P, du) = +8 +� +j=1 +1{du(X) = j} {Uj(O; P) + θu,j(X)} +12 + +where Lj(O; P) and Uj(O; P) are defined in Theorem 2. Therefore, under minor regularity +conditions, an infeasible estimator �L = Pn{ϕℓ(O; �P, dℓ)} would satisfy +√n( �L − L(P)) ⇝ N(0, var{ϕℓ(O; P, dℓ)}) +as long as P{ϕℓ(O; P, dℓ) − ϕℓ(O; �P, dℓ)} = oP(n−1/2). In this regime, �L would be efficient. +5.2 +Estimation & Inference +In light of the discussion in the section above, we propose estimating the bounds with +�L = +8 +� +j=1 +Pn +� +1{�dℓ(X) = j}{Lj(O; �P) + �θl,j(X) +� += Pn{ϕℓ(O; �P, �dℓ)} +�U = +8 +� +j=1 +Pn +� +1{�du(X) = j}{Uj(O; �P) + �θu,j(X) +� += Pn{ϕu(O; �P, �du)} +That is, to estimate the non-smooth part of the bounds, namely the indicators 1{du(X) = j} +and 1{dℓ(X) = j}, we use plug-in estimators 1{�du(X) = j} and 1{�dℓ(X) = j}. A natural +question to ask at this point is under what conditions, if any, the estimators �L and �U behave, +at least asymptotically, like their infeasible counterparts �L and �U. +As shown in the next +theorem, a sufficient condition for the estimators to behave similarly to the oracle ones that +have access to dℓ(x) and du(x) is captured by the following “margin” condition. This additional +assumption controls the probability that the minimum and maximum are near their points of +non-differentiability. +Assumption 5 (Margin condition). There exists α > 0 such that for any t ≥ 0, +P +� +min +j̸=dℓ(X){θℓ,dℓ(X)(X) − θℓ,j(X)} ≤ t +� +≲ tα, +(5) +and +P +� +min +j̸=du(X){θu,j(X) − θu,du(X)(X)} ≤ t +� +≲ tα. +(6) +The margin condition in Assumption 5 is very similar to conditions that have been proposed +and leveraged in the classification literature (Audibert and Tsybakov, 2007), as well as in +dynamic treatment regimes (Luedtke and van der Laan, 2016) and other instrumental variable +problems (Kennedy et al., 2020). In words, condition (5) says that with high probability, θℓ,dℓ +is separated from non-maximal values lower bound quantities θℓ,j. Similarly, condition (6) +limits how close non-minimal upper bound values θu,j are to the actual minimum θu,du. If +minj̸=dℓ(X){θℓ,dℓ(X)(X) − θℓ,j(X)}, for instance, has bounded density near zero, then (5) will +hold with α = 1. This is a relatively weak requirement which we expect to hold in many +cases. Under Assumption 5, we are able to derive sufficient conditions such that �L and �U are +√n-consistent and asymptotically normal. +Theorem 4. Suppose that the nuisance functions �πya.z and �λz are estimated from a separate +independent sample. Moreover, suppose that Assumption 5 holds, P +� +ϵ ≤ �λ1(X) ≤ 1 − ϵ +� += 1, +13 + +for some ϵ > 0, +����λ1 − λ1 +��� = oP(1), and maxy,a,z∈{0,1} ∥�πya.z − πya.z∥ = oP(1). Then, we have +�L − L = (Pn − P)ϕℓ(O; P, dℓ) ++ OP +�����λ1 − λ1 +��� · +max +y,a,z∈{0,1} ∥�πya.z − πya.z∥ + max +1≤j≤8 +����θℓ,j − θℓ,j +��� +1+α +∞ +� ++ oP(n−1/2), +and +�U − U = (Pn − P)ϕu(O; P, du) ++ OP +�����λ1 − λ1 +��� · +max +y,a,z∈{0,1} ∥�πya.z − πya.z∥ + max +1≤j≤8 +����θu,j − θu,j +��� +1+α +∞ +� ++ oP(n−1/2). +The result of Theorem 4 establishes that �L and �U are √n-consistent as long as +����λ1 − λ1 +��� · +max +y,a,z∈{0,1} ∥�πya.z − πya.z∥ + max +1≤j≤8 +����θu,j − θu,j +��� +1+α +∞ += oP(n−1/2) +The first term is the same term arising in the upper bound to the bias of the estimators +from Section 4. The second term denotes a bound on the bias arising when estimating the +indicators 1{dℓ(x) = j} and 1{du(x) = j} and depends on the exponent α from Assumption +5. For example, as long as minj̸=dℓ(x){θℓ,dℓ(X) − θℓ,j(X)} has a bounded density near zero, +then Assumption 5 holds with α = 1. In this case, maxy,a,z∈{0,1} ∥�πya.z − πya.z∥∞ = oP(n−1/4) +is sufficient to imply +max +1≤j≤8 +����θu,j − θu,j +��� +1+α +∞ += oP(n−1/2) +Because estimation in L∞ typically yields the same convergence rate as estimation in L2 up +to a log factor (Tsybakov, 2009), when Assumption 5 holds with α = 1, then the requirements +on the bias of Theorem 4 are essentially the same as those of Theorem 3. Finally, Theorem +4 outlines sufficient conditions for the asymptotic normality of �L and �U so that Wald-type +confidence intervals are straightforward to compute. In particular, the procedure proposed in +Imbens and Manski (2004) can be used to conduct more precise inferences (see also Theorem +3 in Jiang and Ding (2018)). +6 +Direct Estimation with Continuous Outcomes +Thus far, we have only focused on a binary outcome Y ∈ {0, 1}. In this section, we will +extend our approach from the binary outcome case to construct valid bounds on the ATE for +continuous outcomes. In fact, we will assume only that the outcome Y is bounded; without +loss of generality, we may assume Y ∈ [0, 1] (otherwise one can always rescale the outcome). As +observed but not studied in Balke and Pearl (1997), the idea is to replace the binary outcome +in previous analyses with the indicator 1(Y ≤ t). Specifically, for each t ∈ [0, 1], 1(Y ≤ t) is a +binary outcome for which we can apply the bounds proposed in Section 3.1 to obtain pointwise +bounds on the difference in the distribution functions of potential outcomes P(Y (a) ≤ t). We +proceed by integrating the tail probabilities P(Y (a) > t) to bound the mean of potential +outcomes E(Y (a)), and finally arrive at bounds on the ATE, E(Y (a = 1) − Y (a = 0)). +For any a, z ∈ {0, 1} and t ∈ R, define the probabilities π1a.z(t, X) = P(Y ≤ t, A = a | +X, Z = z) and π0a.z(t, X) = P(Y > t, A = a | X, Z = z). Next, we define functions exactly +14 + +as in (1) and (2), replacing with πya.z(X) with πya.z(t, X) (note that θu,j, θℓ,j, γℓ, and γu are +then also functions of t and X). Assuming Z is a valid instrument given covariates X, for +each t ∈ R we view 1(Y ≤ t) as the binary outcome and apply the “Balke-Pearl" bounds to +this new outcome. By Theorem 1, we obtain the following valid (though not necesarily tight) +bounds on the difference in distribution functions of the individual potential outcomes: +γℓ(t, X) ≤ P[Y (a = 1) ≤ t | X] − P[Y (a = 0) ≤ t | X] ≤ γu(t, X). +Integrating with respect to X and noting that E(Y (a)) = +� 1 +0 P[Y (a) > t]dt, we arrive at the +following bounds on the ATE: +− +� 1 +0 +E (γu(t, X)) dt ≤ E(Y (a = 1) − Y (a = 0)) ≤ − +� 1 +0 +E (γℓ(t, X)) dt. +(7) +One natural approach would be to estimate the functions t �→ E (γu(t, X)) and t �→ E (γℓ(t, X)) +by the proposals of Section 4 or Section 5, over a grid of values t ∈ [0, 1], and use these estimates +to approximate the integrated bounds given in (7). Alternatively, one could use Monte Carlo +methods to approximate the integral. Either approach would require computing a binary- +outcome estimator at many different inputs t, which could be quite computationally intensive. +The following theorem characterizes a looser bound that can be computed more efficiently. +Theorem 5. Let W ∼ Uniform(0, 1) be independent of O = (X, Z, A, Y ). Then +� 1 +0 +E (γu(t, X)) dt ≤ E +� +min +1≤j≤8 +�θu,j(X) +� +, +� 1 +0 +E (γℓ(t, X)) dt ≥ E +� +max +1≤j≤8 +�θℓ,j(X) +� +, +where we define �π1a.z(X) = P(Y ≤ W, A = a | X, Z = z), �π0a.z(X) = P(Y > W, A = a | +X, Z = z), and where �θℓ,j(X) and �θu,j(X) are defined exactly as in equations (1) and (2), +replacing π with �π. +Theorem 5 together with (7) implies the following valid bounds on the ATE: +−E +� +min +1≤j≤8 +�θu,j(X) +� +≤ E[Y (1) − Y (0)] ≤ −E +� +max +1≤j≤8 +�θℓ,j(X) +� +. +(8) +Note that the bounds in (8) are of the same form as those in Theorem 1, but with a new binary +outcome 1(Y ≤ W). By introducing an independent uniform random variable, we obtain a +looser bound, but one that lends itself to more computationally feasible estimation. To oper- +ationalize these bounds, we can simulate n i.i.d. Uniform(0, 1) random variables W1, . . . , Wn, +independent of the data, and construct the augmented observation unit �O = (X, Z, A, Y, W). +We then estimate the quantities in the bounds exactly as in the binary case using the methods +proposed in Section 4 or Section 5, with the new binary outcome being 1(Y ≤ W). Note that +the exact value of the final bounds will depend on the realization of W1, . . . , Wn. To reduce +this variability and regain some efficiency, we may repeat this process m times and average +the resulting estimates of the bounds—see Proposition 3 in Appendix D for an illustration of +this behavior in a simplified setting. +15 + +7 +Data Analysis +To illustrate the proposed methods on real data, we re-analyzed data from the randomized +experiment reported in McDonald et al. (1992), and more recently analyzed in Hirano et al. +(2000); Imbens (2014). To study the effect of the seasonal influenza vaccine on flu-related +hospital visits, McDonald et al. (1992) conducted a randomized trial employing an encour- +agement design. In particular, we have data on 2,861 patients that were seen by physicians +who were randomized to receive (Z = 1) or not (Z = 0) reminders to inoculate their patients. +Treatment in this study was whether a patient received (A = 1) or not (A = 0) the flu vac- +cine, and the outcome was whether a flu-related visit to the hospital followed (Y = 1) or not +(Y = 0). In addition to the instrument, treatment, and outcome, (Z, A, Y ), a collection of +baseline covariates X were available on each patient, including age, gender, and indicators +for chronic obstructive pulmonary disease, diabetes mellitus, heart disease, severe renal fail- +ure, and chronic liver failure. As in Hirano et al. (2000) and Imbens (2014), since we do not +have information on which patients were seen by which physicians, we ignore clustering by +physician. +We constructed unadjusted Balke-Pearl bounds nonparametrically based on Theorem 1, +estimating each P[Y = y, A = a | Z = z] with an empirical average, for y, a, z ∈ {0, 1}. We +then computed a 95% confidence interval via the bootstrap. We also computed estimates of the +covariate-adjusted bounds L(P) and U(P) using the direct estimators developed in Section 5, +assuming the margin condition (i.e., Assumption 5). +In particular, we used the marginal +empirical Z = 1 frequency to estimate �λ1 ≡ 0.5145, and used a random forest on the categorical +outcome (Y, A) ∈ {0, 1}2, stratified by the instrument, to fit {�πya.z(X) : y, a ∈ {0, 1}}, for +each z ∈ {0, 1}. For the adjusted bounds, we used cross-fitting with 10 folds. +Summarizing the results, the point estimates for the unadjusted Balke-Pearl bounds were +(−0.239, 0.642), and the resulting bootstrap-based 95% confidence interval for the ATE was +(−0.260, 0.668). Meanwhile, the point estimates for the covariate-adjusted Balke-Pearl bounds +were (−0.105, 0.640), and the resulting Wald-based 95% confidence interval for the ATE was +(−0.135, 0.673). Overall, the confidence interval for the ATE based on the covariate-adjusted +bounds was ∼13% shorter than the interval based on the unadjusted bounds. Interestingly, +the upper bound estimate essentially matched the unadjusted value, while the lower bound +estimate was substantially less negative. +8 +Discussion +In this paper, we proposed estimators of nonparametric bounds on the average treatment +effect using an instrumental variable, avoiding strong structural or parametric assumptions +typically used for point identification. We extended the classic approach of Balke and Pearl +(1997) by incorporating baseline covariate information to (i) control for measured confounders +to render the instrument valid, and/or (ii) construct narrower and hence more informative +bounds. We also proposed a concrete extension to bound the ATE for a general bounded +outcome. Our estimators are based on influence functions, and as a result are robust and +can attain √n-consistency, asymptotic normality, and nonparametric efficiency, under non- +parametric convergence rates of the component nuisance functions—these rates are attainable +under sparsity, smoothness or other structural conditions. +The key difficulty in estimating the covariate-adjusted bound functionals L(P) and U(P) +is that, as means of non-differentiable functions, they are not pathwise-differentiable (Bickel +16 + +et al., 1993). As a result, these parameters do not—at least without further assumptions— +have an influence function to facilitate flexible and efficient estimation (e.g., see Section 5.3 of +Kennedy (2022)). To make progress, in Section 4, we first presented general valid bounds on +the ATE via estimation of pathwise-differentiable approximations to L(P) and U(P). Second, +in Section 5, we invoked a margin condition to render the exact bound functionals pathwise- +differentiable, and proposed influence function-based estimators under this condition. The +first approach has the advantage of not requiring the margin condition, but the resulting +bounds suffer from being slightly conservative, and not converging to the true exact bounds +at parametric rates. The second approach, on the other hand, achieves parametric rates when +the margin condition holds. As a general guideline, we recommend that practitioners use the +direct bound estimators proposed in Section 5, deferring to the approximate bounds only when +there is serious doubt on Assumption 5 arising due to subject matter knowledge. +In our view, it is difficult to imagine concrete scenarios in which Assumption 5 would be +violated. A similar margin condition is invoked in dynamic treatment regime problems, for +example when estimating the mean outcome under the optimal treatment policy in a point +treatment setting: E[max{E(Y (a = 1) | X), E(Y (a = 0) | X)}] (Luedtke and van der Laan, +2016). In that context, one must argue that the CATE is well separated from zero, the value +representing no treatment effect. It is plausible that there is a subgroup for which the CATE +is exactly zero, which would violate the margin condition for that problem. In our setting, +Assumption 5 requires separation of the pointwise maximum (minimum) of the Balke-Pearl +lower (upper) bound functions θℓ(X) (θu(X)) from the second largest (smallest) value. These +lower and upper bound functions are not readily interpretable like the CATE, and it would +seem to necessitate an unlikely confluence of factors to violate the required separation. +It is important to mention some limitations of the proposed methods, as well as possible +extensions and open problems. First, we have throughout assumed a fixed collection of mea- +sured covariates X. In Proposition 1, we provide a general criterion to justify adjusting for +certain covariates, however, it remains to characterize (i) the actual difference in the length of +the bounds based on two valid adjustment sets, and (ii) the effect of the adjustment set on the +variance of the proposed estimators. Second, we have focused primarily on the setting with +instrument, exposure, and outcome all binary variables, since this is the simplest setting for +partial identification with closed form bounds given by Balke and Pearl (1997). In Section 6, +we provided an extension to construct valid ATE bounds for continuous outcomes, though +an important open problem is determining the sharpest possible bounds for such general out- +comes, beyond the binary case. Along the same lines, it of course will also be of interest to +consider multi-valued or continuous instruments and exposures. In these cases, there is no +longer the same closed form solution for the theoretical bounds, and alternative linear pro- +gramming specifications and/or symbolic bounds might be incorporated (Sachs et al., 2022; +Zhang and Bareinboim, 2021). Third, an interesting problem we will explore in future research +is to directly and efficiently estimate the conditional bounds γℓ(X), γu(X) for the CATE given +in Theorem 1, and identify strata for whom we are confident that the CATE is positive or +negative. In doing so, one may extend the ideas in Kennedy et al. (2020) to define “sharpness” +of an instrument—the ability to tightly bound causal effects in certain subgroups—without +relying on the typical monotonicity assumption. +17 + +Acknowledgements +Research in this article was supported by the Patient-Centered Outcomes Research Institute +(PCORI Awards ME-2021C1-22355). All statements in this report, including its findings and +conclusions, are solely those of the authors and do not necessarily represent the views of +PCORI or its Methodology Committee. +18 + +References +Aldous, D. (2005), “Spin Glasses: A Challenge for Mathematicians,” . +Angrist, J. D. and Evans, W. N. 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Berkeley. +21 + +Supporting Information +A +Proofs of Results in Section 2 +A.1 +Proof of Proposition 1 +The bounds based on (X, G) are of the same form as those for X alone, but with underlying +probabilities π† +ya.z(X, G) = P[Y = y, A = a | X, G, Z = z] instead of πya.z(X). The key +observation is that for any y, a, z ∈ {0, 1}, +πya.z(X) = E(π† +ya.z(X, G) | X, Z = z) = E(π† +ya.z(X, G) | X), +where the first equality holds by the tower law, and the second holds by the assumption +Z ⊥⊥ G | X. Therefore, letting θ† +ℓ,j(X, G), θ† +u,j(X, G) be of the same for as θℓ,j(X), θu,j(X), +respectively, for j = 1, . . . , 8, but with π† +ya.z replacing πya.z for all y, a, z ∈ {0, 1}, we have +EP +� +max +1≤j≤8 θℓ,j(X) +� += EP +� +max +1≤j≤8 EP(θ† +ℓ,j(X, G) | X) +� +≤ EP +� +max +1≤j≤8 θ† +ℓ,j(X, G) +� +, +where we used a conditional version of Jensen’s inequality for the pointwise maximum, and +iterated expectations. The proof that the (X, G)-upper bound is lower follows by the same +logic, using concavity of the pointwise minimum. The example in Section 3.2 shows that the +lower (upper) bound based on (X, G) may be strictly greater (smaller). +A.2 +Proof of Corollary 1 +This follows immediately by Proposition 1, replacing X with ∅, and G with X. +B +Proofs of Results in Section 4 +B.1 +Proof of Theorem 2 +Recall an influence function of a pathwise-differentiable functional χ(P), at P in a given sta- +tistical model, is a zero-mean finite-variance function ˙χ(O; P) of observed data O such that +for any regular one-dimensional parametric submodel Pϵ through P0 = P, it holds that +d +dϵχ (Pϵ) +���� +ϵ=0 += EP[ ˙χ(O; P)u(O)] +where u(O) is the score function of the parametric submodel at P. For such a parametric +submodel, invoking the total derivative, we have (let θϵ,ℓ(X) be the nuisance function vector +22 + +at submodel Pϵ) +d +dϵLg (Pϵ) +���� +ϵ=0 += d +dϵEPϵ [g (θϵ,ℓ(X))] +���� +ϵ=0 += d +dϵEPϵ [g (θℓ(X))] +���� +ϵ=0 ++ +8 +� +j=1 +d +dϵEP [g (θℓ,1(X), . . . , θϵ,ℓ,j(X), . . . , θℓ,8(X))] +������ +ϵ=0 += EP [(g (θℓ(X)) − Lg(P)) u(O)] + +8 +� +j=1 +EP +� ∂g (θℓ(X)) +∂θℓ,j(X) +d +dϵθϵ,ℓ,j(X) +���� +ϵ=0 +� +(9) +using the fact that score function u has mean zero to center the first term, and the chain +rule for the remaining eight terms. Next, observe that, for any (y, a, z) ∈ {0, 1}3, (denote in +general u(B | C) as the conditional score function for the distribution of B given C. +d +dϵπϵ,ya.z(X) +���� +ϵ=0 += EP [(1(Y = y, A = a) − πya.z(X)) u(Y, A | Z = z, X) | Z = z, X] += EP +�1(Z = z) +λz(X) +{1(Y = y, A = a) − πya.z(X)} u(Y, A | Z = z, X) | X +� += EP +�1(Z = z) +λz(X) +{1(Y = y, A = a) − πya.z(X)} u(Y, A | Z, X) | X +� += EP (ψya.z(O; P)u(Y, A | Z, X) | X) , +(10) +where the first equation follows from directly taking derivatives and centering with πya.z since +u(Y, A | Z = z, X) has conditional mean zero. The second equation follows from conditioning +on Z, X and using property of conditional expectations. The last equation follows from the +definition of ψya.z. +Since θℓ,j’s are linear combinations of πya.z (and constant 1), we can prove +EP +� ∂g (θℓ(X)) +∂θℓ,j(X) +d +dϵθϵ,ℓ,j(X) +���� +ϵ=0 +� += EP +�∂g (θℓ(X)) +∂θℓ,j(X) Lj(O; P)u(O) +� +. +For illustration, take j = 1 (other j’s can be proved similarly) and we have +EP +� ∂g (θℓ(X)) +∂θℓ,1(X) +d +dϵθϵ,ℓ,1(X) +���� +ϵ=0 +� += EP +� ∂g (θℓ(X)) +∂θℓ,1(X) +d +dϵ {πϵ,11.1(X) + πϵ,00.0(X) − 1} +���� +ϵ=0 +� += EP +�∂g (θℓ(X)) +∂θℓ,1(X) EP [{ψ11.1(O; P) + ψ00.0(O; P)} u(Y, A | Z, X) | X] +� += EP +�∂g (θℓ(X)) +∂θℓ,1(X) {ψ11.1(O; P) + ψ00.0(O; P)} u(Y, A | Z, X) +� += EP +�∂g (θℓ(X)) +∂θℓ,1(X) L1(O; P)u(O) +� +, +where the first equation follows from definition of θϵ,ℓ,1. The second equation follows from (10). +The third equation follows from property of conditional distribution. In the final equation we +23 + +added u(Z, X) and noted u(O) = u(Z, X) + u(Y, A | Z, X) — we are permitted to add this +term as it is a function only of (Z, X), and ψya.z(O; P) has mean zero given (Z, X). +Plugging these equations into (9) we have +d +dϵLg (Pϵ) +���� +ϵ=0 += EP +� +� +� +� +�g (θℓ(X)) − Lg(P) + +8 +� +j=1 +∂g (θℓ(X)) +∂θℓ,j(X) Lj(O; P) +� +� u(O) +� +� +� . +The argument for deriving the influence function of Uh(P) uses the exact same logic. +B.2 +Proof of Theorem 3 +We first prove a proposition that characterizes the conditional bias of the robust estimator. +In the following derivations all the expectations are taken conditioning on training data Dn. +Proposition 2. Suppose g is a twice continuously differentiable function, then we have +E[ �Lg] − Lg = EP +� +∇g +� +�θℓ(X) +�T � +EP(L(O; �P) | X) + �θℓ(X) − θℓ(X) +�� +− 1 +2EP +�� +�θℓ(X) − θℓ(X) +�T +∇2g (θ∗ +ℓ(X)) +� +�θℓ(X) − θℓ(X) +�� +, +where L(O; �P) = +� +L1(O; �P), . . . , L8(O; �P) +�T +and θ∗ +ℓ(X) is a point that lies on the line segment +between θℓ(X) and �θℓ(X). +Proof. By definition of �Lg, +E[ �Lg] − Lg += EP +� +�g +� +�θℓ(X) +� ++ +8 +� +j=1 +∂g +� +�θℓ(X) +� +∂�θℓ,j(X) +Lj(O; �P) − g (θℓ(X)) +� +� += EP +� +�g +� +�θℓ(X) +� ++ +8 +� +j=1 +∂g +� +�θℓ(X) +� +∂�θℓ,j(X) +EP +� +Lj(O; �P) | X +� +− g (θℓ(X)) +� +� += EP +� +∇g +� +�θℓ(X) +�T +EP(L(O; �P) | X) − +� +g (θℓ(X)) − g(�θℓ(X)) +�� += EP +� +∇g +� +�θℓ(X) +�T � +EP[L(O; �P) | X] + �θℓ(X) − θℓ(X) +�� +− 1 +2EP +�� +�θℓ(X) − θℓ(X) +�T +∇2g (θ∗ +ℓ(X)) +� +�θℓ(X) − θℓ(X) +�� +. +The second equality follows from conditioning on X and the last equality follows from second- +order Taylor expansion. +For any (y, a, z) ∈ {0, 1}3, by conditioning on (Z, X) we have +EP +� +ψya.z(O; �P) | X +� += E +� +1(Z = z) +�λz(X) +(P(Y = y, A = a | X, Z = z) − �πya.z(X)) | X +� += λz(X) +�λz(X) +{πya.z(X) − �πya.z(X)} . +24 + +We want to bound each component of EP[L(O; �P) | X] + �θℓ(X) − θℓ(X). We only analyze +the first component as an illustration. All other components can be similarly analyzed. +EP +� +L1(O; �P) | X +� ++ �θℓ,1(X) − θℓ,1(X) += EP +� +ψ11.1(O; �P) + ψ00.0(O; �P) | X +� ++ {�π11.1(X) − π11.1(X)} + {�π00.0(X) − π00.0(X)} += +� +1 − λ1(X) +�λ1(X) +� +{�π11.1(X) − π11.1(X)} + +� +1 − λ0(X) +�λ0(X) +� +{�π00.0(X) − π00.0(X)} . +Note that +1 − λ1(X) +�λ1(X) += +�λ1(X) − λ1(X) +�λ1(X) +, 1 − λ0(X) +�λ0(X) += λ1(X) − �λ1(X) +1 − �λ1(X) +By positivity assumption we have +���EP +� +L1(O; �P) | X +� ++ �θℓ,1(X) − θℓ,1(X) +��� +≤ 1 +ϵ +����λ1(X) − λ1(X) +��� · {|�π11.1(X) − π11.1(X)| + |�π00.0(X) − π00.0(X)|} +Similar inequalities can be obtained for 2 ≤ j ≤ 8. Hence, by Hölder’s inequality, +����EP +� +∇g +� +�θℓ(X) +�T � +EP(L(O; �P) | X) + �θℓ(X) − θℓ(X) +������ +≤ EP +����∇g +� +�θℓ(X) +���� +∞ +���EP(L(O; �P) | X) + �θℓ(X) − θℓ(X) +��� +1 +� +≤ C1 +ϵ EP +� +� +����λ1(X) − λ1(X) +��� +� +(y,a,z)∈R +|�πya.z(X) − πya.z(X)| +� +� , +where R is a multiset of elements in {0, 1}3 such that each (y, a, z) ∈ {0, 1}3 appears in R as +many times as πya.z(X) appears in θℓ(X). By the triangle inequality and Cauchy-Schwarz’s +inequality +����EP +� +∇g +� +�θℓ(X) +�T � +EP(L(O; �P) | X) + �θℓ(X) − θℓ(X) +������ +≲ C1 +����λ1 − λ1 +��� +� +(y,a,z)∈R +∥�πya.z − πya.z∥ +≲ C1 +����λ1 − λ1 +��� +� +max +y,a,z∈{0,1} ∥�πya.z − πya.z∥ +� +It remains to bound the second derivative term in the asymptotic bias expression derived in +Proposition 2. By property of the operator norm we have +����EP +�� +�θℓ(X) − θℓ(X) +�T +∇2g (θ∗ +ℓ(X)) +� +�θℓ(X) − θℓ(X) +������ +≤ +� +sup +θ +∥∇2g(θ)∥ +� +EP +�����θℓ(X) − θℓ(X) +��� +2 +2 +� +≲ C2 +� +(y,a,z)∈R +∥�πya.z − πya.z∥2 +≲ C2 +max +y,a,z∈{0,1} ∥�πya.z − πya.z∥2 +25 + +The bound on conditional bias is established. For a general function f on the sample O, we +have +Pn[ �f] − E[f] = (Pn − E)( �f − f) + (Pn − E)(f) + E( �f − f) +(11) +We apply the decomposition of error (11) to +f(O) = ˙Lg(O; P) + Lg(P) = g (θℓ(X)) + +8 +� +j=1 +∂g (θℓ(X)) +∂θℓ,j(X) Lj(O; P), +Note that Pn[ �f] is exactly the robust estimator. By Lemma 2 in Kennedy et al. (2020), if +∥ �f − f∥2 = oP(1) we have +(Pn − E)( �f − f) = oP(n−1/2). +Also note E( �f −f) is equal to the conditional bias and under the convergence rate assumption +we have E( �f − f) = oP(n−1/2). Finally note that f − Ef = ˙Lg(O; P), the proof is completed. +C +Proofs of Results in Section 5 +C.1 +Proof of Theorem 4 +The proof structures follows that of the proof of Theorem 3. We prove the result for the lower +bound as the result for the upper bound is analogous. By the standard decomposition, we +have +�L − L = (Pn − P){ϕℓ(O; �P, �dℓ) − ϕℓ(O; P, dℓ)} + P{ϕℓ(O; �P, �dℓ) − ϕℓ(O; P, dℓ)} ++ (Pn − P){ϕℓ(O; P, dℓ)} +≡ R1 + R2 + (Pn − P){ϕℓ(O; P, dℓ)} +We will show that R1 = oP(n−1/2) and +R2 = OP +�����λ1 − λ1 +��� · +max +y,a,z∈{0,1} ∥�πya.z − πya.z∥ + max +1≤j≤8 +����θℓ,j − θℓ,j +��� +1+α +∞ +� +under the conditions of the theorem. +C.1.1 +Term R1 +By Lemma 2 in Kennedy et al. (2020), R1 = oP(n−1/2) if +� +{ϕℓ(o; �P, �dℓ) − ϕℓ(o; P, dℓ)}2dP(o) = oP(1) +We have +� +{ϕℓ(o; �P, �dℓ) − ϕℓ(o; P, dℓ)}2dP(o) ≲ +� +{ϕℓ(o; �P, �dℓ) − ϕℓ(o; P, �dℓ)}2dP(o) ++ +� +{ϕℓ(o; P, �dℓ) − ϕℓ(o; P, dℓ)}2dP(o) +For the first term, +� +{ϕℓ(o; �P, �dℓ) − ϕℓ(o; P, �dℓ)}2dP(o) ≲ +8 +� +j=1 +� +{Lj(o; �P) − Lj(o; P) + �θℓ,j − θℓ,j}2dP(o) = oP(1) +26 + +since for example for j = 1, we have +����(�π11.1 − π11.1) +� +1 − 1(Z = 1) +�λ1 +� ++ 1(Z = 1) +�λ1λ1 +{1(Y = 1, A = 1) − π11.1} +� +λ1 − �λ1 +����� ++ +����(�π00.0 − π00.0) +� +1 − 1(Z = 0) +�λ0 +� ++ 1(Z = 0) +�λ0λ0 +{1(Y = 0, A = 0) − π00.0} +� +λ0 − �λ0 +����� +≲ +����λ1 − λ1 +��� + +max +y,a,z∈{0,1} ∥�πya.z − πya.z∥ = oP (1), +by our assumptions, using the fact that �λz and λz are bounded away from zero. +For the second term, we have +� +{ϕℓ(o; P, �dℓ) − ϕℓ(o; P, dℓ)}2dP(o) = +8 +� +j=1 +� ���1{�dℓ(x) = j} − 1{dℓ(x) = j} +��� {Lj(o; P) + θℓ,j(x)}2dP(o) +≲ P +� +θℓ, �d(X)(X) ̸= θℓ,dℓ(X)(X) +� +since θℓ,j(X) and Lj(O; P) are all uniformly bounded. Next, we show that +P +� +θℓ, �d(X)(X) ̸= θℓ,dℓ(X)(X) +� += oP(1) +For any t > 0, we have +P +� +θℓ, �dℓ(X) ̸= θℓ,dℓ(X) +� += P +� +θℓ, �dℓ(X) ̸= θℓ,dℓ(X), +min +j̸=dℓ(X){θℓ,dℓ(X)(X) − θℓ,j(X)} ≤ t +� ++ P +� +θℓ, �dℓ(X) ̸= θℓ,dℓ(X), +min +j̸=dℓ(X){θℓ,dℓ(X)(X) − θℓ,j(X)} > t +� +≤ P +� +min +j̸=dℓ(X){θℓ,dℓ(X)(X) − θℓ,j(X)} ≤ t +� ++ P +� +θℓ,dℓ(X) − θℓ, �dℓ(X) > t +� +≤ Ctα + P +� +θℓ,dℓ(X) − θℓ, �dℓ(X) + �θℓ, �dℓ(X) − �θℓ,dℓ(X) > t +� +≤ Ctα + P +� +� +�2 +8 +� +j=1 +|�θℓ,j − θℓ,j| > t +� +� +� +≤ Ctα + 2 +t +8 +� +j=1 +P|�θℓ,j(X) − θℓ,j(X)| +≤ Ctα + 2 +t +8 +� +j=1 +����θℓ,j − θℓ,j +��� +where C > 0 is the universal constant in Assumption 5. +In the second line, we use that +θℓ, �dℓ(X) ̸= θℓ,dℓ(X) implies �dℓ(X) ̸= dℓ(X), so θℓ,dℓ(X) − θℓ, �dℓ(X) ≥ minj̸=dℓ(X){θℓ,dℓ(X)(X) − +θℓ,j(X)}. In the third line we use Assumption 5 and that �θℓ, �dℓ(X)(X) − �θℓ,dℓ(X)(X) ≥ 0 by +construction of �dℓ(X). The fourth line follows from Markov’s inequality, and the last line from +∥ · ∥L1 ≤ ∥ · ∥L2. Since for each j ∈ {1, . . . , 8} we have +����θℓ,j − θℓ,j +��� = oP (1), as each �θℓ,j − θℓ,j +is a linear combination of the differences {�πya.z −πya.z : y, a, z ∈ {0, 1}}, we obtain the desired +result by invoking Lemma 1. Namely, we set Xn = P +� +θℓ, �dℓ(X) ̸= θℓ,dℓ(X) +� +, and for any ϵ > 0, +choose tϵ = +� ϵ +C +�1/α > 0 and Z(ϵ) +n += 2 +tϵ +�8 +j=1 +����θℓ,j − θℓ,j +���. +27 + +Lemma 1. Suppose that for a given sequence Xn, one can find for any ϵ > 0 another sequence +Z(ϵ) +n +≥ 0 such that |Xn| ≤ ϵ + Z(ϵ) +n +and Z(ϵ) +n += oP (1). Then Xn = oP (1). +Proof. Fixing ϵ > 0, consider a non-negative sequence Z(ϵ/2) +n += oP (1) satisfying |Xn| ≤ ϵ/2 + +Z(ϵ/2) +n +. Then +P[|Xn| > ϵ] ≤ P[ϵ/2 + Z(ϵ/2) +n +> ϵ] = P[Z(ϵ/2) +n +> ϵ/2] → 0 as n → ∞, +since Zn = oP (1), thus proving the result. +C.1.2 +Term R2 +We decompose R2 as +R2 = +8 +� +j=1 +� +P +� +1{�dℓ(X) = j}{Lj(O; �P) + �θℓ,j(X) − θℓ,j(X)} +� ++ P +�� +1{�dℓ(X) = j} − 1{dℓ(X) = j} +� +θℓ,j(X) +�� +We have +P +� +1{�dℓ(X) = j}{Lj(O; �P) + �θℓ,j(X) − θℓ,j(X)} +� +≲ +����λ1 − λ1 +��� · +max +y,a,z∈{0,1} ∥�πya.z − πya.z∥ +For the second term, observe that +���P +� +θℓ, �dℓ(X)(X) − θℓ,dℓ(X)(X) +���� += +���P +� +1{θℓ,dℓ(X)(X) > θℓ, �dℓ(X)(X)} +� +θℓ,dℓ(X)(X) − θℓ, �dℓ(X)(X) +����� +≤ P +� +1 +� +min +j̸=dℓ(X){θℓ,dℓ(X) − θℓ,j} ≤ θℓ,dℓ(X) − θℓ, �dℓ(X) + �θℓ, �dℓ(X) − �θℓ,dℓ(X) +� +× +� +θℓ,dℓ(X) − θℓ, �dℓ(X) + �θℓ, �dℓ(X) − �θℓ,dℓ(X) +�� +≤ 2 max +1≤j≤8 +����θℓ,j − θℓ,j +��� +∞ P +� +min +j̸=dℓ(X){θℓ,dℓ(X) − θℓ,j} ≤ 2 max +1≤j≤8 +����θℓ,j − θℓ,j +��� +∞ +� +≲ max +1≤j≤8 +����θℓ,j − θℓ,j +��� +1+α +∞ +, +by Assumption 5, where we used the fact that θℓ,dℓ(X)(X) ≥ θℓ, �dℓ(X)(X) and �θℓ,dℓ(X)(X) ≤ +�θℓ, �dℓ(X)(X), by construction of dℓ and �dℓ. +D +Proofs of Results in Section 6 +D.1 +Proof of Theorem 5 +First by Fubini’s theorem and Jensen’s inequality we have +� 1 +0 +E +� +min +1≤j≤8 θu,j(t, X) +� +dt = E +�� 1 +0 +min +1≤j≤8 θu,j(t, X)dt +� +≤ E +� +min +1≤j≤8 +� 1 +0 +θu,j(t, X)dt +� +. +28 + +The proof will be completed if we can show +� 1 +0 +θu,j(t, X)dt = �θu,j(X). +Since θu,j’s are linear combinations of π1a.z and π0a.z, we only need to show for any a, z ∈ {0, 1}, +� 1 +0 +π1a.z(t, X)dt = �π1a.z(X), +� 1 +0 +π0a.z(t, X)dt = �π0a.z(X). +We condition on W, by property of conditional expectation, +�π1a.z(X) += P(Y ≤ W, A = a | Z = z, X) += +� 1 +0 +P(Y ≤ t, A = a | Z = z, X, W = t)pw(t | Z = z, X)dt, +where pw denotes the density of W. Since W is independent of the data generating process of +O = (X, Z, A, Y ), we have +P(Y ≤ t, A = a | Z = z, X, W = t) = P(Y ≤ t, A = a | Z = z, X), +pw(t | Z = z, X) = pw(t) = 1(0 < t < 1). +Hence we conclude +� 1 +0 +P(Y ≤ t, A = a | Z = z, X, W = t)pw(t | Z = z, X)dt = +� 1 +0 +P(Y ≤ t, A = a | Z = z, X)dt. +Note that by definition π1a.z(t, X) = P(Y ≤ t, A = a | X, Z = z), which implies +� 1 +0 +π1a.z(t, X)dt = �π1a.z(X). +We can similarly prove +� 1 +0 +π0a.z(t, X)dt = �π0a.z(X). +which completes the proof for the first inequality in Theorem 5. The second inequality can be +proved by the same arguments. +D.2 +Basic Efficiency Result +Proposition 3. Suppose we observe n iid copies of T ∼ P, with T ∈ [0, 1]. Letting µ = +EP (T), VarP (T) = σ2, construct m estimates of µ by sampling m × n independent Unif(0, 1) +variates, {W (j) +i +}j=1,...,m +i=1,...,n , and estimating �µj = Pn[T > W (j)] = 1 +n +�n +i=1 1(Ti > W (j) +i +). Then +the estimator that averages these m estimates, +�µ = 1 +m +m +� +j=1 +�µj, +is unbiased and has variance 1 +n +� +σ2 + 1 +m +� +µ(1 − µ) − σ2�� m→∞ +→ +σ2 +n . +29 + +Proof. Observe that EP (�µj) = P[T > W] = +� 1 +0 P[T > w] dw = EP (T) = µ, for j = 1, . . . , m, +so �µ is unbiased. +Further, VarP (�µj) = +1 +nVarP (1(T > W)) = +1 +nµ(1 − µ), so by identical +distribution of each �µj, +VarP (�µ) = 1 +mVarP (�µ1) + +� +1 − 1 +m +� +CovP (�µ1, �µ2) += +1 +nmµ(1 − µ) + +� +1 − 1 +m +� 1 +nCovP (1(T > W (1)), 1(T > W (2))) +using the fact that {W (j) +i +} ⊥⊥ (T1, . . . , Tn). Next, see that +CovP (1(T > W (1)), 1(T > W (2))) = P[T > max {W (1), W (1)}] − µ2, +and finally, since V = max {W (1), W (1)} has density 2v1(v ∈ (0, 1)), +P[T > V ] = +� 1 +0 +2vP[T > v] dv = +� 1 +0 +P[T > √u] du = +� 1 +0 +P[T 2 > u] du = EP (T 2) = µ2 + σ2, +where we made the substitution u = v2. +30 + diff --git a/pNE0T4oBgHgl3EQf9AKq/content/tmp_files/2301.02796v1.pdf.txt b/pNE0T4oBgHgl3EQf9AKq/content/tmp_files/2301.02796v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8150631cceb4afdef971da9a326245942c0a1c48 --- /dev/null +++ b/pNE0T4oBgHgl3EQf9AKq/content/tmp_files/2301.02796v1.pdf.txt @@ -0,0 +1,1890 @@ +Observation of light thermalization to negative temperature +Rayleigh-Jeans equilibrium states in multimode optical fibers +K. Baudin1,2, J. Garnier2, A. Fusaro3, N. Berti1, C. Michel4, K. Krupa5, G. Millot1,6, A. Picozzi1 +1 Laboratoire Interdisciplinaire Carnot de Bourgogne, +CNRS, Universit´e Bourgogne Franche-Comt´e, Dijon, France +2 CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France +3 CEA, DAM, DIF, F-91297 Arpajon Cedex, France +4 Universit´e Cˆote d’Azur, CNRS, Institut de Physique de Nice, Nice, France +5 Institute of Physical Chemistry Polish Academy of Sciences, Warsaw, Poland and +6 Institut Universitaire de France (IUF), 1 rue Descartes, 75005 Paris, France +Although the temperature of a thermodynamic system is usually believed to be a positive quan- +tity, under particular conditions, negative temperature equilibrium states are also possible. Negative +temperature equilibriums have been observed with spin systems, cold atoms in optical lattices and +two-dimensional quantum superfluids. Here we report the observation of Rayleigh-Jeans thermal- +ization of light waves to negative temperature equilibrium states. The optical wave relaxes to the +equilibrium state through its propagation in a multimode optical fiber, i.e., in a conservative Hamil- +tonian system. The bounded energy spectrum of the optical fiber enables negative temperature +equilibriums with high energy levels (high order fiber modes) more populated than low energy levels +(low order modes). Our experiments show that negative temperature speckle beams are featured, in +average, by a non-monotonous radial intensity profile. The experimental results are in quantitative +agreement with the Rayleigh-Jeans theory without free parameters. Bringing negative tempera- +tures to the field of optics opens the door to the investigation of fundamental issues of negative +temperature states in a flexible experimental environment. +Introduction.- Temperature is a central concept of sta- +tistical mechanics and often reflects a measure of the +amount of disordered motion in a classical ideal gas. Al- +though this intuitive notion is correct for many physical +systems, one should keep in mind that the concept of +temperature is by far more subtle. +A detailed analy- +sis of the concept of temperature, and of its relationship +with energy and entropy shows that, under suitable con- +ditions, the entropy can decrease with the energy, thus +allowing for the existence of equilibrium states at nega- +tive temperatures (NT). Starting from the seminal works +by Onsager [1] and Ramsey [2], who originally conceived +the physical idea and the first theoretical approaches, +during the last decades, many works have been devoted +to the theoretical understanding of these unusual equi- +librium states. Despite the fact that the existence of a +NT equilibrium has created its own share of confusion in +relation with the definition of the entropy [3, 4], NTs are +now broadly accepted in line with different experimental +observations [5–12]. NTs were originally observed experi- +mentally in nuclear spin systems [13]. More recently, NTs +were observed with cold atoms in optical lattices [14]. +Furthermore, NTs originally predicted by Onsager in the +statistical description of point vortices [1] have been re- +cently observed in 2D quantum superfluids [15, 16]. +In this Letter we present an experimental optical setup +in which we report the observation of light thermaliza- +tion to NT equilibrium states. Our system is based on +the nonlinear propagation of speckle beams in a multi- +mode optical fiber (MMF). Because of the presence of +a finite number of modes supported by the MMF, the +spectrum exhibits both lower and upper bounds for the +energy levels. The bounded spectrum, combined to the +nonlinear four-wave interaction, are responsible for the +process of Rayleigh-Jeans thermalization to NT equilib- +rium states [17, 18]. +We stress that, at variance with +other experiments where photon thermalization is driven +by a thermal heat bath [19–22], here light thermaliza- +tion takes place in a conservative Hamiltonian system. +RJ thermalization to usual positive temperature equilib- +riums has been recently demonstrated experimentally in +MMFs [23–26], on the basis of a spatial beam-cleaning ef- +fect [27–31]. As described by the wave turbulence theory +[32–36] applied to MMFs [37–40], the thermalization to +a positive temperature equilibrium is characterized by a +transfer of power (particle number) toward the low-order +modes of the MMF. In marked contrast, here we report +the observation of thermalization to a NT equilibrium +featured by a power transfer to high-order modes (direct +flow of particles), as well as a transfer of energy to low- +order modes (inverse flow of energy). Consequently, the +NT equilibrium is characterized by an inverted modal +population, in which high-order modes are more popu- +lated than low-order modes. +Our experimental optical setup can be used as a simple +and flexible testbed to explore fundamental issues related +to NT states that are discussed in conclusion, e.g., Carnot +cycles operating between temperatures of opposite signs, +or inverted turbulence cascades featured by an analogue +process of condensation at NT. +Experimental system.- The experiment is based on +the single pass propagation of speckle beams through a +MMF. The subnanosecond pulses delivered by a Nd:YAG +laser (λ = 1.06µm) are transmitted through a spiral +arXiv:2301.02796v1 [physics.optics] 7 Jan 2023 + +2 +phase plate and then through a diffuser before injec- +tion of the speckle beam into a 12m long graded-index +MMF (i.e., parabolic-shaped trapping potential), which +guides M = 45 modes, i.e., nine groups of degener- +ate modes. +The energy levels (fiber eigenvalues) are +well approximated by the ones of an harmonic potential +βp = β0(px + py + 1), where {p} labels the two integers +(px, py) that specify a mode (see Supplementary Mate- +rial). We denote by |ap|2 the power in the mode p, with +the total power N = � +p |ap|2 [44]. +The experiment is realized in the weakly nonlinear +regime, where linear effects dominate over nonlinear ef- +fects Llin ∼ β−1 +0 +∼ 0.1mm ≪ Lnl = 1/(γN) ∼ 20cm, +γ being the nonlinear coefficient of the MMF. Accord- +ingly, we do not consider NT states associated to nonlin- +ear coherent structures, e.g., breathers [10, 45, 46]. Since +Llin ≪ Lnl, we only retain the linear contribution to +the Hamiltonian, E = � +p βp|ap|2 [23–25]. We have ver- +ified the conservation of the power N and the energy E +through propagation in the NT region for each realiza- +tion of a speckle beam, which confirms that the coupling +between guided modes and leaky modes of the fiber can +be neglected (see Supplementary Material). +RJ thermalization is driven by the four-wave nonlinear +interaction through the propagation in the MMF. The +speckle beam is expected to relax toward the thermody- +namic equilibrium state described by the RJ distribution +[17, 23–25, 38, 39]: +nRJ +p += T/(βp − µ), +(1) +where T and µ are the temperature and chemical po- +tential, while np = +� +|ap|2� +denotes the modal power av- +eraged over the realizations of the speckle beams. We +have at equilibrium N = T � +p(βp − µ)−1 and E = +T � +p βp/(βp − µ), with (T, µ) uniquely determined by +(N, E) – we deal with a microcanonic description (T is +not defined by a thermostat, it is in units of W·m−1) +[36]. Note that the RJ distribution refers to the classical, +low-energy, limit of the Bose-Einstein distribution [32], +describing highly occupied fiber modes. +Negative temperatures.- The irreversible process of RJ +thermalization is described by the wave turbulence the- +ory [32–36], which provides a nonequilibrium description +of light propagation in MMFs [37–40]. An equilibrium +thermodynamic formulation of multimode optical sys- +tems has been recently developed [17, 47]. We report in +Fig. 1 the relative entropy S = � +p log(nRJ +p ) as a function +of the energy for the MMF used in our experiments with +g = 9 groups of degenerate modes. Because the spec- +trum of the fiber is bounded, β0 ≤ βp ≤ βmax = gβ0, the +system possesses both lower and upper energy bounds +Emin = Nβ0 ≤ E ≤ Emax = Nβmax. +(2) +Starting at minimum energy Emin, where only the fun- +damental mode is populated, an increase in energy leads +FIG. 1: +Negative temperatures and inverted modal +population. (a) RJ equilibrium distribution nRJ +p +for positive +temperature T > 0 (E < E∗) where low-order modes are +more populated, and negative temperatures T < 0 (E > E∗) +featured by an inverted modal population; while for 1/T → 0 +(E = E∗), nRJ +p +=const. (b) Relative entropy S vs energy E, +showing that 1/T = (∂S/∂E)N,M < 0 requires E > E∗. (c) +Temperature T vs energy E. Negative temperatures T < 0 +occur for E > E∗ with E∗/Emin ≃ 6.33 (vertical dashed black +line). The vertical purple lines in (b-c) denote the six values +of E considered in Fig. 2. +to an occupation of a larger number of fiber modes and +therefore an increase in entropy. As the temperature ap- +proaches infinity, all fiber modes become equally popu- +lated nRJ +p +=const, and the entropy reaches a maximum +for E = E∗ = N ⟨βp⟩ = Emin(2g + 1)/3. NT equilibrium +states arise for E > E∗, where the entropy decreases by +increasing the energy, 1/T = (∂S/∂E)M,N < 0. +The +condition E > E∗ can be achieved if high-order modes +are more populated than low-order modes. +Note that +NT equilibrium states persist in the thermodynamic limit +(see Supplementary Material). +RJ thermalization to NT equilibriums.- At variance +with usual experiments of spatial beam cleaning and RJ +thermalization [24–30], here we study the thermalization +for different values of the energy E, while keeping con- +stant the power N. Indeed, by passing the laser beam +through a diffuser before injection into the fiber, we can +vary the amount of randomness of the speckle beam by +keeping N =const – the larger the randomness of the +speckle beam, the higher the energy E. Accordingly, we +study RJ thermalization over a broad range of variation +of the energy. In order to further increase the energy be- +yond the threshold for NT (E > E∗), we pass the beam +through a spiral phase plate before the diffuser, i.e., we +generate a speckle beam from a doughnut-like intensity + +0 +0+ ++818 +(a) +T +T>0 +1/T = 0 +T<0 +p +n +p +p +p +0 +-200 +S +-400 +(b) +5 +2.5 +uIu +0 +-2.5 +(c) +.5 +5 +9 +3 +7 +E/E +min3 +FIG. 2: +Rayleigh-Jeans thermalization to NT equi- +libriums. Experimental modal distributions averaged over +realizations nexp +p += +� +|aexp +p +|2� +, at the fiber input (blue), at the +fiber output (red). Corresponding RJ equilibrium distribu- +tion nRJ +p +(green). Note the quantitative agreement between +nRJ +p +and the experimental output distribution nexp +p +(red). The +six panels correspond to different values of E, or equivalently +different T, see the six vertical purple lines in Fig. 1(b-c). +The modal distribution peaked on the lowest mode for T > 0 +(a), gets inverted for T < 0 (b-f). An average over ≃35 re- +alizations of speckle beams is considered for each panel. The +fiber modes are sorted from the fundamental one (β0) to the +highest mode group (nine-fold degenerate with βmax = 9β0). +Degenerate modes are equally populated at equilibrium, lead- +ing to a staircase distribution nRJ +p . +distribution, which enables the excitation of higher order +fiber modes. +The accurate measurements of the near-field and far- +field intensity distributions allowed us to retrieve the +modal power distribution nexp +p +. To obtain the mode de- +composition, several interferometric approaches based on +use of a reference beam have been exploited to study light +thermalization in MMFs [24–26]. Here, in contrast to the +previous works [23–26], we use a non-interferometric nu- +merical mode decomposition procedure that is based on +the Gerchberg-Saxton algorithm. It allows us to retrieve +the transverse phase profile of the speckle field from the +near-field and far-field intensity distributions measured +in the experiments [48–51]. By projecting the retrieved +complex field over the fiber modes, we get the complete +modal distribution. +The RJ distribution being in essence a statistical dis- +tribution, its comparison with the experiments requires +an average over realizations of speckle beams. We have +recorded 2×300 realizations of the near-field and far-field +intensity distributions for the same power (N = 7kW) +FIG. 3: +Energy flows in mode space. Experimental en- +ergy distributions averaged over 50 realizations εexp +p += βpnexp +p +, +at the fiber input (blue), output (red). The arrow indicates +the energy flow to low-order modes. Corresponding RJ equi- +librium distribution εRJ +p += βpnRJ +p +(green line), which is in +quantitative agreement with the experimental output distri- +bution (red). +and different energies E. For each individual speckle re- +alization, we retrieve the modal distribution |aexp +p +|2. We +partition the ensemble of 300 realizations of {|aexp +p +|2} +within small energy intervals [E − ∆E, E + ∆E] with +∆E = 0.125Emin. We perform an average over the re- +alizations of the modal distributions for each energy in- +terval, which provides the averaged modal distribution +nexp +p += +� +|aexp +p +|2� +. This procedure is applied at the fiber +output (L = 12m), and fiber input (after 20cm of prop- +agation). The error in the procedure has been computed +theoretically and numerically, it decreases with the num- +ber of realizations and has been found remarkably small +(relative standard deviations of ≃ 6%), see Supplemen- +tary Material. +We report in Fig. 2 the averaged modal distributions +nexp +p +at the fiber input (blue) and output (red), for dif- +ferent values of the energy E, or equivalently the tem- +perature T (purple lines in Fig. 1(b-c)). The data are +compared with the theoretical RJ distribution nRJ +p . We +stress that there are no adjustable parameters between +nexp +p +and nRJ +p : The parameters (T, µ) in nRJ +p +are uniquely +determined by N and E measured in the experiments. +We observe in Fig. 2 an excellent agreement between nexp +p +(red circles) and nRJ +p +(green line), for both T > 0 and +T < 0. Fig. 2 then shows that NT equilibriums constitute +attractor states for the random wave, whose robustness +has a thermodynamic origin – maximum entropy state +for a given pair (N, E). +Energy flows in mode space.- The conventional ther- +malization to positive temperatures is characterized by +an energy flow to high-order modes [33, 34, 38]. Thermal- +ization to NTs typically occurs through an inverse energy +flow to low-order modes [18]. This is illustrated in Fig. 3, +which shows that the energy distribution εexp +p += βpnexp +p +at low-order modes increases through propagation in the +MMF and reaches the theoretical RJ equilibrium distri- +bution εRJ +p += βpnRJ +p . + +0.09 +0.04 +DE/E +E/E += 6.623 += 5.196 +RJ +min +min +T/E +T/E += -0.347 += 0.125 +Output +min +min +N +Input +0.02 +(b) +(a) +0 +10 +20 +30 +40 +10 +20 +30 +40 +0 +0 +0.04 +0.05 +E/E +E/E += 6.933 += 7.244 +min +min +T/E += -0.153 +T/E += -0.091 +min +min +0.025 +Q 0.02 +(c) +(d) +0 +0 +10 +20 +30 +40 +10 +20 +30 +40 +0 +0 +0.08 +0.07 +E/E. +E/E +=8.215 += 7.884 +min +T/E += -0.039 +T/E += -0.025 +min +min +0.04 +Q 0.035 +n +(f) +(e) +0 +0 +10 +20 +30 +10 +20 +30 +0 +40 +0 +40 +Modes +ModesE/E += 8.161 +10 +min +T/E += -0.027 +min +10 +RJ +10° +Output +O Input +15 +25 +0 +5 +10 +20 +30 +35 +40 +Modes4 +FIG. 4: +Oscillating radial intensity distribution at +NT. Averaged intensity distribution Iexp(|r|) as a function +of the radial (angle-averaged) distance |r| (red). +Note the +quantitative agreement with the theoretical RJ intensity dis- +tribution IRJ(|r|) in Eq.(3) (dashed green). The oscillating +behavior of the intensity distribution is a signature of the NT +equilibrium. Inset: corresponding 2D intensity averaged over +the realizations (the radius of the circle is the fiber radius). +Oscillating radial intensity distribution.- The intensity +distribution IRJ(|r|) of usual positive temperature equi- +libriums is, in average, a monotonic decreasing function +with the radial distance |r| [23]. This is consistent with +the intuitive idea that low-order modes localized near-by +the fiber center are the most populated ones. In marked +contrast, the inverted modal population of NT equilibri- +ums are characterized by an oscillating behaviour of the +radial intensity distribution. This is illustrated in Fig. 4, +which reports the averaged radial intensity distribution +Iexp(|r|) (with ∆E = 0.25Emin, E/Emin = 7.9). +The +theoretical RJ intensity distribution reads +IRJ(r) = +� +p +nRJ +p u2 +p(r), +(3) +where up(r) denotes the fiber modes [23]. The number of +radial oscillations in Fig. 4 is given by the most oscillating +mode of the fiber, namely the mode LP04 that exhibits 5 +oscillations. +Experiments by increasing power.- We have studied the +optical field at the output of the MMF, with a small +power N = 0.23kW (linear regime), and a high power +N = 7kW (nonlinear regime). Since the MMF length +is kept fixed (L = 12m), the effective number of nonlin- +ear interaction lengths increases by increasing the power. +Fig. 5 reports the fraction of power that populates the +highest group of degenerate modes of the MMF, ˜ng/N for +g = 9. The output field (red) reaches the equilibrium RJ +theory (green line) in the nonlinear regime. The highest +energy level gets macroscopically populated by increasing +the energy, or equivalently by increasing the temperature +of negative sign (see Fig. 1). +Conclusion and perspectives.- We have reported the ob- +servation of RJ thermalization to NT equilibrium states +through light propagation in graded-index MMFs. This +non-equilibrium process of NT thermalization can be de- +scribed by a wave turbulence kinetic equation, which is +FIG. 5: +Macroscopic population of the highest energy +level. Fraction of power ˜ng/N into the highest mode group +g = 9 vs energy E/Emin. +Experimental measurements at +the fiber output: The blue circles refer to the linear regime +(small power), the red circles to the nonlinear regime (high +power). The green line denotes the RJ equilibrium theory. By +increasing the energy, the power goes to the highest energy +level, ˜ng/N → 1 as E/Emin → 9. +found in agreement with the simulations of the nonlin- +ear Schr¨odinger equation (see Supplementary Material). +Our NT experiment then paves the way for the study of +Zakharov-Kolmogorov turbulence cascades [32–34] that +are inverted with respect to those underlying usual posi- +tive temperature thermalization (e.g., inverse energy flow +in Fig. 3). +Along this line, our work suggests a previously un- +recognized process of inverted condensation at NTs: At +variance with usual condensation at positive temperature +where the lowest energy level gets macroscopically pop- +ulated by decreasing the temperature (T → 0+, or E → +Emin) [23, 33–36], at NT an inverted condensation pro- +cess occurs into the highest energy level as the temper- +ature increases to zero (T → 0−, or E → Emax). While +we provide a preliminary study of this effect through +the macroscopic population of the highest energy level +(Fig. 5), the observation of the transition to condensa- +tion requires MMFs with larger number of modes (see +Supplementary Material). +In our work NT states are obtained directly, which is in +contrast with magnetic systems and cold atoms where the +excitation of NT states requires first the creation of a pos- +itive temperature state and then its subsequent inversion +through suitable procedures (magnetic field inversion or +Feshbach resonances). This opens the possibility to study +the physics of NT in a flexible experimental environment. +For instance, the thermalization of two beams at differ- +ent laser wavelengths interacting through the fiber non- +linearity can be exploited to achieve an efficient optical +refrigeration: A highly incoherent speckled beam at NT +can be cooled through its thermalization with a coher- +ent beam towards a highly coherent state without any +power loss. In contrast with usual beam cleaning at posi- +tive temperature where the energy is conserved, here the +cooling process is featured by an energy transfer from +the incoherent to the coherent beam, which significantly +improves the gain of coherence of the incoherent beam. + +0.12 +Exp. +E/E += 7.945 +min +0.1 +-RJ +T/E += -0.036 +min +Intensity +0.08 +0.06 +0.04 +0.02 +0 +5 +10 +15 +0 +[um] +rRJ +High Power +0.8 +Low Power +0.6 +N +n +0.4 +0.2 +0 +6 +6.5 +7.5 +8 +8.5 +9 +E/E +min5 +Following this idea, one can explore the meaning of a +thermostat at NT [10]: If the NT incoherent beam has +a power much larger than the partially coherent beam, +it will play the role of a NT thermal reservoir for such a +partially coherent beam. +The versatile optical experimental environment pro- +posed in this work also opens the possibility to study +controversies about NTs, such as thermodynamic engines +featured by Carnot cycles operating between tempera- +tures of opposite signs, in relation with the generalized +Kelvin-Planck formulation of the second law of thermo- +dynamics stating that it is not possible to completely +transform work into heat at NT [10]. +Acknowledgments.- The authors are grateful to S. +Rica, I. Carusotto and V. Doya for fruitful discus- +sions. Fundings: Centre national de la recherche scien- +tifique (CNRS), Conseil r´egional de Bourgogne Franche- +Comt´e, iXCore Research Fondation, Agence Nationale de +la Recherche (ANR-19-CE46-0007, ANR-15-IDEX-0003, +ANR-21-ESRE-0040). Calculations were performed us- +ing HPC resources from DNUM CCUB (Centre de Cal- +cul, Universit´e de Bourgogne). +SUPPLEMENTARY MATERIAL +EXPERIMENTAL SET-UP +The source is a Nd:YAG laser delivering subnanosec- +ond pulses (400ps) at λ =1064nm. +The laser beam is +passed through a spiral phase plate (Thorlabs) to gener- +ate a doughnut-like ring-shaped beam, and subsequently +through a diffuser before injection of the speckle beam +into the MMF, see Fig. 6. The diffuser plate is placed +in the vicinity of the Fourier plane of a 4f-optical sys- +tem. +The near-field (NF) intensity distribution of the +fiber output beam was magnified and imaged on a first +CCD camera owing to a two lens telescope optical system, +with f2 = 8 mm and f3 = 150 mm. The CCD camera +was placed on a rail orthogonal to the beam propaga- +tion in order to remove or put the camera back on the +beam path. The far-field (FF) intensity distribution of +the magnified image was obtained by placing it in the ob- +ject focal-plan of a lens f4 = 250 mm and using a second +CCD camera positioned in its image (Fourier) focal-plan. +We have computed analytically the propagation of +the optical wave throughout the setup of our detection +scheme, according to Fig. 6 (lower part). If ψ0(r) is the +optical field amplitude at the fiber output (r = (x, y)), +then we have in the NF plane: +ψNF(r) = −ρ−1ψ0(−r/ρ), +with ρ = f3/f2 the magnification factor. In the FF plane, +the wave amplitude reads +ψFF(u) = iρ +λf4 +� +drψ0(r) exp[−i2π(−ρ)r · u/(λf4)], +FIG. 6: +Setup. laser, optical isolator, half-wave plate and +polarizer, lenses for magnification and imaging (fj), spiral +phase plate (V), diffuser (D), graded-index MMF, and cam- +eras for near- and far-field detections (Cam). +which corresponds to the Fourier transform of the field +amplitude at the fiber output (note that the constant +phase prefactor plays no role because the camera records +the intensity). We note that: (i) The optical amplitude +in the NF plane is an exact magnification of the wave +amplitude at the output of the MMF; (ii) the optical +amplitude in the FF detection plane exactly corresponds +to the Fourier transform of the amplitude at the fiber +output. Then, the experimental setup for the detection of +the NF and FF intensities does not introduce detrimental +spurious transverse phases profiles in the optical field, +e.g., related to optical free propagation in air or phase +shifts due to the presence of additional lenses. +Multimode fiber +The refractive index profile of the graded-index MMF +exhibits a parabolic shape in the fiber core with a max- +imum core index (at the center) of nco ≃1.472 and +ncl ≃ 1.457 for the cladding at the pump wavelength +of 1064nm (fiber radius R = 15µm). The fiber length +is L = 12m. +The trapping parabolic potential reads +V (r) = q|r|2 for |r| ≤ R and q = k0(n2 +co −n2 +cl)/(2ncoR2), +k0 = 2π/λ the laser wave-number. +The fundamental +mode energy level is β0 = 2√αq, with α = 1/(2ncok0). +The MMF guides M = 45 modes (i.e. g = 9 groups of +degenerate modes). +The truncation of the potential introduces a frequency +cut-off in the FF spectrum kc = (2π/λ) +� +n2co − n2 +cl [23]. +The conservation of N and E through propagation in the +MMF (see Fig. 7) shows that the coupling from guided +modes to leaky radiation modes in the cladding is negligi- +ble. This is not surprising as the efficiency of such a cou- +pling can be shown to be very small. Indeed, the MMF +has a core (radius 15µm), a cladding (radius 62.5µm), +and a highly absorbing polymer-coating with refractive +index larger than the core. Then leaky radiation modes +in the cladding are rapidly absorbed during propagation +due to their large penetration in the polymer-coating: +we measured a typical absorption length Labs of ≃ 15cm. +Let us consider the coupled amplitude equations describ- +ing the four-wave mixing between four modes including +a leaky mode. The four mode amplitudes a1, a2, a3, a4 + +(+ D) +ND +YAG +P +MMF +FF +NF +Cam +Cam6 +(where a4 stands for the leaky mode amplitude) satisfy +Eqs.(10.2) in [44], in which we need to add an absorption +term of the form −a4/Labs in the equation (10.2.4) in [44] +for a4. Since absorption is strong the overdamped limit is +valid and we get a4 = in2k0Labsf4312a1a2a∗ +3 exp(−i∆βz) +where ∆β = β3 + β4 − β1 − β2, the parameter n2 is the +nonlinear-index coefficient, and f4312 is an overlap inte- +gral between the four mode profiles. By substitution into +one of the first three equations, say (10.2.1) in [44], we +find that the mode amplitude a1 experiences an effective +absorption due to the coupling with the leaky mode that +is given by −4n2 +2k2 +0Labs|f1234|2|a2|2|a3|2a1. This shows +that this absorption is of the order of −Labs/L2 +nla1 times +a coefficient that is of the order of the square overlap in- +tegral between guided and leaky mode profiles. As the +supports of these modes are very different (the guided +modes are essentially supported in the core while the +leaky modes are essentially supported in the cladding +that is much larger), the square overlap integrals are +small (smaller than the respective core-cladding ratio +(15/62.5)4 ≃ 3 10−3) and the effect of the coupling to +the leaky modes onto the guided mode amplitudes can be +neglected when Lnl ≃ 20cm and the propagation distance +is L = 12m. +Measurements of the energy E +From the measurements of the NF and FF intensity +distributions, we have retrieved an accurate measurement +of the power N and the energy E of the speckle beam. +The NF intensity distribution INF(r) = |ψ|2(r) provides +a measurement of the power N = +� +INF(r)dr and of the +potential energy Epot = +� +V (r)|ψ|2(r)dr. The kinetic +energy Ekin = α +� +|∇ψ|2(r)dr is retrieved from the FF +intensity distribution IFF(k) = | ˜ψ|2(k). This provides +the measurement of the (linear) energy (Hamiltonian) +E = Epot + Ekin. +Conservation of N and E through propagation in the +MMF for E > E∗ (negative temperature region) +Power conservation has been verified by keeping fixed +the conditions of injection of the speckle beam into the +MMF: We measured Nout at L = 12m, and then Nin +by cutting the fiber at 20cm, and we always obtained +(Nin − Nout)/Nmoy < 1%. The experimental verification +of energy conservation requires both the NF and FF in- +tensity measurements. The NF and FF intensities are +recorded at the fiber output at L = 12m, which gives +Eout. Without altering the fiber launch conditions, the +fiber is cut to 20cm to record the input NF and FF in- +tensities, which gives Ein. The measurements of Ein and +Eout then refer to an individual realization of the speckle +beam (without average over the realizations). +Fig. 7 +shows that the conservation of the energy is well verified +for E > E∗, i.e. in the negative temperature region. The +FIG. 7: +Conservation of the energy through propaga- +tion in the MMF in the negative temperature region. +(a) Measurements of the energy at the input of the MMF +(blue triangles), and at the output of the MMF (red trian- +gles): The energy E/N is conserved through the propagation +in the MMF over a broad range of variation of E/Emin. We +recall that negative temperatures T < 0 occur for E > E∗ +with E∗/Emin ≃ 6.33, see Fig. 1. +energy E is varied owing to the diffuser before injection +into the MMF, see Fig. 6. +MODAL DECOMPOSITION +Phase retrieval +The procedure of mode decomposition is based on +the well-known Gerchberg-Saxton algorithm [41, 50, 51]. +From the measurements of the NF and FF intensity dis- +tributions in the experiment, it allows us to retrieve the +transverse phase profile of the field. The resulting com- +plex field is subsequently projected onto the fiber modes, +to get the complete modal distribution of the experimen- +tal optical beam. The algorithm is known to be accurate +although it is not efficient in terms of computational cost. +Indeed it is a local search algorithm that updates itera- +tively the unknown phase profile of the field and it is usu- +ally necessary to consider several initial phase guesses. +We have, therefore, carried out a detailed preliminary +analysis of the algorithm by performing numerical simu- +lations that reproduce our experimental configuration in +order to prove that the phase retrieval and modal distri- +bution estimation are reliable. +Error introduced by the Gerchberg-Saxton +algorithm +To evaluate accurately the error in the phase-retrieval +algorithm, we have reproduced in detail the experimental +procedure as follows: +i) We consider a particular value of the energy E (E > E∗ +so that T < 0). Throughout the procedure the power N +is set constant. +The pair (E, N) determines uniquely +(T, µ) and thus the exact RJ distribution at equilibrium +nRJ +p += T/(βp − µ). + +1.8 +indno +Input +1.7 +11.6 +E/N +1.5 +1.4 L +7.5 +8.5 +8 +6.5 +E/E +ulw7 +FIG. 8: +Phase retrieval. Example of numerically gener- +ated near-field intensity distribution (a), and its reconstruc- +tion (b). Original phase field (c) and the corresponding phase +field reconstructed from the Gerchberg-Saxton algorithm (d). +ii) We generate from nRJ +p +a realization of speckle beam +ψ(r) = � +p apup(r), where ap is a complex Gaussian +random variable with variance +� +|ap|2� += nRJ +p +(ap = +a(r) +p ++ ia(i) +p +with a(r) +p +and a(i) +p +real Gaussian independent +random variables with mean zero and variance nRJ +p /2). +We recall that up(r) are the fiber modes. Then ψ(r) is a +particular realization of a complex speckle field at exact +RJ equilibrium. +iii) The particular realization ψ(r) is highly resolved nu- +merically. We mimic the impact of the finite resolution +of the camera used in the experiment. +From ψ(r) we +compute the NF and FF intensity distributions |ψ(r)|2 +and | ˆψ(k)|2. We sample the NF and FF intensity dis- +tributions with the finite number of points available in +the camera (10242) in r-space and k-space and ≃ 950 +points for the dynamics range in intensity. +We apply +the Gerchberg-Saxton algorithm to retrieve the sampled +phase profile. Due to the errors introduced by the sam- +pling of the camera and by the phase-retrieval algorithm, +the resulting complex field ψexp(r) may differ from the +generated speckle beam ψ(r). +We report in Fig. 8 the numerically generated near-field +intensity distribution in one numerical simulation (a) and +its reconstruction (b), the original phase profile (c) and +the reconstructed phase profile (d). It is clear that the re- +construction is very good. We will see below more quan- +titatively that the error is indeed negligible. +iv) We project the complex field ψexp(r) onto the fiber +modes to get the complex modal coefficient aexp +p +and dis- +tribution |aexp +p +|2. Due to the errors introduced by the +sampling of the camera and by the phase-retrieval algo- +rithm, this modal distribution may differ from the mode +FIG. 9: +Error in the modal decomposition. The mode +decomposition is based on the Gerchberg-Saxton algorithm, +whose error is quantified by the distance DQ +err to the exact +distribution, see Eq.(4). (a) DQ +err vs the energy E (for Q = 50 +realizations), by accounting for the sampling due to the lim- +ited resolution of the camera (red), and without the sampling +(blue). Note in (a) that an increase of the randomness of the +speckle beam (i.e., increase of E) does not increase the error. +(b) DQ +err vs number of realizations Q (for E/Emin = 7): The +error decreases with the number Q of realizations of the speck- +les. The green line reports the theoretical estimate of the error +given in Eq.(6). Note that DQ +err is bounded, 0 ≤ DQ +err ≤ 1. +FIG. 10: +Near-field and far-field experimental inten- +sities, and corresponding reconstructed intensity dis- +tributions. Near-field (NF) intensity recorded in the experi- +ment for a single realization of a speckle (a), and correspond- +ing far-field (FF) intensity distribution (c). +Corresponding +NF intensity (b), and FF intensity (d), reconstructed from +the Gerchberg-Saxton algorithm. + +exper. (NF) +reconst. (NF) +10 +10 +[ur] +[wn] +0 +0 +V +-10 +-10 +(b) +(a) +-10 +10 +-10 +10 +0 +0 +x [μum] +x [μum] +exper. (FF) +reconst. (FF) +[,_un] +[,_un]"y +0 +0 +(c) +(d) +-1 +0 +0 +[um-?10 +10 +[un] +y [μum] +0 +0 +-10 +-10 +(a) +(b) +-10 +0 +10 +-10 +0 +10 +x [μum] +x [um] +10 +10 +xy[um] +[um] +0 +0 +-10 +-10 +d) +-10 +10 +-10 +10 +0 +0 +x [um] +x [μum]0.06 +0.04 +Error +0.02 + Without Sampling +With Sampling +(a) +7.5 +6.5 +8.5 +8 +6 +7 +E/E +min +Theory +0.2 +Numerical simulations +Error +0.15 +0.1 +0.05 +(b) +25 +50 +100 +75 +0 +Q8 +distribution |ap|2 used to generate the speckle beam ψ(r). +As we will show below, this error is negligible. +v) We repeat the steps ii)-iv) Q times, each with a +different realization of the speckle beam (i.e., with a +different realization aj +p of ap, for j = 1, . . . , Q). +The +procedure then gives Q distributions |aexp,j +p +|2, j += +1, . . . , Q. We compute the empirical averages nexp,Q +p += +(1/Q) �Q +j=1 |aexp,j +p +|2. +We anticipate that, for Q large +enough, these empirical averages should be close to the +theoretical values nRJ +p . We introduce the estimation er- +ror: +DQ +err = +� +p +��nexp,Q +p +− nRJ +p +�� +� +p nexp,Q +p ++ nRJ +p +. +(4) +Let us imagine for a while that the phase-retrieval al- +gorithm is perfect and that the sampling error due to the +camera is absent. Then, for each realization j = 1, . . . , Q, +we have |aexp,j +p +|2 = |aj +p|2 exactly. Thus, the random vari- +ables |aexp,j +p +|2 are independent and follow exponential dis- +tributions with mean nRJ +p . Consequently, the empirical +quantities ZQ +p = nexp,Q +p +/nRJ +p +are independent and identi- +cally distributed with the gamma probability distribution +Γ(Q, Q) (the law of the sum of Q independent variables +with exponential distribution and mean 1/Q) and the +estimation error is +DQ +err = +� +p nRJ +p |ZQ +p − 1| +� +p nRJ +p (ZQ +p + 1) +, +(5) +which gives when the number of modes is large enough +(ZQ follows the Γ(Q, Q) distribution): +DQ +err ≃ E[|ZQ − 1|] +E[ZQ + 1] = +QQ−1 +(Q − 1)!e−Q. +(6) +For Q ≥ 8 , we have DQ +err ≃ 1/√2πQ. +We have carried out numerical simulations with our +implementation of the phase-retrieval algorithm (using +multiple initial phase guesses) and with the sampling er- +ror of the camera. The results of the distance DQ +err vs +energy E are reported in Fig. 9 with different numbers +Q of realizations per energy. We can see in panel (b) of +Fig. 9 that the errors correspond to the theoretical er- +ror Eq.(6) when the phase-retrieval algorithm makes no +error. +The error introduced by the Gerchberg-Saxton algo- +rithm has been computed by increasing the amount of +complexity in the speckle pattern, i.e., by increasing the +energy E. We can see in Fig. 9(a) that the error does not +increase when the energy E increases. +In the experiments we have typically 35 to 70 indepen- +dent realizations of speckle beams for a given small en- +ergy interval [E − ∆E, E + ∆E] with ∆E = 0.125Emin, +so we can expect that the errors (due to the phase re- +trieval algorithm and the camera sampling) are small. +Error bars with relative standard deviations of the order +of 1/√2πQ ≃ 6% could be added in Fig. 2 but they are +too small to be visible. +FIG. 11: +Experimental attraction to NT RJ equilib- +rium. Distance DRJ [defined in Eq.(7)] to the RJ equilibrium +distribution computed from the experimental data at the fiber +input (blue), and fiber output (red), for different values of the +energy E. The significant reduction of DRJ from input to out- +put measurements shows the attraction to the RJ equilibrium +for T < 0. Note that DRJ in Eq.(7) is bounded, 0 ≤ DRJ ≤ 1. +We recall that negative temperatures T < 0 occur for E > E∗ +with E∗/Emin ≃ 6.33, see Fig. 1. +To complete our study, we report in Fig. 10 the near- +field and far-field intensity distributions recorded during +one of the experiments (left plots) and the correspond- +ing reconstructed intensities from the Gerchberg-Saxton +algorithm (right plots). +Experimental convergence to the NT RJ distribution +We have quantified in our experimental results the at- +traction to the NT equilibrium by using Eq.(4), which +provides a ‘distance’ to the RJ distribution +DRJ = +� +p |nexp +p +− nRJ +p | +� +p nexp +p ++ nRJ +p +. +(7) +We report in Fig. 11 the distance DRJ computed for the +experimental data averaged over the realizations nexp +p +at +the fiber input (blue), and the fiber output (red), for dif- +ferent energies E. The strong reduction of the distance +DRJ from the fiber input to the output confirms the pro- +cess of NT thermalization, which is demonstrated over a +broad range of values of the energy E. +Polarization effects +The polarization state of the optical beam changes as +it propagates through the MMF. The field at the out- +put of the MMF is projected onto a basis of orthogo- +nal linear polarizations. The corresponding NF and FF +intensity distributions are recorded along the orthogo- +nal linear polarizations. +For each polarization, we ap- +ply the mode decomposition procedure based on the +Gerchberg-Saxton presented above. In this way, we re- +trieve the transverse phase profile of the field for the +two orthogonal polarizations. The complex field along +each polarization is subsequently projected over the fiber +modes, to get the modal populations for each polariza- +tion, n(x) +p +and n(y) +p . The modal distribution is obtained + +0.4 +Output +Input +0.3 +0.1 +0 +6.5 +8.5 +7.5 +E/E9 +FIG. 12: +Polarization effects in modal decomposi- +tion. +Experimental modal distribution retrieved from the +Gerchberg-Saxton algorithm, without splitting the orthogonal +polarization (blue line), by splitting the orthogonal polariza- +tion at the fiber output (dashed red line). Note that a single +realization of the speckle beam has been considered, which ex- +plains the discrepancy with the RJ equilibrium distribution +(dashed green line). +by summing the contributions of the two polarizations, +npol +p += (Nxn(x) +p ++ Nyn(y) +p )/(Nx + Ny), where Nx,y denote +the power along the two polarizations. We compare in +Fig. 12 the modal distribution retrieved by following this +procedure (npol +p ), with the modal distribution (np) re- +trieved without separating the polarization states of the +field. This comparison is reported in Fig. 12 for the same +(single) realization of the speckle beam. We can remark +in Fig. 12 that the two distributions are almost identical. +In all cases analyzed, we have always observed the same +good agreement. Given the large number of realizations +of speckle beams (≃300) recorded and analyzed in our +experiments, we didn’t perform the polarization modal +decomposition. +ANALOGUE CONDENSATION AT NT +Thermodynamics relations +We define the thermodynamic relations used to plot +Fig. 1. We consider a field at equilibrium with the RJ +distribution nRJ +p += T/(βp − µ), with N = � +p nRJ +p +and +E = � +p βpnRJ +p , where βp = βpx,py = β0(px + py + 1) are +the eigenvalues of the truncated parabolic potential (βp ≤ +βmax), and the index {p} labels the two integers (px, py) +that specify a mode. Here and below, the sum over the +modes � +p is carried over the set 0 ≤ px + py < g, where +g = βmax/β0 is the number of groups of non-degenerate +modes, with M = g(g + 1)/2 the total number of modes. +Because of the constraint nRJ +p += T/(βp − µ) > 0, a NT +equilibrium state T < 0 requires that µ > βmax. +We +start +from +the +equilibrium +entropy +˜Seq += +� +p log(neq +p ) – note that at equilibrium it coincides with +the nonequilibrium entropy verifying the H theorem of +entropy growth. +It proves convenient to shift the en- +tropy by a constant Seq = ˜Seq − M log N, so that by +FIG. 13: +Analogue effect of condensation at NTs: +(a) Chemical potential µ/β0 − 1 vs energy E/Emin for the +MMF used in the experiments (g = 9), from Eq.(9). Note the +asymptotic behaviors µ → β− +0 for E → Emin, and µ → β+ +max +for E → Emax, which lead respectively to the macroscopic +population of the lowest energy level (β0), and highest energy +level (βmax). The horizontal dashed line denotes µ = βmax +and the vertical one E = E∗. (b) Condensate fraction in the +highest energy level ˜nRJ +g /N vs energy E: As the energy in- +creases E → Emax (or equivalently the temperature increases +T → 0−), the condensate fraction increases ˜nRJ +g /N → 1. The +curves are obtained from Eqs.(14-15), see the text. The crit- +ical behavior of the transition to condensation becomes ap- +parent by increasing the number of modes g. +using T = N/ � +p(βp − µ)−1, we can write +S(µ) = − +� +p +log(βp − µ) − M log +� � +p +1 +βp − µ +� +(8) +E(µ) +Emin += +� +p +βp +βp−µ +� +p +β0 +βp−µ +(9) +T(µ) +Emin += +1 +� +p +β0 +βp−µ +(10) +The parametric plot with resp. to µ of (8) and (9) gives +S(E) in Fig. 1(b); the corresponding parametric plot of +(9) and (10) gives T vs E in Fig. 1(c). +NT states in the thermodynamic limit +The thermodynamic limit is defined by N +→ ∞, +β0 +→ 0 with Nβ2 +0 +=const and βmax +=const. +In +this limit the discrete sums over the modes are re- +placed by continuous integrals, namely N = � +p neq +p → + +10-1 +n +p +nbol +p +-RJ +C +10-3 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +Modes40 +-T>O +0>- +20 +0 +-20 +(a) +-40 +2 +3 +4 +5 +6 +7 +8 +9 +1 +E/E +min +7 +"g=9 +0.8 +-g = 50 +- g = 500 +N +0.6 +n +0.4 +0.2 +(b) +0 +E +min +max +E10 +(T/β2 +0) +� βmax +0 +dx +� βmax−x +0 +dy(x+y+β0−µ)−1, which gives +Nβ2 +0 = Tβmax +� +1 − z log +� +z/(z − 1) +�� +, +(11) +where z = µ/βmax. A negative temperature equilibrium +state (T < 0) is characterized by z > 1. +It can exist +in the thermodynamic limit with the constraint Nβ2 +0 = +const > 0 provided that 1 − z log +� +z/(z − 1) +� +< 0. This +inequality is always verified for z > 1, which confirms the +existence of NT equilibrium states in the thermodynamic +limit. +NT condensation in the highest energy level +Positive temperature T > 0. We start by briefly sum- +marizing the usual positive temperature condensation in +the lowest energy level. This effect of condensation orig- +inates in the singularity of the RJ distribution. Indeed, +the denominator of nRJ +p += T/(βp − µ) vanishes for the +lowest energy level when µ = β0. This gives rise to an +analogue effect of condensation: As the energy decreases +below a critical value Ecrit (or T < Tcrit), then µ → β− +0 +(see Fig. 13(a)) and the singular behavior of the RJ dis- +tribution is regularized by the macroscopic population of +the fundamental mode: +nRJ +0 /N → 1 +as +E → Emin (or T → 0+). +(12) +It has been shown that this condensation-like effect is a +phase transition that occurs in the thermodynamic limit, +see Ref.[23]. +Negative temperature, T < 0. In the NT region, we +reveal an inverted condensation-like effect, which is char- +acterized by a macroscopic population of the highest +energy level. +Aside from the singularity for µ = β0 +discussed here above for T > 0, the RJ distribution +nRJ +p += T/(βp − µ) also exhibits a singularity (vanishing +denominator) for the highest energy level when µ = βmax +(see Fig. 13(a)). We have shown in Fig. 5, that the high- +est energy level g = 9 becomes macroscopically populated +as the energy increases: +˜ng/N → 1 +as +E → Emax (or T → 0−), +(13) +with ˜ng = � +p,px+py+1=9 np. This condensation-like ef- +fect does not occur in the thermodynamic limit. We pose +µ = βmax + ε, with ε > 0. Eq.(11) can be written in +the limit ε → 0+: Nβ2 +0 ≃ Tβmax +� +1 − log(βmax/ε) +� +. By +keeping Nβ2 +0 =const, the chemical potential µ reaches +β+ +max for a vanishing temperature T → 0−, i.e., conden- +sation does not occur in the thermodynamic limit. How- +ever, an analogue effect of condensation occurs through +a macroscopic population of the highest energy level. +By setting βp = βmax in the RJ distribution, then +nRJ +g += −T/(µ − βmax) denotes the power in one mode +of the highest energy level. The total power in the high- +est energy level (g−fold degenerate) ˜nRJ +g += gnRJ +g +then +FIG. 14: +Simulations of NLS equation and wave tur- +bulence kinetic equation: Simulations of the NLS equation +(dashed lines), and wave turbulence kinetic equation (solid +lines) showing the evolutions during the propagtion (in z) +of the power within each of the g =9 groups of degenerate +modes of the fiber that has been used in the experiments, for +E/Emin = 8 (a), E/Emin = 7 (b). +The horizontal dashed +black line denotes the population for the higher mode group +at complete RJ thermal equilibrium. See the text for details +on initial conditions and parameters. +reads +˜nRJ +g +N (µ) = +g +(µ − βmax) � +p(µ − βp)−1 , +(14) +E(µ) +Emin += +� +p +βp +βp−µ +� +p +β0 +βp−µ +. +(15) +The parametric plot of (14) and (15) with respect to +µ provides the condensate fraction reported in Fig. 5. +Fig. 13(b) shows the condensate fraction, ˜nRJ +g /N vs E, +by increasing the number of modes, i.e., by approaching +the thermodynamic limit. The critical behavior of the +condensation curve looks similar to that of a phase tran- +sition, thought strictly speaking phase transitions only +occur in the thermodynamic limit. Nevertheless, if one +takes the macroscopic occupation of an energy level as +the essential characteristic of condensation, then NTs +are characterized by a condensation-like effect into the +highest-energy level. +WAVE TURBULENCE SIMULATIONS +The starting point is the NLS equation governing light +propagation in MMFs [39, 42]. By expanding the random +wave into the fiber modes and considering the dominant +contribution of weak disorder, the polarized modal com- +ponents ap = (ap,x, ap,y)T are governed by [39]: +i∂zap = βpap + Dp(z)ap − γPp(a), +(16) +where Pp(a) = � +l,m,n Splmn +� +1 +3aT +l ama∗ +n + 2 +3a† +namal +� +, +Splmn denoting the spatial overlap among the modes. +The introduction of disorder is important in order to +avoid strong phase correlations among the modes that +are related to Fermi-Pasta-Ulam recurrences [43], which +inhibit the thermalization process [38]. We consider the + +0.8 +modal population +population +(a) +(b) +0.6 +0.3 +0.4 +0.2 +modal +0.2 +0.1 +10 +20 +30 +10 +20 +30 +0 +0 +z[m] +z[m]11 +most general form of disorder that conserves the power: +The Hermitian matrices Dp(z) are expanded into the +Pauli matrices that form a basis for the vector space of +2×2 Hermitian matrices. +The matrices then have the +form Dp(z) = �3 +j=0 νp,j(z)σj, where σj (j = 1, 2, 3) +are the Pauli matrices (σ0 the identity matrix), while +νp,j(z) are independent and identically distributed real- +valued random processes, with variance σ2 and correla- +tion length ℓc. The corresponding characteristic length +scale of disorder is Ld = 1/∆β, with ∆β = σ2ℓc, see +Ref.[39] for details. +The nonequilibrium process of NT thermalization can +be described by a wave-turbulence kinetic equation. The +kinetic equation was derived from the modal NLS Eq.(16) +in the weakly nonlinear regime of the experiment, Llin ∼ +1/β0 ≪ Lnl ∼ 1/(γN). It describes the nonequilibrium +evolution of the averaged modal components np(z) = +� +|ap|2(z) +� +[39]: +∂znp(z) = +γ2 +6∆β +� +l,m,n +|Slmnp|2δK(∆ωlmnp)Mlmnp(n) ++ 4γ2 +9∆β +� +l +|slp(n)|2δK(∆ωlp)(nl − np), (17) +with slp(n) += +� +m′ Slm′m′pnm′, +and Mlmnp(n) += +nlnmnp + nlnmnn − nnnpnm − nnnpnl and ∆ωlp = +βl − βp. The term δK(∆ωlmnp) denotes the four-wave +frequency resonance ∆ωlmnp = βl + βm − βn − βp, with +δK(∆ωlmnp) = 1 if ∆ωlmnp = 0, and zero otherwise. +The kinetic Eq.(17) conserves N = � +p np(z), the energy +E = � +p βpnp(z) and exhibits a H−theorem of entropy +growth, ∂zSkin(z) ≥ 0, for the nonequilibrium entropy +Skin(z) = � +p log +� +np(z) +� +. Accordingly, it describes an +irreversible evolution of the speckle beam to the RJ equi- +librium distribution realizing the maximum of entropy, +nRJ +p += T/(βp − µ) [39]. +We have considered in the simulations the MMF used +in the experiments, see Sec. . The initial condition con- +sists of a speckle beam whose correlation length is varied +in such a way to fix a desired value of the energy E. The +considered parameters are ℓc = 0.3m and 2π/σ = 2.1m +[39]. The results of the simulations of the NLS equation +and the corresponding simulations of the wave turbulence +kinetic equation are reported in Fig. 14. They show the +process of thermalization to the negative temperature RJ +equilibrium state predicted by the theory. +The simu- +lations qualitatively reproduce the experimental results, +although a power of 20kW has been considered to accel- +erate the dynamics. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Universit´e Bourgogne Franche-Comt´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Dijon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' France 2 CMAP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Ecole Polytechnique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Institut Polytechnique de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 91128 Palaiseau Cedex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' France 3 CEA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' DAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' DIF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' F-91297 Arpajon Cedex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' France 4 Universit´e Cˆote d’Azur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Institut de Physique de Nice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Nice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' France 5 Institute of Physical Chemistry Polish Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Warsaw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Poland and 6 Institut Universitaire de France (IUF),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 1 rue Descartes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' France Although the temperature of a thermodynamic system is usually believed to be a positive quan- tity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' under particular conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' negative temperature equilibrium states are also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Negative temperature equilibriums have been observed with spin systems, cold atoms in optical lattices and two-dimensional quantum superfluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Here we report the observation of Rayleigh-Jeans thermal- ization of light waves to negative temperature equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The optical wave relaxes to the equilibrium state through its propagation in a multimode optical fiber, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', in a conservative Hamil- tonian system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The bounded energy spectrum of the optical fiber enables negative temperature equilibriums with high energy levels (high order fiber modes) more populated than low energy levels (low order modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Our experiments show that negative temperature speckle beams are featured, in average, by a non-monotonous radial intensity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The experimental results are in quantitative agreement with the Rayleigh-Jeans theory without free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Bringing negative tempera- tures to the field of optics opens the door to the investigation of fundamental issues of negative temperature states in a flexible experimental environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='- Temperature is a central concept of sta- tistical mechanics and often reflects a measure of the amount of disordered motion in a classical ideal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Al- though this intuitive notion is correct for many physical systems, one should keep in mind that the concept of temperature is by far more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' A detailed analy- sis of the concept of temperature, and of its relationship with energy and entropy shows that, under suitable con- ditions, the entropy can decrease with the energy, thus allowing for the existence of equilibrium states at nega- tive temperatures (NT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Starting from the seminal works by Onsager [1] and Ramsey [2], who originally conceived the physical idea and the first theoretical approaches, during the last decades, many works have been devoted to the theoretical understanding of these unusual equi- librium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Despite the fact that the existence of a NT equilibrium has created its own share of confusion in relation with the definition of the entropy [3, 4], NTs are now broadly accepted in line with different experimental observations [5–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' NTs were originally observed experi- mentally in nuclear spin systems [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' More recently, NTs were observed with cold atoms in optical lattices [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Furthermore, NTs originally predicted by Onsager in the statistical description of point vortices [1] have been re- cently observed in 2D quantum superfluids [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' In this Letter we present an experimental optical setup in which we report the observation of light thermaliza- tion to NT equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Our system is based on the nonlinear propagation of speckle beams in a multi- mode optical fiber (MMF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Because of the presence of a finite number of modes supported by the MMF, the spectrum exhibits both lower and upper bounds for the energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The bounded spectrum, combined to the nonlinear four-wave interaction, are responsible for the process of Rayleigh-Jeans thermalization to NT equilib- rium states [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We stress that, at variance with other experiments where photon thermalization is driven by a thermal heat bath [19–22], here light thermaliza- tion takes place in a conservative Hamiltonian system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' RJ thermalization to usual positive temperature equilib- riums has been recently demonstrated experimentally in MMFs [23–26], on the basis of a spatial beam-cleaning ef- fect [27–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' As described by the wave turbulence theory [32–36] applied to MMFs [37–40], the thermalization to a positive temperature equilibrium is characterized by a transfer of power (particle number) toward the low-order modes of the MMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' In marked contrast, here we report the observation of thermalization to a NT equilibrium featured by a power transfer to high-order modes (direct flow of particles), as well as a transfer of energy to low- order modes (inverse flow of energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Consequently, the NT equilibrium is characterized by an inverted modal population, in which high-order modes are more popu- lated than low-order modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Our experimental optical setup can be used as a simple and flexible testbed to explore fundamental issues related to NT states that are discussed in conclusion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', Carnot cycles operating between temperatures of opposite signs, or inverted turbulence cascades featured by an analogue process of condensation at NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Experimental system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='- The experiment is based on the single pass propagation of speckle beams through a MMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The subnanosecond pulses delivered by a Nd:YAG laser (λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='06µm) are transmitted through a spiral arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='02796v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='optics] 7 Jan 2023 2 phase plate and then through a diffuser before injec- tion of the speckle beam into a 12m long graded-index MMF (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', parabolic-shaped trapping potential), which guides M = 45 modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', nine groups of degener- ate modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The energy levels (fiber eigenvalues) are well approximated by the ones of an harmonic potential βp = β0(px + py + 1), where {p} labels the two integers (px, py) that specify a mode (see Supplementary Mate- rial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We denote by |ap|2 the power in the mode p, with the total power N = � p |ap|2 [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The experiment is realized in the weakly nonlinear regime, where linear effects dominate over nonlinear ef- fects Llin ∼ β−1 0 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='1mm ≪ Lnl = 1/(γN) ∼ 20cm, γ being the nonlinear coefficient of the MMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Accord- ingly, we do not consider NT states associated to nonlin- ear coherent structures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', breathers [10, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Since Llin ≪ Lnl, we only retain the linear contribution to the Hamiltonian, E = � p βp|ap|2 [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We have ver- ified the conservation of the power N and the energy E through propagation in the NT region for each realiza- tion of a speckle beam, which confirms that the coupling between guided modes and leaky modes of the fiber can be neglected (see Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' RJ thermalization is driven by the four-wave nonlinear interaction through the propagation in the MMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The speckle beam is expected to relax toward the thermody- namic equilibrium state described by the RJ distribution [17, 23–25, 38, 39]: nRJ p = T/(βp − µ), (1) where T and µ are the temperature and chemical po- tential, while np = � |ap|2� denotes the modal power av- eraged over the realizations of the speckle beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We have at equilibrium N = T � p(βp − µ)−1 and E = T � p βp/(βp − µ), with (T, µ) uniquely determined by (N, E) – we deal with a microcanonic description (T is not defined by a thermostat, it is in units of W·m−1) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Note that the RJ distribution refers to the classical, low-energy, limit of the Bose-Einstein distribution [32], describing highly occupied fiber modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Negative temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='- The irreversible process of RJ thermalization is described by the wave turbulence the- ory [32–36], which provides a nonequilibrium description of light propagation in MMFs [37–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' An equilibrium thermodynamic formulation of multimode optical sys- tems has been recently developed [17, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We report in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 1 the relative entropy S = � p log(nRJ p ) as a function of the energy for the MMF used in our experiments with g = 9 groups of degenerate modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Because the spec- trum of the fiber is bounded, β0 ≤ βp ≤ βmax = gβ0, the system possesses both lower and upper energy bounds Emin = Nβ0 ≤ E ≤ Emax = Nβmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (2) Starting at minimum energy Emin, where only the fun- damental mode is populated, an increase in energy leads FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 1: Negative temperatures and inverted modal population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (a) RJ equilibrium distribution nRJ p for positive temperature T > 0 (E < E∗) where low-order modes are more populated, and negative temperatures T < 0 (E > E∗) featured by an inverted modal population;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' while for 1/T → 0 (E = E∗), nRJ p =const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (b) Relative entropy S vs energy E, showing that 1/T = (∂S/∂E)N,M < 0 requires E > E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (c) Temperature T vs energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Negative temperatures T < 0 occur for E > E∗ with E∗/Emin ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='33 (vertical dashed black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The vertical purple lines in (b-c) denote the six values of E considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' to an occupation of a larger number of fiber modes and therefore an increase in entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' As the temperature ap- proaches infinity, all fiber modes become equally popu- lated nRJ p =const, and the entropy reaches a maximum for E = E∗ = N ⟨βp⟩ = Emin(2g + 1)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' NT equilibrium states arise for E > E∗, where the entropy decreases by increasing the energy, 1/T = (∂S/∂E)M,N < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The condition E > E∗ can be achieved if high-order modes are more populated than low-order modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Note that NT equilibrium states persist in the thermodynamic limit (see Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' RJ thermalization to NT equilibriums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='- At variance with usual experiments of spatial beam cleaning and RJ thermalization [24–30], here we study the thermalization for different values of the energy E, while keeping con- stant the power N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Indeed, by passing the laser beam through a diffuser before injection into the fiber, we can vary the amount of randomness of the speckle beam by keeping N =const – the larger the randomness of the speckle beam, the higher the energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Accordingly, we study RJ thermalization over a broad range of variation of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' In order to further increase the energy be- yond the threshold for NT (E > E∗), we pass the beam through a spiral phase plate before the diffuser, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', we generate a speckle beam from a doughnut-like intensity 0 0+ +818 (a) T T>0 1/T = 0 T<0 p n p p p 0 200 S 400 (b) 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 uIu 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 (c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 5 9 3 7 E/E min3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 2: Rayleigh-Jeans thermalization to NT equi- libriums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Experimental modal distributions averaged over realizations nexp p = � |aexp p |2� , at the fiber input (blue), at the fiber output (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Corresponding RJ equilibrium distribu- tion nRJ p (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Note the quantitative agreement between nRJ p and the experimental output distribution nexp p (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The six panels correspond to different values of E, or equivalently different T, see the six vertical purple lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 1(b-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The modal distribution peaked on the lowest mode for T > 0 (a), gets inverted for T < 0 (b-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' An average over ≃35 re- alizations of speckle beams is considered for each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The fiber modes are sorted from the fundamental one (β0) to the highest mode group (nine-fold degenerate with βmax = 9β0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Degenerate modes are equally populated at equilibrium, lead- ing to a staircase distribution nRJ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' distribution, which enables the excitation of higher order fiber modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The accurate measurements of the near-field and far- field intensity distributions allowed us to retrieve the modal power distribution nexp p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' To obtain the mode de- composition, several interferometric approaches based on use of a reference beam have been exploited to study light thermalization in MMFs [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Here, in contrast to the previous works [23–26], we use a non-interferometric nu- merical mode decomposition procedure that is based on the Gerchberg-Saxton algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' It allows us to retrieve the transverse phase profile of the speckle field from the near-field and far-field intensity distributions measured in the experiments [48–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' By projecting the retrieved complex field over the fiber modes, we get the complete modal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The RJ distribution being in essence a statistical dis- tribution, its comparison with the experiments requires an average over realizations of speckle beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We have recorded 2×300 realizations of the near-field and far-field intensity distributions for the same power (N = 7kW) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 3: Energy flows in mode space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Experimental en- ergy distributions averaged over 50 realizations εexp p = βpnexp p , at the fiber input (blue), output (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The arrow indicates the energy flow to low-order modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Corresponding RJ equi- librium distribution εRJ p = βpnRJ p (green line), which is in quantitative agreement with the experimental output distri- bution (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' and different energies E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' For each individual speckle re- alization, we retrieve the modal distribution |aexp p |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We partition the ensemble of 300 realizations of {|aexp p |2} within small energy intervals [E − ∆E, E + ∆E] with ∆E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='125Emin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We perform an average over the re- alizations of the modal distributions for each energy in- terval, which provides the averaged modal distribution nexp p = � |aexp p |2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This procedure is applied at the fiber output (L = 12m), and fiber input (after 20cm of prop- agation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The error in the procedure has been computed theoretically and numerically, it decreases with the num- ber of realizations and has been found remarkably small (relative standard deviations of ≃ 6%), see Supplemen- tary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We report in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 2 the averaged modal distributions nexp p at the fiber input (blue) and output (red), for dif- ferent values of the energy E, or equivalently the tem- perature T (purple lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 1(b-c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The data are compared with the theoretical RJ distribution nRJ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We stress that there are no adjustable parameters between nexp p and nRJ p : The parameters (T, µ) in nRJ p are uniquely determined by N and E measured in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 2 an excellent agreement between nexp p (red circles) and nRJ p (green line), for both T > 0 and T < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 2 then shows that NT equilibriums constitute attractor states for the random wave, whose robustness has a thermodynamic origin – maximum entropy state for a given pair (N, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Energy flows in mode space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='- The conventional ther- malization to positive temperatures is characterized by an energy flow to high-order modes [33, 34, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Thermal- ization to NTs typically occurs through an inverse energy flow to low-order modes [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 3, which shows that the energy distribution εexp p = βpnexp p at low-order modes increases through propagation in the MMF and reaches the theoretical RJ equilibrium distri- bution εRJ p = βpnRJ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='04 DE/E E/E = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='623 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='196 RJ min min T/E T/E = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='347 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='125 Output min min N Input 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='02 (b) (a) 0 10 20 30 40 10 20 30 40 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='05 E/E E/E = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='933 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='244 min min T/E = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='153 T/E = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='091 min min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='025 Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='02 (c) (d) 0 0 10 20 30 40 10 20 30 40 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='07 E/E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' E/E =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='215 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='884 min T/E = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='039 T/E = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='025 min min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='04 Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='035 n (f) (e) 0 0 10 20 30 10 20 30 0 40 0 40 Modes ModesE/E = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='161 10 min T/E = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='027 min 10 RJ 10° Output O Input 15 25 0 5 10 20 30 35 40 Modes4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 4: Oscillating radial intensity distribution at NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Averaged intensity distribution Iexp(|r|) as a function of the radial (angle-averaged) distance |r| (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Note the quantitative agreement with the theoretical RJ intensity dis- tribution IRJ(|r|) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (3) (dashed green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The oscillating behavior of the intensity distribution is a signature of the NT equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Inset: corresponding 2D intensity averaged over the realizations (the radius of the circle is the fiber radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Oscillating radial intensity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='- The intensity distribution IRJ(|r|) of usual positive temperature equi- libriums is, in average, a monotonic decreasing function with the radial distance |r| [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This is consistent with the intuitive idea that low-order modes localized near-by the fiber center are the most populated ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' In marked contrast, the inverted modal population of NT equilibri- ums are characterized by an oscillating behaviour of the radial intensity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 4, which reports the averaged radial intensity distribution Iexp(|r|) (with ∆E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='25Emin, E/Emin = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The theoretical RJ intensity distribution reads IRJ(r) = � p nRJ p u2 p(r), (3) where up(r) denotes the fiber modes [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The number of radial oscillations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 4 is given by the most oscillating mode of the fiber, namely the mode LP04 that exhibits 5 oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Experiments by increasing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='- We have studied the optical field at the output of the MMF, with a small power N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='23kW (linear regime), and a high power N = 7kW (nonlinear regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Since the MMF length is kept fixed (L = 12m), the effective number of nonlin- ear interaction lengths increases by increasing the power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 5 reports the fraction of power that populates the highest group of degenerate modes of the MMF, ˜ng/N for g = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The output field (red) reaches the equilibrium RJ theory (green line) in the nonlinear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The highest energy level gets macroscopically populated by increasing the energy, or equivalently by increasing the temperature of negative sign (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Conclusion and perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='- We have reported the ob- servation of RJ thermalization to NT equilibrium states through light propagation in graded-index MMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This non-equilibrium process of NT thermalization can be de- scribed by a wave turbulence kinetic equation, which is FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 5: Macroscopic population of the highest energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Fraction of power ˜ng/N into the highest mode group g = 9 vs energy E/Emin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Experimental measurements at the fiber output: The blue circles refer to the linear regime (small power), the red circles to the nonlinear regime (high power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The green line denotes the RJ equilibrium theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' By increasing the energy, the power goes to the highest energy level, ˜ng/N → 1 as E/Emin → 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' found in agreement with the simulations of the nonlin- ear Schr¨odinger equation (see Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Our NT experiment then paves the way for the study of Zakharov-Kolmogorov turbulence cascades [32–34] that are inverted with respect to those underlying usual posi- tive temperature thermalization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', inverse energy flow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Along this line, our work suggests a previously un- recognized process of inverted condensation at NTs: At variance with usual condensation at positive temperature where the lowest energy level gets macroscopically pop- ulated by decreasing the temperature (T → 0+, or E → Emin) [23, 33–36], at NT an inverted condensation pro- cess occurs into the highest energy level as the temper- ature increases to zero (T → 0−, or E → Emax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' While we provide a preliminary study of this effect through the macroscopic population of the highest energy level (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 5), the observation of the transition to condensa- tion requires MMFs with larger number of modes (see Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' In our work NT states are obtained directly, which is in contrast with magnetic systems and cold atoms where the excitation of NT states requires first the creation of a pos- itive temperature state and then its subsequent inversion through suitable procedures (magnetic field inversion or Feshbach resonances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This opens the possibility to study the physics of NT in a flexible experimental environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' For instance, the thermalization of two beams at differ- ent laser wavelengths interacting through the fiber non- linearity can be exploited to achieve an efficient optical refrigeration: A highly incoherent speckled beam at NT can be cooled through its thermalization with a coher- ent beam towards a highly coherent state without any power loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' In contrast with usual beam cleaning at posi- tive temperature where the energy is conserved, here the cooling process is featured by an energy transfer from the incoherent to the coherent beam, which significantly improves the gain of coherence of the incoherent beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='12 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' E/E = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='945 min 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='1 RJ T/E = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='036 min Intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='02 0 5 10 15 0 [um] rRJ High Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='8 Low Power 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='6 N n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='2 0 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 9 E/E min5 Following this idea, one can explore the meaning of a thermostat at NT [10]: If the NT incoherent beam has a power much larger than the partially coherent beam, it will play the role of a NT thermal reservoir for such a partially coherent beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The versatile optical experimental environment pro- posed in this work also opens the possibility to study controversies about NTs, such as thermodynamic engines featured by Carnot cycles operating between tempera- tures of opposite signs, in relation with the generalized Kelvin-Planck formulation of the second law of thermo- dynamics stating that it is not possible to completely transform work into heat at NT [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='- The authors are grateful to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Rica, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Carusotto and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Doya for fruitful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Fundings: Centre national de la recherche scien- tifique (CNRS), Conseil r´egional de Bourgogne Franche- Comt´e, iXCore Research Fondation, Agence Nationale de la Recherche (ANR-19-CE46-0007, ANR-15-IDEX-0003, ANR-21-ESRE-0040).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Calculations were performed us- ing HPC resources from DNUM CCUB (Centre de Cal- cul, Universit´e de Bourgogne).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' SUPPLEMENTARY MATERIAL EXPERIMENTAL SET-UP The source is a Nd:YAG laser delivering subnanosec- ond pulses (400ps) at λ =1064nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The laser beam is passed through a spiral phase plate (Thorlabs) to gener- ate a doughnut-like ring-shaped beam, and subsequently through a diffuser before injection of the speckle beam into the MMF, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The diffuser plate is placed in the vicinity of the Fourier plane of a 4f-optical sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The near-field (NF) intensity distribution of the fiber output beam was magnified and imaged on a first CCD camera owing to a two lens telescope optical system, with f2 = 8 mm and f3 = 150 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The CCD camera was placed on a rail orthogonal to the beam propaga- tion in order to remove or put the camera back on the beam path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The far-field (FF) intensity distribution of the magnified image was obtained by placing it in the ob- ject focal-plan of a lens f4 = 250 mm and using a second CCD camera positioned in its image (Fourier) focal-plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We have computed analytically the propagation of the optical wave throughout the setup of our detection scheme, according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 6 (lower part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' If ψ0(r) is the optical field amplitude at the fiber output (r = (x, y)), then we have in the NF plane: ψNF(r) = −ρ−1ψ0(−r/ρ), with ρ = f3/f2 the magnification factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' In the FF plane, the wave amplitude reads ψFF(u) = iρ λf4 � drψ0(r) exp[−i2π(−ρ)r · u/(λf4)], FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 6: Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' laser, optical isolator, half-wave plate and polarizer, lenses for magnification and imaging (fj), spiral phase plate (V), diffuser (D), graded-index MMF, and cam- eras for near- and far-field detections (Cam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' which corresponds to the Fourier transform of the field amplitude at the fiber output (note that the constant phase prefactor plays no role because the camera records the intensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We note that: (i) The optical amplitude in the NF plane is an exact magnification of the wave amplitude at the output of the MMF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (ii) the optical amplitude in the FF detection plane exactly corresponds to the Fourier transform of the amplitude at the fiber output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Then, the experimental setup for the detection of the NF and FF intensities does not introduce detrimental spurious transverse phases profiles in the optical field, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', related to optical free propagation in air or phase shifts due to the presence of additional lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Multimode fiber The refractive index profile of the graded-index MMF exhibits a parabolic shape in the fiber core with a max- imum core index (at the center) of nco ≃1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='472 and ncl ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='457 for the cladding at the pump wavelength of 1064nm (fiber radius R = 15µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The fiber length is L = 12m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The trapping parabolic potential reads V (r) = q|r|2 for |r| ≤ R and q = k0(n2 co −n2 cl)/(2ncoR2), k0 = 2π/λ the laser wave-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The fundamental mode energy level is β0 = 2√αq, with α = 1/(2ncok0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The MMF guides M = 45 modes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' g = 9 groups of degenerate modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The truncation of the potential introduces a frequency cut-off in the FF spectrum kc = (2π/λ) � n2co − n2 cl [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The conservation of N and E through propagation in the MMF (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 7) shows that the coupling from guided modes to leaky radiation modes in the cladding is negligi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This is not surprising as the efficiency of such a cou- pling can be shown to be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Indeed, the MMF has a core (radius 15µm), a cladding (radius 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5µm), and a highly absorbing polymer-coating with refractive index larger than the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Then leaky radiation modes in the cladding are rapidly absorbed during propagation due to their large penetration in the polymer-coating: we measured a typical absorption length Labs of ≃ 15cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Let us consider the coupled amplitude equations describ- ing the four-wave mixing between four modes including a leaky mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The four mode amplitudes a1, a2, a3, a4 (+ D) ND YAG P MMF FF NF Cam Cam6 (where a4 stands for the leaky mode amplitude) satisfy Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='2) in [44], in which we need to add an absorption term of the form −a4/Labs in the equation (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='4) in [44] for a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Since absorption is strong the overdamped limit is valid and we get a4 = in2k0Labsf4312a1a2a∗ 3 exp(−i∆βz) where ∆β = β3 + β4 − β1 − β2, the parameter n2 is the nonlinear-index coefficient, and f4312 is an overlap inte- gral between the four mode profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' By substitution into one of the first three equations, say (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='1) in [44], we find that the mode amplitude a1 experiences an effective absorption due to the coupling with the leaky mode that is given by −4n2 2k2 0Labs|f1234|2|a2|2|a3|2a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This shows that this absorption is of the order of −Labs/L2 nla1 times a coefficient that is of the order of the square overlap in- tegral between guided and leaky mode profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' As the supports of these modes are very different (the guided modes are essentially supported in the core while the leaky modes are essentially supported in the cladding that is much larger), the square overlap integrals are small (smaller than the respective core-cladding ratio (15/62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5)4 ≃ 3 10−3) and the effect of the coupling to the leaky modes onto the guided mode amplitudes can be neglected when Lnl ≃ 20cm and the propagation distance is L = 12m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Measurements of the energy E From the measurements of the NF and FF intensity distributions, we have retrieved an accurate measurement of the power N and the energy E of the speckle beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The NF intensity distribution INF(r) = |ψ|2(r) provides a measurement of the power N = � INF(r)dr and of the potential energy Epot = � V (r)|ψ|2(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The kinetic energy Ekin = α � |∇ψ|2(r)dr is retrieved from the FF intensity distribution IFF(k) = | ˜ψ|2(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This provides the measurement of the (linear) energy (Hamiltonian) E = Epot + Ekin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Conservation of N and E through propagation in the MMF for E > E∗ (negative temperature region) Power conservation has been verified by keeping fixed the conditions of injection of the speckle beam into the MMF: We measured Nout at L = 12m, and then Nin by cutting the fiber at 20cm, and we always obtained (Nin − Nout)/Nmoy < 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The experimental verification of energy conservation requires both the NF and FF in- tensity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The NF and FF intensities are recorded at the fiber output at L = 12m, which gives Eout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Without altering the fiber launch conditions, the fiber is cut to 20cm to record the input NF and FF in- tensities, which gives Ein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The measurements of Ein and Eout then refer to an individual realization of the speckle beam (without average over the realizations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 7 shows that the conservation of the energy is well verified for E > E∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' in the negative temperature region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 7: Conservation of the energy through propaga- tion in the MMF in the negative temperature region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (a) Measurements of the energy at the input of the MMF (blue triangles), and at the output of the MMF (red trian- gles): The energy E/N is conserved through the propagation in the MMF over a broad range of variation of E/Emin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We recall that negative temperatures T < 0 occur for E > E∗ with E∗/Emin ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='33, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' energy E is varied owing to the diffuser before injection into the MMF, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' MODAL DECOMPOSITION Phase retrieval The procedure of mode decomposition is based on the well-known Gerchberg-Saxton algorithm [41, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' From the measurements of the NF and FF intensity dis- tributions in the experiment, it allows us to retrieve the transverse phase profile of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The resulting com- plex field is subsequently projected onto the fiber modes, to get the complete modal distribution of the experimen- tal optical beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The algorithm is known to be accurate although it is not efficient in terms of computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Indeed it is a local search algorithm that updates itera- tively the unknown phase profile of the field and it is usu- ally necessary to consider several initial phase guesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We have, therefore, carried out a detailed preliminary analysis of the algorithm by performing numerical simu- lations that reproduce our experimental configuration in order to prove that the phase retrieval and modal distri- bution estimation are reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Error introduced by the Gerchberg-Saxton algorithm To evaluate accurately the error in the phase-retrieval algorithm, we have reproduced in detail the experimental procedure as follows: i) We consider a particular value of the energy E (E > E∗ so that T < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Throughout the procedure the power N is set constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The pair (E, N) determines uniquely (T, µ) and thus the exact RJ distribution at equilibrium nRJ p = T/(βp − µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='8 indno Input 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='7 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='6 E/N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='4 L 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 E/E ulw7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 8: Phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Example of numerically gener- ated near-field intensity distribution (a), and its reconstruc- tion (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Original phase field (c) and the corresponding phase field reconstructed from the Gerchberg-Saxton algorithm (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' ii) We generate from nRJ p a realization of speckle beam ψ(r) = � p apup(r), where ap is a complex Gaussian random variable with variance � |ap|2� = nRJ p (ap = a(r) p + ia(i) p with a(r) p and a(i) p real Gaussian independent random variables with mean zero and variance nRJ p /2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We recall that up(r) are the fiber modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Then ψ(r) is a particular realization of a complex speckle field at exact RJ equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' iii) The particular realization ψ(r) is highly resolved nu- merically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We mimic the impact of the finite resolution of the camera used in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' From ψ(r) we compute the NF and FF intensity distributions |ψ(r)|2 and | ˆψ(k)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We sample the NF and FF intensity dis- tributions with the finite number of points available in the camera (10242) in r-space and k-space and ≃ 950 points for the dynamics range in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We apply the Gerchberg-Saxton algorithm to retrieve the sampled phase profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Due to the errors introduced by the sam- pling of the camera and by the phase-retrieval algorithm, the resulting complex field ψexp(r) may differ from the generated speckle beam ψ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We report in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 8 the numerically generated near-field intensity distribution in one numerical simulation (a) and its reconstruction (b), the original phase profile (c) and the reconstructed phase profile (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' It is clear that the re- construction is very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We will see below more quan- titatively that the error is indeed negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' iv) We project the complex field ψexp(r) onto the fiber modes to get the complex modal coefficient aexp p and dis- tribution |aexp p |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Due to the errors introduced by the sampling of the camera and by the phase-retrieval algo- rithm, this modal distribution may differ from the mode FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 9: Error in the modal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The mode decomposition is based on the Gerchberg-Saxton algorithm, whose error is quantified by the distance DQ err to the exact distribution, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (a) DQ err vs the energy E (for Q = 50 realizations), by accounting for the sampling due to the lim- ited resolution of the camera (red), and without the sampling (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Note in (a) that an increase of the randomness of the speckle beam (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', increase of E) does not increase the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (b) DQ err vs number of realizations Q (for E/Emin = 7): The error decreases with the number Q of realizations of the speck- les.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The green line reports the theoretical estimate of the error given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Note that DQ err is bounded, 0 ≤ DQ err ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 10: Near-field and far-field experimental inten- sities, and corresponding reconstructed intensity dis- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Near-field (NF) intensity recorded in the experi- ment for a single realization of a speckle (a), and correspond- ing far-field (FF) intensity distribution (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Corresponding NF intensity (b), and FF intensity (d), reconstructed from the Gerchberg-Saxton algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' exper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (NF) reconst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (NF) 10 10 [ur] [wn] 0 0 V 10 10 (b) (a) 10 10 10 10 0 0 x [μum] x [μum] exper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (FF) reconst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (FF) [,_un] [,_un]"y 0 0 (c) (d) 1 0 0 [um-?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='10 10 [un] y [μum] 0 0 10 10 (a) (b) 10 0 10 10 0 10 x [μum] x [um] 10 10 xy[um] [um] 0 0 10 10 d) 10 10 10 10 0 0 x [um] x [μum]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='04 Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='02 Without Sampling With Sampling (a) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 8 6 7 E/E min Theory 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='2 Numerical simulations Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='05 (b) 25 50 100 75 0 Q8 distribution |ap|2 used to generate the speckle beam ψ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' As we will show below, this error is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' v) We repeat the steps ii)-iv) Q times, each with a different realization of the speckle beam (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', with a different realization aj p of ap, for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' , Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The procedure then gives Q distributions |aexp,j p |2, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' , Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We compute the empirical averages nexp,Q p = (1/Q) �Q j=1 |aexp,j p |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We anticipate that, for Q large enough, these empirical averages should be close to the theoretical values nRJ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We introduce the estimation er- ror: DQ err = � p ��nexp,Q p − nRJ p �� � p nexp,Q p + nRJ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (4) Let us imagine for a while that the phase-retrieval al- gorithm is perfect and that the sampling error due to the camera is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Then, for each realization j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' , Q, we have |aexp,j p |2 = |aj p|2 exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Thus, the random vari- ables |aexp,j p |2 are independent and follow exponential dis- tributions with mean nRJ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Consequently, the empirical quantities ZQ p = nexp,Q p /nRJ p are independent and identi- cally distributed with the gamma probability distribution Γ(Q, Q) (the law of the sum of Q independent variables with exponential distribution and mean 1/Q) and the estimation error is DQ err = � p nRJ p |ZQ p − 1| � p nRJ p (ZQ p + 1) , (5) which gives when the number of modes is large enough (ZQ follows the Γ(Q, Q) distribution): DQ err ≃ E[|ZQ − 1|] E[ZQ + 1] = QQ−1 (Q − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='e−Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (6) For Q ≥ 8 , we have DQ err ≃ 1/√2πQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We have carried out numerical simulations with our implementation of the phase-retrieval algorithm (using multiple initial phase guesses) and with the sampling er- ror of the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The results of the distance DQ err vs energy E are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 9 with different numbers Q of realizations per energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We can see in panel (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 9 that the errors correspond to the theoretical er- ror Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (6) when the phase-retrieval algorithm makes no error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The error introduced by the Gerchberg-Saxton algo- rithm has been computed by increasing the amount of complexity in the speckle pattern, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', by increasing the energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 9(a) that the error does not increase when the energy E increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' In the experiments we have typically 35 to 70 indepen- dent realizations of speckle beams for a given small en- ergy interval [E − ∆E, E + ∆E] with ∆E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='125Emin, so we can expect that the errors (due to the phase re- trieval algorithm and the camera sampling) are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Error bars with relative standard deviations of the order of 1/√2πQ ≃ 6% could be added in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 2 but they are too small to be visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 11: Experimental attraction to NT RJ equilib- rium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Distance DRJ [defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (7)] to the RJ equilibrium distribution computed from the experimental data at the fiber input (blue), and fiber output (red), for different values of the energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The significant reduction of DRJ from input to out- put measurements shows the attraction to the RJ equilibrium for T < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Note that DRJ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (7) is bounded, 0 ≤ DRJ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We recall that negative temperatures T < 0 occur for E > E∗ with E∗/Emin ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='33, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' To complete our study, we report in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 10 the near- field and far-field intensity distributions recorded during one of the experiments (left plots) and the correspond- ing reconstructed intensities from the Gerchberg-Saxton algorithm (right plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Experimental convergence to the NT RJ distribution We have quantified in our experimental results the at- traction to the NT equilibrium by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (4), which provides a ‘distance’ to the RJ distribution DRJ = � p |nexp p − nRJ p | � p nexp p + nRJ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (7) We report in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 11 the distance DRJ computed for the experimental data averaged over the realizations nexp p at the fiber input (blue), and the fiber output (red), for dif- ferent energies E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The strong reduction of the distance DRJ from the fiber input to the output confirms the pro- cess of NT thermalization, which is demonstrated over a broad range of values of the energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Polarization effects The polarization state of the optical beam changes as it propagates through the MMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The field at the out- put of the MMF is projected onto a basis of orthogo- nal linear polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The corresponding NF and FF intensity distributions are recorded along the orthogo- nal linear polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' For each polarization, we ap- ply the mode decomposition procedure based on the Gerchberg-Saxton presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' In this way, we re- trieve the transverse phase profile of the field for the two orthogonal polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The complex field along each polarization is subsequently projected over the fiber modes, to get the modal populations for each polariza- tion, n(x) p and n(y) p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The modal distribution is obtained 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='4 Output Input 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='1 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='5 E/E9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 12: Polarization effects in modal decomposi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Experimental modal distribution retrieved from the Gerchberg-Saxton algorithm, without splitting the orthogonal polarization (blue line), by splitting the orthogonal polariza- tion at the fiber output (dashed red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Note that a single realization of the speckle beam has been considered, which ex- plains the discrepancy with the RJ equilibrium distribution (dashed green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' by summing the contributions of the two polarizations, npol p = (Nxn(x) p + Nyn(y) p )/(Nx + Ny), where Nx,y denote the power along the two polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We compare in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 12 the modal distribution retrieved by following this procedure (npol p ), with the modal distribution (np) re- trieved without separating the polarization states of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This comparison is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 12 for the same (single) realization of the speckle beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We can remark in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 12 that the two distributions are almost identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' In all cases analyzed, we have always observed the same good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Given the large number of realizations of speckle beams (≃300) recorded and analyzed in our experiments, we didn’t perform the polarization modal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' ANALOGUE CONDENSATION AT NT Thermodynamics relations We define the thermodynamic relations used to plot Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We consider a field at equilibrium with the RJ distribution nRJ p = T/(βp − µ), with N = � p nRJ p and E = � p βpnRJ p , where βp = βpx,py = β0(px + py + 1) are the eigenvalues of the truncated parabolic potential (βp ≤ βmax), and the index {p} labels the two integers (px, py) that specify a mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Here and below, the sum over the modes � p is carried over the set 0 ≤ px + py < g, where g = βmax/β0 is the number of groups of non-degenerate modes, with M = g(g + 1)/2 the total number of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Because of the constraint nRJ p = T/(βp − µ) > 0, a NT equilibrium state T < 0 requires that µ > βmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We start from the equilibrium entropy ˜Seq = � p log(neq p ) – note that at equilibrium it coincides with the nonequilibrium entropy verifying the H theorem of entropy growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' It proves convenient to shift the en- tropy by a constant Seq = ˜Seq − M log N, so that by FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 13: Analogue effect of condensation at NTs: (a) Chemical potential µ/β0 − 1 vs energy E/Emin for the MMF used in the experiments (g = 9), from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='(9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Note the asymptotic behaviors µ → β− 0 for E → Emin, and µ → β+ max for E → Emax, which lead respectively to the macroscopic population of the lowest energy level (β0), and highest energy level (βmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The horizontal dashed line denotes µ = βmax and the vertical one E = E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (b) Condensate fraction in the highest energy level ˜nRJ g /N vs energy E: As the energy in- creases E → Emax (or equivalently the temperature increases T → 0−), the condensate fraction increases ˜nRJ g /N → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The curves are obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (14-15), see the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The crit- ical behavior of the transition to condensation becomes ap- parent by increasing the number of modes g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' using T = N/ � p(βp − µ)−1, we can write S(µ) = − � p log(βp − µ) − M log � � p 1 βp − µ � (8) E(µ) Emin = � p βp βp−µ � p β0 βp−µ (9) T(µ) Emin = 1 � p β0 βp−µ (10) The parametric plot with resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' to µ of (8) and (9) gives S(E) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 1(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' the corresponding parametric plot of (9) and (10) gives T vs E in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' NT states in the thermodynamic limit The thermodynamic limit is defined by N → ∞, β0 → 0 with Nβ2 0 =const and βmax =const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' In this limit the discrete sums over the modes are re- placed by continuous integrals, namely N = � p neq p → 10-1 n p nbol p RJ C 10-3 0 5 10 15 20 25 30 35 40 45 Modes40 T>O 0>- 20 0 20 (a) 40 2 3 4 5 6 7 8 9 1 E/E min 7 "g=9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='8 g = 50 g = 500 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='6 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='2 (b) 0 E min max E10 (T/β2 0) � βmax 0 dx � βmax−x 0 dy(x+y+β0−µ)−1, which gives Nβ2 0 = Tβmax � 1 − z log � z/(z − 1) �� , (11) where z = µ/βmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' A negative temperature equilibrium state (T < 0) is characterized by z > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' It can exist in the thermodynamic limit with the constraint Nβ2 0 = const > 0 provided that 1 − z log � z/(z − 1) � < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This inequality is always verified for z > 1, which confirms the existence of NT equilibrium states in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' NT condensation in the highest energy level Positive temperature T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We start by briefly sum- marizing the usual positive temperature condensation in the lowest energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This effect of condensation orig- inates in the singularity of the RJ distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Indeed, the denominator of nRJ p = T/(βp − µ) vanishes for the lowest energy level when µ = β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This gives rise to an analogue effect of condensation: As the energy decreases below a critical value Ecrit (or T < Tcrit), then µ → β− 0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 13(a)) and the singular behavior of the RJ dis- tribution is regularized by the macroscopic population of the fundamental mode: nRJ 0 /N → 1 as E → Emin (or T → 0+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (12) It has been shown that this condensation-like effect is a phase transition that occurs in the thermodynamic limit, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Negative temperature, T < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' In the NT region, we reveal an inverted condensation-like effect, which is char- acterized by a macroscopic population of the highest energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Aside from the singularity for µ = β0 discussed here above for T > 0, the RJ distribution nRJ p = T/(βp − µ) also exhibits a singularity (vanishing denominator) for the highest energy level when µ = βmax (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 13(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We have shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 5, that the high- est energy level g = 9 becomes macroscopically populated as the energy increases: ˜ng/N → 1 as E → Emax (or T → 0−), (13) with ˜ng = � p,px+py+1=9 np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' This condensation-like ef- fect does not occur in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We pose µ = βmax + ε, with ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (11) can be written in the limit ε → 0+: Nβ2 0 ≃ Tβmax � 1 − log(βmax/ε) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' By keeping Nβ2 0 =const, the chemical potential µ reaches β+ max for a vanishing temperature T → 0−, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', conden- sation does not occur in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' How- ever, an analogue effect of condensation occurs through a macroscopic population of the highest energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' By setting βp = βmax in the RJ distribution, then nRJ g = −T/(µ − βmax) denotes the power in one mode of the highest energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The total power in the high- est energy level (g−fold degenerate) ˜nRJ g = gnRJ g then FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 14: Simulations of NLS equation and wave tur- bulence kinetic equation: Simulations of the NLS equation (dashed lines), and wave turbulence kinetic equation (solid lines) showing the evolutions during the propagtion (in z) of the power within each of the g =9 groups of degenerate modes of the fiber that has been used in the experiments, for E/Emin = 8 (a), E/Emin = 7 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The horizontal dashed black line denotes the population for the higher mode group at complete RJ thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' See the text for details on initial conditions and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' reads ˜nRJ g N (µ) = g (µ − βmax) � p(µ − βp)−1 , (14) E(µ) Emin = � p βp βp−µ � p β0 βp−µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (15) The parametric plot of (14) and (15) with respect to µ provides the condensate fraction reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 13(b) shows the condensate fraction, ˜nRJ g /N vs E, by increasing the number of modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=', by approaching the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The critical behavior of the condensation curve looks similar to that of a phase tran- sition, thought strictly speaking phase transitions only occur in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Nevertheless, if one takes the macroscopic occupation of an energy level as the essential characteristic of condensation, then NTs are characterized by a condensation-like effect into the highest-energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' WAVE TURBULENCE SIMULATIONS The starting point is the NLS equation governing light propagation in MMFs [39, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' By expanding the random wave into the fiber modes and considering the dominant contribution of weak disorder, the polarized modal com- ponents ap = (ap,x, ap,y)T are governed by [39]: i∂zap = βpap + Dp(z)ap − γPp(a), (16) where Pp(a) = � l,m,n Splmn � 1 3aT l ama∗ n + 2 3a† namal � , Splmn denoting the spatial overlap among the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The introduction of disorder is important in order to avoid strong phase correlations among the modes that are related to Fermi-Pasta-Ulam recurrences [43], which inhibit the thermalization process [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We consider the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='8 modal population population (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='2 modal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='1 10 20 30 10 20 30 0 0 z[m] z[m]11 most general form of disorder that conserves the power: The Hermitian matrices Dp(z) are expanded into the Pauli matrices that form a basis for the vector space of 2×2 Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The matrices then have the form Dp(z) = �3 j=0 νp,j(z)σj, where σj (j = 1, 2, 3) are the Pauli matrices (σ0 the identity matrix), while νp,j(z) are independent and identically distributed real- valued random processes, with variance σ2 and correla- tion length ℓc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The corresponding characteristic length scale of disorder is Ld = 1/∆β, with ∆β = σ2ℓc, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' [39] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The nonequilibrium process of NT thermalization can be described by a wave-turbulence kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The kinetic equation was derived from the modal NLS Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (16) in the weakly nonlinear regime of the experiment, Llin ∼ 1/β0 ≪ Lnl ∼ 1/(γN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' It describes the nonequilibrium evolution of the averaged modal components np(z) = � |ap|2(z) � [39]: ∂znp(z) = γ2 6∆β � l,m,n |Slmnp|2δK(∆ωlmnp)Mlmnp(n) + 4γ2 9∆β � l |slp(n)|2δK(∆ωlp)(nl − np), (17) with slp(n) = � m′ Slm′m′pnm′, and Mlmnp(n) = nlnmnp + nlnmnn − nnnpnm − nnnpnl and ∆ωlp = βl − βp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The term δK(∆ωlmnp) denotes the four-wave frequency resonance ∆ωlmnp = βl + βm − βn − βp, with δK(∆ωlmnp) = 1 if ∆ωlmnp = 0, and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The kinetic Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' (17) conserves N = � p np(z), the energy E = � p βpnp(z) and exhibits a H−theorem of entropy growth, ∂zSkin(z) ≥ 0, for the nonequilibrium entropy Skin(z) = � p log � np(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' Accordingly, it describes an irreversible evolution of the speckle beam to the RJ equi- librium distribution realizing the maximum of entropy, nRJ p = T/(βp − µ) [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' We have considered in the simulations the MMF used in the experiments, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The initial condition con- sists of a speckle beam whose correlation length is varied in such a way to fix a desired value of the energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The considered parameters are ℓc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='3m and 2π/σ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content='1m [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The results of the simulations of the NLS equation and the corresponding simulations of the wave turbulence kinetic equation are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' They show the process of thermalization to the negative temperature RJ equilibrium state predicted by the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The simu- lations qualitatively reproduce the experimental results, although a power of 20kW has been considered to accel- erate the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' The purely spatial model considered here then captures the essential features of the conden- sation process reported experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' At variance with NLS simulations that are stochas- tic and thus exhibit fluctuations, the simulations of the wave turbulence kinetic equation are deterministic (free of fluctuations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf'} +page_content=' A good agreement between 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Szabó∗ +6th January 2023 +Abstract +We present a structural description of finite nilpotent groups of class at most +2 using a specified number of subdirect and central products of 2-generated such +groups. As a corollary, we show that all of these groups are isomorphic to a subgroup +of a Heisenberg group satisfying certain properties. +The motivation for these results is of topological nature as they can be used to +give lower bounds to the nilpotently Jordan property of the birational automorphism +group of varieties and the homeomorphism group of compact manifolds. +1 +Introduction +A finite non-abelian p-group G is special if its Frattini subgroup Φ(G), derived subgroup +G′ and centre Z(G) all coincide and is isomorphic to (Z/pZ)r for some r. A special group +G is extra-special if r = 1. The structure of these groups is described by the following +classical result. +Theorem 1.1. Every special p-group is a subdirect product of groups of the form: central +product of an extra-special p-group and an abelian group [Suz82, (4.16)/(ii)]. Every extra- +special p-group H of order p2n+1 is the central product of n extra-special p-subgroups of +order p3 [Suz82, Theorem 4.18]. For every prime p, there are exactly two extra-special +groups of order p3 (up to isomorphism). +We present a generalisation to finite nilpotent groups of class at most 2. +Theorem A. Every finite nilpotent group G is a subdirect product of d(Z(G)) groups +each with cyclic centre, see Proposition 2.7. +Every finite ≤2-step nilpotent group G with cyclic commutator subgroup is the +internal central product of t many suitable nilpotent 2-generated subgroups of class 2 +and an abelian subgroup A satisfying d(G) = 2t + d(A) and some further properties +discussed at Theorem 3.7. +∗The project leading to this application has received funding from the European Research Council +(ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement +No 741420). The author was supported by the National Research, Development and Innovation Office +(NKFIH) Grant K138596. +1 +arXiv:2301.01863v1 [math.GR] 5 Jan 2023 + +The p-groups of class 2 with 2 generators are classified in [AMM12, Theorem 1.1]. +The central product decomposition of Theorem A is a generalisation of [BBC69, The- +orem 2.1] where groups with cyclic centre were considered. The argument presented in +the current paper is more structural and gives some invariants needed for a topological +application below. +As an application of (the proof of) Theorem A, we embed all of the groups above to +matrix groups with a very specific form. Note that the number of isomorphism classes of +groups of order pn is p +2 +27 n3+O(n8/3) as n → ∞ [Sim65, Theorem, p. 153] of which at least +p +2 +27 n3− 12 +27 n2 is ≤2-step nilpotent [Hig60, Theorem 2.3]. The main statement of the paper is +the following. +Theorem B. Every finite ≤2-step nilpotent group G is isomorphic to a subgroup of +a non-degenerate Heisenberg group of the form +� 1 A C +0 1 B +0 0 1 +� +for suitable abelian groups +A, B, C whose number of generators and exponents are bounded by concrete invari- +ants of G. See Theorem 5.5 for the precise statement and details. +See [Mag98, Corollary 2.21] for a weaker statement in a much more general setup. We +remark here that both Theorem A and Theorem B are true in the finitely generated +setup, see the thesis of the author [Sza21, Theorems A, B]. The ideas presented in the +current paper are heavily polished and simplified compared to the thesis. Bounding the +invariants in both statements is essential for the following topological application (that +shall not be proved in this paper). +Theorem C ([Sza21, Theorem C]). For every natural number d, there exists an +algebraic variety Xd, respectively a compact manifold Md, such that every finite ≤2- +step nilpotent ≤d-generated group acts faithfully on Xd via birational automorphisms, +respectively on Md via diffeomorphisms. +We briefly discuss the relevance of this statement. Let N be a class of finite groups. +A group G is called N -Jordan, if there exists an integer JG such that for every finite +subgroup F of G sits in a short exact sequence 1 → N → F → B → 1 with N ∈ N and +|B| ≤ JG; informally if every finite subgroup of G ‘almost’ belong to N . +Theorem 1.2 (Guld, [Gul20, Theorem 2]). The birational automorphism group of any +variety over a field of characteristic zero is N -Jordan where N = {≤2-step nilpotent}. +Theorem 1.3 (Csikós, Pyber and Szabó, [CPS22, Theorem 1.3]). The homeomorphism +group of a compact topological manifold is N -Jordan where N = {nilpotent}. +Theorem C shows that N essentially has to contain the ≤2-step nilpotent groups for +both Theorem 1.2 and Theorem 1.3, thereby the sharpness of Theorem 1.2. +Structure of the paper +In Section 2, we prove the first part of Theorem A using +induction and find the smallest number of factors needed. In Section 3, we turn our +attention to ≤2-step nilpotent groups G with cyclic centre. The commutator map on +these groups induce an alternating bilinear map on the Z-module on G/G′. We prove the +second part of Theorem A by finding a suitable generating set imitating the Darboux basis +of symplectic vector spaces. In Section 4, we define a complex structure and isotropic +2 + +real structure on G/ Z(G). This enables to assign a Heisenberg group to G using the +same method independently on the prime divisors of |G|. Then we discuss the general +idea to modify the previous construction by extending the centre so that G embeds to +the resulting Heisenberg group. Finally, in Section 5 we prove Theorem B in two steps. +First, we consider the special case when the group has cyclic centre and apply the method +of the previous two sections. Second, we use the reduction of Section 2 to handle the +general case. +G : G′ ⊆ Z(G) +� 1 +A +C +0 +1 +B +0 +0 +1 +� +G1 : G′ +1 ⊆ Z(G1) +d(Z(G1)) = 1 +Hermitian form with +isotropic real structure +� 1 +A1 +C1 +0 +1 +A1 +0 +0 +1 +� +reduction +Proposition 2.7 +embedding +Theorem 5.5 +Lemma 3.10 +[−,−] +embedding +Proposition 5.3 +Lemma 4.8 +Proposition 2.7 � +Notation +|X| denotes the cardinality of a set X. N+ ⊂ Z is the set of positive integers, +N0 = {0}∪N+. n +�� k denotes divisibility. (Note that this symbol is slightly taller than the +one denoting the cardinality.) lcm(n1, . . . , nk) is the least common multiple of integers +n1, . . . , nk. We apply functions from the left. For f : X → Y and X0 ⊆ X, we write +f|X0 : X0 → Y for restriction and f(X0) := {f(x0) : x0 ∈ X0}. For maps fi : Xi → Yi, +denote by f1×f2 : X1×X2 → Y1×Y2, (x1, x2) �→ (f1(x1), f2(x2)) their direct product. The +arrow +indicates a monomorphism (or an injective map), +means an epimorphism +(or a surjective map), and these arrow notations can be combined. We write +for the +identity map. In bigger diagrams, we use +, +or +to indicate the ‘chronological +order’: the further it is from a solid one, the later it appears in the construction. +Let G denote a group. We denote the identity element of G by 1, or sometimes by +0 when G is an additive abelian group. By abuse of notation, we also write 1 or 0 for +the trivial group. For a subset S ⊆ G, ⟨S⟩ denotes the subgroup generated by S, and +write ⟨g1, . . . , gn⟩ := ⟨{g1, . . . , gn}⟩. Hom(X, Y ) is the set of morphisms X → Y . N ◁ G +means that N is a normal subgroup of G. Write [−, −]: G × G → G′, (g, h) �→ [g, h] for +the commutator map where we use the convention [g, h] := g−1h−1gh. The commutator +subgroup (or derived subgroup) is denoted by G′ := [G, G]. Z(G) is the centre of G. Syl(G) +is the set of Sylow subgroups of G, Sylp(G) consists of Sylow p-subgroups. We denote by +exp(G) := inf{n ∈ N+ : ∀g ∈ G gn = 1} the exponent of a group G. Write d(G) for the +cardinality of the smallest generating set. We say G is ≤d-generated, if d ≤ d(G); and G +is d-generated if d = d(G). +2 +Subdirect product decomposition +The goal of this section is to prove the first part of Theorem A from page 1 by +passing to abelian groups. To show the existence of such a subdirect product, we +recursively factor by the invariant factors of the centre. To attain minimal number +of factors, we consider intersections with the centre. +Definition 2.1 (C). Let C denote the class of groups with cyclic centre. +A C- +decomposition in a group G is a finite set D of normal subgroups of G such that +G/N ∈ C for every N ∈ D and � D = 1. (Use the convention � D = G if D = ∅.) Let +3 + +mC(G) denote the minimal |D| amongst all C-decomposition D in G, or ∞ is no such +decomposition exists. +Remark 2.2. This is a reformulation of subdirect products, as the associated (central) +embedding µD : G ↣ G/D := � +N∈D G/N, gK �→ (gN)N∈D makes G a subdirect product +of groups from C. +Lemma 2.3. There is a C-decomposition in every finite group G. +Furthermore, +d(Z(G)) ≤ mC(G). +Proof. Let l(G) be the maximal length of a strictly increasing subgroup series consisting +of normal subgroups of G. Note that l(G/N) < l(G) for any non-trivial normal subgroup +N of G as a series K0/N < K1/N < · · · < Kn/N of normal subgroups of G/N induces +1 < N < K1 < · · · < Kn in G. Write Z(G) = �d +i=1 Ci where Ci are non-trivial cyclic +groups. If d ≤ 1 (for example when l(G) = 0), then D = {1} is a C-decomposition in G. +Otherwise, by induction of l(G), there are C-decompositions Di in G/Ci. Lift Di to a set +of normal subgroups ¯Di of G containing N. i.e. Di = {K/Ci : K ∈ ¯Di}. We claim that +D := �d +i=1 ¯Di is a C-decomposition in G. Indeed, it is a finite set of normal subgroups of +G. For every K ∈ D, we have K/Ci ∈ Di for some i, hence G/K ∼= (G/Ci)/(K/Ci) ∈ C +as Di is a C-decomposition in G/Ci. Finally, note that � D = �d +i=1 +� ¯Di = �d +i=1 Ci = 1. +For the second part, suppose D is a C-decomposition in G. +Then µD(Z(G)) ⊆ +Z(G/D) = � +N∈D Z(G/N), so d(Z(G)) ≤ � +N∈D d(Z(G/N)) ≤ |D| since G/N has cyclic +centre by assumption and using that d is a monotone function on abelian groups. +Lemma 2.4. Let A be an additive finite abelian p-group, X be a trivially intersecting set +of subgroups of A. Then there exists Y ⊆ X with |Y | ≤ d(A) and � Y = 0. +Proof. We prove this by induction on d(A). If d(A) = 0, then A is trivial, and Y = ∅ +works by convention. Else assume that d(A) > 0. For any subgroup K ≤ A, define +V (K) = {g ∈ K : gp = 1}. Note that this is an Fp-vector space of dimension d(K). +Assume by contradiction that V (K) = V (A) for all K ∈ X. Then V (A) ⊆ K, hence +V (A) ⊆ � X = 0, but this contradicts that V (A) has positive dimension. So we may +pick B ∈ X so that d(B) < d(A). Now XB := {B ∩K : K ∈ X} is a trivially intersecting +set of subgroups of B, so by induction, there is YB ⊆ XB of size at most d(B) with trivial +intersection. Lift back YB to Z ⊆ X. Then |Z| = |YB| and YB = {B ∩ K : K ∈ Z}. We +show that Y := {B}∪Z ⊆ X satisfies the claim. Indeed, |Y | ≤ 1+|YB| ≤ 1+d(B) ≤ d(A) +by construction, and � Y = B ∩ � Z = � +K∈Z(B ∩ K) = � YB = 0. +The next statement is motivated by an idea of Endre Szabó. +Lemma 2.5. mC(P) = d(Z(P)) for any finite p-group P. +Proof. There is a C-decomposition D in P by Lemma 2.3. We claim the existence of a +C-decomposition S ⊆ D of size at most d(Z(P)). This then proves the statement as no +smaller C-decomposition may exist by Lemma 2.3. +Let A := Z(P) and consider X := {N ∩ A : N ∈ D}, a trivially intersecting set of +subgroups of the abelian group A. Let Y ⊆ X with |Y | ≤ d(A) and � Y = 1 be given +by Lemma 2.4. Lift Y back to S ⊆ D. Then 1 = � Y = Z(P) ∩ � S, so we must have +� S = 1 since in a nilpotent group, there is no non-trivial normal subgroup intersecting +the centre trivially. Also by construction, |S| = |Y | ≤ d(A) = d(Z(P)), so S is indeed a +C-decomposition of P with the stated properties. +4 + +Lemma 2.6. Let G be a finite nilpotent group. +Then mC(G) = max{mC(P) : P ∈ +Syl(G)} where Syl(G) is the set of Sylow subgroups of G. +Proof. Let D be a C-decomposition in G and P ∈ Syl(G). We claim that DP := {N ∩P : +N ∈ D} is a C-decomposition in P. Indeed, P is a normal subgroup of G because G is +finite nilpotent, so N ∩ P is a normal subgroup of G, hence of P. On the other hand +� NP = P ∩ � D = 1. This shows that mC(G) ≥ mC(P) for all P ∈ Syl(G). +For the other direction, let Dp be a C-decomposition for Gp ∈ Sylp(G) for all prime +divisors p of |G|. Let D be a partition of � +p Dp of size max{|Dp| : p} such that |S∩Dp| ≤ 1 +for every S ∈ D and p. We claim that D := {� +N∈S N : S ∈ D} is a C-decomposition +in G. Indeed, as above, every Np ∈ Dp is normal in G, so every N ∈ D is also a normal +subgroup of G being the product of such groups in a finite nilpotent group. On the other +hand, Gp ∩ � D = �{Gp ∩ � +N∈S N : S ∈ D} = �{K : K ∈ Dp} = � Dp = 1 using the +assumption on the partition. This shows that mC(G) ≤ max{mC(P) : P ∈ Syl(G)}. +Proposition 2.7. mC(G) = d(Z(G)) for any finite nilpotent group G. In particular, +G is a subdirect product of d(Z(G)) groups each having cyclic centre. +Proof. Using Lemma 2.6, Lemma 2.5 and the Chinese remainder theorem, we get +mC(G) = max{mC(P) : P ∈ Syl(G)} = max{d(Z(P)) : P ∈ Syl(G)} += max{d(Q) : Q ∈ Syl(Z(G))} = d(Z(G)) +after noting that G = � +P∈Syl(G) P implies Syl(Z(G)) = {Z(P) : P ∈ Syl(G)}. +Remark 2.8. The statement holds when G is finitely generated using a similar reasoning. +See [Sza21, §3.1.2]. +3 +Alternating modules +In this section, we introduce alternating modules to generalise the notion of sym- +plectic vector spaces prominently to the abelianisation of ≤2-step nilpotent groups. +We show that, under some conditions, they possess an analogue of the Darboux +basis. Using this, on one hand we prove the second part of Theorem A from page 1, +on the other hand we endow the module with a non-canonical complex structure +having an isotropic real structure. +We start with an elementary statement. +Lemma 3.1 (‘Alternating Smith’ normal form). Let R be a principal ideal domain, +W ∈ Rn×n be an alternating matrix (i.e. W ⊤ = −W and has 0’s at the main diagonal). +Then for s = 1 +2 rk(W), there exist elements d1 | d2 | · · · | ds ̸= 0 in R (unique up to unit +multiples) and B ∈ SLn(R) such that +B⊤WB = diag +�� 0 +d1 +−d1 +0 +� +, +� 0 +d2 +−d2 +0 +� +, . . . , +� 0 +ds +−ds +0 +� +, 0, . . . , 0 +� +. +(1) +5 + +Proof. The idea is similar to the standard proof of Smith normal form [Jac85, §3.7], but +instead of focusing on the main diagonal, we consider the superdiagonal entries. At each +step, we choose different pivots and apply the base change W �→ X⊤WX (to respect the +alternating property) for a series of well-chosen matrices X ∈ SLn(R) until the stated +form for W = (wi,j) is obtained. Once the existence is verified, the uniqueness statement +follows from the fact that for each 1 ≤ k ≤ n, the ideal generated by the k × k minors is +unchanged under these transformations. See [Sza21, §2.3] for details. +A key notion is the following analogue of symplectic vector spaces. +Definition 3.2. We call (M, ω, C) an alternating R-module, if M and C are R-modules +and ω: M × M → C is a R-bilinear map that is alternating, i.e. ω(m, m) = 0 for every +m ∈ M. N ≤ M is N isotropic if ω(N N) = 0. The orthogonal complement of N is +N ⊥ := {m ∈ M : ω(m, N) = 0}. Call ω and (M, ω, C) non-degenerate if M ⊥ = 0. +For simplicity, in this paper we will further require the extra conditions of M being +finitely generated, C cyclic and R a principal ideal domain. +Lemma 3.3 (Darboux-generators). Let (M, ω, C) be an alternating R-module. +Then +there exists a minimal R-module generating set B of M, and a subset {x1, y1, . . . , xt, yt} ⊆ +B such that ω(M, M) = Rω(x1, y1) ≥ Rω(x2, y2) ≥ · · · ≥ Rω(xt, yt) ̸= 0 and ω(b1, b2) = 0 +for all other pairs (b1, b2) ∈ B2. +In any such generating set, the chain of submodules of C above is invariant. More +concretely, t = 1 +2d(M/M ⊥) and M/M ⊥ ∼= �t +i=1(Rω(xi, yi))⊕2. +Proof. Let (M, ω, C) be a Darboux module. Pick a minimal R-module generating set +{b1, . . . , bn} of M, and let c be a fixed generator of ω(M, M). Pick wi,j ∈ R such that +ω(bi, bj) = wi,jc. +(Note that these are not necessarily unique, but wi,j + annR(c) ∈ +R/ annR(c) are.) Without loss of generality, we may assume that W := (wi,j)i,j ∈ Rn×n +is an alternating matrix. Let B ∈ SLn(R) given by Lemma 3.1. Since B is an invertible +square matrix, B := {�n +j=1 Bj,ibj : 1 ≤ i ≤ n} = {x1, y1, . . . , xs, ys, . . . } is also a (min- +imal) generating set of M in which ω can be expressed at the matrix from (1). Then +the statement follows by setting t ∈ N0 so that dic ̸= 0 for 1 ≤ i ≤ t, and djc = 0 for +t < j ≤ s. +For the second part, let xi, yi be as above. We claim that +0 +M ⊥ +M +�t +i=1(Rω(xi, yi))⊕2 +0 +m +(ω(m, yi), ω(xi, m))t +i=1 +⊆ +ω♭ +(2) +is a short exact sequence of R-modules. Indeed, ω♭ is well-defined using the bilinearity of +ω and the orthogonality elements of B. ker(ω♭) = {m ∈ M : ∀i ω(xi, m) = ω(m, yi)} = +{m ∈ M : ∀b ∈ B ω(b, m) = 0} = M ⊥. On the other hand, using orthogonality once +more, ω♭(�t +i=1 rixi + siyi) = (ri, yi)i for arbitrary ri, si ∈ R, thus ω♭ is surjective. Hence +M/M ⊥ ∼= �t +i=1(Rω(xi, yi))⊕2, so the isomorphism class of the R-modules Rω(xi, yi) +are invariant by the structure theorem of finitely generated modules over a principal +ideal domain and t = +1 +2d(M/M ⊥). If annR(c) ̸= 0, then ω(M, M) is a cyclic torsion +R-module, so its isomorphic submodules are necessarily equal. +This means that the +submodules Rω(xi, yi) ≤ ω(M, M) are themselves invariant. Otherwise, ω(M, M) is a +free R-module of rank 1 generated by, say, c. +Thus ω(xi, yi) = dic for some unique +di(c) ∈ R. By Lemma 3.1, Rdi is independent of the choice the generators of M. Hence +Rdic = Rω(xi, yi) may depend only on c, but the right-hand side does not. +6 + +Our main resource of alternating Z-modules is the followings. +Definition 3.4. A short exact sequence ϵ : 1 → C +ι−→ G +π−→ M → 1 of groups is called +a central-by-abelian extension, if ι(C) ⊆ Z(G) and M is abelian. This extension ϵ is +non-degenerate, if ι(C) = Z(G). +Lemma 3.5 (The alternating functor A). Every central-by-abelian extension of finitely +generated groups ϵ : 1 → C +ι−→ G +π−→ M → 1 induces an alternating Z-bilinear map +ω: M × M → C defined by (m1, m2) �→ ι−1([g1, g2]) for arbitrary gi ∈ π−1(mi). +In +particular, when the extra conditions on M and C from Definition 3.2 are satisfied, then +A(ϵ) := (M, ω, C) is an alternating Z-module. +Proof. We consider M and C as Z-modules. +First note that G′ ⊆ ι(C) ⊆ Z(G), so +G is necessarily ≤2-step nilpotent. Then the general commutator identities [g1g2, h] = +[g1, h][g1, h, g2][g2, h] and [g, h1h2] = [g, h2][g, h1][g, h1, h2] imply that [−, −] : G×G → G′ +is a group morphism in both coordinates. +Next, we check that ω is well-defined. Pick gi, g′ +i ∈ π−1(mi). Then g−1 +i g′ +i ∈ ker(π) = +Im(ι), so there are ci ∈ C with ι(ci) = g−1 +i g′ +i. +Then [g′ +1, g′ +2] = [g1ι(c1), g2ι(c2)] = +[g1, g2][g1, ι(c2)][ι(c1), g2][ι(c1), ι(c2)] = [g1, g2] by above as ι(C) ⊆ Z(G). Finally G′ ⊆ +ι(C) implies that we can apply ι−1 to this element. +Z-bilinearity of ω follows directly from the previously mentioned fact. The alternating +property follows as every group element commutes with itself. +Remark 3.6. Lemma 3.5 gives the following dictionary between subgroups H, Hi of G +and submodules of M. The commutator map corresponds to ω ([g, g′] = ι◦ω(π(g), π(g′)) +and [H1, H2] = ι ◦ ω(π(H1), π(H2))), commutes to being orthogonal ([H1, H2] = 1 ⇐⇒ +π(H1) ⊥ π(H2)), the centraliser to the orthogonal complement (π(CG(H)) = π(H)⊥, in +particular π(Z(G)) = M ⊥), abelian to isotropic ([H, H] = 1 ⇐⇒ π(H) ⊥ π(H)), and +the notion of non-degeneracy coincide. +The dictionary can be extended to Darboux-generators as the following generalisation +of Theorem 1.1 and [BBC69, Theorem 2.1] shows. +Theorem 3.7 (Central product decomposition). Let G be a finitely generated ≤2- +step nilpotent group with cyclic commutator subgroup G′. Then it contains pairwise +commuting subgroups A and E1, . . . , Et such that G = AE1 . . . Et (a central product) +where A ≤ Z(G), Ei are 2-generated and of class exactly 2, d(G) = d(A) + 2t and +G′ = E′ +1 ⊋ E′ +2 ⊋ · · · ⊋ E′ +t ̸= 1. +In any such case, t = +1 +2d(G/ Z(G)) and E′ +i ⊆ G′ are invariants given by +G/ Z(G) ∼= �t +i=1 E′2 +i . +Proof. Let (M, ω, C) = A(1 → G′ +⊆−→ G +π−→ G/G′ → 1). This is an alternating Z-module +by Lemma 3.5. Consider the minimal generating set B = {x1, y1, . . . , xt, yt, o1, . . . , ok} +of M = G/G′ as in Lemma 3.3. For every b ∈ B, fix an arbitrary lift ¯b ∈ π−1(b) ⊆ G, +and set ¯B := {¯g : g ∈ B}. We show that the subgroups Ei := ⟨¯xi, ¯yi⟩ ≤ G and A := +⟨¯o1, . . . , ¯ok⟩ ≤ G satisfy the statement Indeed, A ⊆ π−1(M ⊥) = Z(G) using Remark 3.6. +Moreover, [Ei, Ej] = 1 if and only if (Zxi + Zyi) ⊥ (Zxj + Zyj) if and only if i ̸= j by +Remark 3.6 and Lemma 3.3. By Remark 3.6, G′ = [G, G] = ω(M, M) and E′ +i = [Ei, Ei] = +ω(Zxi + Zyi, Zxi + Zyi) = Zω(xi, yi). Then all parts about the derived subgroups follow +7 + +from Lemma 3.3. In particular, G′ = E′ +1 = Zω(xi, yi) = ⟨[¯x1, ¯y1]⟩, so considering central- +by-abelian extension, we see that G = ⟨ ¯B ∪ G′⟩ = ⟨ ¯B ∪ {[¯x1, ¯y1]}⟩ = ⟨ ¯B⟩ = AE1 . . . Et So +d(G) ≤ d(A) + �t +i=1 d(Ei) ≤ | ¯B| = |B| = d(M) ≤ d(G). This forces equality everywhere, +so d(Ei) = 2 and 2t + d(A) = d(G). Finally Remark 3.6 shows that M ⊥ = π(Z(G)) = +Z(G)/G′, so M/M ⊥ = (G/G′)/(Z(G)/G′) ∼= G/ Z(G). +Remark 3.8. The isomorphism class of the subgroups E1, . . . , Et are not unique, which is +demonstrated by the classical decomposition of extra-special p-groups Theorem 1.1. For +example, the extra-special p-group G of order p2t+1 of exponent p2 has many different +internal central product decompositions. For any 1 ≤ s ≤ t, G = E1E2 . . . Et such that +Ei ∼= M for 1 ≤ i ≤ s and Ei ∼= E for s < i ≤ t, where E and M are the non-abelian +groups of order p3 and of exponent p and p2, respectively [Suz82, Theorem 4.18]. +If the alternating module is non-degenerate, we can endow it with additional struc- +tures. +Definition 3.9. For a commutative ring Q, define a ring Q[i] := Q[x]/(x2 + 1) with +i := x + (x2 + 1) ∈ Q[i] and maps σ: Q[i] → Q, q + iq′ �→ q − iq′ (conjugation) and +ℑ: Q[i] → Q, q + iq �→ q′ (the imaginary part). For a Q[i]-module M, we call a map +h: M ×M → Q[i] a Hermitian form on M over Q[i] if h is Q[i]-linear in the first argument +and h is σ-conjugate symmetric (i.e. h(m, m′) = σ(h(m′, m))). +Lemma 3.10. Let (M, ω, C) be a non-degenerate alternating R-module. +Set Q := +R/ annR(C), and pick a generator of C, i.e. a Q-module isomorphism ϕ: Q → C. Then +there is a Hermitian form h on M over Q[i] making the following diagram commute. +M × M +C +Q[i] +Q +ω +h +∃ +ℑ +ϕ +∼ +Furthermore, M has an a non-canonical isotropic real structure, i.e. an isotropic Q- +submodule MQ of M such that M = MQ ⊕ iMQ (as Q-modules). +Remark 3.11. The Q[i]-module structure of M is non-canonical, but is compatible +with R → Q → Q[i]. +The Q-module iMQ is automatically isotropic as ω(ia, ia′) = +ϕ(ℑ(h(ia, ia′))) = ϕ(ℑ(h(a, a′))) = ω(a, a′) = 0. +Remark 3.12. While the structures themselves from the statement depend on the choice +of the generators of M, their isomorphism class do not. More concretely, let ¯ +M be an +arbitrary Q[i]-module structure on M together with a Hermitian form ¯h and an isotropic +real structure ¯ +M = ¯ +MQ⊕i ¯ +MQ as at the statement. Then Lemma 3.1 implies the existence +of a Q[i]-module isomorphism f : M → ¯ +M such that f(MQ) = ¯ +MQ and ¯h ◦ (f × f) = h +(hence ω ◦ (f × f) = ω). +Proof. First, we claim that annR(C) ⊆ annR(M). Indeed, for arbitrary r ∈ annR(C) +and m ∈ M, ω(rm, m′) = rω(m, m′) = 0 for every m′ ∈ M, hence rm ∈ M ⊥ = 0. +This then means that M can naturally be considered as a Q-module. +To define the +Q[i]-module structure, let B = {x1, y1, . . . , xt, yt, o1, . . . , ok} be a Darboux-generating set +as in Lemma 3.3. The isomorphism ω♭ : M ∼= �t +j=1(Rω(xj, yj))⊕2 from (2) shows that +k = 0 and M = �t +j=1(Rxi ⊕ Ryi) = �t +j=1(Qxi ⊕ Qyi). In particular, MQ := �t +j=1 Qxi +and MiQ := �t +j=1 Qyi are (non-canonical) isotropic submodules of M giving a Q-module +8 + +decomposition M = A ⊕ B. Define a Q-module automorphism ιj of (Qω(xj, yj))⊕2 by +ιj : (n, n′) �→ (−n′, n). Pulling back �t +j=1 ιj along ω♭ gives a non-canonical automorphism +ι of M such that ι(xj) = yj and ι(yj) = −xj. Thus ι ◦ ι = −idM, ι(MQ) = MiQ and +ι(MiQ) = MQ. Thus defining (q + iq′) · m := qm + ι(q′m) gives the Q[i]-module structure +in which MiQ = iMQ. +Set ωQ := ϕ−1 ◦ ω. We claim that +h: M × M → C[i], +(m, m′) �→ ωQ(im, m′) + iωQ(m, m′) +(3) +is the Hermitian form with the stated properties. Indeed, Q-linearity in the first argument +is inherited from ω, and +h(im, m′) = ωQ(−m, m′) + iωQ(im, m′) = i(ωQ(im, m′) + iωQ(m, m′)) = ih(m, m′) +then implies Q[i]-linearity. +For the conjugate symmetry, first note that ω(ixj, iyj) = +ω(yj, −xj) = ω(xj, yj) using the alternating property, and for all other pairs (b1, b2) ∈ B2, +we also have ω(ib1, ib2) = 0 = ω(b1, b2). +Hence the Q-bilinearity of ωQ implies and +ω(im, im′) = ω(m, m′) for every m, m′ ∈ M. This together with the alternating property +of ωQ gives ωQ(im, m′) = −ωQ(m′, im) = ωQ(i2m′, im) = ωQ(im′, m), thus +h(m, m′) = ωQ(im, m′) + iωQ(m, m′) = ωQ(im′, m) − iωQ(m′, m) = σ(h(m′, m)). +Remark 3.13. Given Hermitian form and the alternating map determine each other +uniquely via (3) and ℑ ◦ h = ωQ. Furthermore, given the isotropic real structure, these +maps are determined by the restriction µ: MQ × iMQ → C, (a, b) �→ ω(a, b). Indeed, +ω(a + ib, a′ + ib′) = µ(a, ib′) − µ(a′, ib) using the bilinearity of ω and that MQ is isotropic. +4 +Heisenberg groups +In this section, associate a (polarised) Heisenberg group for every Z-bilinear map, in +particular to alternating modules, or to finite ≤2-step nilpotent groups G with cyclic +centre. We show that upon extending the centre of the Heisenberg group suitably +using an extended polarisation, it will contain a normal subgroup isomorphic to G. +Definition 4.1 (Heisenberg group). Let A, B and C be Z-modules and µ: A × B → C +a Z-bilinear map. We call µ non-degenerate, if µ(a, B) = 0 implies a = 0 and µ(A, b) = 0 +implies b = 0. Define the associated Heisenberg group as H(µ) := A ⋉ϕ (B × C) where +ϕ: A → Aut(B×C), a �→ ((b, c) �→ (b, µ(a, b)+c). Call H(µ) non-degenerate if Z(H(µ)) = +{(0, 0, c) : c ∈ C}. Define a central-by-abelian extension +H(µ) : 1 → C +ιµ +−→ H(µ) +πµ +−→ A × B → 1 +by ι := ιµ : c �→ (0, 0, c) and π := πµ : (a, b, c) �→ (a, b). +Remark 4.2. More explicitly, the group structure on H(µ) is given by +(a, b, c) ∗ (a′, b′, c′) = (a + a′, b + b′, c + µ(a, b′) + c′) +with (0, 0, 0) being the identity and (a, b, c)−1 = (−a, −b, µ(a, b) − c) the inverse. So +formally H(µ) ∼= +� 1 A C +0 1 B +0 0 1 +� +with matrix multiplication induced by µ. +In particular, +[(a, b, c), (a′, b′, c′)] = (0, 0, µ(a, b′) − µ(a′, b)), i.e. the commutator coincides with ιµ ◦ ω ◦ +(πµ × πµ) using the notation of Remark 3.13. Note that Z(H(µ)) = {(a, b, c) : µ(a, B) = +µ(A, b) = 0} ⊇ ιµ(C). The notion of non-degeneracy for µ, ω, H(µ) and H(µ) all coincide. +9 + +Example 4.3 (Heisenberg group of alternating modules). Let (M, ω, C) be a non- +degenerate alternating R-module. +Apply Lemma 3.10, set A := MQ and B := iMQ +considered as Z-modules. Then the restriction µ: A × B → C as at Remark 3.13 is +non-degenerate and produces the non-degenerate central-by-abelian extension H(µ). +Note that while this construction depends on isotropic real structure, the isomorphism +class H(µ) does not because f from Remark 3.12 induce an isomorphism of short exact +sequences, cf. +[Sza21, §4.1] In particular, the isomorphism class of H(µ) is invariant +which we call the Heisenberg group of the alternating module. +Note that (M, ω, C), µ, H(µ) and H(µ) basically decode the same information as they +mutually determine each other. +Remark 4.4. Alternatively, if 2 ∈ R has a multiplicative inverse (for example C is finite +of odd order), then there is a canonical way to define this Heisenberg group. As for +symplectic vector spaces, we can define a group on the set H := M × C with binary +operation (m, c) · (m′, c′) := (m + m′, c + c′ + 2−1ω(m, m′)). Then H → H(µ), (a + b, c) �→ +(a, b, c − 2−1µ(a, b)) is a group isomorphism to the group from Example 4.3. We shall use +the construction of Example 4.3 to treat all cases uniformly independently whether the +group has 2-torsion or not. +Example 4.5. Let G be a finite ≤2-step nilpotent group with cyclic centre. +Apply +Lemma 3.5 to the non-degenerate central-by-abelian extension +Z(G) +1 +Z(G) +G +G/ Z(G) +1 +: +⊆ +πZ +giving a non-degenerate alternating Z-module (G/ Z(G), ω, Z(G)) := A(Z(G)). Hence +Example 4.3 gives a non-degenerate Z-bilinear map µG : A × B → Z(G) (where actually +A ∼= B) and +H(µG) +1 +Z(G) +H(µG) +A × B +1. +: +ιµg +πµg +In this way, we assign a Heisenberg group H(µG) to G. These groups share many proper- +ties: the order, isomorphism class of centre and commutator subgroup, nilpotency class. +However, they are non-isomorphic for example if G = Q8 (the quaternion group) or the +extra-special p-group of order p3 of exponent p2. Thus in general, we cannot even ex- +pect to have a morphism between the two short exact sequences, as that would imply +G ∼= H(µG) by the 5-lemma. In fact, this (iso)morphism exists if and only if G ∼= H(µG) +since then A ⊕ B gives an isotropic real structure by Remark 3.13. +Our goal in to establish a monomorphism Z(G) → H(ˆµ) for a suitable (non- +degenerate) ˆµ. In fact, we will show that ˆµ = ζ ◦ µG works for a suitable ζ : Z(G) ↣ ˆC. +For this, we generalise the notion of isotropic real structure from Lemma 3.10. +Definition 4.6. An extended polarisation of a central-by-abelian extension ϵ is the pair +of the following commutative diagrams (j ∈ {1, 2}) +ϵj +1 +Cj +Gj +Lj +1 +ϵ +1 +C +G +M +1 +ˆC +L1 × L2 +: +ιj +κj +πj +γj +ζj +λj +: +ι +ζ +π +∼ +λ:=λ1⊕λ2 +(4) +10 + +such that ϵj is a central-by-abelian extension, λ: L1 × L2 → M, (l1, l2) �→ λ1(l1) + λ2(l2) +is an isomorphism and ˆC is an abelian group. +Lemma 4.7. Every extended polarisation as in (4) induces a decomposition G = +γ2(G2)ι(C)γ1(G1). +Proof. Pick g ∈ G arbitrarily. +Set (l1, l2) := λ−1(π(g)) ∈ L1 × L2. +From the sur- +jectivity of πj, pick gj ∈ Gj such that πj(gj) = lj. +Then π(γ2(g2)−1gγ1(g1)−1) = +−λ2(π2(g2)) + (λ1(l1) + λ2(l2)) − λ1(π1(g1)) = 0 using the commutativity of the dia- +gram (4). So γ2(g2)−1gγ1(g1)−1 ∈ ker(π) = Im(ι), hence there is a c ∈ C such that +ι(c) = γ2(g2)−1gγ1(g1)−1. Rearranging this gives the decomposition as claimed. +Lemma 4.8 (Key). Every extended polarisation as at (4) can be completed to a +commutative diagram +ϵ +1 +C +G +M +1 +H(ˆµ) +1 +ˆC +H(ˆµ) +L1 × L2 +1 +: +ι +ζ +π +δ +∼ +λ−1 +: +ιˆµ +πˆµ +(5) +where ˆµ is defined by +M × M +C +L1 × L2 +ˆC +ω +ζ +ˆµ +λ1×λ2 +for the alternating Z-bilinear map ω from Lemma 3.5 when applied to ϵ. +Remark 4.9. The 4-lemma implies that δ is injective if and only if ζ is. In this case, H(ˆµ) +is the external central product of ˆC and G amalgamating C along ι and ζ; in particular, +G is isomorphic to a normal subgroup of a Heisenberg group. +Remark 4.10. The central-by-abelian extension ϵ is non-degenerate if and only if H(ˆµ) is +non-degenerate. +Proof. First note that ˆµ is indeed an alternating Z-bilinear map by Lemma 3.5, so H(ˆµ) +is well-defined. We show that δ := (δ1, δ2, δ3) satisfies the statement for j ∈ {1, 2} and +δj : G → Lj, g �→ πj(gj), +δ3 : G → ˆC, g �→ ζ2(g2)ζ(c)ζ1(g1), +for any decomposition g = γ2(g2)ι(c)γ1(g1) from Lemma 4.7. The map δj is actually the +natural composition of group morphisms G +π−→ M +λ−1 +−−→ L1 × L2 → Lj, in particular, δj is +independent of the choice of the decomposition. To show that δ3 is independent of the +choice of the decomposition, let γ2(g2)ι(c)γ1(g1) = g = γ2(g′ +2)ι(c′)γ1(g′ +1). Then on one +hand, πj(gj) = δj(g) = πj(g′ +j) by above, hence by the exactness of ϵj, there are cj ∈ Cj +such that ι1(c1) = g′ +1g−1 +1 +and ι2(c2) = g−1 +2 g′ +2. On the other hand, using ι(C) ⊆ Z(G), +rearranging the original equation gives ι(cc′−1) = γ2(g−1 +2 g′ +2)γ1(g′ +1g−1 +1 ) = ι(κ2(c2)κ1(c1)), +hence cc′−1 = κ2(c2)κ1(c1) as ι is injective. Putting these together gives +ζ2(g2)ζ(c)ζ1(g1) = ζ2(g′ +2ι2(c2)−1) · ζ(κ2(c2)c′κ1(c1)) · ζ1(ι1(c1)−1g′ +1) +11 + += ζ2(g′ +2) · (ζ2(ι2(c2))−1ζ(κ2(c2))) · ζ(c′) · (ζ(κ1(c1))ζ1(ι1(c1))−1) · ζ1(g′ +1) += ζ2(g′ +2)ζ(c′)ζ1(g′ +1) +using commutativity of (4). Thus δ3 is indeed well defined. +Note that unlike the other maps, δ3 is just a map of sets, not a group morphism. +Its failure to be a group morphism is measured by ˆµ. Indeed, pick decompositions g = +γ2(g2)ι(c)γ1(g1) and g′ = γ2(g′ +2)ι(c′)γ1(g′ +1). +Set x := ω(π(γ1(g1)), π(γ2(g′ +2))) ∈ C, so +ι(x) = [γ1(g1), γ2(g′ +2)]. Use this to find a decomposition of the product as +gg′ = γ2(g2)ι(c)γ1(g1)γ2(g′ +2)ι(c′)γ1(g′ +1) += γ2(g2)γ2(g′ +2)ι(cc′)[γ1(g1), γ2(g′ +2)]γ1(g1)γ1(g′ +1) += γ2(g2g′ +2)ι(cc′x)γ1(g1g′ +1). +Then by definitions and using the commutativity of the diagram, +δ3(gg′) = ζ2(g2g′ +2)ζ(cc′x)ζ1(g1g′ +1) += ζ2(g2)ζ(c)ζ1(g1) · ζ2(g′ +2)ζ(c′)ζ1(g′ +1) · ζ(ω(π(γ1(g1)), π(γ2(g′ +2)))) += δ3(g)δ3(g′)ˆµ(π1(g1), π2(g′ +2)) += δ3(g)δ3(g′)ˆµ(δ1(g), δ2(g′)). +This property together with Remark 4.2 imply that δ is a group morphism: +δ(gg′) = (δ1(gg′), δ2(gg′), δ3(gg′)) += (δ1(g)δ1(g′), δ2(g)δ2(g′), δ3(g)δ3(g′)ˆµ(δ1(g), δ2(g′))) += (δ1(g), δ2(g), δ3(g)) ∗ (δ1(g′), δ2(g′), δ3(g′)) += δ(g) ∗ δ(g′). +We check that the diagram (5) is commutative. Indeed, if c ∈ C, then using the +decomposition ι(c) = γ2(1)ι(c)γ1(1) gives δ ◦ ι = (c �→ (0, 0, ζ(c))) = ιˆµ ◦ ζ by defin- +itions. Similarly, the decomposition g = γ2(g2)ι(c)γ1(g1) ∈ G gives λ−1 ◦ π = (g �→ +λ−1(π(γ2(g2)γ1(g1))) = δ1(g) + δ2(g)) = πˆµ ◦ δ. +5 +Heisenberg embeddings +In this section, we put together the pieces from earlier sections to prove Theorem B +from 2, the main statement of the paper. First, we handle the cyclic centre case +by constructing a suitable extended polarisation using the isotropic real structure +from Section 3. For the general case, we take the direct product of the resulting +Heisenberg groups (which itself is of this type) and use the first part of Theorem A. +We start with an elementary statement. +Lemma 5.1. The solid arrows of the following diagram can be completed with suitable +dashed arrows making a commutative diagram of groups where K is a finite abelian group +and l = lcm(m, exp(K)). +Z/nZ +K +Z/mZ +Z/lZ +ι +κ +ϕ +θ +12 + +Proof. First, we prove the case K = Z/kZ. The map κ is defined by some b ∈ Z such +that κ(1+nZ) = m +n b+mZ. Similarly, ι is given by some a ∈ Z with ι(1+nZ) = k +na+kZ. +Since ι is injective, we have gcd(a, n) = 1, hence we may pick x ∈ Z so that ax ≡ b +(mod n). Define ϕ(i + kZ) := i l +kx + lZ and θ(i + mZ) := i l +m + lZ. Short computation +shows that ϕ ◦ ι = θ ◦ κ with these definitions. +In the general case, K = � +C∈S C for some suitable set S of cyclic subgroups of K +of prime power order. For C ∈ S, let πC : K → C be the natural projection, and write +o(C) := | Im(πC ◦ ι)|. For every prime p dividing n, pick a Cp ∈ S so that o(Cp) = +max{o(C) : C ∈ S, p +�� |C|}. Define the composition +¯ι: Z/n +ι−→ K ↠ +� +p|n +Cp ∼= Z/dZ → Z/kZ +where the second map is � +p|n πCp, the isomorphism is given by the Chinese remainder +theorem, and the last one is any embedding where d +�� k := exp(K). This map is injective, +because | Im(¯ι)| = lcm{|Cp| : p | n} = lcm{o(C) : C ∈ S} = n. So replacing ι by ¯ι reduces +to the special case K = Z/kZ discussed above. +Remark 5.2. Essentially this statement replaces the usage of the second part of Theorem A +from 1 and the central product approach of [Sza21] and [Sza19] with a much shorter +argument. The statement could be extracted when tracking down the behavior of centre +the group during taking maximal central products [Sza21, proof of Proposition 4.2.18] and +the construction of extended polarisation for 2-generated groups [Sza21, Lemma 4.2.13]. +More concretely, writing G = �t +i=1 Ci as a product of cyclic groups, we take iteratively +the (so called maximal) central product of Z/mZ, C1, . . . , Ct at each step amalgamating +the largest possible subgroup compatible with the given maps. This maximality condition +ensures that the resulting groups remain cyclic. +Proposition 5.3. For every finite ≤2-step nilpotent group G with cyclic centre, there +exists a monomorphism +Z(G) +1 +Z(G) +G +G/ Z(G) +1 +H(ˆµ) +1 +ˆC +H(ˆµ) +A × A +1 +: +f +⊆ +ζ +πZ +δ +∼ +ν +: +⊆ +πˆµ +(6) +of non-degenerate central-by-abelian extensions for a suitable ˆµ: A × A → ˆC where +ˆC is cyclic and the exp(A) = |G′| +�� | Z(G)| +�� | ˆC| +�� exp(G) divisibility conditions hold. +Remark 5.4. Actually, the Heisenberg group from the statement can be replaced with a +canonical one as follows. Define a Z-bilinear map ν : Hom(A, ˆC) × A → ˆC by α(a) �→ +α(a), and write H(A, ˆV ) := H(ν) for the corresponding Heisenberg group. Then the map +H(ˆµ) → H(A, ˆC) given by (a, a′, c) �→ (x �→ ˆµ(a, x), a′, c) is an isomorphism because ˆµ +is non-degenerate and exp(A) +�� |C|. In particular, every G as above is isomorphic to a +normal subgroup of H(A, ˆC) of index at most exp(G)/| Z(G)|. +Proof. By Remark 3.6, Lemma 3.10 is applicable to (G/ Z(G), ω, Z(G)) := A(Z(G)) +giving an isotropic real structure Z(G) = L1 ⊕ L2. Note that L2 = iL1 ∼= L1. Since Lj is +isotropic, π−1 +Z (Lj) is abelian by Remark 3.6. +13 + +Consider the inclusion maps from the following diagram. +Z(G) ∩ π−1 +Z (L1) +π−1 +Z (L1) +Z(G) +C +ˆC +Z(G) ∩ π−1 +Z (L2) +π−1 +Z (L2) +⊆ +ι1 +⊆ +κ1 +ϕ1 +ζ1 +θ1 +ζ +θ2 +⊆ +κ2 +⊆ +ι2 +ϕ2=ζ2 +Applying Lemma 5.1 to the inclusions ι1 and κ1 gives a cyclic group C with a morphisms +ϕ1, θ1. Then we can apply Lemma 5.1 to the inclusion ι2 and the composition κ2 giving +another cyclic ˆC with ϕ2, θ2. By construction, θ1 and θ2 are both injective, hence so is +ζ := θ2 ◦θ1. This diagram is commutative by construction. Set ζ1 := θ2 ◦ϕ1 and ζ2 := ϕ2. +The maps above give the following extended polarisation of Z(G) +ϵj +1 +Z(G) ∩ π−1 +Z (Lj) +π−1 +Z (Lj) +Lj +1 +Z(G) +1 +Z(G) +G +G/ Z(G) +1 +ˆC +L1 × L2 +: +⊆ +⊆ +πj +⊆ +ζj +⊆ +: +⊆ +ζ +πZ +where πj := πZ|π−1 +Z (Lj). Then Lemma 4.8 gives the diagram (6) upon setting A := L1 ∼= L2. +Finally, we check that the stated properties hold. +Note that the injectivity of ζ +together with the 4-lemma implies the injectivity of δ. +exp(A) = exp(L1 × L2) = +exp(G/ Z(G)) = |(G′)| follows from Theorem 3.7. Since G is of nilpotency class at most +2, G′ ⊆ Z(G) implies |G′| +�� | Z(G)|. By Lemma 5.1, +| ˆC| = lcm(| Z(G)|, exp(π−1 +Z (L1)), exp(π−1 +Z (L2))) +�� exp(G). +Theorem 5.5. For every finite ≤2-step nilpotent group G, there exists +Z(G) +1 +Z(G) +G +G/ Z(G) +1 +H(µ) +1 +C +H(µ) +A × B +1 +: +f +•⊆ +ζ +πZ +δ +ν +: +•⊆ +πµ +for a suitable non-degenerate µ: A × B → C such that +d(G/ Z(G)) ≥ d(A), d(B), +d(Z(G)) = d(C), +exp(G/ Z(G)) +�� exp(A × B) +�� exp(G′) +�� exp(Z(G)) +�� exp(C) +�� exp(G). +Furthermore, there is a monomorphism H(µ) ↣ �d(Z(G)) +i=1 +H(µi) for suitable +µi : Ai×Ai → Ci where each H(µi) is non-degenerate, Ci is cyclic and d(Ai) ≤ 1 +2d(G). +14 + +Remark 5.6. The monomorphism ζ shows that d(C) is as small as possible. On the other +hand, ν gives d(G/ Z(G)) ≤ d(A × B) ≤ 2d(G/ Z(G)). The lower bound is attained in +the d(C) ≤ 1 case, see Proposition 5.3. It is a natural question to ask for the smallest +possible value of d(A×B) in general. To give a better upper bound than above, one may +need to develop some version of Lemma 3.1 for matrices with entries from Zn. +Proof. Using Proposition 2.7, write ϕ: G ↣ �n +i=1 Gi as a subdirect product where n := +d(Z(G)) and each Z(Gi) is cyclic. Then Gi are all ≤2-step nilpotent as this class is closed +under taking quotients, so Proposition 5.3 gives fi = (ζi, δi, νi): Z(Gi) → H(µi) for some +µi : Ai × Bi → Ci (where Bi = Ai). Let ¯A := �n +i=1 Ai, ¯B := �n +i=1 Bi, ¯C := �n +i=1 Ci and +¯µ := �n +i=1 µi : ¯A × ¯B → ¯C, a non-degenerate Z-bilinear map by construction. We obtain +the following diagram of central-by-abelian extensions where � is a shorthand for �n +i=1. +Z(G) +1 +Z(G) +G +G/ Z(G) +1 +� Z(Gi) +1 +� Z(Gi) +� Gi +� Gi/ Z(Gi) +1 +� +H(µ) +1 +� Ci +� H(µi) +� Ai × Bi +1 +H(¯µ) +1 +¯C +H(¯µ) +¯A × ¯B +1 +: +Proposition 2.7 +⊆ +ϕ|Z(G) +πZ +ϕ +[ϕ] +: +� fi +Proposition 5.3 +⊆ +� ζi +� πZ +� δi +∼ +� νi +: +∼ +⊆ +� πµi +∼ +∼ +: +⊆ +π¯µ +Denote by ¯f = (¯ζ, ¯δ, ¯ν): Z(G) → H(¯µ) the resulting monomorphism. This may have +more generators than stated, so we take a suitable subobject of H(¯µ). Let A ≤ ¯A be +the image of G/ Z(G) +¯ν−→ ¯A × ¯B → ¯A, and B ≤ ¯B be that of G/ Z(G) +¯ν−→ ¯A × ¯B → ¯B. +Then d(A) and d(B) are at most d(G/ Z(G)). Let C := ⟨¯ζ(Z(G)), ¯µ(A, B)⟩ ≤ ¯C. Then +d(Z(G)) = d(¯ζ(Z(G))) ≤ d(C) ≤ d( ¯C) = �n +i=1 d(Ci) ≤ n = d(Z(G)), hence comparing +the two ends give d(C) = d(Z(G)). +Define µ: A × B → C, (a, b) �→ ¯µ(a, b), an abelian bihomomorphism. +The image +of ¯f lies in H(µ) by definition, so restricting the domain to H(µ) gives a map f = +(ζ, δ, ν): Z(G) → H(µ), i.e. ¯f = (H(µ) ↣ H(¯µ)) ◦ f for the natural inclusion map. We +show that this f satisfies the statement. +We check that µ is non-degenerate. Pick 0 ̸= a ∈ A and write a = (a1, . . . , an) ∈ +�n +i=1 Ai. Then without loss of generality, a1 ̸= 0. Then by the non-degeneracy of µ1, +there is a b′ +1 ∈ B1 such that 0 ̸= µ1(a1, b′ +1) ∈ C1. By the diagram above, there is g′ +1 ∈ G1 +such that ν1(g′ +1 Z(G1)) = (0, b′ +1). As ϕ is a subdirect product, there is g′ ∈ G such that +the 1st factor of [ϕ](g′ Z(G)) is g′ +1 Z(G1). Write b′ = (b′ +1, . . . , b′ +n) for the image of g′ Z(G) +under G/ Z(G) +¯ν−→ ¯A × ¯B → ¯B. By construction, b′ +1 coincides with the above choice. +By definition, b ∈ B, and µ(a, b) = (¯µ1(a1, b′ +1), . . . , ¯µn(an, b′ +n)) ̸= 0 as the first factor is +non-trivial by construction. This argument remains valid when the roles of A and B are +swapped, hence µ is non-degenerate. +Assume that G is finite and consider the statement on the exponents. exp(G/ Z(G)) +�� +exp(A×B) follows from ν being a monomorphism of abelian groups. For every i, exp(Ai× +Bi) exp(G′ +i) +�� exp(G′) using Proposition 5.3 and the fact that as Gi is a quotient of G. +Thus exp(A × B) +�� exp( ¯A × ¯B) = lcm{exp(Ai × Bi) : 1 ≤ i ≤ n} +�� exp(G′). Since +G is ≤2-step nilpotent, we have G′ ⊆ Z(G), so exp(G′) +�� exp(Z(G)). The embedding +ζ : Z(G) ↣ C shows exp(Z(G)) +�� exp(C). Once again using Proposition 5.3, we see +15 + +that exp(Ci) +�� exp(Gi) +�� exp(G) as Gi is a quotient of G. Then exp(C) +�� exp( ¯C) = +lcm{exp(Ci) : 1 ≤ i ≤ n} +�� exp(G) as stated. +References +[AMM12] +Azhana Ahmad, Arturo Magidin and Robert Fitzgerald Morse. ‘Two generator p- +groups of nilpotency class 2 and their conjugacy classes’. In: Publicationes Mathem- +aticae Debrecen 81 (2012), pp. 145–166 (cit. on p. 2). +[BBC69] +J.M. Brady, R.A. Bryce and John Cossey. ‘On certain abelian-by-nilpotent varieties’. +In: Bulletin of the Australian Mathematical Society 1.3 (1969), pp. 403–416. doi: +10.1017/S0004972700042325 (cit. on pp. 2, 7). +[CPS22] +Balázs Csikós, László Pyber and Endre Szabó. Finite subgroups of the homeomorph- +ism group of a compact topological manifold are almost nilpotent. 2022. arXiv: 2204. +13375 [math.GT] (cit. on p. 2). +[Gul20] +Attila Guld. Finite subgroups of the birational automorphism group are ’almost’ +nilpotent of class at most two. 2020. arXiv: 2004.11715 [math.AG] (cit. on p. 2). +[Hig60] +Graham Higman. ‘Enumerating p-Groups. I: Inequalities’. In: Proceedings of the +London Mathematical Society s3-10.1 (Jan. 1960), pp. 24–30. issn: 0024-6115. doi: +10.1112/plms/s3-10.1.24. url: https://doi.org/10.1112/plms/s3-10.1.24 +(cit. on p. 2). +[Jac85] +Nathan Jacobson. Basic algebra. New York: W.H. Freeman, 1985. isbn: 0-7167-1480- +9 (cit. on p. 6). +[Mag98] +Arturo Magidin. Bilinear maps and central extensions of abelian groups. 1998. arXiv: +9802066 [math.GR] (cit. on p. 2). +[Sim65] +Charles C. Sims. ‘Enumerating p-Groups’. In: Proceedings of the London Mathemat- +ical Society s3-15.1 (Jan. 1965), pp. 151–166. issn: 0024-6115. doi: 10.1112/plms/ +s3-15.1.151. url: https://doi.org/10.1112/plms/s3-15.1.151 (cit. on p. 2). +[Suz82] +Michio Suzuki. Group theory II. 1st ed. Grundlehren der mathematischen Wis- +senschaften. Springer-Verlag Berlin Heidelberg, 1982. isbn: 9783642868870 (cit. on +pp. 1, 8). +[Sza21] +Dávid R Szabó. ‘Jordan Type Problems via Class 2 Nilpotent and Twisted Heisen- +berg Groups’. PhD thesis. Central European University, 2021 (cit. on pp. 2, 5, 6, 10, +13). +[Sza19] +Dávid R. Szabó. Special p-groups acting on compact manifolds. 2019. arXiv: 1901. +07319 [math.DG] (cit. on p. 13). +Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13–15, H–1053, Bud- +apest, Hungary +E-mail address: szabo.david@renyi.hu +16 + diff --git a/ptFPT4oBgHgl3EQf7zXe/content/tmp_files/2301.13206v1.pdf.txt b/ptFPT4oBgHgl3EQf7zXe/content/tmp_files/2301.13206v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..715bf4eecc40485bb72347a827fb0ec235398b97 --- /dev/null +++ b/ptFPT4oBgHgl3EQf7zXe/content/tmp_files/2301.13206v1.pdf.txt @@ -0,0 +1,4596 @@ +arXiv:2301.13206v1 [math.AG] 29 Jan 2023 +PURITY AND TORSORS OVER PRÜFER BASES +NING GUO AND FEI LIU +Abstract. We establish Zariski–Nagata purity theorem concerning finite étale covers on smooth schemes +over Prüfer rings by proving Auslander’s flatness criterion in this non-Noetherian context. Inspired by +Gabber–Ramero’s upper bound of projective dimensions over Prüfer bases, we present an Auslander– +Buchsbaum formula. On the basis of the analysis of reflexive sheaves, we prove various purity theorems +for torsors under reductive group algebraic spaces. Specifically, by parafactorial results in [EGA IV4] on +smooth schemes over normal bases, we prove the purity for cohomology groups of multiplicative type +groups at this level of generality. +Subsequently, we take advantage of aforementioned purity results +to give affirmative answer to the Grothendieck–Serre conjecture for torsors on smooth schemes over +semilocal Prüfer rings in certain cases. Along the way, inspired by the recent preprint of ˇCesnaviˇcius +[Čes22c], we also prove several versions of Nisnevich conjecture in our context. +1. Purity and the Grothendieck–Serre on schemes smooth over Prüfer bases . . . . . . . . +2 +Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2. Auslander–Buchsbaum formula over valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +3. Geometry of schemes over Prüfer bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3.1. Geometric properties and reduction methods +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3.2. Reflexive sheaves on schemes over Prüfer bases with regular fibers . . . . . . . . . . . . . . . . . . . . . +9 +4. Auslander’s flatness criterion on schemes smooth over valuation rings . . . . . . . . . . . . . 12 +5. Generalities on torsors over algebraic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 +6. Purity for torsors and finite étale covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +6.1. Purity for reductive torsors on relative curves +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 +6.2. Local variants of purity results +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 +6.3. Extending generically trivial torsors . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 +6.4. Purity for finite locally free torsors and the Zariski–Nagata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 +7. Geometric lemmata for the Grothendieck–Serre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 +7.1. Geometric presentation lemma over Prüfer bases +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 +7.2. A variant of Lindel’s lemma . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 +8. Cohomology of groups of multiplicative type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +8.1. Geometrically parafactorial pairs +. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 +8.2. Purity for groups of multiplicative type +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +8.3. Grothendieck–Serre type results for groups of multiplicative type +. . . . . . . . . . . . . . . . . . . . . . 35 +9. Grothendieck–Serre on a semilocal Prüfer domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +37 +9.1. Lifting maximal tori of reductive group schemes over semilocal rings +. . . . . . . . . . . . . . . . . . +37 +9.2. Harder’s weak approximation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +38 +9.3. Product formula over semilocal Prüfer domains, passage to the local case . . . . . . . . . . . . . . 40 +10. Torsors on a smooth affine relative curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 +11. Torsors under a reductive group scheme over a smooth projective base . . . . . . . . . . +46 +12. Torsors under a constant reductive group scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 +13. Torsors under a quasi-split reductive group scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 +References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 +Date: February 1, 2023. +2010 Mathematics Subject Classification. Primary 14F22; Secondary 14F20, 14G22, 16K50. +Key words and phrases. purity, Zariski–Nagata, Auslander–Buchsbaum, Grothendieck–Serre, vector bundles, principal +bundles, Prüfer rings, torsors, homogeneous spaces, group schemes, valuation rings. +1 + +1. Purity and the Grothendieck–Serre on schemes smooth over Prüfer bases +1.1. Purity and regularity. In algebraic geometry, purity refers to a diverse range of phenomena in +which certain invariants or categories associated to geometric objects are insensitive to the removal of +closed subsets of large codimensions. In the classical Noetherian world, purities, say, for vector bundles +(and even torsors), or for finite étale covers, are intimately related to the regularities measured by lengths +of regular sequences of geometric objects. For a concrete instance, the Auslander–Buchsbaum formula +depthR M ` proj.dimRM “ depthR R +([AB57, Theorem 3.7]) +controls the projective dimension of the finite type module M over the Noetherian local ring R via depths, +leading to the purity for vector bundles on regular local rings of dimension two ([Sam64, Proposition 2]). +Granted this, Colliot-Thélène and Sansuc [CTS79, Théorème 6.13] established the purity for reductive +torsors over arbitrary regular local ring R of dimension two by bootstrapping from the vector bundle +case: +the restriction +H1 +´etpSpec R, Gq +„ +ÝÑ H1 +´etpSpec RztmRu, Gq +is bijective +for every reductive R-group scheme G. Nevertheless, not only does the term ‘regularity’ make sense for +Noetherian rings, its non-Noetherian generalization can still enlighten us to contemplate purity problems. +1.2. Regularity of Prüfer rings. Originally formulated by Bertin [Ber71], [Ber72, Définition 3.5] +for coherent local rings, we say that a ring R is regular if every finitely generated ideal of R has finite +projective dimension. This coincides with the classical notion of regularity when restricting to Noetherian +rings by Serre’s homological characterization [Ser56, Théorème 3]. A typical non-Noetherian example +can be sought in Prüfer rings, namely, the rings whose all local rings are valuation rings. By definition, an +integral domain V is a valuation ring if every x P pFrac V qzV satisfies x´1 P V . Beyond fields, Noetherian +valuation rings are exactly discrete valuation rings. The regularity of Prüfer rings thus follows from the +fact that all finitely generated ideals of valuation rings are principal. In addition to the regularity and +other nature (Lemma 3.1.1), the ubiquity of Prüfer rings in the study of nonarchimedean geometry, +Zariski–Riemann spaces, among others, motivates us to investigate their algebro-geometric properties. +1.3. Basic setup I. The purity part of the present article focuses on a semilocal affine Prüfer scheme S +with dim S ą 0 (and with dim S ă 8 if necessary), an S-flat finite type algebraic space X with regular +S-fibers, and a closed subset Z Ă X such that j : XzZ ãÑ X is quasi-compact. For a point x P X lying +in an open subscheme, the local ring of X at x makes sense and we denote A :“ OX,x. When involving +torsors on X, we let G be an X-group algebraic space that étale-locally permits an embedding G ãÑ GLn +such that GLn {G is X-affine. This condition is fulfilled if G is X-reductive1, or finite and locally free. +1.4. Auslander–Buchsbaum over Prüfer bases. Gabber–Ramero’s upper bound of projective di- +mensions of coherent modules over X unveils a glimpse of the Prüferian Auslander–Buchsbaum formula +Theorem 2.8.1: if x P X lies over a closed point s P S, then every finitely presented A-module M satisfies +proj. dimA M ` depthAM “ depthA A “ d ` 1, +where d “ dim OXs,x. +Here proj. dimAp0q “ ´8 and depthA M is the smallest i such that the i-th local cohomology of M +be nonzero (§2.4). +Our proof is significantly different from the classical case [AB57, Theorem 3.7]. +Specifically, taking Gabber–Ramero’s boundness [GR18, Proposition 11.4.1] as an input, we bypass +the interpretation of projective dimensions in terms with Tor functors, which is a crucial ingredient in +Auslander–Buchsbaum’s argument. In the sequel, we will only use Gabber–Ramero’s part of Proposi- +tion 3.2.7(i). +1.5. Purity for torsors on smooth relative curves over Prüfer rings. Once the projective di- +mensions of reflexive sheaves on X are controlled, by imposing codimensional constraints on Z, we may +extend vector bundles on XzZ to X, as in Noetherian scenarios. Subsequently, this allows us to obtain +the purity Theorem 6.1.4 for G-torsors: if Z satisfies +Zη “ H +for each generic point η P S +and +codimpZs, Xsq ě 1 for all s P S, +and X is an S-curve, then restriction induces the following equivalence of categories of G-torsors +TorspX´et, Gq +„ +ÝÑ TorsppXzZq´et, Gq. +1By this we mean a smooth affine X-group algebraic space G whose X-geometric fibers are (connected) reductive algebraic +groups. Then, étale-locally on X, G splits so admits a closed immersion G ãÑ GLn,X for some integer n; by [Alp14, 9.4.1], +the reductivity of G implies that the quotient GLn,X{G is X-affine of finite type. +2 + +In particular, passing to isomorphism classes of objects, we have the following bijection of pointed sets +H1 +´etpX, Gq » H1 +´etpXzZ, Gq. +Meanwhile, a local version Theorem 6.2.1 allows us to loose constraints on the relative dimension of X: +if +either x P Xη with dim OXη,x “ 2, +or x P Xs with s ‰ η and dim OXs,x “ 1, +then every G-torsor over Spec OX,xztxu extends uniquely to a G-torsor over Spec OX,x. This permits us +to iteratively extend reductive torsors beyond a closed subset of higher fiberwise codimensions. +1.6. Zariski–Nagata over Prüfer bases. The Zariski–Nagata purity, known as “purity of branch +locus”, states that every finite extension A Ă B of rings with A regular Noetherian and B normal is +unramified if and only if so it is in codimension one on Spec B. This purity was settled by Zariski [Zar58] +in a geometric context, and more algebraically by Nagata [Nag59] based on Chow’s local Bertini theorem. +In contrast to them, Auslander gave an alternative proof [Aus62, Theorem 1.4] by skillful homological +methods leading to a criterion for flatness. In [SGA 2new, Exposé X, §3], Grothendieck reformulated their +results into a purity concerning finite étale covers and proved this purity on Noetherian local rings that is +a complete intersection of dimension ě 3 by reducing the assertion to hypersurfaces via several passages +involving formal completions. Nevertheless, a practical deficiency of the later argument is that, even over +a rank-one valuation ring V with pseudo-uniformizer ̟, the coherence of the ̟-adic completion pA of A +is unknown to us, not to mention the primary decomposition on it. To circumvent this technical obstacle, +we revert to Auslander’s argument by establishing a Prüferian counterpart Theorem 4.1 of the criterion +for flatness [Aus62, Theorem 1.3]. Granted this, we acquire the Prüferian Zariski–Nagata Theorem 6.4.2: +the pullback +FÉtX +„ +ÝÑ FÉtXzZ +is an equivalence +for every closed subset Z Ă X in the basic setup §1.3 that satisfies the following condition +codimpZη, Xηq ě 2 +for each generic point η P S +and +codimpZs, Xsq ě 1 for all s P S. +In particular, for every geometric point x: Spec Ω Ñ XzZ with a separably closed field Ω, the map +π´et +1 pXzZ, xq Ñ π´et +1 pX, xq +is an isomorphism. +1.7. Grothendieck–Serre on semilocal Prüfer rings. The Grothendieck–Serre conjecture predicts +that, for a regular local ring R and a reductive R-group scheme G, every generically trivial G-torsor is +trivial, that is, the following restriction map of nonabelian cohomology pointed sets has trivial kernel: +ker pH1 +´etpR, Gq Ñ H1 +´etpFrac R, Gqq “ t˚u. +The conjecture was settled in the equicharacteristic case and in certain unramified mixed characteristic +cases, see the histrical summary below. Thanks to the purity for cohomology of groups of multiplicative +type, we prove the non-Noetherian counterpart of Colliot-Thélène–Sansuc’s result for tori and then obtain +a product formula for tori. Based on this, the similar argument in [Guo20] leads to a passage from the +semilocal case to the local case. Hence, we settle the Grothendieck–Serre on semilocal Prüfer rings in §9. +1.8. Basic setup II. The second half of this article deals mainly with the following. For a semilocal +Prüfer ring R, an irreducible R-smooth scheme X, the semilocalization A :“ OX,x of X at a finite +subset x Ă X contained in a single affine open of X, and a reductive A-group scheme G, we study the +trivialization behaviour of G-torsors. +1.9. Grothendieck–Serre on smooth projective schemes. This result was proved by the second +author and simultaneously by an unpublished work of Panin and the first author in the Noetherian case. +We show that, when X is R-projective in §1.8 and G has a reductive model over X, every generically +trivial G-torsor on A is trivial, that is, +ker pH1pA, Gq Ñ H1pFrac A, Gqq “ t˚u. +To prove this, we use crucially our purity Theorem 6.1.4 after spreading out to extend the domain of the +torsor in question to an open subset as large as possible: according to that purity, a generically trivial +torsor on OX,x extends to a torsor on an open neighbourhood of x whose complementary closed has +codimension ě 3 (resp., ě 2) in the generic (resp., non-generic) R-fibers of X, see Corollary 6.3.2. This +codimension bound is sharp enough for us to apply the geometric presentation Lemma 7.1.1 and glueing +techniques to reduce the problem to studying torsors on relative affine lines that we treat in detail in +§10. +3 + +1.10. Grothendieck–Serre under constant reductive groups. Assume that G is ‘constant’, namely, +it is a pullback from the Prüfer base ring R. Then every generically trivial G-torsor on A is trivial, that +is, +ker pH1pA, Gq Ñ H1pFrac A, Gqq “ t˚u. +For this, we first devise a variant (in some aspect, a stronger form) of Lindel’s lemma (Proposition 7.2.1), +which states that, for a closed subscheme Y Ă X that avoids all the maximal points of the R-fibers of X, +the pair pY, Xq Zariski-locally on X can be presented as an elementary étale neighbourhood of a similar +pair pY 1, X1q, where X1 is an open of some projective R-space. This allows us to use glueing techniques +to reduce to studying generically trivial torsors on opens of projective R-spaces, which is done in §1.9. +1.11. Grothendieck–Serre under quasi-split groups. As for the quasi-split case of the Grothendieck– +Serre, we will follow a similar strategy of [Čes22a] (with its earlier version given by Fedorov [Fed22b]), +where the key input is our toral version of purity Proposition 8.2.5 and Grothendieck–Serre type Propo- +sition 8.3.2 in this context. Precisely, by the valuative criterion of properness, a generically trivial torsor +on X, say, reduces to a generically trivial torsor under a Borel B away from a closed subset Z of X +that has codimension ě 2 (resp., ě 1) in the generic (resp., non-generic) R-fiber. Further, utilizing the +aforementioned toral purity and Grothendieck–Serre type results, one shows that the above torsor further +reduces to a radupBq-torsor on XzZ. In conclusion, when G is quasi-split, we prove Theorem 13.1 that +ker +` +H1pA bR K, Gq Ñ H1pFrac A, Gq +˘ +“ t˚u; +if R has Krull dimension 1, then every generically trivial G-torsor is trivial, that is, +ker +` +H1pA, Gq Ñ H1pFrac A, Gq +˘ +“ t˚u. +1.12. Nisnevich’s purity conjecture. Now, we turn to Nisnevich’s purity conjecture, where we require +the total isotropicity of group schemes. A reductive group scheme G defined over a scheme S is totally +isotropic at s P S if every Gi in the decomposition [SGA 3III new, Exposé XXIV, Proposition 5.10 (i)] +Gad +OS,s – ś +i ResAi{OS,spGiq +contains a Gm,Ri. If this holds for all s P S, then G is totally isotropic. Proposed by Nisnevich [Nis89, +Conjecture 1.3] and modified due to the anisotropic counterexamples of Fedorov [Fed22b, Proposition 4.1], +the Nisnevich conjecure predicts that, for a regular semilocal ring R, a regular parameter r P R (that is, +r P mzm2 for every maximal ideal m Ă R), and a reductive R-group scheme G such that GR{rR is totally +isotropic, every generically trivial G-torsor on Rr 1 +rs is trivial, that is, the following map +H1pRr 1 +rs, Gq Ñ H1pFrac R, Gq +has trivial kernel. +The case when R is a local ring of a regular affine variety over a field and G “ GLn was settled by +Bhatwadekar–Rao in [BR83] and was subsequently extended to arbitrary regular local rings containing +fields by Popescu [Pop02, Theorem 1]. Nisnevich in [Nis89] proved the conjecture in dimension two, +assuming that R is a local ring with infinite residue field and that G is quasi-split. For the state of +the art, the conjecure was settled in equicharacteristic case and in several mixed characteristic case by +Česnavičius in [Čes22c, Theorem 1.3] (previously, Fedorov [Fed21] proved the case when R contains +an infinite field). Besides, the toral case and some low dimensional cases are known and surveyed in +[Čes22b, Section 3.4.2 (1)] including Gabber’s result [Gab81, Chapter I, Theorem 1] for the local case +dim R ď 3 when G is either GLn or PGLn. +In this article, we prove several variants of Nisnevich +conjecture over Prüfer bases, see Theorem 11.1 (ii) and Theorem 12.1 (ii). +1.13. The Grothendieck–Serre conjecture: a history. Since proposed by Serre [Ser58, page 31] and +Grothendieck [Gro58, pages 26–27, Remarques 3], [Gro68a, Remarques 1.11 a)], the Grothendieck–Serre +conjecture has already various known cases beyond the trivial dim R “ 0 case for fields, as listed below. +(i) The case when G is a torus is proved by Colliot-Thélène and Sansuc in [CTS87]. +(ii) The case when dim R “ 1, namely, R is a discrete valuation ring, was addressed by Nisnevich in +[Nis82] and [Nis84], then is improved and generalized to the semilocal Dedekind case in [Guo22]. +Several special cases were proved in [Har67], [BB70], [BTIII] over discrete valuation rings, and in +[PS16], [BVG14], [BFF17], [BFFH20] for the semilocal Dedekind case. +(iii) The case when R is Henselian was settled in [BB70] and [CTS79, Assertion 6.6.1] by reducing +the triviality of G-torsors to residue fields then inducting on dim R to reach Nisnevich’s resolved +case. +4 + +(iv) The equicharacteristic case, namely, when R contains a field k, was established by Fedorov and +Panin [FP15] when k is infinite (see also [PSV15], [Pan20b] for crucial techniques) and by Panin +[Pan20a] when k is finite, which was later simplified by [Fed22a]. Before these, several equichar- +acteristic subcases were proved in [Oja80],[CTO92], [Rag94], [PS97], [Za˘ı00], [Oja01], [Oja04], +[Pan05], [Zai05], [Che10], [PSV15]. +(v) When R is of mixed characteristic, Česnavičius [Čes22a] settled the case when G is quasi-split +and R is unramified (that is, for p :“ charpR{mRq, the ring R{pR is regular). Prior to this, +Fedorov [Fed22b] proved the split case under additional assumptions on R. Recently, Česnavičius +[Čes22c, Theorem 1.3] settled a generalized Nisnevich conjecture under certain conditions, which +specializes to the equal and mixed characteristic cases of the Grothendieck–Serre proved in [FP15], +[Pan20a], [Čes22a]. +(vi) There are sporadic cases where R or G are speical (with possible mixed characteristic condition), +see [Gro68a, Remarque 1.11 a)], [Oja82], [Nis89], [Fed22b], [Fir22], [BFFP22], [Pan21]. +1.14. Notations and conventions. All rings in this paper are commutative with units, unless stated +otherwise. For a point s of a scheme (resp., for a prime ideal p of a ring), we let κpsq (resp., κppq) denote +its residue field. For a global section s of a scheme S, we write Sr 1 +ss for the open locus where s does not +vanish. For a ring A, we let Frac A denote its total ring of fractions. For a morphism of algebraic spaces +S1 Ñ S, we let p´qS1 denote the base change functor from S to S1; if S “ Spec R and S1 “ Spec R1 are +affine schemes, we will also write p´qR1 for p´qS1. +Let S be an algebraic space, and let G be an S-group algebraic space. For an S-algebraic space T , by +a G-torsor over T we shall mean a GT :“ G ˆR T -torsor (see Definition 5.2). Denote by TorspSfppf, Gq +(resp., TorspS´et, Gq) the groupoid of G-torsors on S that are fppf-locally (resp., étale-locally) trivial; +specifically, if G is S-smooth (e.g., G is S-reductive, see below), then every fppf-locally trivial G-torsor +is étale-locally trivial, so we have +TorspSfppf, Gq “ TorspS´et, Gq. +For an algebraic space S, a reductive S-group algebraic space is a smooth affine S-group algebraic space +whose geometric S-fibers are (connected) reductive algebraic groups. For a scheme S this coincides with +the definition of reductive S-groups schemes given in [SGA 3III new]. +Acknowledgements. The authors would like to thank Kęstutis Česnavičius and Ivan Panin for their +constant encouragements. We thank Matthew Morrow and Colliot-Thélène for proposing the Grothendieck– +Serre on smooth schemes over semilocal Prüfer rings during the defense of the first author. On several +occasions during the past few months, we talked about some aspects of this article with Kęstutis Čes- +navičius, Arnab Kundu, Shang Li, and Ivan Panin. We thank them for these conversations. We thank +Kęstutis Česnavičius for helping us to remove the assumptions on finite residue fields in our original for- +mulation of the Theorem 12.1. After an earlier version of this paper was finished, Kęstutis Česnavičius +kindly sent to us his note which contained a sketch of a different proof of the Theorem 12.1(i) in the +Noetherian case. We thank Jiandi Zou for useful suggestions about the article. This project has received +funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research, +the innovation programme (grant agreement No. 851146), the grant 075-15-2022-289, and the excellent +environment for research of the Euler International Mathematical Institute. +2. Auslander–Buchsbaum formula over valuation rings +The goal of this section is to establish Theorem 2.8.1, the Auslander–Buchsbaum formula over finite rank +valuation rings as an analogue of the classical regular case [AB57, Theorem 3.7]. Based on the upper- +bound of projective dimensions [GR18, Proposition 11.4.1], we induct by using the notion of depths. +2.1. Coherent rings and schemes. For a ring A, a finitely generated A-module M is coherent if its +any finitely generated A-submodule is finitely presented. A ring A is coherent if it is a coherent A-module. +For a scheme X, an OX-module F is coherent if, for every affine open U Ă X, ΓpU, Fq is a coherent +ΓpU, OUq-module. A scheme X is locally coherent if OX is a coherent OX-module. A locally coherent +scheme is coherent if it is quasi-compact quasi-separated. +Example 2.2. Noetherian rings and Prüfer rings are coherent rings ([SP, 05CY, 0EWV]). Although +Noetherian schemes are coherent, open subschemes of affine integral Prüfer schemes are not coherent +5 + +in general: there exists a valuation ring V such that Spec V ztmV u has no closed points and is not +quasi-compact. +Lemma 2.3. Let A be a coherent ring. +(i) For every multiplicative subset S Ă A, the localization S´1A is a coherent ring. +(ii) Any A-module M is coherent if and only if it is finitely presented over A. +Further, the full +subcategory of coherent A-modules is an abelian subcategory of the category of A-modules and is +closed under taking extensions. +Proof. For (i), see [Gla89, Theorem 2.4.2]. For the first assertion of (ii), see [FK18, Chapter 0, Corol- +lary 3.3.5]. +□ +2.4. Depth. For a local ring A and the closed point x P Spec A, consider the following functor +Γtxu : A-Mod Ñ A-Mod +M ÞÑ ker +´ +ΓpSpec A, Ă +Mq Ñ ΓpSpec Aztxu, Ă +Mq +¯ +sending M to its largest A-submodule supported on txu. The functor Γtxu is left exact so gives rise to a +right derived functor RΓtxu : D`pA-Modq Ñ D`pA-Modq. The depth of M P D`pA-Modq is +depthApMq :“ suptn P Z | RiΓtxuM “ 0 +for all i ă nu P Zě0 Y t`8u, +For an A-module N supported on txu, we also consider the following closely related quantity +τNpMq :“ suptn P Z | Exti +ApN, Mq “ 0 +for all i ă nu P Zě0 Y t`8u. +Lemma 2.5. For a local ring A, an A-module M, and an M-regular sequence pf1, ¨ ¨ ¨ , frq in mA, +depthApMq “ depthApM{ řr +i“1 fiMq ` r +and +τNpMq “ τNpM{ řr +i“1 fiMq ` r. +Proof. The two equalities are proved similarly, so we only treat the one concerning depth. By induction +on r, we are reduced to the case when r “ 1 and f1 “ f is a nonzerodivisor of M in mA. From the short +exact sequence 0 Ñ M +fÝÑ M Ñ M{fM Ñ 0, we derive the following long exact sequence +¨ ¨ ¨ Ñ Ri´1ΓtxuM +fÝÑ Ri´1ΓtxuM Ñ Ri´1ΓtxupM{fMq Ñ RiΓtxuM +fÝÑ RiΓtxuM Ñ ¨ ¨ ¨ . +If depthA M “ `8, then M “ 0 so it suffices to assume that depthA M “ d for an integer d ě 0. If +d “ 0, then there is a nontrivial A-submodule of M supported on txu, contradicting to the assumption +that f P mA is a nonzerodivisor of M. Therefore, we have d ě 1 and RiΓtxuM “ 0 for every 0 ď i ď d´1. +The displayed long exact sequence implies that RiΓtxupM{fMq “ 0 for every 0 ď i ď d ´ 2 (if d ´ 2 ă 0 +then such i does not exist). If Rd´1ΓtxupM{fMq “ 0, then the map RdΓtxuM ãÑ RdΓtxuM induced by +multiplication by f is injective. However, since the nonzero A-module RdΓtxuM is supported on txu and +f P mA, we deduce that RdΓtxuM “ 0, that is, depthAM ą d, a contradiction. Consequently, we have +Rd´1ΓtxupM{fMq ‰ 0 and depthApM{fMq “ d ´ 1 “ depthA M ´ 1. +□ +Example 2.6. Assume that A is Noetherian, and take N “ A{I for an ideal I of A (for instance, +N “ A{mA). Then for any finitely generated A-module M we have +depthA M “ τNpMq. +Indeed, utilizing Lemma 2.5, one verifies easily that both of them equals the length of any maximal +M-regular sequence in mA (so the length is independent of all choices). However, this is false when +A is non-Noetherian. For instance, we let A :“ V be a non-discrete valuation ring of finite rank and +let N :“ V {mV be its residue field. +Take M “ V {fV for a nonzero f P mV . +Then one checks +immediately that depthV pV {fV q “ 0, but there are no nonzero elements of V {fV annihilated by mV . +Thus HomV pV {mV , V {fV q “ 0, and so τV {mV pV {fV q ě 1 ą 0 “ depthV pV {fV q. +Lemma 2.7. For a valuation ring V of finite rank, a V -flat finitely presented scheme X, and a point +x P X with image s P Spec V such that OXs,x is regular, +(i) we have depthApAq “ d ` 1, where d “ dim OXs,x; +(ii) for any A-module N supported on txu, we have Exti +ApN, Aq “ 0 for all i ď d. +6 + +Proof. +(i) Since by assumption V has nonzero finite rank, we can pick an element f P mV such that +dim V {pfq “ 0. Let pg1, ¨ ¨ ¨ , gdq be a sequence in mA such that their images in the regular local +ring A{mV A forms a regular system of parameters, and hence also forms a regular sequence. By +the flatness criterion [EGA IV3, Théorème 11.3.8], pg1, ¨ ¨ ¨ , gdq is a regular sequence of A, and +the quotient ring A :“ A{pg1, ¨ ¨ ¨ , gdq is V -flat with mV A “ mA. Therefore, pg1, ¨ ¨ ¨ , gd, fq is a +regular sequence of A in mA such that dim A{pfA ` řd +i“1 giAq “ 0. Applying Lemma 2.5 yields +depthApAq “ depthApA{pfA ` řd +i“1 giAqq ` d ` 1 “ d ` 1. +(ii) Repeating the preceding argument involving Lemma 2.5, we deduce the following inequality +τNpAq “ τNpA{pfA ` řd +i“1 giAqq ` d ` 1 ě d ` 1. +By the definition of τNpAq, this is equivalent to the displayed vanishing. +□ +Lemma 2.8. For a local ring pA, mAq, a nonzero A-module M supported on tmAu, and a matrix H P +MmˆnpAq, if the A-linear map HM : M ‘n Ñ M ‘m induced by H (via left multiplication) is injective, +then H admits a left inverse, or, equivalently, H exhibits A‘n as a direct summand of A‘m. +Proof. Recall [SP, 0953] that the assumption on the support of M means that, for any w P M and any +finitely generated ideal I Ă A, we have INw “ 0 for large enough N. Let H “ phijq, then McCoy’s +theorem [Gla89, page 211] implies that the ideal generated by the minors of order n of H does not +annihilate a nonzero element of M. Indeed, the invertibility of minors already yields a left inverse of +H and we are done. Precisely, since M is supported at tmAu, there exist some i, j for which hij P Aˆ. +We may assume that h11 P Aˆ. By subtracting suitable multiples of the first row of H to other rows +(resp. the first column of H to other columns), we may also assume that h1j “ 0 for 1 ă j ď n and +hi1 “ 0 for 1 ă i ď m (the assumption and conclusion of the lemma are preserved if we replace H by +H1HH2, where H1 P MmˆmpAq and H2 P MnˆnpAq). In other words, we have H “ ph11q ‘ H1, where +H1 P Mpm´1qˆpn´1qpAq. Then the map H1 +M : M ‘pn´1q Ñ M ‘pm´1q induced by H1 is also injective. So +we may assume by induction that H1 admits a left inverse H2 P Mpn´1qˆpm´1qpAq. Then ph´1 +11 q ‘ H2 is +a left inverse of H. +□ +Now, we acquire the Prüferian analogy of the Auslander–Buchsbaum formula [AB57, Theorem 3.7]. +Theorem 2.8.1 (Auslander–Buchsbaum formula over valuation rings). For a valuation ring V of finite +nonzero rank, a V -flat finite type scheme X, a point x P X lying over the closed point s P Spec V such +that OXs,x is regular, and the local ring A :“ OX,x, every finitely presented A-module M satisfies +proj. dimApMq ` depthApMq “ depthApAq “ d ` 1, +where d “ dim OXs,x. +(By convention, proj. dimAp0q “ ´8) +Proof. Let M be a finitely presented nonzero A-module. We will induct on proj. dimApMq to verify the +formula. Note that, by [GR18, Proposition 11.4.1], we have proj. dimApMq ď d`1. If proj. dimApMq “ 0, +or, M is A-free, then by Lemma 2.7 we have depthApMq “ d`1, so proj. dimApMq`depthApMq “ d`1. +Next, assume that proj. dimApMq ě 1, so every partial resolution 0 Ñ M 1 +ιÝÑ A‘n Ñ M Ñ 0 is non-split +and satisfies proj. dimApM 1q “ proj. dimApMq ´ 1. We exploit the associated long exact sequence +¨ ¨ ¨ Ñ Ri´1ΓtxuM 1 Ñ Ri´1ΓtxuA‘n Ñ Ri´1ΓtxuM Ñ RiΓtxuM 1 Ñ RiΓtxuA‘n Ñ ¨ ¨ ¨ . +If proj. dimApMq “ 1, then M 1 » A‘m for some m ě 1, and the map A‘m » M 1 +ιÝÑ A‘n is given by an +n ˆ m matrix H P MnˆmpAq. We have known that proj. dimApM 1q “ d ` 1, and so RiΓtxuM 1 “ 0 for all +i ď d. It follows from the above long exact sequence that RiΓtxuM “ 0 for all i ď d´1. If RdΓtxuM “ 0, +then left multiplication by H induces an injection +` +Rd`1ΓtxuA +˘‘m “ Rd`1ΓtxupA‘mq » Rd`1ΓtxuM 1 ãÑ Rd`1ΓtxupA‘nq “ +` +Rd`1ΓtxuA +˘‘n . +Since Rd`1ΓtxuA is a nonzero A-module supported on txu, we deduce from Lemma 2.8 that H admits a +left inverse. This implies that ι splits, and so M is A-free, contradicting our assumption proj. dimApMq “ +1. Hence, depthApMq “ d, and we thus obtain the desired formula proj. dimApMq ` depthApMq “ d ` 1. +7 + +If proj. dimApMq ą 1, then proj. dimApM 1q “ proj. dimApMq ´ 1. Applying the induction hypothesis to +M 1, we have +depthApM 1q “ d ` 1 ´ pproj. dimApMq ´ 1q “ d ` 2 ´ proj. dimApMq, +which is +ď d. +It follows from the above long exact sequence and the fact depthApA‘nq “ d ` 1 that Ri´1ΓtxuM » +RiΓtxuM 1 for all i ď d. Combining this with the bound depthApM 1q ď d, we deduce that depthApMq “ +depthApM 1q ´ 1. Therefore, by induction hypothesis, we have +proj. dimApMq ` depthApMq “ +` +proj. dimApM 1q ` 1 +˘ +` +` +depthApM 1q ´ 1 +˘ +“ d ` 1. +This finishes the induction step. +□ +3. Geometry of schemes over Prüfer bases +In this section, we recollect useful geometric properties on scheme over Prüfer bases. +3.1. Geometric properties and reduction methods +Lemma 3.1.1. For a valuation ring V with spectrum S, a finite type irreducible S-scheme X, a point +x P X and its image s P S, the following assertions hold +(i) all nonempty S-fibers have the same dimension; +(ii) if X is S-flat , then X is finitely presented over S; +(iii) if X is S-flat, then for any maximal point ξ P Xs, the local ring OX,ξ is a valuation ring such +that the extension OS,s ãÑ OX,ξ of valuation rings induces an isomorphism of value groups; +(iv) for x1 P X that is distinct with x whose image is denoted by s1, if x P tx1u, then +‚ either htpsq “ htps1q (i.e., s “ s1) and then dimpOXs1 ,x1q ă dimpOXs,xq; +‚ or htps1q ă htpsq and then dimpOXs1 ,x1q ď dimpOXs,xq. +Proof. For (i), see [EGA IV3, Lemme 14.3.10]. For (ii), see [Nag66, Theorem 3’]. For (iii), see [MB22, +Théorème A]. Now, to prove (iv), we may assume that X is affine and of some pure relative dimension, +say, n, over V . +By assumption, we have s P ts1u. +Assume that we are not in the first case, then +htps1q ă htpsq. The schematic closure tx1u is a finite type dominant scheme over ts1u (the spectrum of +a valuation ring), so by (i), all its non-empty fibers have the same dimension. Thus, we deduce from +tx1u Ą txu that +dimptx1us1q “ dimptx1usq ě dimptxusq. +Hence, we have +dimpOXs1 ,x1q “ n ´ dimptx1us1q ď n ´ dimptxusq “ dimpOXs,xq. +□ +The following Lemma 3.1.2 provides us a passage to the case when there is a section. +Lemma 3.1.2. For a valuation ring V , an essentially finitely presented (resp., essentially smooth) V - +local algebra A, there are an extension of valuation rings V 1{V with trivial extension of value groups, +and an essentially finitely presented (resp., essentially smooth) V -map V 1 Ñ A with finite residue fields +extension. +Proof. Assume A “ OX,x for an affine scheme X finitely presented over V and a point x P X lying over the +closed point s P SpecpV q. Let t “ tr.degpκpxq{κpsqq. As κpxq is a finite extension of l :“ κpsqpa1, ¨ ¨ ¨ , atq +for a transcendence basis paiqt +1 of κpxq{κpsq, we have t “ diml Ω1 +l{κpsq ď dimκpxq Ω1 +κpxq{κpsq. +Choose +sections b1, ¨ ¨ ¨ , bt P ΓpX, OXq such that db1, ¨ ¨ ¨ , dbt P Ω1 +κpxq{κpsq are linearly independent over κpxq, +where the bar stands for their images in κpxq. Define p : X Ñ At +V by sending the standard coordinates +T1, ¨ ¨ ¨ , Tt of At +V to b1, ¨ ¨ ¨ , bt, respectively. Since db1, ¨ ¨ ¨ , dbt P Ω1 +κpxq{κpsq are linearly independent, the +image η :“ ppxq is the generic point of At +κpsq, so V 1 :“ OAt +V ,η is a valuation ring whose value group is +ΓV 1 » ΓV . Note that κpxq{κpηq is finite, the map V 1 Ñ A induces a finite residue fields extension. +When V Ñ A is essentially smooth, the images of db1, ¨ ¨ ¨ , dbt under the map Ω1 +X{V b κpxq Ñ Ω1 +κpxq{κpsq +are linearly independent, so are their images in Ω1 +X{V b κpxq. Hence, p is essentially smooth at x. +□ +8 + +In the sequel, we will use the following limit argument repeatedly. +Lemma 3.1.3. Every semilocal Prüfer domain R is a filtered direct union of its subrings Ri such that: +(i) for every i, Ri is a semilocal Prüfer domain of finite Krull dimension; and +(ii) for i large enough, Ri Ñ R induces a bijection on the sets of maximal ideals hence is fpqc. +Proof. Write FracpRq “ YiKi as the filtered direct union of the subfields of FracpRq that are finitely +generated over its prime field K. Let Ri :“ R X Ki. Then R “ YiRi. It remains to see that every Ri +is a semilocal Prüfer domain whose local rings have finite ranks. Let tpju1ďjďn be the set of maximal +ideals of R. Then R “ Ş +1ďjďn Rpj is the intersection of the valuation rings Rpj. Thus we have +Ri “ Ş +1ďjďn +` +Ki X Rpj +˘ +. +Since Ki{K has finite transcendence degree, by Abhyankar’s inequality, every Ki X Rpj is a valuation +ring of finite rank. By [BouAC, VI, §7, Proposition 1–2], Ri is a semilocal Prüfer domain, and its local +rings at maximal ideals are precisely the minimal elements of the set tKi X Rpju1ďjďn under inclusion. +This implies (i). For (ii), it suffices to show that for i large enough there are no strict inclusion relation +between Ki X Rpj1 and Ki X Rpj2 for j1 ‰ j2. Indeed, if πj P pjz Ť +j1‰j pj1 for 1 ď j ď n, then (ii) holds +for any i for which tπju1ďjďn Ă Ki. +□ +3.2. Reflexive sheaves on schemes over Prüfer bases with regular fibers +3.2.1. Reflexive sheaves. Assume that X is a locally coherent scheme, see 2.1. For an OX-module F, +its dual is denote by F _ :“ HomOXpF, OXq. A coherent OX-module F is reflexive if the canonical map +F Ñ F __ is an isomorphism. Since every coherent OX-module G is Zariski-locally finitely presented +O‘m +X +Ñ O‘n +X +Ñ G Ñ 0, by taking dual, G _ is finitely copresented as 0 Ñ G _ Ñ O‘n +X +Ñ O‘m +X +. In +particular, the dual G _ of a coherent OX-module G is also coherent (equivalently, finitely presented). +Moreover, Lemma 3.2.2 shows that for integral X and every coherent OX-module G , the double dual +G __ is OX-reflexive, hence G __ is the reflexive hull of G . +Lemma 3.2.2 (reflexive hull). For a locally coherent integral scheme X and two OX-modules F and +G , if F is coherent and G is reflexive, then H :“ HomOXpF, G q is reflexive. In particular, the double +dual +F __ :“ HomOXpHomOXpF, OXq, OXq +is a reflexive OX-module. +Proof. For the coherence of H , it suffices to take a presentation O‘m +X +Ñ O‘n +X +Ñ F Ñ 0 of F and its +sheaf homomorphism with G so that H “ kerpG ‘n Ñ G ‘mq which is coherent by [SP, 01BY]. +Claim 3.2.3. For a domain R, a finitely presented R-module M, and an exact sequence 0 Ñ M Ñ M 1 Ñ +M 2 of finite R-modules, if M 1 is reflexive and M 2 is torsion-free, then M is reflexive. +Proof of the claim. Denote p´q_ “ HomRp´, Rq and consider the following commutative diagram +0 +M +M 1 +M 2 +M __ +M 1__ +M 2__. +By [SP, 0AV0], M 1 is torsion-free, so is M, hence the map M ãÑ M __ is injective. It remains to show +that this map is surjective. For the map u: M 1_ Ñ M _, consider the exact sequence M 1_ Ñ M _ Ñ +cokerpuq Ñ 0. As M 1 is reflexive, it is finitely presented, so [SP, 0583] applies, yielding the exact sequence +HomRpM 1 bR K, Kq Ñ HomRpM bR K, Kq Ñ cokerpuq bR K Ñ 0, +where K :“ Frac R. Since K is R-flat, the injectivity of M ãÑ M 1 implies that cokerpuqK “ 0, hence +cokerpuq is R-torsion and cokerpuq_ “ 0. Therefore, M __ ãÑ M 1__ is injective. Because M 2 is torsion- +free, the map M 2 ãÑ M 2__ is injective. By snake lemma, M ։ M __ is surjective so M is reflexive. +□ +Since H is coherent, it is finitely presented. The desired reflexivity follows from Claim 3.2.3. +□ +By reflexive hull, reflexive sheaves extend from quasi-compact open (cf. [GR18, Proposition 11.3.8(i)]). +9 + +Corollary 3.2.4. For a coherent reduced scheme X with an quasi-compact open U Ă X, the restriction +OX-Rflx Ñ OU-Rflx +is essentially surjective. +Proof. It suffices to assume that X is irreducible, so X is integral. Every reflexive OU-module F, by +[GR18, Lemma 10.3.24 (ii)], extends to a finitely presented quasi-coherent OX-module Ă +F, which is +coherent. Then by Lemma 3.2.2, the reflexive hull Ă +F __ is a reflexive extension of F on X. +□ +Corollary 3.2.5. For a locally coherent integral scheme X and two OX-modules F and G , if F is coher- +ent and G is reflexive, then the natural map HomOXpF __, G q +„ +ÝÑ HomOXpF, G q is an isomorphism. +Proof. Locally on X the reflexive OX-module G fits into an exact sequence 0 Ñ G Ñ O‘m +X +Ñ O‘n +X , +hence we have the following commutative diagram of OX-modules with exact rows +0 +HomOXpF __, G q +HomOXpF __, O‘m +X +q +HomOXpF __, O‘n +X q +0 +HomOXpF, G q +HomOXpF, O‘m +X +q +HomOXpF, O‘n +X q +By Lemma 3.2.2, F _ is reflexive, hence the two rightmost vertical arrows are bijective and so is the +leftmost vertical arrow, as desired. +□ +Lemma 3.2.6. Let X Ñ S be a finite type morphism with regular fibers between topologically Noetherian +schemes, let j : U ãÑ X be a quasi-compact open immersion with complement Z :“ XzU satisfying +codimpZs, Xsq ě 1 for every s P S +and +codimpZη, Xηq ě 2 for every generic point η P S, +and let F be a reflexive OX-module. Assume that S is a cofiltered inverse limit of integral schemes +pSλqλPΛ with generic point ηλ and surjective transition maps. Then, there is a λ0 P Λ, a finite type +morphism Xλ0 Ñ Sλ0 with regular fibers such that Xλ0 ˆSλ0 S » X, a closed subscheme Zλ0 Ă Xλ0 such +that Zλ0 ˆSλ0 S » Z, the open immersion jλ0 : Xλ0zZλ0 ãÑ Xλ0 is quasi-compact, and the following +codimppZλ0qs, pXλ0qsq ě 1 for every s P Sλ0 +and +codimppZλ0qη0, pXλ0qη0q ě 2 +is satisfied. Also, there is a reflexive OXλ0 -module Fλ0 whose inverse image on X is F. +Proof. The condition that X has regular S-fibers descends to Xλ0 by [EGA IV2, Proposition 6.5.3]. +The reflexive OX-module F descends thanks to [EGA IV3, Théorème 8.5.2] and by applying [EGA IV3, +Corollaire 8.5.2.5] to F +„ +ÝÑ F __. Because Z is contructible closed, by [EGA IV3, Théorème 8.3.11], it +descends to Zλ such that p´1 +λ pZλq “ Z. For fλ : Xλ Ñ Sλ, by the transversity of fibers and [EGA IV2, +Corollaire 4.2.6], Zλ does not contain any irreducible components of f ´1 +λ psλq for any sλ P Sλ. Finally, +the image of the generic point η P S is the generic point ηλ P Sλ. By [EGA IV2, Corollaire 6.1.4], we +have codimppZλqηλ, pXλqηλq “ codimpZη, Xηq ě 2. +□ +Proposition 3.2.7. For a valuation ring V with spectrum S and a flat, locally of finite type morphism +f : X Ñ S of integral schemes with regular fibers, the following assertions hold. +(i) For every x P X and every coherent OX-module F that is reflexive at x, we have +proj.dimOX,xFx ď maxp0, n ´ 1q, +where +n “ dim Of ´1pfpxqq,x. +(ii) For a closed subset Z Ă X such that j : XzZ ãÑ X is quasi-compact and satisfies the following +codimpZs, Xsq ě 1 for all s P S +and +codimpZη, Xηq ě 2 for the generic point η P S, +the restriction functors induce the following equivalences of categories. +OX-Rflx +„ +ÝÑ OXzZ-Rflx +Pic X +„ +ÝÑ Pic XzZ +(3.2.7.1) +In particular, for every X-affine finite type algebraic space Y , we have a bijection of sets +Y pXq » Y pXzZq. +10 + +(iii) For a closed subset Z Ă X such that j : XzZ ãÑ X is quasi-compact and XzZ contains all the +associated points of the generic fiber of X and every X-separated algebraic space Y , the map +Y pXq ãÑ Y pXzZq +is injective. +(iv) For a closed subset Z Ă X satisfying the assumption in (ii) and a quasi-compact quasi-separated +morphism p: W Ñ XzZ such that p˚OW is a reflexive OXzZ-module, we have the Cartesian +square +AffXzZW +AffXW +XzZ +X, +paff +ν +j +where AffXzZW “ SpecXzZpp˚OW q and AffXW “ SpecXpj˚p˚OW q, such that paff and ν are +finite, paff is the relative normalization [SP, 035H] of XzZ in W and ν is the relative normalization +of X in W. In particular, ν˚pOAffpW{Xqq is a reflexive OX-module. +(v) For a closed subset Z Ă X satisfying the assumption in (ii) and a finite flat locally finitely +presented morphism p: W Ñ XzZ, the morphism ν : AffXW Ñ X is the relative normalization +of X in W such that pAffXWqXzZ “ W. In particular, ν˚pOAffXW q is a reflexive OX-module. +Proof. The assertion (i) is [GR18, Proposition 11.4.1 (iii)]. For (ii), by Lemmata 3.1.3 and 3.2.6, we +may assume that V has finite rank. Since |X| is the finite disjoint union of its S-fibers Xs, which are +Noetherian spaces, we know that X is topologically Noetherian. In particular, every open subset of X is +quasi-compact. By Corollary 3.2.4, the functors (3.2.7.1) are essentially surjective. For the faithfulness, +consider two morphisms α, β : F Ñ G between reflexive OX-modules such that α|XzZ “ β|XzZ. To show +that α “ β, since it is a local problem, it suffices to check that αx “ βx : Fx Ñ Gx for every x P Z. Take +a presentation O‘m +X,x Ñ O‘n +X,x Ñ Fx Ñ 0 and copresentation 0 Ñ Gx Ñ O‘m1 +X,x Ñ O‘n1 +X,x, then αx and βx +induce two morphisms between these copresentations. Then we are reduced to the case when Fx and Gx +are free. We may assume that Fx “ O‘r +X,x and Gx “ O‘s +X,x, so the following isomorphisms lead to α “ β +HomOX,xpFx, Gxq “ HomOX,xpO‘r +X,x, O‘s +X,xq » Homj˚OX,xpj˚O‘r +X,x, j˚O‘s +X,xq. +It remains to show that (3.2.7.1) are full. If F and G are two reflexive OX-modules with a morphism +φ: j˚F Ñ j˚G , then by [GR18, Corollary 11.3.9], taking j˚p´q induces the following morphism +rφ: F » j˚j˚F Ñ j˚j˚G » G . +For the second assertion of (ii), by the sheaf property, the problem is étale local on X, so we can +assume that X is affine. Choose an embedding Y ãÑ An +X for some integer n. The assumption implies +that XzZ is scheme-theoretically dense in X. Hence, for every morphism φ: XzZ Ñ Y , if φ extends +uniquely to a morphism rφ: X Ñ An +X, then rφ´1pY q is a closed subscheme of X containing XzZ and by +[EGA IV4, Lemme 20.3.8.8], coincides with X. In other words, if rφ exists uniquely, then it factorises as +X +ψÑ Y ãÑ An +X such that ψ is the unique extension of φ. This reduces us to the case Y “ An +X. Now, by +the reflexivity of OX and the full faithfulness of OX-Rflx +„ +ÝÑ OXzZ-Rflx, we have the desired bijections +An +XpXq “ HomOXpOX, O‘n +X q » HomOXpOXzZ, O‘n +XzZq “ An +XpXzZq. +To prove (iii), we first prove that XzZ Ă X is scheme-theoretically dense in the sense of [SP, 0834]. +By [SP, 0836], we need to show that OX Ñ j˚OXzZ is injective, which through the terminology of +[GR18, 10.4.2 and 10.4.19], signifies that δpz, OXq ą 0 for all z P Z. It suffices to take étale coverings of +X by schemes and use the depth formula [GR18, Corollary 10.4.46] for flat morphisms to deduce that +all z P Z satisfies δpx, OXq ą 0. Since j is quasi-compact, by [SP, 0835], the schematic image of XzZ is +X. Therefore, we apply [SP, 084N] to conclude. The (iv) follows from (ii). For (v), note that p˚OW is +OXzZ-reflexive since by [SP, 02KB], p is finite locally free, hence it is a special case of (iv). +□ +11 + +4. Auslander’s flatness criterion on schemes smooth over valuation rings +The goal is to establish Theorem 4.1 as a counterpart of Auslander’s flatness criterion [Aus62, Theo- +rem 1.3] on schemes smooth over valuation rings. As expected, our criterion leads to a Zariski–Nagata +purity. +Theorem 4.1. For a valuation ring V with spectrum S and closed point s P S, an S-smooth finite type +scheme X, a point x P X lying over s with local ring A :“ OX,x, and a reflexive A-module M, +EndApMq is isomorphic to a direct sum of copies of M +if and only if +M is A-free. +As Auslander’s proof, our strategy relies on an estimate of the length of cohomology groups of M. To +begin with, we introduce the length function on torsion modules over valuation rings. +4.2. Lengths of torsion modules. For a nontrivial valuation ring V with fraction field K, value group +Γ and a valuation map ν : K Ñ Γ, every finitely presented torsion V -module M is of the form +M » À +i V {aiV +for finitely many ai P V zt0u. +Define the length of M as δpMq “ ř +i νpaiq P Γě0. The element δpMq is well defined, and δpMq “ 0 if +and only if M “ 0. Every acyclic, bounded complex M ‚ of torsion, finitely presented V -modules satisfies +ř +jp´1qjδpM jq “ 0. +Lemma 4.3. For a nontrivial valuation ring V , an essentially smooth V -local algebra pA, mAq, and the +collection A-Modtor,fp of all finitely presented A-modules M such that SupppMq Ă tmAu, there exist a +totally ordered abelian group Γ and a map l: A-Modtor,fp Ñ Γě0 satisfying the following properties: +‚ for A-module M P A-Modtor,fp, we have lpMq “ 0 if and only if M “ 0; +‚ for every acyclic, bounded complex M ‚ such that M j P A-Modtor,fp for each j, one has +ř +jp´1qjlpM jq “ 0. +Proof. First we assume that the structural map V Ñ A admits a section A ։ V . In this case we claim +that M is finitely presented over V and is V -torsion, so we can simply let Γ be the valuation group of V +and set lpMq :“ δpMq, where δ is delivered from 4.2. Indeed, it is clear that M is V -torsion. Any section +Spec V Ñ Spec A is a regular immersion [SP, 067R], so there is a finitely generated ideal J Ă A such +that V » A{J. Hence, since M P A-Modtor,fp, we see that JnM “ 0 for a large n. On the other hand, +the essential smoothness of A over V implies that J{J2 is a free V » A{J-module whose rank equals the +rank of the free A-module Ω1 +A{V , and there is a natural isomorphism of graded V » A{J-algebras +À +ně0 Jn{Jn`1 » Sym‚ +A{JpJ{J2q. +In particular, A{Jn is a finite free V -module for every n ě 1. Therefore, by tensoring a presentation +AN Ñ AN Ñ M Ñ 0 +of M with A{Jn for a large enough n, we get a desired finite presentation of the V -module M. +In the general case, we first use Lemma 3.1.2 to reduce to the case when the residue fields extension of +V Ñ A is finite. Then, if B is the integral closure of V in an algebraic closure of FracpV q, we let V 1 be a +valuation ring of FracpBq centered at a maximal ideal of B. It’s clear that V 1 is absolutely integral closed, +so it is strictly Henselian and there exists a V -map φ : A{mA Ñ V 1{mV 1. Let A1 :“ A bV V 1. Then φ +induces a V 1-map φ1 : A1 Ñ V 1{mV 1; let p Ă A1 be its kernel. Then A1 +p is essentially smooth over V 1 and +φ1 induces a V 1-map A1 +p Ñ V 1{mV 1, which, by the Henselianity of V 1, lifts to a V 1-map A1 +p Ñ V 1. By the +previous paragraph, the lemma is true for A1 +p, say, with corresponding map l1 valued in Γ, where Γ is +the valuation group of V 1. Since A Ñ A1 +p is faithfully flat, it suffices to define lpMq :“ l1pM bA A1 +pq. +□ +Lemma 4.4. For a valuation ring V , a V -smooth finite type scheme X, a point x P X that lies over a +non-generic point s P SpecpV q, and a map of finitely presented OX,x-modules M Ñ N that induces an +isomorphism over SpecpOX,xqztxu, we have an isomorphism Exti +OX,xpN, OX,xq „ +ÝÑ Exti +OX,xpM, OX,xq for +every i ă d and a monomorphism Extd +OX,xpN, OX,xq ãÑ Extd +OX,xpM, OX,xq, where d :“ dim OXs,x. +12 + +Proof. Let ker, coker, and im be the kernel, cokernel, and image of M Ñ N, respectively. By assumption +and the coherence of OX,x, ker and coker are coherent, or, equivalently, finitely presented OX,x-modules +([SP, 05CX]) supported at txu. Consider the following short exact sequences +0 Ñ ker Ñ M Ñ im Ñ 0, +0 Ñ im Ñ N Ñ coker Ñ 0. +By applying HomOX,xp´, OX,xq, we get two long exact sequences concerning Ext’s, and the lemma follows +from the vanishing Exti +OX,xpker, OX,xq “ 0 and Exti +OX,xpcoker, OX,xq “ 0 for i ď d (Lemma 2.7). +□ +Lemma 4.5. For finitely presented A :“ OX,x-modules M and N, Exti +ApM, Nq and TorA +i pM, Nq are +finitely presented over A for all i ě 0 and are zero for i ą d ` 1, where d “ dim OXs,x. +Proof. By [GR18, Proposition 11.4.1 (i)], since A is coherent, the coherent A-module ([SP, 05CX]) M +has a resolution by finite free A-modules of length ď d ` 1: F‚ Ñ M, Fi “ 0 for i ą d ` 1. Then +Exti +ApM, Nq “ HipHompF‚, Nqq +and +TorA +i pM, Nq “ HipF‚ b Nq +are all coherent, or equivalently, finitely presented A-modules, and are zero for i ą d ` 1. +□ +Lemma 4.6. For a finitely presented A :“ OX,x-module M, we have a natural isomorphism +EndApMq__ +„ +ÝÑ EndApM __q. +Proof. First, we define a natural map EndApMq__ Ñ EndApM __q. Note that M __ is A-reflexive due +to Lemma 3.2.2. By Corollary 3.2.5, where M __ plays the role of G , we get a natural isomorphism +HomApM, M __q +„ +ÐÝ EndApM __q. +It suffices to consider the natural maps EndApMq Ñ HomApM, M __q +„ +ÐÝ EndApM __q. By Lemma 3.2.2, +the two rightmost modules are reflexive. +Taking double dual yields the desired map of reflexive A- +modules. +It remains to check that the map EndApMq__ Ñ EndApM __q is an isomorphism. The equivalence of +categories of reflexive modules in Proposition 3.2.7(ii) reduces us to checking this at x P X that is either +a one-codimensional point of the generic V -fiber or a maximal point of a non-generic V -fiber, where, by +Lemma 3.1.1(iii), A is a valuation ring, so there is an N P Zě0 and finitely many ai P mAzt0u such that +M » A‘N À p‘iA{aiAq . +Consequently, we conclude by the isomorphisms EndApMq__ » EndApA‘Nq » EndApM __q. +□ +Proof of Theorem 4.1. The proof proceeds as the following steps. +Preliminary cases and reductions. First, since X is locally of finite presentation over S and M +is finitely presented over A, by a standard limit argument involving Lemmata 3.1.3 and 3.2.6, we are +reduced to the case when V is a finite-rank valuation ring. Secondly, if V 1 is a valuation ring of an +algebraic closure of FracpV q that dominates V and if x1 P X1 :“ X ˆV V 1 is a point lying over x P X, +then MA1 :“ M bA A1 is a finitely presented reflexive A1-module and EndA1pMA1q » EndApMq bA A1 is +isomorphic to a direct sum of copies of MA1, where A1 :“ OX1,x1 (because A1 is faithfully flat over A). By +faithfully flat descent [SP, 08XD, 00NX], the freeness of M over A is equivalent to the freeness of MA1 +over A1. Therefore, by replacing V by V 1, A by A1, and M by MA1, we are reduced to the case when +FracpV q is algebraically closed (this assumption will be only used in the very end of the proof). +Set dx :“ dimpOXs,xq and r :“ rankpV q. The case r “ 0 and dx arbitrary is classical. The case r +arbitrary and dx “ 0 is trivial, because A is a valuation ring (Lemma 3.1.1(iii)). The case r arbitrary +and dx “ 1 follows from Proposition 3.2.7(i). Subsequently, we may assume dx ě 2 in the sequel. +Case 1: r is arbitrary and dx “ 2. Now, we deal with the crucial case when r arbitrary and dx “ 2 +by induction on r. The induction hypothesis is that the assertion holds for dx “ 2 and r1 ď r ´1. Notice +that, for any proper generalization x1 P X of x that lies over, say, s1 P SpecpV q, by Lemma 3.1.1(iv), +we have either s1 “ s and dx1 ă 2, or htps1q ă r and dx1 ď 2. Hence, by induction hypothesis and +13 + +the preliminary cases above, the assertion holds for OX,x1. Since Mx1 is a finitely presented reflexive +OX,x1-module and +EndOX,x1 pMx1q “ EndOX,xpMq bOX,x OX,x1 » p +à +Mq bOX,x OX,x1 “ +à +Mx1, +the induction hypothesis applies to the OX,x1-module Mx1, implying that Mx1 is OX,x1-free. In other +words, Ă +M is locally free over Spec Aztxu. Consider the following evaluation map +M _ bA M Ñ HomApM, Mq, +f b m ÞÑ rm1 ÞÑ fpm1qms, +which, by the local freeness of Ă +M over Spec Aztxu, is an isomorphism over Spec Aztxu. Since dx “ 2 ą 1, +by Lemma 4.4, we apply Ext1 +Ap´, Aq to the above map to obtain the following isomorphism +Ext1 +ApM _ b M, Aq » Ext1 +ApEndApMq, Aq » Ext1 +ApM, Aq‘rkM +(4.6.1) +of A-modules that are supported on txu by the local freeness of Ă +M over Spec Aztxu, where rkM “ +dimFrac A M bA Frac A. By Lemma 4.5, the modules in (4.6.1) are also finitely presented over A. +For the adjunction HomApM, HomApM _, ´qq » HomApM b M _, ´q, we take their derived functors +valued at A, so the E2-page of the associated Grothendieck spectral sequence yields a monomorphism +Ext1 +ApM, Mq ãÑ Ext1 +ApM b M _, Aq +p4.6.1q +» +Ext1 +ApM, Aq‘rkM , +where we have used M __ » M; again, by the local freeness of Ă +M over Spec Aztxu and Lemma 4.5, they +are finitely presented supported on txu. In particular, the map l from Lemma 4.3 applies so we have +lpExt1 +ApM, Mqq ď rkM ¨ lpExt1 +ApM, Aqq. +(4.6.2) +Since M is reflexive, by Proposition 3.2.7(i), we have proj.dimAM ď dx´1 “ 1. We prove proj.dimpMq “ +0 by contradiction. If proj.dimpMq “ 1, then M has a free resolution 0 Ñ F1 Ñ F0 Ñ M Ñ 0 by finite +A-modules. As M is not free, the sequence is nonsplit, corresponding to a nontrivial extension class in +Ext1 +ApM, F1q » Ext1 +ApM, AqrankpF1q. +In particular, we have C :“ Ext1 +ApM, Aq ‰ 0. Applying HomAp´, Aq to F‚ Ñ M yields an exact sequence +0 Ñ M _ Ñ F _ +0 Ñ F _ +1 Ñ Ext1 +ApM, Aq Ñ 0. +Tensoring it with M, we get an exact sequence F _ +0 bA M Ñ F _ +1 bA M Ñ Ext1 +ApM, Aq bA M Ñ 0. Since +coker pF _ +0 b M Ñ F _ +1 b Mq » coker pHomApF0, Mq Ñ HomApF1, Mqq “ Ext1 +ApM, Mq, +we deduce that Ext1 +ApM, Mq » Ext1 +ApM, Aq bA M “ C bA M. +By tensoring 0 Ñ F1 Ñ F0 Ñ M Ñ 0 with C “ Ext1 +ApM, Aq (which is a nonzero finitely presented +A-module supported at txu, by the locally freeness of Ă +M over Spec Aztxu), we get an exact sequence +0 Ñ TorA +1 pC, Mq Ñ C bA F1 Ñ C bA F0 Ñ C bA M Ñ 0 +of finitely presented A-modules supported on txu. Applying the map l from Lemma 4.3, we obtain +lpC bA Mq “ lpC bA F0q ´ lpC bA F1q ` lpTorA +1 pC, Mq “ rkM ¨ lpCq ` lpTorA +1 pC, Mqq, +(4.6.3) +where rkM “ rankpF0q ´ rankpF1q ą 0. On the other hand, since C bA M » Ext1 +ApM, Mq, we deduce +lpC bA Mq +p4.6.2q +ď +rkM ¨ lpCq. +(4.6.4) +The combination of (4.6.3) and (4.6.4) leads to lpTorA +1 pC, Mqq “ 0. So, we have an exact sequence +0 Ñ C bA F1 Ñ C bA F0 Ñ C bA M Ñ 0, +which combined with Lemma 2.8 implies that the map F1 Ñ F0 splits, that is, M is A-free, contradicting +our assumption that proj.dimApMq “ 1. This completes the case when r is arbitrary and dx “ 2. +Case 2: r is arbitrary and dx ą 2. We deduce by double induction on the pair pr “ htpsq, dxq. By +induction hypothesis, the assertion holds for all smooth V -scheme X1 and all points x1 P X1 such that +htps1q ď htpsq and dx1 ď dx, where s1 P SpecpV q lies below x1, and at least one of equalities is strict. In +particular, by Lemma 3.1.1(iv), the induction hypothesis applies to OX,x1 for all proper generalization +x1 P X of x. Since Mx1 is a finitely presented reflexive OX,x1-module and +EndOX,x1 pMx1q “ EndOX,xpMqx1 » +à +Mx1, +the induction hypothesis gives that Mx1 is OX,x1-free. In other words, Ă +M is locally free over Spec Aztxu. +14 + +Claim 4.6.5 ([SP, 057F]). Assume that the residue field extension of V Ñ A is separable (e.g., this holds +if κpsq :“ V {mV is perfect), then there exists an a P A such that A :“ A{paq is essentially V -smooth and +dimpA{mV Aq “ dx ´ 1. +Since our V has algebraically closed fraction field (by the first paragraph), all of its primes have alge- +braically closed residue fields, so we can choose a P A as in the above claim. Since a is nonzerodivisor in +A and M “ HomApM _, Aq, we see that a is M-regular. Set M :“ M{aM. Applying HomApM, ´q to +the short exact sequence 0 Ñ M +aÝÑ M Ñ M Ñ 0, we get an exact sequence +0 Ñ HomApM, Mq aÝÑ HomApM, Mq Ñ HomApM, Mq Ñ Ext1 +ApM, Mq. +Substituting our assumption HomApM, Mq – M ‘rkM into it yields an exact sequence of A-modules +0 Ñ M +‘rkM Ñ HomApM, Mq Ñ T Ñ 0, +where T Ă Ext1 +ApM, Mq is a finitely presented A-submodule (Lemma 4.5), which, by the locally freeness +of Ă +M over Spec Aztxu, is supported on txu. Since dimpA{mV Aq “ dx´1 ě 2, taking dual (as A-modules) +of the above short exact sequence and using Lemma 4.4, we see that +pM +_q‘rkM » HomApM, Mq_. +Taking dual further and invoking Lemma 4.6, we get the following isomorphism +pM +__q‘rkM » HomApM +__, M +__q. +Since the double dual M +__ is finitely presented over A and is reflexive (Lemma 3.2.2), we can apply our +induction hypothesis to the A-module M +__ and conclude that it is A-free. The same lemma also implies +that M +_ is A-reflexive, so M +_ » M +___ is A-free. +Finally, we show that M is A-free. Since Ă +M is locally free over Spec Aztxu, the natural map M Ñ M +__ +is an isomorphism over Spec Aztxu, and, since dimpA{mV Aq “ dx ´ 1 ą 1, we may apply Lemma 4.4 to +see that Ext1 +ApM, Aq » Ext1 +ApM +__, Aq “ 0. Since a is M-regular, we deduce that +Ext1 +ApM, Aq » Ext1 +ApM bL +A A, Aq » Ext1 +ApM, Aq “ 0. +Applying HomApM, ´q to the short exact sequence 0 Ñ A aÝÑ A Ñ A Ñ 0 we get an exact sequence +0 Ñ M _ +aÝÑ M _ Ñ HomApM, Aq Ñ Ext1 +ApM, Aq aÝÑ Ext1 +ApM, Aq Ñ Ext1 +ApM, Aq. +As all modules are finitely presented over A and Ext1 +ApM, Aq “ 0, Nakayama’s lemma gives that +Ext1 +ApM, Aq “ 0. Therefore, M _{aM _ » HomApM, Aq “ M +_ is A-free (by the previous paragraph). +From this we can deduce that M is A-free. Indeed, the A-free module M _{aM _ has projective dimension +1 over A, thus, for any finitely presented A-module N, we can derive from +0 Ñ M _ +aÝÑ M _ Ñ M _{aM _ Ñ 0 +an exact sequence of finitely presented A-modules +Ext1 +ApM _, Nq aÝÑ Ext1 +ApM _, Nq Ñ Ext2 +ApM _{aM _, Nq. +As Ext2 +ApM _{aM _, Nq “ 0, by Nakayama’s lemma, we have Ext1 +ApM _, Nq “ 0. In particular, for any +surjection A‘n ։ M _ with, say, kernel N, the extension class of the short exact sequence 0 Ñ N Ñ +A‘n Ñ M _ Ñ 0 is zero. This implies that M _ is A-free, hence so is M “ M __. +□ +5. Generalities on torsors over algebraic spaces +5.1. Setup. Throughout this section, we let S denote a base scheme, X an algebraic space over S, and +G an X-group algebraic space. +Definition 5.2. +(1) A (right) G-torsor (for the fppf topology) is an X-algebraic space P equipped with a G-action +a : P ˆX G Ñ P such that the following conditions hold: +(i) the induced morphism P ˆX G +„ +ÝÑ P ˆX P, pp, gq ÞÑ pp, app, gqq, is an isomorphism; and +(ii) there exists a fppf covering tXi Ñ XuiPI of algebraic spaces [SP, 03Y8] such that PpXiq ‰ H +for every i P I. +15 + +(2) For G-torsors P1 and P2, a morphism P1 Ñ P2 is a G-equivariant morphism P1 Ñ P2 of X- +algebraic spaces. +(3) By a trivialization of a G-torsor P we mean a G-equivariant isomorphism t : G +„ +ÝÑ P, where G +acts on itself via right multiplication; this amounts to the choice of a section tp1Gq P PpXq (if +exists). A G-torsor P is trivial if there exists a trivialization, or, equivalently, if PpXq ‰ H. +Note that every morphism of two G-torsors is an isomorphism. To see this, one may pass to a fppf +covering of X to reduce to the case when both torsors are trivial, in this case the assertion is trivial. +Remark 5.3. One can also define a sheaf torsor for an X-group algebraic space G. It is a sheaf +P : pSch{Sqopp +fppf Ñ Set +equipped with a map P Ñ X of sheaves and a G-action a : P ˆX G Ñ P such that the above two condi- +tions (i) and (ii) in (1) hold. However, it turns out that such a sheaf torsor is necessarily representable by +an algebraic space, so working with sheaf torsors adds no more generality. To see this, let tXi Ñ XuiPI +be a fppf covering as in (ii) that trivializes P. Then every P ˆX Xi » G ˆX Xi is an algebraic space, +and the map +Ů +i P ˆX Xi Ñ P +is representable by algebraic spaces and is a fppf covering, because it is the base change of the fppf +covering Ů +i Xi Ñ X of algebraic spaces via P Ñ X. Here, all coproducts are taken in the category of +sheaves on pSch {Sqfppf. It follows from (3) of [SP, 04S6] that P is an algebraic space, as desired. +Let P1, P2 be two G-torsors. Define a functor +IsomXpP1, P2q : pSch{Xqopp Ñ Set +which associates to any scheme T over X the set of GT -equivariant isomorphisms P1,T Ñ P2,T over T . +Lemma 5.4. For two G-torsors P1 and P2, IsomXpP1, P2q is an algebraic space over S. +Further, +G Ñ X is quasi-compact (resp., étale, smooth, flat, separated, (locally) of finite type, (locally) of finite +presentation, quasi-affine, affine, or finite) if and only if IsomXpP1, P2q Ñ X is so. +Proof. Since IsomXpP1, P2q is fppf locally on X isomorphic to G, it admits a representable fppf covering +by algebraic spaces, hence it is an algebraic space by [SP, 04S6]. +The list properties of morphisms of algebraic spaces are all stable under base changes and are fppf local on +the target, see [SP, 03KG] (resp., [SP, 03XT, 03ZF, 03MM, 03KM, 040Y, 0410, 03WM, 03WG, 03ZQ]). +Consequently, since IsomXpP1, P2q is fppf locally on X isomorphic to G, the properties of G are inherited +by and can be detected from IsomXpP1, P2q. +□ +Since every G-torsor P Ñ X trivializes over a fppf covering tXi Ñ Xu, one may try to obtain P by +glueing the trivial GXi-torsors PXi using the canonical isomorphisms +φij : pPXiqXij » PXij » PXjqXji, +where +Xij “ Xi ˆX Xj. +It turns out that, unlike the case of schemes, this is always possible in the framework of algebraic spaces, +see Lemma 5.6. Note that, by taking U :“ Ů Xi, we may assume that PU is trivial for a fppf covering +U Ñ X with U an algebraic space. +Definition 5.5 (Descent datum for torsors). Let S, X and G be as in 5.1. Let U Ñ X be a fppf covering +of algebraic spaces over S. For every integer n ě 0, denote by U pnq the n-fold fiber product of U over X. +The category of descent datum for G-torsors relative to U Ñ X, denoted +Tors +´ +pU p2q Ñ Uqfppf, G +¯ +, +has pairs pQ, φq as objects, where +‚ Q Ñ U is a GU-torsor; and +16 + +‚ φ : pr˚ +1Q +„ +ÝÑ pr˚ +2Q is an isomorphism of GUp2q-torsors such that the following diagram commutes +(i.e., the cocycle condition holds) +pr˚ +12pr˚ +1Q +pr˚ +12pr˚ +2Q +pr˚ +23pr˚ +1Q +pr˚ +23pr˚ +2Q +pr˚ +13pr˚ +1Q +pr˚ +13pr˚ +2Q. +pr˚ +12pφq +» +» +pr˚ +23pφq +» +pr˚ +13pφq +A morphism from a pair pQ, φq to another pair pQ1, φ1q is a morphism θ : Q Ñ Q1 of GU-torsors +compatible with φ and φ1, that is, pr˚ +2pθqφ “ φ1pr˚ +1pθq. +To every G-torsor P one can associate a pair ΨpPq :“ pPU, canq via base changes, where can is the +canonical isomorphism pr˚ +1pPUq » PUp2q » pr˚ +2pPUq. Thus we obtain a functor +Ψ : TorspXfppf, Gq Ñ TorsppU p2q Ñ Uqfppf, Gq. +Lemma 5.6 (Descent G-torsors). Ψ is an equivalence of category. +In other words, every descent datum pQ, φq for G-torsors are effective in the sense that there exists a +G-torsor P and an isomorphism Q » PU compatible with θ and the canonical descent datum for PU. +Proof. The full faithfulness of Ψ follows from the sheaf property of the functor IsomXpP1, P2q for any +G-torsors P1 and P2. To show that Ψ is essential surjective, we pick a descent datum pQ, φq, and we +need to show that there exists a G-torsor P such that pPU, canq » pQ, φq. +When both X and U are schemes, this is proven in [SP, 04U1]. The case of algebraic spaces can be +proved similarly, and we repeat the argument for convenience. First we view Q as a sheaf on the site +pAS{Uqfppf (by the natural equivalence of the topoi associated to pAS{Uqfppf and pSch {Uqfppf). Since +descent datums for sheaves on any site are always effective [SP, 04TR], we may find a sheaf P on the site +pAS{Xqfppf and an isomorphism of sheaves PU » Q compatible with the descent datums. Further, since +maps of sheaves on any site can be glued [SP, 04TQ], the GU-action on Q descent to a G-action on P. +All the assumptions (i) and (ii) of Definition 5.2 hold, because they can be checked on the fppf covering +U Ñ X. It remains to see that P is representable by an algebraic space over X. However, this follows +from (3) of [SP, 04S6], in view of the fact that the map Q Ñ P is representable by algebraic spaces and +is a fppf covering (being a base change of the fppf covering U Ñ X). +□ +We end this section with the following result, which will be used repeatedly in the sequel. +Lemma 5.7. Let S be a scheme, X an algebraic space over S, and G an X-group algebraic space. Let +f : Y Ñ X be a morphism of algebraic spaces over S. Assume the following conditions hold: +(i) for every fppf covering T Ñ X with T a scheme, the pullback functor +f ˚ +T : TorspTfppf, GT q Ñ TorsppYT qfppf, GYT q +is fully faithful, where YT :“ Y ˆX T , and fT :“ f ˆX T ; and +(ii) for every GY -torsor P, there is a fppf covering T Ñ X with T a scheme such that PYT lies in +the essential image of f ˚ +T . +Then pullback induces an equivalence f ˚ : TorspXfppf, GT q +„ +ÝÑ TorspYfppf, GY q. +Similarly, if G Ñ X is smooth, then we have an equivalence +f ˚ : TorspX´et, GT q +„ +ÝÑ TorspY´et, GY q, +provided that one replaces ‘fppf’ by ‘étale’ everywhere in the above assumptions. +Proof. We prove the Lemma for fppf torsors. It remains to check that f ˚ is essentially surjective. Let P +be a GY -torsor. By assumption (ii) there is a fppf covering T Ñ X with T a scheme and a GT -torsor Q +such that f ˚ +T Q » PYT . Using this isomorphism we can transform the canonical descent datum on PYT +to a descent datum +θ : pr˚ +1f ˚ +T Q +„ +ÝÑ pr˚ +2f ˚ +T Q +17 + +on f ˚ +T Q (relative to the covering YT Ñ Y ). For every integer n ě 0, denote by T pnq the n-fold fiber +product of T over X. Using the canonical identifications +pr˚ +1f ˚ +T Q “ f ˚ +T p2qpr˚ +1Q +and +pr˚ +2f ˚ +T Q “ f ˚ +T p2qpr˚ +2Q, +the full faithfulness of fT p2q implies that there is a unique isomorphism +τ : pr˚ +1Q +„ +ÝÑ pr˚ +2Q +of GT p2q-torsors such that f ˚ +T p2qpτq “ θ. Since +pr˚ +13pθq “ pr˚ +13pf ˚ +T p2qpτqq “ f ˚ +T p3qpr˚ +13pτq +and +pr˚ +13pθq “ pr˚ +23pθqpr˚ +12pθq +“ pr˚ +23 +` +f ˚ +T p2qpτq +˘ +pr˚ +12 +` +f ˚ +T p2qpτq +˘ +“ f ˚ +T p3q ppr˚ +23pτqq f ˚ +T p3q ppr˚ +12pτqq +“ f ˚ +T p3q ppr˚ +23pτqpr˚ +12pτqq , +the full faithfulness of f ˚ +T p3q implies that pr˚ +13pτq “ pr˚ +23pτqpr˚ +12pτq, that is, τ is a descent datum on Q +relative to T Ñ X. By Lemma 5.6, there is a G-torsor R and an isomorphism pQ, φq » pRT , canq of +descent datums. Pulling back to YT , we get an isomorphism of descent datums +pPYT , canq » f ˚ +T pQ, τq » pRYT , canq. +By Lemma 5.6 again (applied to the covering YT Ñ Y ), we see that f ˚pRq “ RY » P. +□ +6. Purity for torsors and finite étale covers +We begin with generalities about linear groups that will be fundamental in multiple types of purities +for reductive torsors, where the overall argument is bootstrapped from that for vector bundles. Hence, +in this process, controlling on the projective dimensions of extended reflexive sheaves leads to relative- +dimensional constraints. In particular, we obtain the purity for reductive torsors on relative curves §6.1. +We then present local variants of the acquired purity results §6.2, where the constraints on dimensions +are more flexible. By virtue of this, we shrink complements of domains of reductive torsors to a higher- +codimensional closed subset §6.3, laying the groundwork for later proofs of the Grothendieck–Serre. +Finally, by our Auslander’s flatness criterion, we present a Prüferian counterpart of the Zariski–Nagata +purity in §6.4. +6.0.1. Coaffine locally linear groups. Let X be an algebraic space. An X-group algebraic space G is +linear if there exists a group monomorphism G ãÑ GLpV q for a locally free OX-module V of finite rank; +it is fppf (resp., étale) locally linear if there exists a fppf covering (resp., an étale covering) X1 Ñ X such +that GX1 is linear. A locally linear X-group algebraic space G is coaffine, if it locally has an X-affine +coset GLpV q{G. For instance, if a linear group G Ă GLpV q is reductive or finite locally free, then +GLpV q{G is X-affine. In the sequel, we mainly consider locally linear coaffine X-group algebraic spaces +G. +6.1. Purity for reductive torsors on relative curves +Now we study the extension behavior of torsors over relative curves. Motivated by [EGA IV4, Proposi- +tion 21.9.4] that every invertible sheaf on a curve over a field extend across finitely many closed points, +Proposition 6.1.2 concerns relative curves over valuation rings and generalizes [Guo20, Lemma 7.3]. +6.1.1. Torsors on relative curves. For a valuation ring V with spectrum S, a V -flat finite type scheme +X with regular one-dimensional V -fibers, and a closed subscheme D Ă X such that +(i) D is finite locall free over V ; and +(ii) D factors through an affine open Spec R Ă X, +we consider the completion pR :“ lim +ÐÝn R{In, where I Ă R is the ideal determined by D. +Denote +BD :“ Spec pR as the formal neighborhood of D and UD :“ BDzD for the punctured formal neighborhood. +18 + +Proposition 6.1.2. For a valuation ring V with spectrum S, an S-flat finite type scheme X with regular +one-dimensional S-fibers, an S-finite locally free closed subscheme D Ă X inside an affine open X0 Ă X +with complementary open j : XzD ãÑ X, then the restriction functor between the categories +VectX Ñ VectXzD +is essentially surjective. +In particular, for the formal neighborhood BD :“ p +XD, the punctured neighborhood UD :“ BDzD, we +have +H1 +ZarpUD, GLnq “ H1 +´etpUD, GLnq “ t˚u. +Proof. Every vector bundle E on XzD by Corollary 3.2.4 extends to a reflexive sheaf rE on X. Hence +Proposition 3.2.7(i) implies that rE is a vector bundle. Now let V be a vector bundle on UD and denote +the Henselization of the pair pX0, Dq by pBh +D, Dq with U h +D :“ Bh +DzD. Then [BČ22, Corollary 2.1.22] +descends V to a vector bundle V h on U h +D. Since Bh +D is the limit of elementary étale neighorhoods of +D Ă X0, by a limiat argument, V h descends to a vector bundle V 1 on an S-flat finite type scheme X1 +with regular one-dimesensional S-fibers and the open X1zD. Since VectX1 Ñ VectX1zD is essentially +surjective, V 1 extends to a vector bundle r +V 1 on X1. Consequently, there exists a vector bundle r +V on BD +extending V . Since pBD, Dq is a Henselian pair, by [Čes22b, Proposition 6.1.1], we have an isomorphism +VectBD » VectD. Note that D is semilocal and affine, so r +V is trivial, in particular, V is trivial. +□ +Lemma 6.1.3. For a semilocal affine Prüfer scheme S, an S-flat finite type algebraic space X with +regular one-dimensional S-fibers, and its closed subset Z such that j : XzZ ãÑ X is quasi-compact and +Zη “ H +for each generic point η P S +and +codimpZs, Xsq ě 1 for all s P S, +the pushforward j˚p´q and restriction as inverse induce an equivalence between categories of vector +bundles +VectXzZ +„ +ÝÑ VectX. +Proof. We simply verify the assumptions of Lemma 5.7 for G “ GLn,X. For vector bundles E1 and E2, +Y :“ IsomXpE1, E2q +is X-affine of finite type (Lemma 5.4), so Y pXzZq “ Y pXq by Proposition 3.2.7(ii). The same holds when +we base change to every étale X-scheme. For (ii), by taking étale atlas, we may assume that X is a scheme. +By Proposition 3.2.7(ii), every vector bundle E on XzZ extends to a reflexive OX-module j˚E . To show +that the reflexive OX-module j˚E is a vector bundle, it suffices to exploit Proposition 3.2.7(i). +□ +Theorem 6.1.4 (cf. [CTS79, Théorème 6.13]). For a semilocal affine Prüfer scheme S, an S-flat finite +type algebraic space X with regular one-dimensional S-fibers, an X-group algebraic space G that is étale- +locally linear and coaffine2, and a closed subset Z Ă X such that j : XzZ ãÑ X is quasi-compact and +Zη “ H +for each generic point η P S +and +codimpZs, Xsq ě 1 for all s P S, +restriction of torsors induces the following equivalence of categories of G-torsors +TorspX´et, Gq +„ +ÝÑ TorsppXzZq´et, Gq. +In particular, passing to isomorphism classes of objects, we have an isomorphism +H1 +´etpX, Gq » H1 +´etpXzZ, Gq. +Proof. We simply verify the assumptions of Lemma 5.7. +(i) Since the assumption on the fiber codimension still holds when we base change to every étale +scheme over X, it suffices to verify that the restriction functor +TorspX´et, Gq Ñ TorsppXzZq´et, Gq +is fully faithful. Indeed, for any G-torsors P1 and P2, by Lemma 5.4, +Y :“ IsomXpP1, P2q +is an X-affine algebraic space of finite type, so Y pXzZq “ Y pXq by Proposition 3.2.7(ii). +2A special case is when X is an affine scheme and G is X-reductive, as explained in a footnote of the introduction. +19 + +(ii) Étale locally on X, every G-torsor on XzZ extends to a G-torsor on X. To see this, we may +assume that X is affine and G Ă GLn,X, then exploit the commutative diagram with exact rows +pGLn,X{GqpXq +H1 +´etpX, Gq +H1 +´etpX, GLn,Xq +pGLn,X{GqpXzZq +H1 +´etpXzZ, Gq +H1 +´etpXzZ, GLn,Xq, +» +where the bijectivity of the left vertical arrow follows from Proposition 3.2.7(ii) and our assump- +tion GLn,X{G being affine over X. For every G-torsor P on XzZ, by Lemma 6.1.3, we may +replace X by an affine open cover to ensure that the induced GLn,XzZ-torsor P ˆGXzZ GLn,XzZ +is trivial. A diagram chase implies that there exists a G-torsor Q on X such that Q|XzZ » P, as +claimed. +□ +6.2. Local variants of purity results +The following is a variant of Theorem 6.1.4. +Theorem 6.2.1. For a finite-rank valuation ring R with spectrum pS, ηq, an S-flat finite type scheme X +with regular fibers, an X-group scheme G that is étale-locally linear and coaffine, and a point x that is +(i) either x P Xη with dim OXη,x “ 2, or +(ii) x P Xs with s ‰ η and dim OXs,x “ 1, +every G-torsor over Spec OX,xztxu extends uniquely to a G-torsor over Spec OX,x. +Proof. The argument of Theorem 6.1.4 reduces us to the case of vector bundles, namely, G “ GLn. +Then the assertion (i) follows from the classical purity (see for instance, [Gab81, §1, Lemma 1]). For +(ii), by the quasi-compactness of SpecpOX,xqztxu and Proposition 3.2.7(ii), every vector bundle E on +Spec OX,xztxu, extends to a reflexive sheaf j˚pE q on Spec OX,x, which, by the assumption dim OXs,x “ 1 +and Proposition 3.2.7(i), is projective, hence the assertion follows. +□ +Lemma 6.2.2. For an algebraic space S with a finitely presented closed subspace Z Ă X and an affine +morphism of algebraic spaces f : X1 Ñ X, denote Z1 :“ Z ˆX X1, U :“ XzZ, and U 1 :“ U ˆX X1, +consider the following Cartesian square +U 1 +X1 +U +X, +fU +j1 +f +j +where j and j1 are open immersions. If f is faithfully flat and induces an isomorphism Z » Z1, then +(i) The restriction Ψ: FÉtX +„ +ÝÑ FÉtU ˆFÉtU1 FÉtX1 is an equivalence of categories. In particular, +if FÉtX1 Ñ FÉtU1 is essentially surjective (resp., an equivalence), then so is FÉtX Ñ FÉtU. +(ii) If X, X1 are schemes, then for a quasi-affine, flat, finitely presented X-group scheme G, the +following base change functor is an equivalence of categories of G-torsors +TorspXfppf, Gq +„ +ÝÑ TorspX1 +fppf, Gq ˆTorspU1 +fppf,Gq TorspUfppf, Gq. +Proof. +(i) Consider the fibered category AFF over the category of algebraic spaces such that every algebraic +space T has the fiber category AFFpT q, the category of T -affine algebraic spaces. By [MB96, +Théorème 1.1], then base change induces the following equivalence of categories +ΦAFF : AFFpXq +„ +ÝÑ AFFpX1q ˆAFFpU1q AFFpUq. +Hence Ψ is fully faithful. For the essential surjectivity, it suffices to patch finite étale covers over +U and X1 to an X-affine algebraic space, and conclude by using faithfully flat descent for finite +étale properties. +(ii) See [Čes22a, Lemma 7.1]. +□ +20 + +Corollary 6.2.3. For a local scheme X, the closed point x and punctured spectrum U :“ Xztxu, if for +the Henselization Xh of X at x with punctured spectrum U h, +FÉtXh +„ +ÝÑ FÉtUh +is an equivalence if and only if so is +FÉtX +„ +ÝÑ FÉtU. +Proposition 6.2.4. Let X1 Ñ X be a flat morphism of affine schemes that are smooth over a semilocal +Prüfer domain R with spectrum pS, ηq such that there is a closed subscheme Z Ă X satisfies the following +(i) codimpZs, Xsq ě 1 for every s P S and codimpZη, Xηq ě 2 for the generic point η P S; and +(ii) X1 Ñ X induces an isomorphism between Z and its preimage Z1 :“ Z ˆX X1. +Denote U :“ XzZ and U 1 :“ U ˆX X1. For an affine, smooth X-group (resp., U-group) F with a +filtration +F “ F0 Ą F1 Ą ¨ ¨ ¨ Ą Fn “ 0 +by affine smooth S-normal subgroups (U-normal subgroups) such that every subquotient Fi{Fi`1 is a +vector group on X (resp., such that Fi{Fi`1 is a vector group on S and is central in F{Fi`1), the map +H1 +´etpU, Fq Ñ H1 +´etpU 1, Fq +has trivial kernel (resp., is surjective). +Proof. When F is an X-group, since X and X1 are affine, both H1pX, Fq and H1pX1, Fq vanish. Then, +for every F-torsor P on U that becomes trivial over U 1, we utilize Lemma 6.2.2 to patch trivial torsors on +X1 and U to obtain a trivial F-torsor rP on X such that rP|U » P. Hence, P is trivial and the displayed +map has trivial kernel. +Now assume that F is a U-group and we induct on n. When n “ 1, then F is associated to a vector bundle +F on U. Let j : U ãÑ X denote the open immersion, then for j˚F we apply [GR18, Lemma 10.4.17 (iii)] +to deduce that RΓZpX, j˚Fq » RΓZ1pX1, j˚Fq. Consequently, we have HipU, Fq +„ +ÝÑ HipU 1, Fq. When +n ą 1, we invoke the nonabelian cohomology sequences [Gir71, Chapitre IV, Remarque 4.2.10] for a +central extension to acquire the following commutative diagram with exact rows +H1pU, Fn´1q +H1pU, Fq +H1pU, F{Fn´1q +H2pU, Fn´1q +H1pU 1, Fn´1q +H1pU 1, Fq +H1pU 1, F{Fn´1q +H2pU 1, Fn´1q +„ +„ +by a diagram chase up to twist technique [Gir71, Chapitre III, Proposition 2.6.1(i)], we conclude. +□ +Theorem 6.2.5. For a semilocal Prüfer domain R with spectrum S and generic point η, an S-smooth +algebraic space X, and a point x P X that is not any maximal point of S-fibers of X such that dim OX,x ě +2, then pullback induces an equivalence of categories of finite étale covers +FÉtSpec OX,x +„ +ÝÑ FÉtSpec OX,xztxu. +Further, for a qc open immersion j : U ãÑ X such that every z P XzU satisfies the condition for x, +FÉtX Ñ FÉtU +is essentially surjective. +Proof. If x R Xη and dim OXs,x “ 1, then the assertion is due to Theorem 6.1.4. The remained case +is proved below. To show that FÉtX Ñ FÉtU is essentially surjective, let f : rU Ñ U be a finite étale +cover and we use Noetherian induction to reduce to showing that the finite étale cover f extends to an +open subset of U strictly containing U. Pick a maximal point of XzU so U ˝ :“ U ˆX Spec OX,x “ +Spec OX,xztxu. Restricting f over U ˝ to f ˝ : rU ˝ Ñ U ˝, the equivalence FÉtSpec OX,x +„ +ÝÑ FÉtSpec OX,xztxu +yields a finite étale cover W Ñ U. A spreading out [SP, 0BQ5, 0EY3] provides an open neighborhood +x P U 1 Ą U with a finite étale cover W 1 Ñ U 1 extending rU Ñ U, as desired. +□ +Remark 6.2.6. Let S be a semilocal affine geometrically unibranched scheme with total ring of fractions +K. For an étale locally constant group scheme E over S of finite type3, the map H1 +´etpS, Eq ãÑ H1 +´etpK, Eq +has trivial kernel. Let T be an E-torsor that trivializes over K. This signifies that T pKq ‰ H. Since +S is geometrically unibranched, by [SGA 3II, Exposé X, Théorème 5.16] (the Noetherian assumption is +3This means that after a finite étale covering, the constant group is a finite type abelian group, see [SGA 3II, X, 5.1] +21 + +removable), E is isotrivial, so there is a finite étale covering S1 Ñ S with total ring of fractions K1 such +that ES1 is a constant group in finite type abelian group. Therefore, we have the commuative diagram +T pSq +T pS1q +T pS1 ˆ S1q +T pKq +T pK1q +T pK1 ˆ K1q +so descent yields the equality T pSq “ T pKq. (If S is the spectrum of a Prüfer domain and E is S-finite, +then this is simplier by valuative criterion for properness) In particular, we have T pSq ‰ H so T is +trivial. +Remark 6.2.7. For a valuation ring V with fraction field K, every reductive K-group scheme G has +at most one reductive V -model. To see this, we let G be a reductive V -model of G and consider the +commutative diagram with exact rows +H0 +´etpV, OutpGqq +H1 +´etpV, Gadq +H1 +´etpV, AutpGqq +H1 +´etpV, OutpGqq +H0 +´etpK, OutpGqq +H1 +´etpK, Gadq +H1 +´etpK, AutpGqq +H1 +´etpK, OutpGqq. +f0 +f1 +f2 +f3 +The map f1 is injective by [Guo20]. By diagram chase, f2 has trivial kernel, so we are done. +6.3. Extending generically trivial torsors +Granted the purity Theorem 6.2.1, we extend reductive torsors outside a closed subset of higher codi- +mension. +Proposition 6.3.1. For a semilocal affine Prüfer scheme S, an S-flat finite type scheme X with regular +S-fibers, a closed subset Z Ă X such that XzZ Ă X is quasi-compact and satisfies the following condition +codimpZη, Xηq ě 2 for each generic point η P S +and +codimpZs, Xsq ě 1 for all s P S, +and a reductive X-group scheme G, there is a closed subset Z1 Ă Z satisfying the following condition +codimpZ1 +η, Xηq ě 3 for each generic point η P S +and +codimpZ1 +s, Xsq ě 2 for all s P S, +such that every G-torsor on XzZ extends to a G-torsor on XzZ1. +Proof. Write R “ colimλPΛRλ as in Lemma 3.1.3. By a standard limit argument ([SP, 0EY1, 0C0C]), +for large enough λ P Λ, the scheme X, the open XzZ Ă X, and the reductive X-group scheme G descend +to a quasi-compact quasi-separated Rλ-smooth scheme Xλ, a quasi-compact open pXzZqλ Ă Xλ, and a +reductive Xλ-group scheme Gλ, respectively. Also, up to enlarging λ, the G-torsor over XzZ in question +descends to a Gλ-torsor over pXzZqλ. By Lemma 3.2.6 that descends the fiberwise codimension of Z, +we are reduced to the case when all local rings of R are valuation rings of finite ranks. +Let PXzZ be a G-torsor over XzZ. Since S has finitely many points and each fiber Xs is Noetherian, +there are finitely many points x P Z satisfying one of the assumptions (i)-(ii) of Theorem 6.2.1; among +these points we pick a maximal one under the generalization, say x. The maximality of x implies that +pXzZqXSpecpOX,xq “ SpecpOX,xqztxu, so, by Theorem 6.2.1, the G-torsor PXzZ|XzZXSpecpOX,xq extends +to a G-torsor Px over SpecpOX,xq. As X is topologically Noetherian, we may spread out Px to obtain a +G-torsor PUx over an open neighbourhood Ux of x such that PXzZ|pXzZqXUx » PUx|pXzZqXUx as G-torsors +over pXzZqXUx. Consequently, we may glue PXzZ and PUx to obtain a G-torsor over U1 :“ pXzZqYUx. +Since Z1 :“ XzU1 contains strictly fewer points x satisfying the assumptions (i) or (ii) of Theorem 6.2.1, +we extend P iteratively to find the desired closed subset Z1 Ă X such that PXzZ extends over XzZ1. +□ +Corollary 6.3.2. For a semilocal Prüfer affine scheme S, an S-flat finite type scheme X with regular +S-fibers, finitely many points x Ă X contained in an affine open, a nonzero element r P OX,x, and a +reductive X-group scheme G, every generically trivial G-torsor over OX,xr 1 +rs extends to a G-torsor over +an open neighbourhood U of SpecpOX,xr 1 +rsq whose complementary closed Z :“ XzU satisfies the following +codimpZη, Xηq ě 3 for each generic point η P S +and +codimpZs, Xsq ě 2 for all s P S. +22 + +Proof. As in the proof of Proposition 6.3.1, we may assume that S has finite Krull dimension; in particular, +X is topologically Noetherian. Let P be a generically trivial G-torsor over OX,xr 1 +rs. By spreading out, P +extends to a G-torsor PU over U :“ Spec Rr 1 +rs for a subring R Ă OX,x. It remains to extend U and PU +to ensure that Z :“ XzU satisfies the assumptions of Proposition 6.3.1. Let z P Z be such that either +(i) z P Xη and dim OX,z “ 1, in which case SpecpOX,zq X U is a maximal point of X, or +(ii) z is a maximal point of Xs with s ‰ η, in which case SpecpOX,zq, and hence also SpecpOX,zq X U, +is the spectrum of a valuation ring (Lemma 3.1.1(iii)). +By the Grothendieck–Serre over valuation rings [Guo20], the generically trivial G-torsor PU|SpecpOX,zqXU +is trivial. Thus, as in the proof of Proposition 6.3.1, we can glue PU with the trivial G-torsor over a +small enough open neighbourhood of z to extend PU across such a point z P Z. Note that Z contains +finitely many points z satisfying the above assumption (i) or (ii). Therefore, iteratively extend U and +PU, we may assume that Z does not contain any point z satisfying (i) or (ii), when Proposition 6.3.1 +applies. +□ +6.4. Purity for finite locally free torsors and the Zariski–Nagata +With the purity for reflexive sheaves and Auslander’s flatness criterion Theorem 4.1 in hand, we obtain +the purity theorem for finite locally free torsors and establish our non-Noetherian Zariski–Nagata. +Theorem 6.4.1 (Purity for finite locally free groups). +(i) For a semilocal affine Prüfer scheme S, an S-smooth algebraic space X, an X-finite locally free +group algebraic space G, and a closed subset Z Ă X such that j : XzZ ãÑ X is quasi-compact and +codimpZη, Xηq ě 2 +for each generic point η P S +and +codimpZs, Xsq ě 1 for all s P S, +the restriction functor induces the following equivalence of categories of G-torsors. +TorspXfppf, Gq +„ +ÝÑ TorsppXzZqfppf, Gq. +In particular, passing to isomorphism classes of objects, we have the following isomorphism +H1 +fppfpX, Gq » H1 +fppfpXzZ, Gq. +(ii) For a finite-rank valuation ring R with spectrum S, an S-smooth scheme X, an X-finite locally +free group scheme G, and a point x P X that is not a maximal point of S-fibers of X such that +dim OX,x ě 2, the restriction functor induces the following equivalence of category of G-torsors +TorsppSpec OX,xqfppf, Gq +„ +ÝÑ TorsppSpec OX,xztxuqfppf, Gq. +In particular, passing to isomorphism classes of objects, we have the following isomorphism +H1 +fppfpSpec OX,x, Gq » H1 +fppfpSpec OX,xztxu, Gq. +Proof. (i) We simply verify the assumptions of Lemma 5.7. By considering the space IsomX of isomor- +phisms of two torsors (see Lemma 5.4), we deduce from Proposition 3.2.7(ii) that the restriction functor +is fully faithful. The same holds when we base change to every étale X-scheme over S. +Next, we show that, étale locally on X, any G-torsor on XzZ extends to a G-torsor over X. For this, we +may assume that X is an affine scheme and S is the spectrum of a valuation ring. By a standard limit +argument involving Lemma 3.1.3, we reduce to the case when S has finite Krull dimension. Since every +R-fiber of X is Noetherian and S has finitely many points, X is topologically Noetherian. +Let P be a GXzZ-torsor. Then j˚OP by Proposition 3.2.7(iv) is a reflexive OX-module. First, we prove +the OX-flatness of j˚OP. Since X is topologically Noetherian, we use Noetherian induction to reduce +to the case when X is local and essentially smooth over R and Z “ txu is its closed point. Then, our +Auslander’s flatness criterion Theorem 4.1 reduces us to showing that the following is an isomorphism +HomOXpj˚OP, j˚OPq » pj˚OPq‘r , +where +r “ rankOXOG. +Note that in such local case, we have OG » O‘r +X , consider the following map of reflexive OX-modules +HomOXpOG, j˚OPq Ñ HomOXpj˚OP, j˚OPq, f ÞÑ +´ +j˚OP +j˚ρ +ÝÝÑ OG bOX j˚OP +pf,idq +ÝÝÝÑ j˚OP +¯ +. +23 + +This is an isomorphism: by Proposition 3.2.7(ii), it suffices to argue over XzZ, then its explicit inverse +is +g ÞÑ +´ +OGXzZ +idb1 +ÝÝÝÑ OGXzZ bOXzZ OP +pρ,idq´1 +ÝÝÝÝÝÑ OP bOXzZ OP +pg,idq +ÝÝÝÑ OP +¯ +. +Then, we prove that the G-torsor structure of P extends uniquely to that of SpecXpj˚OPq. As G is finite +locally free, by projection formula [SP, 01E8], taking j˚ of the co-action ρ : OP Ñ j˚OG bOXzZ OP yields +j˚ρ: +j˚OP Ñ OG bOX j˚OP. +To check that j˚ρ is a co-action, we verify the associativity, the commutativity of the following diagram +j˚OP +OG bOX j˚OP +OG bOX j˚OP +OG bOX OG bOX j˚OP, +j˚pρq +j˚pρq +idbj˚pρq +µGbid +where µG : OG Ñ OG bOX OG is the co-multiplication of G. Since all sheaves involved are OX-reflexive, +the commutativity over XzZ by Proposition 3.2.7(ii) extends over X. Finally, the following map +pj˚ρ, 1 b idq: +j˚OP bOX j˚OP Ñ OG bOX j˚OP, +by the OX-flatness of j˚OP and Proposition 3.2.7(ii), is an isomorphism since so is its restriction on +XzZ. +(ii) This can be proved similarly. For instance, for the essential surjectivity of the restriction functor, +the finite rank assumption on V guarantees j : Spec OX,xztxu ãÑ Spec OX,x to be quasi-compact quasi- +separated, and so j˚OP is a reflexive OX,x-module (by Proposition 3.2.7(ii)) for any G-torsor P over +Spec OX,xztxu. Then one uses Auslander’s flatness criterion Theorem 4.1 to show that j˚OP is OX,x-free +and inherits the G-torsor structure on P, giving the desired extension of P to Spec OX,x. +□ +Theorem 6.4.2 (Zariski–Nagata: purity for finite étale covers). +(i) For a semilocal affine Prüfer scheme S, an S-smooth algebraic space X, and a closed subset +Z Ă X such that XzZ ãÑ X is quasi-compact and satisfies the following condition +codimpZη, Xηq ě 2 +for each generic point η P S +and +codimpZs, Xsq ě 1 for all s P S, +the pullback functor induces the following equivalence between categories of finite étale covers +FÉtX +„ +ÝÑ FÉtXzZ. +In particular, for every geometric point x: Spec Ω Ñ XzZ with a separably closed field Ω, the +map +π´et +1 pXzZ, xq Ñ π´et +1 pX, xq +is an isomorphism. +(ii) For a finite-rank valuation ring R with spectrum S and generic point η P S, an S-smooth scheme +X, and a point which is either x P Xη with dim OXη,x “ 2, or x P Xs with s ‰ η and dim OXs,x “ +1, the pullback functor induces the following equivalence of categories of finite étale covers +FÉtSpec OX,x +„ +ÝÑ FÉtSpec OX,xztxu. +Proof. (i) Full faithfulness. For two finite étale covers πi : Xi Ñ X, i “ 1, 2, consider the X-functor +Y :“ HomXpX1, X2q +that parameterizes X-morphisms from X1 to X2; it is a subfunctor of HomXpπ2,˚OX2, π1,˚OX1q con- +sisting of sections compatible with algebraic structures of π2,˚OX2 and π1,˚OX1, which amount to the +commutativity of a certain diagram of OX-modules. So Y Ă HomXpπ2,˚OX2, π1,˚OX1q is a closed sub- +functor Zariski-locally. Hence, Y is an algebraic space finite over X. (Using the infinitesimal criterion +for formal smoothness, one can check that Y Ñ X is even finite étale, but we will not need this.) By +Proposition 3.2.7(ii), we have Y pXq » Y pXzZq, thereby proving the full faithfulness. +Essential surjectivity. Let V Ñ XzZ be a finite étale cover. We need to show that it extends to a finite +étale cover of X. By the full faithfulness, we may use glueing in the étale topology to reduce to the case +that X is an affine scheme. By the S-smoothness of X, X and also V is normal, so, by breaking X and V +into connected components, we may assume that both X and V are integral schemes. Let rV Ñ XzZ be +24 + +a connected finite étale Galois cover dominating V Ñ XzZ, say with Galois group G :“ GalprV {pXzZq. +Let H :“ GalprV {V q Ă G. By Theorem 6.4.1(i), the G-Galois cover rV Ñ XzZ extends (uniquely) to a G- +Galois cover ˜W Ñ X. By Grothendieck–Galois correspondence, the subcover Ă +W{H Ñ X corresponding +to the subgroup H Ă G is a finite étale cover that extends V Ñ XzZ. +(ii) This is proved in the same way as (i), using Theorem 6.4.1(ii) in place of Theorem 6.4.1(i). +□ +7. Geometric lemmata for the Grothendieck–Serre +7.1. Geometric presentation lemma over Prüfer bases +In both of the works of Fedorov and ˇCesnaviˇcius on mixed charateristic Grothendieck–Serre, a certain +type geometric results in the style of Gabber-Quillen play a prominent role, see [Fed22b, Proposition 3.18] +and [Čes22a, Variant 3.7], respectively. This is also true in our context, and we begin with an analog of +[Čes22a, Variant 3.7]. +Lemma 7.1.1. Let R be a semilocal Prüfer ring, X a projective, flat R-scheme with fibers of pure +dimension d ą 0, OXp1q a R-ample line bundle on X, W Ă Xsm an open, x Ă W finitely many points, +and Y Ă X a closed subscheme that is R-fiberwise of codimension ą 0. Upon replacing OXp1q by any +large power, there exists nonzero +h0 P ΓpX, OXp1qq, h1 P ΓpX, OXpw1qq, ¨ ¨ ¨ , hd´1 P ΓpX, OXpwd´1qq +with +w1, ¨ ¨ ¨ , wd´1 ą 0, +such that +(i) the hypersurface H0 :“ V ph0q Ă X is disjoint from x; +(ii) the hypersurfaces Hi :“ V phiq Ă X satisfy Y X H0 X ¨ ¨ ¨ X Hd´1 “ H; +(iii) in the following commutative diagram with vertical maps determined by the h0, ¨ ¨ ¨ , hd´1: +XzH0 +XzpH0 X ¨ ¨ ¨ X Hd´1q +X :“ BlXph0, ¨ ¨ ¨ , hd´1q +Ad´1 +R +PRp1, w1, ¨ ¨ ¨ , wd´1q +PRp1, w1, ¨ ¨ ¨ , wd´1q, +π +π +π +the map π is smooth of relative dimension 1 at x; +(iv) we have Y X H0 X π´1pπpxqq “ H; +(v) if Y zXsm is R-fiberwise of codimension ě 2 in X, then π is smooth at Y X π´1pπpxqq; +(vi) if Y zW is R-fiberwise of codimension ě 2 in X, then pY zWq X π´1pπpxqq “ H; +(vii) if Y zW is R-fiberwise of codimension ě 2 in X, then there are affine opens +S Ă Ad´1 +R +and +x Ă U Ă W X π´1pSq Ă XzH0 +such that π : U Ñ S is smooth of relative dimension 1 and Y X U “ Y X π´1pSq is S-finite. +Proof. This can be proved similarly as [Čes22a, Variant 3.7]. +□ +7.2. A variant of Lindel’s lemma +According to a lemma of Lindel [Lin81, Proposition 1 et seq Lemma], an étale extension of local rings +A Ñ B with trivial extension of residue fields automatically induces isomorphisms +A{rnA +„ +ÝÑ B{rnB, +where +n ě 1, +for a well-chosen non-unit r P A. In our context in which the prescribed B is essentially smooth over a +valuation ring, we will prove the following variant of loc. cit. by allowing to fix the r P B in advance, at +the cost of that A is a carefully-chosen local ring of an affine space over that valuation ring. This result +will be the key geometric input for dealing with torsors under a reductive group scheme that descends to +the Prüfer base ring, and, as the cited work of Lindel on the Bass–Quillen conjecture for vector bundles, +it reduces us to studying torsors on opens of affine spaces. +25 + +Proposition 7.2.1. Let Λ be a semilocal Prüfer domain, X an irreducible, Λ-smooth affine scheme of +pure relative dimension d ą 0, Y Ă X a finitely presented closed subscheme that avoids all the maximal +points of the Λ-fibers of X, and x Ă X a finite subset. Assume that for every maximal ideal m Ă Λ with +finite residue field, there are at most maxp# κpmq, dq ´ 1 points of x lying over m. There are an affine +open neighbourhood W Ă X of x, an affine open subscheme U Ă Ad +Λ, and an étale surjective Λ-morphism +f : W Ñ U such that the restriction f|WXY : W X Y Ñ U is a closed immersion and f induces a +Cartesian square: +W X Y +W +W X Y +U. +f +Moreover, if Y is a Cartier divisor on X, then W X Y is a Cartier divisor on U. +Remark 7.2.2. The assumption on the cardinality of x holds, for instance, either if x is a singleton or +if d ą # x. The latter will be critical to settle the general semilocal case of Theorem 12.1. On the other +hand, the following finite field obstruction shows a certain assumption on #x is necessary: if d “ 1 and +Λ “ k is a finite field, then the map f delivered from Proposition 7.2.1 gives a closed immersion x ãÑ A1 +k, +which is impossible as soon as # x ą # k. +To prove Proposition 7.2.1 we begin with the following reduction: +Lemma 7.2.3. The proof of Proposition 7.2.1 reduces to the case when x consists of closed points of the +closed Λ-fibers of X. +Proof. As an initial step, by a standard limit argument involving Lemma 3.1.3, we can reduce to the +case when SpecpΛq has a finite underlying space (which we will assume from now on). +If for each x P x the closure txu contains a closed point x1 of the closed Λ-fibers of X and if the new +collection tx1 : x P xu satisfies the same cardinality assumption on x, we can simply replace each x by +x1 to complete the reduction process. However, it may happen that txu does not contain any point of +the closed Λ-fibers of X, and even if it does, the new collection tx1 : x P xu may destroy the cardinality +assumption on x. To overcome this difficulty, we will use a trick by adding auxiliary primes to SpecpΛq +(and adding the corresponding fibers to X and Y ) so that txu contains closed points of the closed Λ-fibers +of X for all x P x. More precisely, we will show that there are a semilocal Prüfer domain Λ1, an open +embedding SpecpΛq Ă SpecpΛ1q, an irreducible, affine, Λ1-smooth scheme X1 of pure relative dimension +d, a closed Λ1-subscheme Y 1 Ă X1 that avoids all the maximal points of the Λ1-fibers of X1, and a +Λ-isomorphism X1 +Λ » X that identifies Y 1 +Λ with Y such that the assumptions of the first sentence of this +paragraph hold for our new X1 and Y 1. +To construct the desired Λ1 (and X1, Y 1), we can first use the specialization technique to reduce to the +case when all points of x are closed in the corresponding Λ-fibers of X, that is, if x P x lies over p Ă Λ, +then x is κppq-finite. For the rest of proof we will assume, without lose of generality, that there is exactly +one point of x, say x, that lies over some non-maximal prime of Λ, say p. Write Λp “ Ť A as a filtered +union of its finitely generated Z-subalgebras A. By a standard limit argument ([SP, 0EY1, 0C0C]), for +large enough A, +(a) XΛp descends to an irreducible, affine, A-smooth scheme X of pure relative dimension d; +(b) the finitely presented closed subscheme YΛp Ă XΛp descends to a closed A-subscheme Y Ă X +which, upon enlarging A, avoids all the maximal points of the A-fibers of X: by [EGA IV3, +Proposition 9.2.6.1], +the subset +ts P SpecpA : dim Ys “ du Ă SpecpA +is constructible, +and its pullback to SpecpΛpq “ limA SpecpA is empty, hence after enlarging A we can assume +that it is already empty; +(c) the κppq-finite point x descends to a A{pA-finite closed subscheme rx Ă XA{pA, where pA :“ AXp; +For any prime Λ Ą q Ą p with htpqq “ htppq ` 1, choose an element aq P qzp. We assume that +(d) a´1 +q +P A for all such q. (This guarantees the equality A ¨ Λm “ Λp for every maximal ideal m Ă Λ +containing p.) +26 + +Since a maximal ideal m Ă Λ containing p gives rise to a non-trivial valuation ring Λm{pΛm of κppq, the +field κppq is not finite. As κppq “ Ť +A A{pA, by enlarging A we may assume that A{pA is also not a finite +field, and therefore we can find a nonzero prime p1 Ă A{pA. (We have used the following fact: for a finite +type Z-algebra, a prime ideal is maximal if and only if its residue field is finite.) Choose a valuation +ring of κppAq with center p1 in A{pA, and then extend it to a valuation ring Vp1 of κppq. Let V be the +composite of Λp and Vp1; explicitly, V is the preimage of Vp1 under the reduction map Λp ։ κppq. Then +V is a valuation ring of FracpΛq, and, by the above assumption (d), the equality V ¨ Λm “ Λp holds for +any maximal ideal m Ă Λ containing p. Therefore, by [BouAC, VI, §7, Proposition 1-2], +Λ1 :“ Λ X V +is a semilocal Prüfer domain whose spectrum is obtained by glueing SpecpΛq with SpecpV q along their +common open SpecpΛpq. Consequently, we may glue X with XV along XΛp to extend X to an irreducible, +affine, Λ1-smooth scheme X1 of pure relative dimension d, with a closed Λ1-subscheme Y 1 Ă X1 obtained +by glueing Y with YV along YΛp; by construction, Y 1 avoids all the maximal points of the Λ1-fibers of +X1. Since the closed subscheme rxV Ă XV is V -finite, we may specialize x to a point of rxV Ă X1 that lies +over the closed point of SpecpV q. Hence, by replacing Λ by Λ1, X by X1 and Y by Y 1, we can reduce to +the already treated case when all points of x specialize to closed points of the closed Λ-fibers of X. +□ +Henceforth, we may assume that x consists of closed points of the closed Λ-fibers of X. Then, since the +relative dimension of X{Λ is d ą 0, the closed subset x Ť Y does not contain any maximal points of the +R-fibers of X, and so, by prime avoidance, there is an a P ΓpX, OXq such that a vanishes on x Ť Y but +does not vanish at any maximal points of Λ-fibers of X. Since for the proof of Proposition 7.2.1 we are +free to enlarge Y to a closed subscheme of X that still avoids all the maximal points of the Λ-fibers of +X, by replacing Y by V paq Ă X, we reduce to the case +‚ x consists of closed points of the closed Λ-fibers of X, and +‚ x Ă Y “ V paq for some a P ΓpX, OXq. +For the rest of the proof we will assume this throughout. +Lemma 7.2.4. For a field k, an affine k-variety X, a closed subscheme Y Ă X of pure dimension +e ą 0, a finite subset of closed points x Ă Y X Xsm, and an arbitrary element ptpxqq P ś +xPx κpxq, there +is a morphism h : X Ñ A1 +k that is smooth at x such that h|Y has fiber dimension e ´ 1 and such that +hpxq “ tpxq for every x P x. +Proof. Choose a finite subset of closed points y Ă Y that is disjoint from x and that contains precisely 1 +point of every irreducible component of Y . For every integer n ą 0 denote by xpnq (resp., ypnq) the nth +infinitesimal neighbourhood of x (resp., y) in X. Let hx P H0pxp1q, Oxp1qq be such that +hxpxq “ tpxq +and +dhxpxq ‰ 0 P mx{m2 +x +for every +x P x. +(7.2.1) +By prime avoidance, there exists a hy P H0pX, OXq whose restriction to every irreducible component of +Yred is not identically zero. By the faithfully flatness of +OYred,y “ +ź +yPy +OYred,y Ñ +ź +yPy +{ +OYred,y “ lim +n H0pypnq X Yred, OypnqXYredq, +we see that for large enough n, the restriction of hy to every component of ypnq X Yred is nonzero. Let +h P H0pX, OXq be any element whose restriction to xp1q is hx and whose restriction to ypnq is congruent +to hy for large n. Since X is smooth at x, (7.2.1) implies that the morphism h : X Ñ A1 +k (obtained +by sending the standard coordinate of A1 +k to h) is smooth at x and hpxq “ tpxq for every x P x. Since +the restriction of h to every irreducible component of ypnq X Yred and hence also to Yred is nonzero, the +morphism h is non-constant on every irreducible component of Y , so h|Y has fiber dimension e ´ 1. +□ +Lemma 7.2.5. There exists a Λ-morphism g : X Ñ Ad´1 +Λ +such that +(i) it smooth of relative dimension 1 at x; +(ii) the restriction g|Y is quasi-finite at x; and +(iii) for x P x lying over m, one has κpmq “ κpgpxqq. +27 + +In addition, if d ą #px X Xκpmqq for every maximal ideal m Ă Λ with finite residue field, then we may +find such a g under which all points of x have pairwise distinct images. +Proof. We first reduce the lemma to the case when Λ “ k is a field. Assume that for every maximal +ideal m Ă Λ there exists a κpmq-morphism gm : Xκpmq Ñ Ad´1 +κpmq that is smooth at x X Xκpmq such that +the restriction gm|Yκpmq is quasi-finite at x X Xκpmq. We then use Chinese remainder theorem to lift the +maps tgmum simultaneously to obtain a Λ-morphism g : X Ñ Ad´1 +Λ +which would verify the first assertion +of the lemma: only the flatness of g at x need to be checked, but this follows from the fibral criterion +of flatness [EGA IV3, Théorème 11.3.10]. In addition, if all points of x X Xκpmq have pairwise distinct +images under gm, then the resulting morphism g verifies the second assertion of the lemma. +In case Λ “ k being a field, our assumptions become that X is a k-smooth affine variety of pure dimension +d ą 0 and Y “ V paq is a closed k-subvariety of pure codimension 1 that contains x, and, for the second +assertion, our assumption becomes that d ą # x. +For a collection of maps t1, ¨ ¨ ¨ , td´1 : x Ñ k, taking products yields maps pt1, ¨ ¨ ¨ , tiq : x Ñ Ai +kpkq “ ki +for 1 ď i ď d ´ 1. We now apply Lemma 7.2.4 inductively to show: +Claim 7.2.1. For 1 ď i ď d ´ 1, there exists a k-morphism gi : X Ñ Ai +k such that +‚ gi is smooth at x with gi|x “ pt1, ¨ ¨ ¨ , tiq; and +‚ every irreducible component of gi|´1 +Y pgipxqq intersecting x has dimension d ´ 1 ´ i. +Proof of the claim. Assume the morphism gi´1 has been constructed. We apply Lemma 7.2.4, with k +being the ring k1 of global sections of gi´1pxq here, X being g´1 +i´1pgi´1pxqq here, Y being the union Y 1 +of all the irreducible components of gi´1|´1 +Y pgi´1pxqq meeting x here, and t being ti|k1, to obtain a k1- +morphism h : g´1 +i´1pgi´1pxqq Ñ A1 +k1 that is smooth at x such that h|Y 1 has fiber dimension d ´ 1 ´ i and +such that h|x “ ti|k1, where ti|k1 : x +ti +ÝÑ k Ñ k1. It remains to take gi :“ pgi´1,rhq : X Ñ Ai +k “ Ai´1 +k +ˆk A1 +k +for any lifting rh P H0pX, OXq of +h P H0 ´ +g´1 +i´1pgi´1pxqq, Og´1 +i´1pgi´1pxqq +¯ +. +□ +Starting from a map pt1, ¨ ¨ ¨ , td´1q : x Ñ kd´1, the map g :“ gd´1 of the Claim 7.2.1 immediately settles +the first assertion of the lemma. For the second assertion, it suffices to note that, under the stated +assumption, there always exists an injection x ãÑ kd´1: for an infinite field k, kd´1 is infinite, and, for a +finite field k, # kd´1 ě d ´ 1. +□ +Consider the map pg, aq : X Ñ Ad +Λ “ Ad´1 +Λ +ˆΛ A1 +Λ. +By construction, it is quasi-finite at x, and, +by the openness of the quasi-finite locus of a finite type morphism, we may shrinking X if needed +to assume that it is already quasi-finite; since the generic Λ-fibers of its domain and codomain are +irreducible varieties of the same dimension d, it is also dominant. +Consequently, by Zariski’s main +theorem [EGA IV4, Corollaire 18.12.13], pg, aq factors as +X +jÝÑ X +h1 +ÝÑ Ad +Λ, +where X is an integral affine scheme, j is an open immersion, and h1 is finite, dominant. (Unless Λ is +a DVR, ΓpX, OXq is, in general, only a finite type Λ-subalgebra of the integral closure of Λrt1, ¨ ¨ ¨ , tds +in the function field of X.) Denote g :“ pr1 ˝ h1, where pr1 : Ad +Λ Ñ Ad´1 +Λ +is the projection onto the first +pd ´ 1q-coordinates, and let a P ΓpX, OXq be the pullback of the last standard coordinate of Ad +Λ. Then +h1 “ pg, aq, and g (resp., a) restricts to g (resp., a) on X. In what follows, we shall identify the points +of jpxq with the corresponding points of x via j. +Write S Ă Spec Λ for the union of the closed points of Spec Λ (with the reduced structure). +Lemma 7.2.6. There exists an element b P ΓpX, OXq such that the morphism +h2 :“ pg, bq : X Ñ Ad +Λ “ Ad´1 +Λ +ˆΛ A1 +Λ +has the following properties: +28 + +(i) set-theoretically we have h´1 +1 ph1pxqq X h´1 +2 ph2pxqq “ x; +(ii) h2 is étale around x and induces a bijection x „ +ÝÑ h2pxq; and +(iii) h2 induces an isomorphism of residue fields κph2pxqq „ +ÝÑ κpxq for every x P x. +Proof. Since h1 is finite, surjective, g´1pgpxqq is an S-curve that contains g´1pgpxqq as an open subcurve, +so it is S-smooth around x. For a point x P x lying over a maximal ideal m Ă Λ, its first infinitesimal +neighbourhood in g´1pgpxqq is isomorphic to Specpκpxqruxs{pu2 +xqq, where ux is an uniformizer of g´1pgpxqq +at x. Recall the fact that the residue field of a point on a smooth curve over a field is a simple extension +of that field, see [Čes22a, Lemma 6.3]. It follows that, for x P x lying over m, there exists a closed +κpmq-immersion xp1q ãÑ A1 +κpmq “ A1 +gpxq. For a maximal ideal m Ă Λ with finite residue field, under our +assumption that #px XXκpmqq ă maxp# κpmq, dq, either x contains at most # κpmq´1 points lying over +m or the fiber of gκpmq contains at most 1 point of x (Lemma 7.2.5). Consequently, we may arrange the +above immersions so that they jointly give a closed immersion over Ad´1 +Λ +: +ğ +xPx +xp1q ãÑ A1 +gpxq Ă A1 +Ad´1 +Λ +“ Ad +Λ, +(7.2.1) +where we regard gpxq Ă Ad´1 +Λ +as a closed subscheme. +Note that the complement of the image of +the morphism (7.2.1) in Ad +Λ has at least 1 rational point Ad´1 +Λ +-fiberwisely. Thus, by sending any y P +ph´1 +1 ph1pxqqzxq to a suitable rational point of A1 +gpyq, we may further extend (7.2.1) to a Ad´1 +Λ +-morphism +u: Z :“ +`Ů +xPx xp1q˘ Ů ´Ů +yPh´1 +1 +ph1pxqqzx y +¯ +Ñ Ad +Λ +such that upxq X uph´1 +1 ph1pxqqzxq “ H, or, what amounts to the same, +h´1 +1 ph1pxqq X u´1pupxqq “ x. +(7.2.2) +As Z is a closed subscheme of the affine scheme X, we can lift u˚ptq P ΓpZ, OZq to obtain an element +b P ΓpX, OXq, where t is the standard coordinate on A1 +Λ. +Consider the morphism h2 :“ pg, bq : X Ñ Ad +Λ “ Ad´1 +Λ +ˆΛ A1 +Λ. Viewing X as a Ad´1 +Λ +-scheme via g, the +base change of h2 to gpxq Ă Ad´1 +Λ +restricts to u on Z, so h2 is unramified at x. Now (i) follows from +(7.2.2), (iii) is a consequence of our choice of the morphism (7.2.1). For (ii), it suffices to argue that h2 +is flat at x; however, since the domain and the codomain of h2 are Λ-flat of finite presentation, the fibral +criterion of flatness [EGA IV3, Théorème 11.3.10] reduces us to checking the flatness of the Λ-fibers of +h2 at x, while the latter follows from the flatness criterion [EGA IV2, Proposition 6.1.5]. +□ +Let Λrh˚ +1pt1q, ¨ ¨ ¨ , h˚ +1ptd´1q, a, bs Ă ΓpX, OXq be the Λ-subalgebra generated by a, b and h˚ +2ptiqp“ h˚ +1ptiq “ +g˚ptiqq for 1 ď i ď d ´ 1. We introduce the following notations. +‚ Let V :“ SpecpΛrh˚ +1pt1q, ¨ ¨ ¨ , h˚ +1ptd´1q, a, bsq, and let h3 : X Ñ V be the morphism induced by +the inclusion Λrh˚ +1pt1q, ¨ ¨ ¨ , h˚ +1ptd´1q, a, bs Ă ΓpX, OXq. +‚ Let v1 : V Ñ Ad +Λ be the map such that v˚ +1 ptiq “ h˚ +1ptiq for 1 ď i ď d ´ 1 and v˚ +1 ptdq “ a. +‚ Let v2 : V Ñ Ad +Λ be the map such that v˚ +2 ptiq “ h˚ +1ptiq for 1 ď i ď d ´ 1 and v˚ +2 ptdq “ b. Note +that there is a natural surjection +Λrh˚ +1pt1q, ¨ ¨ ¨ , h˚ +1ptd´1q, bs ։ Λrh˚ +1pt1q, ¨ ¨ ¨ , h˚ +1ptd´1q, a, bs{paq “ ΓpV, OV q{paq; +this implies that v2 induces a closed immersion +v2 : SpecpΓpV, OV q{paqq ãÑ V +v2 +ÝÑ Ad +Λ. +29 + +We have the following commutative diagram of morphisms of affine schemes: +X +X +V +Ad +Λ +Ad +Λ +j +h2 +h1 +h3 +v2 +v1 +. +Lemma 7.2.7. The map h3 induces a bijection x +„ +ÝÑ h3pxq with h´1 +3 ph3pxqq “ x. Further, h3 induces +an isomorphism of semilocal rings +OV,h3pxq » OX,x “ OX,x. +Proof. By Lemma 7.2.6(ii)-(iii), we see that h3 induces a bijection x +„ +ÝÑ h3pxq and an isomorphism of +residue fields κph3pxqq „ +ÝÑ κpxq for every x P x. Chasing the above diagram we see that +h´1 +3 ph3pxqq Ă h´1 +1 ph1pxqq X h´1 +2 ph2pxqq “ x, +where the last equality is Lemma 7.2.6(i). As h3 is finite, surjective, we see that h´1 +3 ph3pxqq “ x. By +Lemma 7.2.6(ii), h3 is unramified at x. It follows that the base change of h3 to Spec OV,h3pxq is +Spec OX,x Ñ Spec OV,h3pxq, +and it is actually an isomorphism: letting J be the Jacobson radical of the semilocal ring OV,h3pxq, since +the natural map +ź +xPx +κph3pxqq » OV,h3pxq{J +h˚ +3 +ÝÝÑ OX,x{JOX,x » +ź +xPx +κpxq +is an isomorphism (in particular, surjective), an application of Nakayama lemma shows +h˚ +3 : OV,h3pxq » OX,x “ OX,x. +□ +End of the proof of Proposition 7.2.1. Define f :“ h2 ˝ j : X Ñ Ad +Λ, which we may assume to be étale +upon replacing X by an affine open neighbourhood of x. By Lemma 7.2.7, there exists an affine open +neighbourhood W 1 +0 Ă V of h3pxq such that W0 :“ h´1 +3 pW0q Ă jpXq and h3|W0 : W0 +„ +ÝÑ W 1 +0. We shall +identify W0 as an open subscheme of X via j. As noted above, v2 induces a closed immersion +v2 : Y 1 :“ SpecpΓpV, OV q{paqq ãÑ Ad +Λ. +In particular, the topology of Y 1 is induced from that of Ad +Λ via v2. Note also that, since a vanishes on x, +h3pxq Ă Y 1 Ă V . Consequently, there exists an affine open neighbourhood U Ă Ad +Λ of fpxq “ v2ph3pxqq +such that v´1 +2 pUq Ă W 1 +0. Therefore, f induces a closed immersion of affine schemes +YU :“ f ´1pUq X Y “ ph3 ˝ jq´1pv´1 +2 pUq X Y 1q “ ph3 ˝ jq´1pv´1 +2 pUqq » v´1 +2 pUq ãÑ U. +Since f is separated and étale, any section of f ˆAd +Λ,f YU, such as the one induced by the inclusion +YU ãÑ X, is an inclusion of a clopen, so +X ˆAd +Λ,f YU “ rY1 \ rY2 +with +rY1 +„ +ÝÑ YU. +Let W Ă f ´1pUq be an affine open whose preimage in X ˆAd +Λ,f YU is rY1. Then f|W : W Ñ U is an étale +morphism such that f|WXY : W XY ãÑ U is a closed immersion and such that W ˆU,f pW XY q „ +ÝÑ W XY . +As any étale map is open, we may shrink U around fpxq to ensure that f|W : W Ñ U is also surjective. +This proves the first assertion of Proposition 7.2.1. +The second assertion follows from descent theory, because the ideal sheaf of W X Y on U pulls back to +that of W X Y on W. +□ +8. Cohomology of groups of multiplicative type +Inspired by the purity results in [ČS21, Theorem 7.2.8], we investigate the parafactoriality over Prüfer +bases and then present the purity for cohomology of group schemes of multiplicative type. +30 + +8.1. Geometrically parafactorial pairs +8.1.1. Parafactorial pairs. Let pX, OXq be a ringed space with a closed subspace Z Ă X and open +immersion j : XzZ ãÑ X, if for every open subspace U Ă X the restriction +Pic X +„ +ÝÑ Pic XzZ, +L ÞÑ L |UXpXzZq +is an equivalence of categories, +then the pair pX, Zq is parafactorial. In particular, we have L » j˚j˚L . A local ring A is parafactorial +if the pair pSpec A, Spec A{mAq is parafactorial. We list several parafactorial pairs pX, Zq and local rings. +(i) when A is a Noetherian factorial local ring, by [EGA IV4, Exemples 21.13.9 (ii)], it is parafactorial; +(ii) by [EGA IV4, Proposition 21.13.8], a local ring A is parafactorial if and only if +Pic pSpec Aztxuq “ 0 +and +A » ΓpSpec Aztxu, rAq +for the closed point x P Spec A; +(iii) when X is a locally Noetherian and locally complete intersection and Z satisfies codimpZ, Xq ě 4, +by [SGA 2new, Exposé XI, Théorème 3.13 (ii)], the pair pX, Zq is parafactorial; +(iv) for a normal scheme S, an S-smooth scheme X and a closed subset Z Ă X satisfying +codimpZη, Xηq ě 2 for each generic point η P S +and +codimpZs, Xsq ě 1 for every s P S, +by [EGA IV4, Proposition 21.14.3], the pair pX, Zq is parafactorial. +Now we assume that X is a scheme. A parafactorial pair pX, Zq is geometrically parafactorial, if for every +X-étale X1 with the base change Z1 :“ ZˆX X1, the pair pX1, Z1q is parafactorial. For a local ring A, if its +strict Henselization Ash is parafactorial, then A is geometrically parafactorial (cf. [ČS21, Theorem 7.2.8]). +Lemma 8.1.2. For a topologically locally Noetherian scheme X and a closed subscheme Z Ă X, +(i) the pair pX, Zq is parafactorial if and only if OX,z is parafactorial for every z P Z; +(ii) the pair pX, Zq is geometrically parafactorial if and only if Osh +X,z is parafactorial for every z P Z. +Proof. The assertion (ii) follows the same argument of (i), except viewing Osh +X,z as the inverse limit of étale +neighborhoods of z P X. Assume that pX, Zq is parafactorial and for each z P Z, denote Uz :“ Spec OX,z +and U ˝ +z :“ Uzztzu. To show that OX,z is parafactorial, we prove that every invertible OUz-module L0 +is isomorphic to OU˝ +z . Then by [EGA IV3, Proposition 8.2.13] and [EGA I, Proposition 2.4.2], U ˝ +z is the +inverse limit of B˝ :“ BzpB X tzuq where B ranges over all open affine neighborhoods of z P X. Since +every B˝ is topologically Noetherian and separated, by a limit argument [SP, 0B8W], there exists an +open affine neighborhood B of z P X and an invertible OB˝-module LB˝ such that L0 » LB˝|U˝ +z . By +assumption and [EGA IV4, Corollaire 21.13.6 (i)(ii)], the pair pB, B Xtzuq is parafactorial. In particular, +there exists an invertible OB-module Ă +LB such that Ă +LB|B˝ » LB˝. Shrinking B if necessary, we have +Ă +LB » OB hence L0 » OU˝ +z . +For the other side, assume that OX,z are parafactorial for all z P Z, which, combined with [EGA IV4, +Proposition 21.13.5], reduces us to showing that for every invertible OXzZ-module L , the pushforward +j˚L is an invertible OX-module. For this, we use Noetherian induction. Namely, consider the subset +Ω :“ tx P X | j˚L is invertible on an open neighborhood of xu +Then Ω Ă X is a non-empty open whose complementary closed is XzΩ “: Y Ă Z. +By [EGA IV2, +Lemme 2.3.1], the quasi-compact quasi-separated morphism j guarantees that the formation of j˚L +commutes with arbitrary flat base changes (in particular, localizations). Pick a maximal point y P Y Ă Z +so OX,y is parafactorial. The maximality of y P Y implies that Ω X Uy “ U ˝ +y, so L0 :“ pj˚L q|U˝ +y is +an invertible OU˝ +y -module. The parafactoriality of OX,y yields an extension of L0 to an invertible OUy- +module Ă +L0, which, by the limit argument [SP, 0B8W] again, descends to an invertible OW -module Ă +LW +for an open neighborhood W of y P X. Shrinking W if necessary, we may assume that the restrictions +of j˚L and Ă +LW on Ω X W are equal. With this gluing datum, let Ω1 :“ Ω Y W, so there is an invertible +Ω1-module L 1 such that L 1|W “ Ă +LW and L 1|Ω “ pj˚L q|Ω. Since XzZ Ă Ω1 and L 1|XzZ “ L , hence +OX » j˚OXzZ and pj˚L q|Ω1 » L 1, which leads to a desired contradiction with the definition of Ω. +□ +Proposition 8.1.3. For a normal scheme S and an S-scheme X satisfying one of the following +31 + +(i) either X Ñ S is a smooth morphism of topologically locally Noetherian schemes; or +(ii) S is semilocal Prüfer of finite dimension and X is S-flat locally of finite type with regular S-fibers +then every x P X that does not contain any maximal point of S-fibers of X and dim OX,x ě 2 satisfies +OX,x is geometrically parafactorial, +namely, +Osh +X,x is parafactorial. +Proof. The parafactoriality of Osh +X,x is that of pSpec Osh +X,x, txuq, which by Lemma 8.1.2(ii), is equivalent +to the parafactoriality of pSpec OX1,x1, tx1uq for all X-étale X1. Since all X1 and x1 satisfy the conditions +in the statement above ([BS15, Lemma 6.6.10 (3)]), we are reduced to showing that OX,x is parafacto- +rial. For the Zariski closure Z :“ txu, by Lemma 8.1.2 again, we are reduced to finding a small open +neighborhood U of x P X such that pU, Z X Uq is a parafactorial pair. Now, take an arbitrary open +neighborhood U of x P X, by [EGA IV3, Proposition 9.5.3] applied to Z Ă X, shrinking U, we may +assume that U X Z does not contain any irreducible components of S-fibers of X. If a z P Z lies over a +maximal point η P S, since x specializes to z, then we have dim OXη,z “ dim OX,z ě 2. Consequently, +we have codimpXη X Z, Xηq ě 2 and by §8.1.1(iv) and Proposition 3.2.7(ii), the desired parafactoriality +of pU, Z X Uq follows. +□ +8.2. Purity for groups of multiplicative type +Now we study purity for groups of multiplicative type in the situation of higher relative dimension. We +start with the following generalization of Theorem 6.1.4 when G “ M is a X-group algebraic space of +multiplicative type. +Lemma 8.2.1. For an algebraic space X with a closed subspace Z Ă X such that for every geometric +point z Ñ Z, the strict local ring OX,z is parafactorial, the open immersion j : XzZ ãÑ X and a finite +type multiplicative type X-group algebraic space M , the following map between fppf sheaves on X +M +„ +ÝÑ j˚j˚M +is an isomorphism. +In particular, we have H0 +ZpX, M q “ H1 +ZpX, M q “ 0 and ΓpX, Pq » ΓpU, Pq for every M -torsor P on +X. +Proof. For an M -torsor P, to show that ΓpX, Pq » ΓpU, Pq, it suffices to prove that P » j˚j˚P, +which can be checked fppf locally. Hence, it suffices to prove the first assertion in the case when X is a +scheme. By [SGA 3II, Exposé X, Corollaire 4.5], M is quasi-isotrivial, namely, there is an étale surjective +morphism r +X Ñ X such that M ˆX r +X splits. We need to show that the morphism M Ñ j˚j˚M is an +isomorphism fppf locally at all z P Z. Suppose f : X1 Ñ X is a flat morphism inducing g: X1zZ1 Ñ XzZ, +where Z1 :“ Z ˆX X1 with the open immersion j1 : X1zZ1 ãÑ X1 . Taking inverse image of M Ñ j˚j˚M , +we obtain f ˚M Ñ f ˚j˚j˚M . By [EGA IV2, Lemme 2.3.1], the formation of j˚p´q commutes with flat +base change, hence f ˚j˚j˚M » j1 +˚g˚j˚M “ j1 +˚pj1q˚f ˚M and the inverse image of M Ñ j˚j˚M is +f ˚M Ñ j1 +˚pj1q˚f ˚M . We may assume that X1 “ Spec Osh +X,z and Z1 “ tzu, so the desired isomorphism +is reduced to an isomorphism M +„ +ÝÑ j1 +˚pj1q˚M for a split finite type multiplicative group sheaf M . For +an X1-group µn, we have the following short exact sequence +0 Ñ µn Ñ Gm +ˆn +Ñ Gm Ñ 0, +hence j1 +˚pj1q˚µn “ kerpj1 +˚pj1q˚Gm +ˆn +Ñ j1 +˚pj1q˚Gmq, reducing us to the case when MX1 “ Gm. +Since +pX1, Z1q is parafactorial, we have Oˆ +X1 +„ +ÝÑ j1 +˚pj1q˚Oˆ +X1, so the assertion follows. +□ +Proposition 8.2.2. For a finite-rank valuation ring R with spectrum S and generic point η P S, an +S-flat finite type scheme X with regular S-fibers, a point x P X, and an OX,x-torus T , +(1) if either x P Xη with dim OXη,x ě 2, or x P Xs with s ‰ η and dim OXs,x ě 1, then we have +Hi +txupOX,x, T q “ 0 +for +0 ď i ď 3; +(2) otherwise, OX,x is a valuation ring, then if T is flasque we have +H2 +txupOX,x, T q “ 0. +32 + +Proof. (1) Notice that the finite-rank assumption on R guarantees X being topologically locally Noe- +therian. By the local-to-global E2 spectral sequence [SGA 4II, Exposé V, Proposition 6.4], +Hp +´etpOX,x, Hq +txupT qq ñ Hp`q +txu pOX,x, T q, +where Hq +txupT q is the sheafification of the étale presheaf +´ +U +hÝÑ SpecpOX,xq +¯ +ÞÑ Hq +h´1pxqpU, T q. +Therefore, it suffices to prove the vanishing of the sheaves Hq +txupT q for 0 ď q ď 2. We calculate their +stalks at a geometric point x lying over x: +Hq +txupT qx “ Hq +txupOsh +X,x, T q. +Now, since TOsh +X,x » Gdim T +m,Osh +X,x, and, since by Proposition 8.1.3 Osh +X,x is parafactorial, we have +Hq +´etpSpecpOsh +X,xq, T q » Hq +´etpSpecpOsh +X,xqztxu, T q +for 0 ď q ď 1; +as Osh +X,x is strictly Henselian, we have +H2 +´etpSpecpOsh +X,xq, T q “ 0. +Looking at the local cohomology exact sequence for the pair pSpecpOsh +X,xq, ¯xq and T , we see that +Hq +t¯xupOsh +X,x, T q “ 0 +for 0 ď q ď 2. +This implies Hq +txupT q “ 0 for 0 ď q ď 2, as desired. +(2) In this case, either x P Xη with dim OXη,x ď 1, then OX,x is a discrete valuation ring, or x is a +maximal point of some fiber of X Ñ S, then, by Lemma 3.1.1(iii), OX,x is a valuation ring. The desired +vanishing is proven in [Guo20, Lemma 2.3]. +□ +Lemma 8.2.3 (cf. [ČS21, Lemma 7.1.1]). For an algebraic space X, an open subspace U Ă X with +complement i : Z :“ XzU ãÑ X, and an abelian sheaf F on pSch{Xqfppf, if for any integer q ě 0, +Hq +ZpFq denotes the étale-sheafification of the presheaf X1 ÞÑ Hq +Z1pX1, Mq where Z1 :“ Z ˆX X1, then we +have a convergent spectral sequence +Epq +2 “ Hp +´etpX, Hq +ZpFqq ñ Hp`q +Z +pX, Mq. +Theorem 8.2.4. +(i) (cf. [ČS21, Theorem 7.2.8 (a)]) For an algebraic space X, a quasi-compact open immersion +j : U ãÑ X with complement Z :“ XzU, and an X-group algebraic space M of multiplicative +type, if for every geometric point z Ñ Z, the strict local ring OX,z is parafactorial, then restric- +tion functor +TorspXfppf, Mq +„ +ÝÑ TorspUfppf, Mq +induces an equivalence of categories of M-torsors. +In particular, passing to isomorphism classes of objects, we have the following isomorphisms +Hi +fppfpX, Mq +„ +ÝÑ Hi +fppfpU, Mq +for i ď 1 +and +H2 +fppfpX, Mq ãÑ H2 +fppfpU, Mq. +(ii) For a semilocal Prüfer domain R with spectrum S, a quasi-compact quasi-separated S-smooth +scheme X, a quasi-compact open U Ă X with complement Z :“ XzU, and an X-torus T such +that TOX,z is flasque for every z P Z for which OX,z is a valuation ring, then we have +H1 +´etpX, T q ։ H1 +´etpU, T q +and +H2 +´etpX, T q ãÑ H2 +´etpU, T q. +Proof. (i) By the local cohomology exact sequence for the pair pX, Zq and the sheaf M, everything +reduces to show the vanishings Hq +ZpX, Mq “ 0 for 0 ď q ď 2. By the spectral sequence in Lemma 8.2.3, +it suffices to show the vanishings of Hq +ZpMq, the étale-sheafification of the presheaf X1 ÞÑ Hq +Z1pX1, Mq +where Z1 :“ Z ˆX X1. Further, the quasi-compactness of j allows us to identify the stalk of Hq +ZpMq +at a geometric point z Ñ Z as Hq +tzupOX,z, Mq. Hence we may assume that M split as µn or Gm, and +since µn “ kerpGm +ˆn +Ñ Gmq, it suffices to show that Hq +tzupOX,z, Gmq “ 0 for 0 ď q ď 2. Since OX,z is +parafactorial, we have +HqpSpecpOX,zq, Gmq » HqpSpecpOX,zqztzu, Gmq +for 0 ď q ď 1; +33 + +as OX,z is strictly Henselian, we have +H2pSpecpOX,zq, Gmq “ 0. +Looking at the local cohomology exact sequence for the pair pSpecpOX,zq, zq and T , we deduce the desired +vanishings +Hq +tzupOX,z, Gmq “ 0 +for 0 ď q ď 2. +(ii) By the local cohomology exact sequence +¨ ¨ ¨ Ñ H1pX, T q Ñ H1pU, T q Ñ H1 +ZpX, T q Ñ H2pX, T q Ñ H2pU, T q Ñ ¨ ¨ ¨ , +the assertion is equivalent to the vanishing H2 +ZpX, T q “ 0. Since X is quasi-compact quasi-separated +and U Ă X is a quasi-compact open, by a limit argument involving Lemma 3.1.3, we reduce to the case +R having finite Krull dimension, so X is topologically Noetherian. Recall the coniveau spectral sequence +[Gro68b, §10.1] +Epq +2 “ +à +zPZppq +Hp`q +tzu pT q ñ Hp`q +Z +pX, T q; +the topological Noetherianness of X allows us to identify +Hp`q +tzu pT q :“ colim Hp`q +tzuXUpU, T q +as Hp`q +tzu pOX,z, T q, where U runs over the open neighbourhoods of z in X. Therefore, it is enough to +show H2 +tzupOX,z, T q “ 0, which has been solved by Proposition 8.2.2. +□ +Proposition 8.2.5. For a normal scheme S and an S-algebraic space X satisfying one of the following +(i) either X Ñ S is a smooth morphism of topologically Noetherian algebraic spaces; or +(ii) S is semilocal Prüfer of finite dimension and X is S-flat locally of finite type with regular S-fibers, +a quasi-compact open U Ă X with complementary closed Z :“ XzU satisfying the following condition +codimpZη, Xηq ě 2 for every generic point η P S +and +codimpZs, Xsq ě 1 for all s P S, +and a finite type X-group algebraic space M of multiplicative type, the following restriction functor +TorspXfppf, Mq +„ +ÝÑ TorspUfppf, Mq +induces an equivalence of categories of M-torsors. +In particular, passing to isomorphism classes of objects, we have the following isomorphisms +H0pX, Mq » H0pU, Mq, +H1 +fppfpX, Mq » H1 +fppfpU, Mq, +H2 +fppfpX, Mq ãÑ H2 +fppfpU, Mq. +Proof. We simply verify the assumptions of Lemma 5.7. First, the restriction functor is fully faithful, +because M is X-affine so is Y :“ IsomXpP1, P2q for arbitrary M-torsors P1 and P1 on X (Lemma 5.4), +which implies that Y pXq » Y pUq (note that Y is an AutGpP1q » M-torsor, so we have Y pXq » Y pUq +by Lemma 8.2.1). The same holds when we base change to every scheme étale over X. Next, we show +that, fppf locally on X, every M-torsor on U extends on X. For this we may assume that X is affine. +Since X is normal, M is isotrivial, so there is an X-torus T and a finite X-group µ of multiplicative type +fitting into the short exact sequence +1 Ñ T Ñ M Ñ µ Ñ 1, +From which we leverage the following commutative diagram with exact rows +µpXq +H1 +fppfpX, T q +H1 +fppfpX, Mq +H1 +fppfpX, µq +µpUq +H1 +fppfpU, T q +H1 +fppfpU, Mq +H1 +fppfpU, µq, +where µpXq » µpUq follows from the X-affineness of µ. A diagram chase reduces us to showing that +H1 +fppfpX, T q +„ +ÝÑ H1 +fppfpU, T q +and +H1 +fppfpX, µq +„ +ÝÑ H1 +fppfpU, µq +are isomorphisms. +Since the extension problem is fppf local, we may assume that M splits, without loss of generalities, say +M » Gm or M “ µn. By Proposition 3.2.7(ii), the X-affineness of M implies that MpX1q » MpU 1q. +34 + +When M “ Gm, we have Pic X +„ +ÝÑ Pic U because pX, Zq is a parafactorial pair. It remains to prove +that H1 +´etpX, µnq » H1 +´etpU, µnq, for which we consider the commutative diagram +0 +OpXqˆ{OpXqˆn +H1 +´etpX, µnq +n PicpXq +0 +0 +OpUqˆ{OpUqˆn +H1 +´etpU, µnq +n PicpUq +0 +A diagram chase leads to the desired isomorphism H1 +´etpX, µnq +„ +ÝÑ H1 +´etpU, µnq. Finally, all fppf local +extension data glue together. Hence we obtain the desired essential surjectivity. +□ +8.3. Grothendieck–Serre type results for groups of multiplicative type +Lemma 8.3.1. Let φ : X Ñ Y be a morphism of schemes. Let L be an invertible OX-module. If +(1) Y is quasi-compact quasi-separated, integral, and normal, +(2) there exist a smooth projective morphism φ : X Ñ Y , with geometrically integral fibers, and a +quasi-compact open immersion X ãÑ X over Y , and +(3) L is trivial when restricted to the generic fiber of φ, +then L » φ˚N for some invertible OY -module N . +Proof. When Y is Noetherian, this follows from a much more general result [SP, 0BD6]; for instance, (2) +can be replaced by the assumption that X Ñ Y is faithfully flat of finite presentation, with integral fibers. +The general case can be deduced from this via Noetherian approximations. More precisely, we first use +[SP, 01ZA] to write Y “ limi Yi for a filtered inverse system tYiu of finite type integral Z-schemes with +affine transition morphisms. Since the normalization of a finite type integral Z-scheme is finite, we may +assume that each Yi is normal. Next, by [SP, 01ZM, 0C0C], for some i0 there exist a finite type smooth +morphism φi0 : Xi0 Ñ Yi0 such that X » Xi0 ˆYi0 Y as Y -schemes, an open subscheme Xi0 Ă Xi0 +whose pullback to X identifies with X, and, by [SP, 0B8W], there is an invertible OXi0 -module Li0 +whose pullback to X is isomorphic to L . For any i ě i0 denote by φi : Xi :“ Xi0 ˆYi0 Yi Ñ Yi the +base change of φi0|Xi0 to Yi, and denote by Li the pullback of Li0 to Xi. By [SP, 01ZM, 01ZP], any +projective embedding of X over Y descends to a projective embedding of Xi over Yi for large enough i; +in particular, φi is projective for large enough i. +Since Y is normal, the assumption (3) implies that the Stein factorization of φ is itself; in particular, +OY +„ +ÝÑ φ˚OX. This implies that the finite extension OYi0 ãÑ φi0,˚OXi0 is an isomorphism, because its +base change to the function field of Y is so and Yi0 is normal. In particular, by Zariski’s main theorem, φi0 +has connected geometric fibers; as it is also smooth, all its fibers are even geometrically integral. By limit +formalism, for large enough i, Li is trivial when restricted to the generic fiber of φi. Consequently, for +large enough i, the morphism φi : Xi Ñ Yi and the invertible OXi-module Li satisfy all the assumptions +of the Lemma, so Li » φ˚ +i Ni for some invertible OYi-module Ni. Then L » φ˚N where N is the +pullback of Ni to Y . +□ +Proposition 8.3.2 (cf. [CTS87, 4.1–4.3]). For a Prüfer domain R with spectrum pS, ηq, an irreducible +scheme X essentially smooth over S with function field KpXq, an X-group scheme M of multiplicative +type, and a connected finite étale Galois covering X1 Ñ X splitting M 4, the restriction maps +H1 +fppfpX, Mq Ñ H1 +fppfpKpXq, Mq +and +H2 +fppfpX, Mq Ñ H2 +fppfpKpXq, Mq +are injective in each of the following cases: +(i) X “ SpecpAq and A is a semilocal ring essentially smooth over R; +(ii) For some essentially smooth semilocal R-algebra A, there exists a quasi-compact open immersion +X ãÑ X, where X is a smooth projective A-scheme, with geometrically integral fibers, such that +PicpXLq “ 0 for any finite separable fields extension L{FracpAq, and M “ NX for N an A-group +of multiplicative type (for instance, X could be any quasi-compact open subscheme of PN +A ); +4Such a covering always exists, because X is normal and so M is isotrivial. +35 + +(iii) any subcovering X2 Ñ X of X1 Ñ X satisfies PicpX2q “ 0. +Further, if M is a flasque X-torus, then in all cases piq-piiiq the restriction map +H1 +´etpX, Mq +„ +ÝÑ H1 +´etpKpXq, Mq +is bijective. +Proof. It is clear that (i) is a particular case of (ii). Let us show that (ii) is a particular case of (iii). +Let A Ñ B be a connected finite étale Galois covering that splits N. Take X1 :“ X ˆA B. By the +normality of A and the smoothness of X Ñ SpecpAq, X is also normal. Then, since X Ñ SpecpAq +has geometrically integral generic fiber, the natural map π´et +1 pXq Ñ π´et +1 pSpec Aq is surjective. +This +implies that any subcovering X2 Ñ X of X1 Ñ X is of the form X2 “ X ˆA C for some subcovering +A Ñ C of A Ñ B. By assumption, PicpXFracpCqq “ 0, so we may apply Lemma 8.3.1 to the morphism +X ˆA C Ñ SpecpCq to deduce that the pullback map +PicpSpecpCqq Ñ PicpX ˆA Cq +is surjective. +Since C is semilocal, we conclude that PicpSpecpCqq “ 0 “ PicpX ˆA Cq. +It is thus enough to prove all assertions only for (iii). Assume first that M “ T is an X-torus. Take a +flasque resolution +1 Ñ F Ñ P Ñ T Ñ 1, +where F is a flasque X-torus and P is a quasitrivial X-torus. This yields a commutative diagram +H1 +´etpX, Pq +H1 +´etpX, T q +H2 +´etpX, Fq +H1 +´etpKpXq, T q +H2 +´etpKpXq, Fq +ρ1 +ρ2 +with exact rows. Now the quasitrivial torus P is isomorphic to a finite direct product of tori ResX2{XGm,X2 +for finite étale subcoverings X2 Ñ X of X1 Ñ X. Hence, assumption (iii) implies that H1 +´etpX, Pq “ 0, +and so the injectivity of ρ1 reduces to that of ρ2. To prove that ρ2 is injective we pick a P H2 +´etpX, Fq for +which a|KpXq “ 0. By spreading out, we may assume that X is a localization of an irreducible, smooth, +affine R-scheme r +X, F “ rFX for a flasque r +X-torus rF, and a “ ra|X for some class ra P H2p r +X, rFq. Since +ra|KpXq “ 0, for a large enough hypersurface Z Ă r +X, +ra|Ă +XzZ “ 0 P H2 +´etp r +XzZ, rFq. +By Theorem 8.2.4(ii), ra “ 0, so a “ ra|X “ 0. This proves the injectivity of ρ2 and hence also of ρ1. Now +let M be an arbitrary X-group of multiplicative type, then there is an X-subtorus T Ă M such that +µ :“ M{T is X-finite. Consequently, for any generically trivial M-torsor P, the µ-torsor P{T is finite over +X; as X is normal, this implies pP{T qpXq “ pP{T qpKpXqq. Therefore, P{T Ñ X has a section that lifts +to a generic section of P Ñ X, that is, P reduces to a generically trivial T -torsor PT . By the injectivity +of ρ1, PT and hence also P is trivial. This proves the injectivity of H1 +´etpX, Mq Ñ H1 +´etpKpXq, Mq. +On the other hand, there is a short exact sequence +1 Ñ M Ñ F Ñ P Ñ 1 +of X-groups of multiplicative type, where F is flasque and P is quasitrivial, both split after base change +by X1 Ñ X. This yields the following commutative diagram with exact rows +H1 +fppfpX, Pq +H2 +fppfpX, Mq +H2 +fppfpX, Fq +H2 +fppfpKpXq, Mq +H2 +fppfpKpXq, Fq +ρ3 +ρ2 +Since we have already shown that H1 +fppfpX, Pq “ 0 and ρ2 is injective, the injectivity of ρ3 follows from +a diagram chase. +Finally, if M is a flasque X-torus, the bijectivity of H1 +fppfpX, Mq Ñ H1 +fppfpKpXq, Mq will follow if one +proves its surjectivity, but the latter follows from Theorem 8.2.4(ii) via a limit argument. +□ +36 + +9. Grothendieck–Serre on a semilocal Prüfer domain +The main result of this section is the following mild generalization of [Guo20]. +Theorem 9.0.1. For a semilocal Prüfer domain R with fraction field K, and a reductive R-group scheme +G, the following restriction map has trivial kernel: +ker +` +H1 +´etpR, Gq Ñ H1 +´etpK, Gq +˘ +“ t˚u. +9.0.2. Setup. We fix the following notations. For a semilocal Prüfer domain R of finite Krull dimension, +all the maximal ideals pmiqr +i“1 of R, the local rings Oi :“ Rmi, an element a P R such that V paq “ +tmiur +i“1, let pR (resp., p +Oi) denote the a-adic completion of R (resp., of Oi). +Then p +Oi is an a-adic +complete valuation ring of rank 1, and we have pR » śr +i“1 p +Oi, compatibly with the topologizes. Denote +pKi :“ Frac pOi “ p +Oir 1 +as. Topologize Rr 1 +as by declaring timpanR Ñ Rr 1 +asquně1 to be a fundamental system +of open neighbourhood of 0; the associated completion is +Rr 1 +as Ñ pRr 1 +as » śr +i“1 pOir 1 +as “ śr +i“1 pKi, +where each pKi is a complete valued field, with pseudo-uniformizer (the image of) a. In particular, for an +R-scheme X, we have a map +ΦX : XpRr 1 +asq Ñ śr +i“1 Xp pKiq. +If X is locally of finite type over R, we endow the right hand side with the product topology where each +Xp pKiq, by, for example, Conrad, has a natural topology induced from that of pKi, which we will call the +a-adic topology. If moreover X is affine, we can canonically topologize XpRr 1 +asq by choosing a closed +embedding X ãÑ AN +R and endowing XpRr 1 +asq ãÑ Rr 1 +asN with the subspace topology (this is independent +of the choices of the embeddings), then ΦX is a continuous map. +9.1. Lifting maximal tori of reductive group schemes over semilocal rings +Lemma 9.1.1. For a scheme S, an S-smooth finitely presented group scheme G whose S-fibers are +connected and affine, and a finite subset I Ă S. If I satisfies the following conditions +(i) I is contained in an affine open subset of S; +(ii) for each residue field κi of S at i P I, the fiber Gκi is a κi-reductive group; and +(iii) 7κi ě dimpGκi{Ziq for the center Zi Ă Gκi, +then there is an open neighborhood U of I such that the following map is surjective +TorpGqpUq ։ ś +iPI TorpGqpκiq. +Proof. By [SGA 3II, Exposé XVI, Théorème 5.2], there is an open neighborhood U of I such that G|U is a +U-reductive group scheme, so we may replace S by U. By [SGA 3II, Exposé XII, Théorème 4.7 c)], G has +a reductive center Z and we have Zi “ pZqκi for every i P I. Since TorpGq » TorpG{Zq, we may replace +G by G{Z. By [SGA 3II, Exposé XIV, Théorème 3.18], the maximal tori of G are exactly the subgroups +of type (C), which are bijectively assigned by D ÞÑ LiepDq to the Cartan subalgebras of g :“ LiepGq +([SGA 3II, Exposé XIV, Théorème 3.9]). It suffices to lift a Cartan subalgebra c0 Ă ś +iPI gκi to that of g. +Denote ci :“ pc0qκi. Since for each i P I, we have 7κi ě dimpG{Zq “ dimpGq, by [Bar67, Theorem 1], ci +is of the form Nilpaiq :“ Ť +n kerpadpan +i qq for some ai P ci. Hence [SGA 3II, Exposé XIII, Corollaire 5.7] +implies that each ai P ci is a regular element of gκi. We take a section a of g passing through all ai and +claim that V :“ ts P Spec R such that as P gs is regularu is an open subset of Spec R. We may assume +that R is reduced. Since the nilpotent rank of g is locally constant, there is an open neighborhood U +of I such that the nilpotent rank of g is constant on each connected component Uα of U. On each +Uα, the Killing polynomial of g at every s P Uα is uniformly Pα,gsptq “ trαptn´rα ` pc1qstn´rα´1 ` +¨ ¨ ¨ ` pcn´rαqsq such that pcn´rαqs is nonzero. Thus, the regular locus in g is the principle open subset +Ş +αtcn´rα ‰ 0u Ă Wpgq so V is nonempty and open, hence shrinking U if necessary, we have V “ U. +In particular, the regular elements pai P ciqiPI are lifted to a quasi-regular section a P g, which by +[SGA 3III new, Exposé XIV, Corollaire 3.7], is regular. By definition of regular sections, there is a Cartan +subalgebra of g containing a and is the desired lifting of c0. +□ +37 + +Lemma 9.1.2. For a semilocal Prüfer domain R of finite Krull dimension, we use the notations in the +setup §9.0.2. For a reductive R-group scheme G, the scheme TorpGq of maximal tori of G, and the a-adic +topology on TorpGqp p +Kiq, the image of the following map is dense: +TorpGqpRr 1 +asq Ñ śr +i“1 TorpGqp p +Kiq. +Proof. The proof proceeds in the following steps. +Step 1. The ring A :“ lim +ÝÑkě0 CauchyěkpRr 1 +asq is a semilocal ring with residue fields Frac pOi. Let I +be the kernel of the surjection A ։ śr +i“1 Frac pOi. Since A{I is a product of fields, it suffices to show +that 1 ` I Ă Aˆ. For a sequence pbNqN P I, its tail lies in impakR Ñ Rr 1 +asq for all k ą 0, so the tail +of p1 ` bNqN is invertible in Rˆ. Since Rr 1 +as is semilocal, the tail of p1 ` bNqN is termwise invertible in +Rr 1 +as and the inverses form a Cauchy sequence. +Step 2. We combine the Step 1 and Lemma 9.1.1 to obtain the following surjective map +lim +ÝÑmě0 +´ +TorpGq +` +CauchyěmpRr 1 +asq +˘¯ +» TorpGq +´ +lim +ÝÑmě0 +` +CauchyěmpRr 1 +asq +˘¯ +։ śr +i“1 TorpGqpFrac pOiq, +which signifies that every Cauchy sequence in the image of TorpGqpRr 1 +asq converges in śr +i“1 TorpGqFrac pOi, +hence the assertion follows. +□ +9.2. Harder’s weak approximation +Lemma 9.2.1. For a semilocal Prüfer domain R of finite Krull dimension, we use the setup §9.0.2. For +a Rr 1 +as-torus T , let Li{ pKi be minimal Galois field extensions splitting Tx +Ki and consider the norm map +Ni : T pLiq Ñ T p pKiq. +Then, the image U of śr +i“1 Ni is a-adically open and is contained in impT pRr 1 +asq Ñ śr +i“1 T p pKiqq. +Proof. The proof proceeds as the following steps. +Step 1. The image U is a-adically open. For each i, there is a short exact sequence of tori +1 Ñ Ti Ñ ResLi{x +Ki TLi Ñ Tx +Ki Ñ 1 +and the norm map Ni : ResLi{x +Ki TLip pKiq Ñ pResLi{x +Ki TLi{Tiqp p +Kiq, which by [Čes15, Proposition 4.3 (a) +and §2.8 (2)] is a-adically open. As a product of open subsets, U is open in śr +i“1 T p pKiq. +Step 2. We prove that U is contained in the closure of impT pRr 1 +asqq. Equivalently, we show that every +u P U and every a-adically open neighborhood Bu Ă U satisfy that Bu X impT pRr 1 +asqq ‰ H. Let rR{Rr 1 +as +be a minimal Galois cover splitting T . Consider the following commutative diagram +T p ˜Rq +śr +i“1 T pLiq +T pRr 1 +asq +śr +i“1 T p pKiq. +NĂ +R{Rr 1 +a s +śr +i“1 Ni +Take a preimage v P pśr +i“1 Niq´1puq Ă śr +i“1 T pLiq and let Bv Ă śr +i“1 T pLiq be the preimage of Bu. +Since T rR splits, the image of T p rRq in śr +i“1 T pLiq is a-adically dense, hence T p rRq ˆśr +i“1 T pLiq Bv ‰ H, +namely, there is r P T p rRq whose image is in Bv. Let s :“ N rR{Rr 1 +a sprq P T pRr 1 +asq, then the image of s +under the map T pRr 1 +asq Ñ śr +i“1 T p pKiq is contained in Bu, so the assertion follows. +□ +Lemma 9.2.2. For a semilocal Prüfer domain R of finite Krull dimension, we use the setup §9.0.2. For +a reductive R-group scheme G and for each i a fixed maximal torus Ti Ă Gx +Ki with minimal Galois field +extension Li{ pKi splitting Ti, consider the following norm map +Ni : T pLiq Ñ T p pKiq. +38 + +Then the image U of the map śr +i“1 Ni is an a-adically open subgroup of śr +i“1 T p pKiq and is contained in +the closure of impGpRr 1 +asq Ñ śr +i“1 Gp pKiqq. +Proof. By the same arguement in Lemma 9.2.1, the image U is a-adically open in ś +i“1 T p pKiq. It remains +to show that U Ă impGpRr 1 +asqq, which proceeds as the following steps. +Step 1. The map φi : Gp pKiq Ñ TorpGqp p +Kiq defined by g ÞÑ gT g´1 is a-adically open for each i. Since +the image of T pRr 1 +asq Ñ śr +i“1 T p pKiq is a-adically dense, for every open neighborhood W Ă śr +i“1 Gp pKiq +of id, we have ppśr +i“1 φiqpWqq X ImpTorpGqpRr 1 +asq Ñ śr +i“1 TorpGqp p +Kiqq ‰ H. Therefore, there exist a +torus T 1 P TorpGqpRr 1 +asq and a pgiqr +i“1 P W such that giTig´1 +i +“ T 1 +x +Ki for all i. +Step 2. For any u P U, consider the map śr +i“1 Gp pKiq Ñ śr +i“1 Gp pKiq defined by g ÞÑ g´1ug. Then, +we apply the Step 1 to the preimage W of U under this map: there is a γ “ pγiqr +i“1 P W and a torus +T 1 P TorpGqpRr 1 +asq such that γ´1 +i +Tiγi “ T 1 +x +Ki for each i. Then, u P γUγ´1 “ γppśr +i“1 NiqpTipLiqqqγ´1, +which by transport of structure, is pśr +i“1 NiqpT 1 +x +KipLiqq. By Lemma 9.2.1, the last term is contained in +the closure of impT 1pRr 1 +asq Ñ śr +i“1 T p pKiqq, so is contained in impGpRr 1 +asqq. +□ +Proposition 9.2.3. For a semilocal Prüfer domain R of finite Krull dimension, we use the setup §9.0.2. +For a reductive group scheme G over R, the closure GpRr 1 +asq of the image of GpRr 1 +asq Ñ śr +i“1p pKiq, +GpRr 1 +asq +contains an open normal subgroup N of śr +i“1 Gp pKiq. +Proof. The proof proceeds in the following steps. +(i) For each i, we fix a maximal torus Ti Ă Gx +Ki. Then Lemma 9.2.2 provides the open subgroup +U Ă śr +i“1 Tip pKiq. Since each component of the norm map defining U is the image of the pKi- +points of ResLi{x +KipTiqLi Ñ Ti, and ResLi{x +KipTiqLi is a Zariski dense open subset of an affine +space over pKi, we have U X śr +i“1 T reg +i +p pKiq ‰ H. +(ii) Fix an element τ P U X śr +i“1 T reg +i +p pKiq, by [SGA 3II, Exposé XIII, Corollaire 2.2], for each i, +fi : Gx +Ki ˆ Ti Ñ Gx +Ki, +pg, tq ÞÑ gtg´1 +is smooth at pid, τq. +Hence, there is a Zariski open neighborhood B of pid, τq such that pśr +i“1 fiq|B : B Ñ śr +i“1 Gx +Ki +is smooth. By [GGMB14, Proposition 3.1.4], the map Bpśr +i“1 pKiq Ñ śr +i“1 Gp p +Kiq is open. Then +the image of W :“ Bpśr +i“1 pKiq X pśr +i“1 Gp p +Kiq ˆ Uq under f “ śr +i“1 fi is open. Subsequently, +all śr +i“1 Gp pKiq translations of W have open images, so there is an open subset U0 Ă U such that +E :“ fpśr +i“1 Gp pKiq ˆ U0q is open. Now we define N as the subgroup of śr +i“1 Gp pKiq generated +by E, then E is an open subgroup. Further, by construction, E is stable under conjugations by +śr +i“1 Gp pKiq, thus N is normal. +(iii) We prove that N is contained in the closure of impGpRr 1 +asq Ñ śr +i“1 Gp pKiqq. Since E is the union +of all conjugates of U0, which are contained in GpRr 1 +asq by Lemma 9.2.2, so E is in this closure, +and so is N. +□ +Corollary 9.2.4. For a semilocal Prüfer domain R of finite Krull dimension, we use the setup §9.0.2. +For a reductive group scheme G over R, a maximal torus Ti Ă G p +Oi for each i, and any a-adically open +neighborhood W of id P śr +i“1 Gp pKiq such that W Ă GpRr 1 +asq X śr +i“1 Gp p +Oiq, there exist g “ pgiqi P W +and a maximal torus T P TorpGqpRq such that for every i, we have +Tx +Ki “ gipTiqx +Kig´1 +i +. +Proof. By Proposition 9.2.3, GpRr 1 +asqXśr +i“1 Gp p +Oiq is an a-adically open neighborhood of id P śr +i“1 Gp pKiq, +so it makes sense to take its subset W such that W is a neighborhood of id. Now consider the a-adically +open map φ: śr +i“1 Gp pKiq Ñ śr +i“1 TorpGqp p +Kiq defined by gi ÞÑ gipTiqx +Kig´1 +i +. Then φpWq is an a-adically +open neighborhood of pTiqi P śr +i“1 TorpGqp p +Kiq. Since śr +i“1 TorpGqp p +Oiq Ă śr +i“1 TorpGqp p +Kiq is also an +39 + +a-adically open neighborhood of pTiqi, we have an open intersection φpWq X śr +i“1 TorpGqp p +Oiq ‰ H. +Then the density of the image of TorpGqpRr 1 +asq Ñ śr +i“1 TorpGqp pKiq provided by Lemma 9.1.2 yields an +element +T P TorpGqpRq +„ +ÝÑ TorpGqpRr 1 +asq ˆśr +i“1 TorpGqpx +Kiq +śr +i“1 TorpGqp p +Oiq. +Therefore, T is a maximal torus of G over R satisfying the conditions. +□ +Corollary 9.2.5. With the notations in Proposition 9.2.3, we have +GpRr 1 +asq ¨ śr +i“1 Gp p +Oiq “ impGpRr 1 +asq Ñ śr +i“1 Gp pKiqq ¨ śr +i“1 Gp p +Oiq. +9.3. Product formula over semilocal Prüfer domains, passage to the local case +Lemma 9.3.1. For a semilocal Prüfer domain R of finite Krull dimension, we use the notations in the +setup §9.0.2. For an R-torus T , we have the following product formula +śr +i“1 T p pKiq “ impT pRr 1 +asq Ñ śr +i“1 T p pKiqq ¨ śr +i“1 T p pOiq. +Proof. Let Rh denote the Henselization of the pair pR, aRq. Then we have the commutative digram +0 +T pRq +T pRr 1 +asq +H1 +ta“0upR, T q +H1pR, T q +H1pRr 1 +as, T q +0 +T pRhq +T pRhr 1 +asq +H1 +ta“0upRh, T q +H1pRh, T q +H1pRhr 1 +as, T q, +whose exact rows are the local cohomology exact sequences. Since the case of tori for Theorem 9.0.1 is +proved, the two horizontal arrows of the rightmost squares are injective, hence the coset T pRhr 1 +asq{T pRhq » +H1 +ta“0upRh, T q. By excision [Mil80, III, 1.28], we have an isomorphism H1 +ta“0upR, T q – H1 +ta“0upRh, T q, +which leads to a surjection T pRr 1 +asq ։ H1 +ta“0upRh, T q. Therefore, we obtain the product formula +T pRhr 1 +asq “ impT pRr 1 +asq Ñ T pRhr 1 +asqq ¨ T pRhq. +(9.3.1) +On the other hand, by [BČ22, 2.2.17], the image of T pRhr 1 +asq Ñ śr +i“1 T p pKiq is dense in śr +i“1 T p pKiq +with respect to the topology fixed in §9.0.2. Since each T p pOiq Ă T p pKiq is an open subgroup, we have +im +` +T pRhr 1 +asq Ñ śr +i“1 T p pKiq +˘ +¨ śr +i“1 T p pOiq “ śr +i“1 T p pKiq. +(9.3.2) +Consequently, the combination of (9.3.1) and (9.3.2) leads to the assertion. +□ +Proposition 9.3.2. For a semilocal Prüfer domain R of finite Krull dimension, we use the notations in +the setup §9.0.2. For a reductive R-group scheme G, we have +śr +i“1 Gp pKiq “ im +´ +GpRr 1 +asq Ñ śr +i“1 Gp pKiq +¯ +¨ śr +i“1 Gp p +Oiq. +Proof. We will proceed verbatim as in [Guo20, §4]. We choose a minimal parabolic pOi-subgroup Pi for +each Gi :“ G ˆR pOi. Denote Ui :“ radupPiq. +(i) for the maximal split torus Ti Ă Pi, we have śr +i“1 Tip pKiq Ă impGpRr 1 +asq Ñ śr +i“1 Gp pKiqq ¨ +śr +i“1 Gp p +Oiq. By [SGA 3III new, Exposé XXVI, Corollaire 6.11], there is a maximal torus rTi of Gi +containing Ti. In particular, p rTiqx +Ki is a maximal torus of Gx +Ki. Then we apply Corollary 9.2.4 +to all rTi: there are a g “ pgiqi P GpRr 1 +asq ¨ śr +i“1 Gp p +Oiq and a maximal torus T0 Ă G such that +pT0qx +Ki “ gip rTiqx +Kig´1 +i +for every i, which combined with the product formula Lemma 9.3.1 for T0 +yields +śr +i“1 rTp pKiq “ śr +i“1 g´1 +i +T0p pKiqgi Ă śr +i“1 impGpRr 1 +asqq ¨ Gp p +Oiqgi. +Since g P impGpRr 1 +asqq X śr +i“1 Gp p +Oiq, the inclusion displayed above implies that śr +i“1 rTip p +Kiq Ă +impGpRr 1 +asqq ¨ śr +i“1 Gp p +Oiq. Therefore, we obtain the following desired inclusion +śr +i“1 Tip pKiq Ă śr +i“1 rTip pKiq Ă impGpRr 1 +asqq ¨ śr +i“1 Gp p +Oiq. +40 + +(ii) we have śr +i“1 Uip pKiq Ă impGpRr 1 +asq Ñ śr +i“1 Gp p +Kiqq. Consider the Ti-action on Gi defined by +Ti ˆ Gi Ñ Gi, +pt, gq ÞÑ tgt´1. +Recall the open normal subgroup N Ă śr +i“1 Gp p +Kiq constructed in Proposition 9.2.3, then each +N XUip pKiq is open in Uip pKiq. The dynamic argument in [Guo20] shows that Uip pKiq “ N XUip pKiq, +hence Uip pKiq Ă N for each i. Therefore, we have śr +i“1 Uip pKiq Ă impGpRr 1 +asq Ñ śr +i“1 Gp pKiqq. +(iii) we have śr +i“1 Pip p +Kiq Ă impGpRr 1 +asq Ñ śr +i“1 Gp p +Kiqq ¨ śr +i“1 Gp p +Oiq. The quotient Hi :“ Li{Ti is +anisotropic, therefore we have Hip pKiq “ Hip p +Oiq for every i. Consider the commutative diagram +0 +Tip p +Oiq +Lip p +Oiq +Hip pOiq +H1p p +Oi, Tiq “ 0 +0 +Tip pKiq +Lip p +Kiq +Hip pKiq +H1p pKi, Tiq “ 0 +with exact rows. By diagram chase, we have Lip pKiq “ Tip pKiq ¨ Lip p +Oiq for every i. Subsequently, +the combination of (i) and (ii) yields the inclusion +śr +i“1 Pip pKiq Ă impGpRr 1 +asq Ñ śr +i“1 Gp pKiqq ¨ śr +i“1 Gp pOiq. +(iv) Recall [SGA 3III new, Exposé XXVI, Théorème 4.3.2 and Corollaire 5.2] that for each Pi, there is +a parabolic subgroup Qi of Gi such that Pi X Qi “ Li fitting into the following surjection +radupPiqp pKiq ¨ radupQiqp pKiq ։ Gp pKiq{Pip pKiq. +This surjection, combined with the result of (ii) gives an inclusion +śr +i“1 Gp pKiq Ă impGpRr 1 +asq Ñ śr +i“1 Gp pKiqq ¨ śr +i“1 Pip pKiq. +Now we further use the result of (iii) to obtain śr +i“1 Gp pKiq Ă impGpRr 1 +asq Ñ śr +i“1 Gp p +Kiqq ¨ +śr +i“1 Gp p +Oiq. Hence, we have the following product formula +śr +i“1 Gp pKiq “ im +´ +GpRr 1 +asq Ñ śr +i“1 Gp pKiq +¯ +¨ śr +i“1 Gp p +Oiq. +□ +10. +Torsors on a smooth affine relative curve +In this section we prove the following result concerning triviality of torsors on a smooth affine relative +curve. The idea of the proof ultimately depends on the geometry of affine Grassmannians developed by +Fedorov, who proved Theorem 10.1 (i) for C “ A1 +R. A similar result can also be found in the recent +preprint [Čes22c, Theorem 4.4]. +Theorem 10.1 (Section theorem). Let R be a semilocal domain whose local rings at primes are geomet- +rically unibranch5, C a smooth, affine, relative R-curve, and G a reductive C-group scheme. Let A be a +R-algebra. Let P be a G-torsor over CA :“ C ˆR A that trivializes over CAzZA for some R-finite closed +subscheme Z Ă C. For a section s P CpRq, if either +(i) A is semilocal, or +(ii) s˚ +ApGq is totally isotropic, +then the pullback s˚ +ApPq is trivial as an s˚ +ApGq-torsor, where sA stands for the image of s in CApAq. +To prove Theorem 10.1, we first use Lemma 10.2 to reduce to the case when G is the base change of a +reductive R-group scheme, and then to the case when C “ A1 +R, see Lemma 10.3. As for the latter, one +can approach it via the geometry of affine Grassmannians. +We start with the following result concerning equating reductive group schemes, which was already known +to experts, see also [Čes22c, Lemma 3.5]. +5According to [SP, 0BPZ], a local ring A is geometrically unibranch if its reduction Ared :“ A{ +a +p0q is a domain, and if +the integral closure of Ared in its fraction field is a local ring whose residue field is purely inseparable over that of A. By +[SP, 06DM], A is geometrically unibranch iff its strict Henselization Ash has a unique minimal prime. +41 + +Lemma 10.2 (Equating reductive group schemes). Let B be a semilocal ring whose local rings are +geometrically unibranch, and let G1, G2 be two reductive B-group schemes whose geometric B-fibers are +of the same type. Let T1 Ă G1, T2 Ă G2 be maximal B-tori. Assume that, for some ideal I Ă B, there is +an isomorphism of B{I-group schemes +ι : pG1qB{I » pG2qB{I +such that +ιppT1qB{Iq “ pT2qB{I. +There are a faithfully flat, finite, étale B-algebra B1, a section s : B1 ։ B{I, and an isomorphism of +B1-groups ι1 : pG1qB1 » pG2qB1 such that ιppT1qB1q “ pT2qB1 and whose s-pullback is ι. +Proof. According to [SGA 3III new, Exposé XXIV, Corollaire 2.2], the condition on the geometric B-fibers +ensures that the functor +X :“ IsomBppG1, T1q, pG2, T2qq +parameterizing the isomorphisms of the pairs pG1, T1q and pG2, T2q is representable by a B-scheme and +is a H :“ AutBppG1, T1qq-torsor. We need to show that, for any ι P XpB{Iq, there are a faithfully flat, +finite, étale B-algebra B1, an ι1 P XpB1q, and a section s : B1 ։ B{I such that spι1q “ ι P XpB{Iq. +By loc. cit., H is an extension of an étale locally constant B-group scheme by T ad +1 , the quotient of T1 +by the scheme-theoretic center of G1. According to [SGA 3III new, Exposé XXIV, Proposition 2.6], T ad +1 +acts freely on X and the quotient +X :“ X{T ad +1 +is represented by a faithfully flat B-scheme that is étale locally constant on B. As B is geometrically +unibranch, by [SGA 3III new, Exposé X, Corollaire 5.14], every connected component of X is finite, étale +over B. As the image of ι : SpecpB{Iq Ñ X Ñ X intersects only finitely many connected components +of X, the union of these components is the spectrum of a finite étale B-algebra A, and there are an +ι P XpAq and a section t : A ։ B{I such that tpιq “ ι. By adding more connected components of X +into SpecpAq if needed, we may assume that A is faithfully flat over B. Let +Y :“ X ˆX,ι SpecpAq; +it is a T ad +A -torsor equipped with a point ι P Y pA{Jq Ă XpA{Jq, where J :“ ker pA ։ B{Iq. +By +[Čes22b, Corollary 6.3.2], there are a faithfully flat, finite, étale A-algebra B1, a section +s1 : B1 ։ A{J » B{I, +and an ι1 P Y pB1q Ă XpB1q such that s1pι1q “ ι. +□ +Lemma 10.3. The proof of the Theorem 10.1 reduces to the case when C “ A1 +R and G is the base change +of a reductive R-group scheme. +Proof. Let B be the semilocal ring of C at the closed points of impsqYZ; its local rings are geometrically +unibranch. By abuse of notation, we may view s : B ։ R as a section of the R-algebra B. As B +is semilocal, by [SGA 3II, Exposé XIV, Corollaire 3.20], GB admits a maximal B-torus TB. Since the +pullbacks of the paris pGB, T q and pps˚pGqqB, ps˚pT qqBq along s are the same, by Lemma 10.2, there are +a faithfully flat, finite, étale B-algebra B1, a section s1 : B1 ։ R that lifts s, and a B1-isomorphism +ι : pGB1, TB1q » pps˚pGqqB1, ps˚pT qB1q +whose s-pullback is the identity. We may spread out SpecpB1q Ñ SpecpBq to obtain a finite étale covering +C1 Ñ U of a small enough affine open neighbourhood U of impsq Y Z in C. By shrinking U if necessary, +we may assume that the isomorphism ι is defined over C1. In both cases of Theorem 10.1 we may replace +C by C1, Z by C1 ˆC Z, s by s1, and P by P|C1 +A to reduce to the case when G is the base change of the +reductive R-group scheme s˚pGq. +Next, in order to apply glueing Lemma 6.2.2(ii) to achieve that C “ A1 +R, we need to modify C so that +Z embeds into A1 +R. For this, we first replace Z by Z Y impsq to assume that s factors through Z. Then +we apply Panin’s ‘finite field tricks’ [Čes22a, Proposition 7.4] to obtain a finite morphism rC Ñ C that is +étale at the points in rZ :“ rC ˆC Z such that s lifts to rs P rCpRq, and there are no finite fields obstruction +to embedding rZ into A1 +R in the following sense: for every maximal ideal m Ă R, +7 +! +z P rZκpmq : rκpzq : κpmqs “ d +) +ă 7 +! +z P A1 +κpmq : rκpzq : κpmqs “ d +) +for every +d ě 1. +42 + +Then, by [Čes22a, Lemma 6.3], there are an affine open C2 Ă rC containing imprsq, a quasi-finite, flat +R-map C2 Ñ A1 +R that maps Z isomorphically to a closed subscheme Z1 Ă A1 +R with +Z » Z1 ˆA1 +R C2. +(Actually, by shrinking C2 around imprsq, one can show that C2 Ñ A1 +R is étale.) +For both cases of +Theorem 10.1, since P|C2 +A is a G-torsors that trivializes over C2 +Az rZA, we may use Lemma 6.2.2(ii) to glue +PC2 +A with the trivial G-torsor over A1 +A to obtain a G-torsor P1 over A1 +A that trivializes over A1 +AzZ1 +A. Let +s1 P A1 +RpRq be the image of rs; then s1˚pP1q » s˚pPq. It remains to replace C by A1 +R, Z by Z1, s by s1, +and P by P1. +□ +The analysis of torsors on A1 +R ultimately depends on the geometry of affine Grassmannians. A nice +summary of and complement on the relevant techniques can be found in [Čes22b, §5.3]. In particular, +we will use the following result; it is a slight variant of [Čes22b, Proposition 5.3.6], which in turn is a +mild generalization of [Fed22b, Theorem 6]. +Proposition 10.4. For a semilocal ring R with connected spectrum and a reductive R-group scheme G, +let +Gad » +ź +i +ResRi{RpGiq +be the canonical decomposition of the adjoint quotient Gad [SGA 3III new, Exposé XXIV, Proposition 5.10], +where Gi is an adjoint simple Ri-group scheme, and Ri is a finite, étale R-algebra with connected spectra. +Let Y Ă A1 +R be a R-finite, étale, closed subscheme with the following properties: +(i) for every i, there is a clopen Yi Ă Y ˆR Ri such that pGiqYi contains a copy of Gm,Yi; +(ii) for every maximal ideal m Ă Ri such that pGiqκpmq is isotropic, the line bundle OP1 +κpmqp1q is trivial +over P1 +κpmqzpYiqκpmq; +(iii) the line bundle OP1 +Rp1q is trivial over P1 +RzY . +Let P be a G-torsor over P1 +R that trivializes over P1 +RzZ for some R-finite closed subscheme Z Ă A1 +RzY . +Assume that for every maximal ideal m Ă R the Gad-torsor over P1 +κpmq induced by P lifts to a generically +trivial pGadqsc-torsor over P1 +κpmq. Then the restriction P|P1 +RzY is trivial. +Recall that, by [SGA 3III new, Exposé XXVI, Corollaire 6.12], (i) is equivalent to that the base change of +pGiqYi to every connected component of Yi contains a proper parabolic subgroup scheme. For instance, if +G is quasi-split, we can just take Yi “ Y ˆR Ri to ensure (i). In practice, we achieve (i) by guaranteeing +base change of pGiqYi to connected components of Yi contain proper parabolics. For (ii), we can take Yi +so that Yipκpmqq ‰ H for every maximal ideal κpmq Ă Ri with pGiqκpmq isotropic. For (iii), we just need +to choose Y so that it contains finite étale R-schemes of degrees d and d ` 1 for some d ě 1, because +Opdq and Opn ` 1q are both trivial on P1 +RzY , and so is Op1q. +Proof. We will deduce Proposition 10.4 from (the proof of) a particular case of [Čes22b, Proposition 5.3.6]. +(We remind that the assumption (ii) of loc. cit. +should read as ‘pGiqYi contains a copy of Gm,Yi’, as its +proof shows.) +The R-finite étale Y is the vanishing locus of a monic polynomial t in the standard coordinate of A1 +R; +namely, t is the characteristic polynomial of this standard coordinate acting on rR :“ ΓpY, OY q. The +formal completion of P1 +R along Y has coordinate ring rRrrtss. Recall that, by formal glueing, a G-torsor +over P1 +R can be viewed as the glueing of its restriction to P1 +RzY and to rRrrtss along the ‘intersection’ rRpptqq; +since our torsor P is trivial over an open neighbourhood U Ă P1 +R of Y , both of the restriction P|UzY and +P| rRrrtss are trivial, and once a trivialization of the former was chosen, all such glueings are parameterized +by elements of Gp rRpptqqq{Gp rRrrtssq. In particular, since Gp rRpptqqq acts on Gp rRpptqqq{Gp rRrrtssq (via left +multiplication), an element of Gp rRpptqqq yields a modification of P along Y : it is the G-torsor over P1 +R +whose restriction to P1 +RzY and to rRrrtss are the same as P, but their corresponding glueings, viewed as +elements of Gp rRpptqqq{Gp rRrrtssq, differ by a left translation by the element of rRpptqq we choose. +Denote by Pad the Gad-torsor over P1 +R induced by P. +Since the formation of H1pP1 +R, ´q commutes +with taking products, Pad corresponds to a collection pPad +i q, where Pad +i +is a ResRi{RpGiq-torsor over P1 +R +43 + +satisfying the analogous assumptions (i)-(iii) of the Proposition 10.4. Since R Ñ Ri is finite étale and +Gi is Ri-smooth, we have R1f˚Gi “ 1 for the map f : SpecpRiq Ñ SpecpRq induced by R Ñ Ri. By the +exact sequence from [Gir71, Chapitre V, Proposition 3.1.3], +1 Ñ H1pP1 +R, ResRi{RpGiqq Ñ H1pP1 +Ri, Giq Ñ H1pP1 +R, R1f˚Giq. +Thus Q ÞÑ ResRi{RpQq defines a bijection of pointed sets H1pP1 +Ri, Giq +„ +ÝÑ H1pP1 +R, ResRi{RpGiqq. In par- +ticular, each Pad +i +corresponds to a Gi-torsor Qi over P1 +Ri. As one can see immediately, the assumptions +(i)-(iii) of Proposition 10.4 for the ResRi{RpGiq-torsor Pad +i +translate into the assumptions [Čes22b, Propo- +sition 5.3.6] (i)-(iv) for the Gi-torsor Qi over P1 +Ri. By the proof of loc. cit., for some element +αi P im +´ +Gsc +i pp rR bR Riqpptqqq Ñ Gipp rR bR Riqpptqqq +¯ +, +the corresponding modification of Qi along Y ˆR Ri is trivial. We can view +α :“ pαiq P im +´ +pGadqscp rRpptqqq Ñ Gadp rRpptqqq +¯ +; +as pGadqsc Ñ Gad factors through pGadqsc Ñ G, α lifts to rα P Gp rRpptqqq. Denote by Q the modification +of P along Y using rα. By our construction, the Gad-torsor Qad over P1 +R induced by Q corresponds +to the collection of modifications of the Pad +i +“ ResRi{RpQiq along Y using αi P Gipp rR bR Riqpptqqq “ +ResRi{Rp rRpptqqq, which is trivial, so that Qad is trivial, to the effect that Q reduces to a torsor over P1 +R +under the center ZG of G. Now, as the last paragraph of the proof of [Čes22b, Proposition 5.3.6] shows, +any ZG-torsor over P1 +R is the sum of a constant torsor (i.e., the pullback of a ZG-torsor over R) and +λ˚pOp1qq for a unique cocharacter λ of ZG. Therefore, by our assumption (iii), Q is a constant torsor, +and, by checking along the infinity section, it is even trivial, so is P|P1 +RzY “ Q|P1 +RzY , as desired. +□ +The following result will help us to construct the desired R-finite, étale schemes Yi and Y from the +previous theorem. +Lemma 10.5. Let R be a semilocal ring with connected spectrum, let R1 be a finite, étale R-algebra with +connected spectrum, let W Ă A1 +R be a R-finite closed scheme, and let G1 be a simple R1-group scheme. +There is a R1-finite, étale scheme Y1, and a closed immersion Y1 Ă A1 +RzW over R such that pG1qY1 +contains a copy of Gm,Y1, and, for every maximal ideal m Ă R1 with pG1qκpmq isotropic, the line bundle +OP1 +κpmqp1q is trivial over P1 +κpmqzpY1qκpmq. (Notice that Y1 is a clopen of Y1 ˆR R1, thus naturally embeds +into A1 +R1.) +In addition, there is a R1-finite, étale scheme Y 1 and a closed immersion Y 1 ãÑ A1 +RzW over R such that +the line bundle OP1 +Rp1q is trivial over P1 +RzY 1. +Proof. Let Par1 Ñ SpecpR1q be the scheme parameterizing proper parabolic subgroup schemes of the +reductive R1-group scheme G1; it is smooth projective over R1 (cf. [SGA 3III new, Exposé XXVI, Corol- +laire 3.5]). Fix an embedding Par1 ãÑ PN +R1 over R1. Write Par1 “ Ůt +i“1 Pt as a disjoint union of its +connected components; every Pt has a constant relative dimension dt over R1. For every maximal ideal +m Ă R1 with pG1qκpmq isotropic, a proper parabolic subgroup of pG1qκpmq gives a point bm P Par1pκpmqq. +Fix an i “ 1, ¨ ¨ ¨ , t. For every maximal ideal m Ă R1, by Bertini theorem (including Poonen’s version +over finite fields), one can find a hypersurface in PN +κpmq of large enough degree such that it passes through +all points bm that lies in Pi and it intersects pPiqκpmq transversally. We may assume that the above +hypersurfaces have the same degree for all m. By the Chinese Remainder theorem, one can lift these +simultaneously to get a hypersurfaces H Ă PN +R1. Then H X Pi is a smooth projective R1-scheme of pure +relative dimension di ´ 1, and bm P H X Pi whenever bm P Pi. The same argument can be applied to +the hypersurface section H X Pi. Continuing in this way, we finally arrive at a R1-finite, étale, closed +subscheme Yi Ă Pi such that bm P Yi whenever bm P Pi. Denote Y 1 +1 :“ Ůt +i“1 Yi. Unfortunately, Y 1 +1 may +not embed into A1 +RzW. So let’s first modify Y 1 +1 using Panin’s ‘finite field tricks’. +Let d ą 0 be a large enough integer such that, for every maximal ideal n Ă R, +(1) we have d ą dimκpnq ΓpWκpnq, OWκpmqq; +(2) for every maximal ideal n1 Ă ΓpY 1 +1, OY 1 +1q lying over n and every n ě d, there are at least degpY 1 +1{Rq +(resp., at least one) closed point(s) on A1 +κpnq (resp., on A1 +κpn1q) of exact degree n. +44 + +For every maximal ideal n1 Ă ΓpY 1 +1, OY 1 +1q we choose a monic polynomial hn1 P κpn1qrus of degree 2d ` 1 +such that: +(i) if κpn1q is finite, hn1 is a product of two irreducible polynomials of degrees d and d`1, respectively +(which is possible by (2)); +(ii) if κpn1q is infinite, hn1 is a separable polynomial and has at least one root in κpn1q. +Let h P ΓpY 1 +1, OY 1 +1qrus be a common monic lifting of hn1 for all n1 Ă ΓpY 1 +1, OY 1 +1q, and define +Y1 “ Spec +ˆΓpY 1 +1, OY 1 +1qrus +phq +˙ +; +it is finite, étale over Y 1 +1, and hence also over R1. By (1)-(2), there is a closed immersion +ğ +nĂR +pY1qκpnq ãÑ A1 +RzW +over R; +by Nakayama’s lemma, any of its lifting Y1 ãÑ A1 +RzW over R (which exists by Chinese Remainder +theorem) is also a closed immersion. +By construction, the restriction of pG1qY 1 +1 to every connected +component of Y 1 +1 contains a proper parabolic subgroup scheme. Thus, by [SGA 3III new, Exposé XXVI, +Corollaire 6.12], pG1qY 1 +1 contains Gm,Y 1 +1, and so pG1qY1 contains Gm,Y1. By (i)-(ii), for m Ă R1 with +pG1qκpmq isotropic, the line bundle OP1 +κpmqp1q is trivial over P1 +κpmqzpY1qκpmq. +To construct Y 1, it suffices to produce, for a large enough d, a R-finite, étale, closed subschemes Y2 Ă A1 +R +of R-degrees d and d ` 1 which are disjoint from W, and then take Y 1 :“ Y1 +Ů Y2. To achieve this, one +just need to imitate the above procedure for constructing Y1 from Y 1 +1. Details are omitted. +□ +Proof of Theorem 10.1. By the reduction Lemma 10.3, we may assume throughout that C “ A1 +R and G +is a reductive R-group scheme. Up to shifting we may assume that s “ 0R P A1 +RpRq is the zero section, +and base changing to A reduces us further to the case A “ R at the cost that R need not be a domain or +geometrically unibranch. Thus, in case (i), our R is semilocal, and, in case (ii), our G is totally isotropic +(but R need not be semilocal). By decomposing SpecpRq into connected components, we can assume +that R has connected spectrum. +For both cases (i)-(ii), by glueing P with the trivial G-torsor over P1 +RzZ we extend P to a G-torsor Q over +P1 +R. By [Fed22b, Proposition 2.3] or [Čes22b, Lemma 5.3.5], up to replacing Q and Z by their pullbacks +by P1 +R Ñ P1 +R, t ÞÑ td, where d is divisible by the R-fibral degree of the simply-connected central cover +pGadqsc Ñ Gad, we may assume that for every maximal ideal m Ă R the Gad-torsor over P1 +κpmq induced +by Q lifts to a generically trivial pGadqsc-torsor over P1 +κpmq. +Claim 10.5.1. In both cases (i)-(ii), assume that R is semilocal. +For any R-finite closed subscheme +W0 Ă A1 +R, there exists a R-finite, étale, closed subscheme Y Ă A1 +RzW0 such that Q|P1 +RzY is trivial. +Proof of the claim. We write the canonical decomposition of Gad as in Proposition 10.4. Replacing W0 +by W0 Y Z, we may assume that Z Ă W0. Applying Lemma 10.5 separately to each simple Ri-group +scheme Gi (with appropriate choices of W’s), we get Ri-finite, étale schemes Yi such that pGiqYi is totally +isotropic, a closed immersion Ů +i Yi ãÑ A1 +RzW0 over R such that for every maximal ideal m Ă Ri with +pGiqκpmq isotropic, the line bundle OP1 +κpmqp1q is trivial over P1 +κpmqzpYiqκpmq. Applying the second part of +Lemma 10.5 to W :“ p\iYiq Ů W0, we get a R-finite, étale, closed subscheme +Y 1 Ă A1 +Rz +´ +p\iYiq +ğ +W0 +¯ +such that OP1 +Rp1q is trivial over P1 +RzY 1. Let Y :“ Y 1 Ůp\iYiq. Then all the assumptions (i)-(iii) of +Proposition 10.4 are verified, so we conclude that Q|P1 +RzY is trivial. +□ +For (i), we take W0 “ Z Y 0R, then the above Claim 10.5.1 gives a R-finite, étale, closed subscheme +Y Ă A1 +RzW0 such that Q|P1 +RzY is trivial. Since Y X 0R “ H, we deduce that the pullback of Q along +s “ 0R is also trivial, as wanted. +For (ii), we will follow [Čes22c, Lemma 4.3] to show that both P “ Q|A1 +R and Q|P1 +Rz0R descend to G- +torsors over R, and then we are done: both of these descendants agree with the restriction of Q along +45 + +1R P A1 +RpRq, so they agree with the restriction of Q along 8R, which is trivial, and hence they must be +trivial. By Quillen patching [Čes22b, Corollary 5.1.5 (b)], for the descent claim we may replace R by its +localizations at maximal ideals to assume that R is local. +Now, since R is local, we may apply the above Claim 10.5.1 to W0 “ 0R to find a R-finite, étale, closed +subscheme Z1 Ă A1 +Rz0R such that Q|P1 +RzZ1 is trivial. It remains to apply Proposition 10.4 twice, with +Y “ 0R and Y “ 8R respectively, to show that both Q|P1 +Rz0R and Q|P1 +Rz8R are trivial. +□ +11. +Torsors under a reductive group scheme over a smooth projective base +The main result of this section is the following: +Theorem 11.1. For a semilocal Prüfer domain R, an r P Rzt0u, an irreducible, smooth, projective +R-scheme X, a finite subset x Ă X with semilocal ring A :“ OX,x, and a reductive X-group scheme G, +(i) any generically trivial G-torsor over A is trivial, that is, +ker pH1pA, Gq Ñ H1pFrac A, Gqq “ t˚u; +(ii) if GAr 1 +r s is totally isotropic, then any generically trivial G-torsor over Ar 1 +rs is trivial, that is, +ker pH1pAr 1 +rs, Gq Ñ H1pFrac A, Gqq “ t˚u +The case (i) is a version of the Grothendieck–Serre conjecture in the case the relevant reductive group +scheme GA has a reductive model over some smooth projective compactification of SpecpA. The case (ii) +provides a version of Nisnevich conjecture for such ‘nice’ reductive groups satisfying the total isotropicity +assumption: if R is a discrete valuation ring with uniformizer r and if R Ñ A is a local homomorphism +of local rings, then r P mAzm2 +A, and (ii) says that any generically trivial G-torsor over Ar 1 +rs is trivial (the +isotropicity assumption on GA is essential, see, for instance, [Fed21]). +Remark 11.2. An inspection of the proof below shows that, if Xns Ă X denotes the loci where a finitely +presented morphism X Ñ SpecpRq is non-smooth, then Theorem 11.1 still holds provided that X is only +a flat projective R-scheme such that Xns is R-fiberwise of codimension ě 2 in X, x X Xns “ H, and G +is a reductive XzXns-group scheme. +To prove Theorem 11.1, we first derive from Corollary 6.3.2 and Lemma 7.1.1 the following key result, +which reduces the proof of Theorem 11.1 to studying torsors on a smooth affine relative curve. +Lemma 11.3. For a semilocal Prüfer domain R of finite Krull dimension, an irreducible, smooth, pro- +jective R-scheme X of pure relative dimension d ą 0, a finite subset x Ă X, and a reductive X-group +scheme G, the following assertions hold. +(i) Given a generically trivial G-torsor P over A :“ OX,x, there are +- a smooth, affine A-curve C, an A-finite closed subscheme Z Ă C, and a section s P CpAq; +- a reductive C-group scheme G satisfying s˚G » GA and a G -torsor F such that F|CzZ is +trivial and s˚F » P. +(ii) Given an r P Rzt0u and a generically trivial G-torsor rP over Ar 1 +rs, there are +- a smooth, affine A-curve C, an A-finite closed subscheme Z Ă C, and a section s P CpAq; +- a reductive C-group scheme G such that s˚G » GA, a G -torsor rF over Cr 1 +rs :“ C ˆA Ar 1 +rs +such that rF|Cr 1 +r szZr 1 +r s is trivial and ps|Ar 1 +r sq˚p r +Fq » rP. +Proof. By Corollary 6.3.2, P (resp., rP) extends to a G-torsor P0 (resp., Ă +P0) over an open neighbourhood +W Ă X of x (resp., an open neighbourhood Ă +W Ă X of SpecpAr 1 +rsq) such that +codimppXzWqK, XKq ě 3 +and +codimppXzWqs, Xsq ě 2 for all s P SpecpRq; +and +codimppXzĂ +WqK, XKq ě 3 +and +codimppXzĂ +Wqs, Xsq ě 2 for all s P SpecpRq. +Here, K is the fraction field of R. Let z Ă X be the set of maximal points of the R-fibers of X; the +above codimension bounds implies z Ă W (resp., z Ă Ă +W). By Lemma 3.1.1(iii), the semilocal ring OX,z, +46 + +and hence also OX,zr 1 +rs, is a Prüfer domain. By the Grothendieck–Serre on semilocal Prüfer schemes +(Theorem 9.0.1), the generically trivial G-torsor pP0q|OX,z (resp., pĂ +P0q|OX,zr 1 +r s) is actually trivial. Thus +there exists a closed subscheme Y Ă X (resp., rY Ă X) that avoids all the maximal points of R-fibers of +X such that the restriction pP0q|XzY (resp., pĂ +P0q|pXz rY qr 1 +r s) is trivial; such a Y (resp., rY ) is R-fiberwise +of codimension ą 0 in X. Now, we treat the two cases (i)–(ii) separately. +(i) By the above, XzW is R-fiberwise of codimension ě 2 in X; a fortiori, the same codimension +bound holds for Y zW in X. Consequently, we can apply Lemma 7.1.1 (vii) to obtain an affine +open S Ă Ad´1 +R +, an affine open neighbourhood U Ă W of x, and a smooth morphism π: U Ñ S +of pure relative dimension 1 such that U X Y is S-finite. +Let τ : C :“ U ˆS Spec A Ñ Spec A be the base change of π to Spec A. Let Z and F be the +pullbacks of U X Y and pP0q|U under pr1 : C Ñ U, respectively. Then, via τ, C is a smooth +affine A-curve, Z Ă C is a A-finite closed subscheme, and F is a G :“ pr˚ +1pGUq-torsor that +trivializes over CzZ. Finally, the diagonal in C induces a section s P CpAq with s˚F » P (as +s˚G “ GA-torsors). +(ii) Since SpecpAr 1 +rsq consists of points of Xr 1 +rs :“ X ˆR Rr 1 +rs that specializes to some point of x, we +deduce from the inclusion SpecpAr 1 +rsq Ă Ă +W that no points of pXzĂ +Wqr 1 +rs “ Xr 1 +rszĂ +Wr 1 +rs specializes +to any points of x. Hence, the closure pXzĂ +Wqr 1 +rs (in X) is disjoint from x, so Ă +W 1 :“ XzpXzĂ +Wqr 1 +rs +is an open neighbourhood of x. Notice that X is topological Noetherian, because its R-fibers are +projective varieties over fields, and by our assumption SpecpRq has a finite underlying space. Since +by the above pXzĂ +Wqr 1 +rs is Rr 1 +rs-fiberwise of codimension ě 2 in Xr 1 +rs, by Lemma 3.1.1(i) applied +to the closures of the (finitely many) maximal points of pXzĂ +Wqr 1 +rs, the closure pXzĂ +Wqr 1 +rs “ XzĂ +W 1 +is R-fiberwise of codimension ě 2 in X; a fortiori, the same holds for rY zĂ +W 1 in X. Consequently, +we can apply Lemma 7.1.1 (vii) to obtain an affine open rS Ă Ad´1 +R +, an affine open neighbourhood +rU Ă Ă +W 1 of x, and a smooth morphism rπ : rU Ñ rS of pure relative dimension 1 such that rU X rY +is rS-finite. Notice that rUr 1 +rs Ă Ă +W 1r 1 +rs “ Ă +Wr 1 +rs, so we have the restriction pĂ +P0q| rUr 1 +r s. +Let τ : C :“ rU ˆ rS SpecpA Ñ SpecpA be the base change of rπ to SpecpA. Let Z be the pullback +of rU X rY under pr1 : C Ñ rU. Let rF be the pullback of pĂ +P0q| rUr 1 +r s under pr1 : Cr 1 +rs Ñ rUr 1 +rs. +Then, via τ, C is a smooth affine A-curve, Z Ă C is a A-finite closed subscheme, and r +F is +a G :“ pr˚ +1pG rUq-torsor over Cr 1 +rs that trivializes over Cr 1 +rszZr 1 +rs. Finally, the diagonal in C +induces a section s P CpAq with s˚ +Ar 1 +r sp r +Fq » rP, and s˚G “ GA. +□ +Proof of Theorem 11.1. By a standard limit argument involving Lemma 3.1.3, one easily reduces to the +case when R has finite Krull dimension. Now, let P (resp., rP) be a generically trivial G-torsor over +A :“ OX,x (resp., over Ar 1 +rs) which we want to trivialize. Let d be the relative dimension of X over R. If +d “ 0, then A and Ar 1 +rs are semilocal Prüfer domains, so, by the Grothendieck–Serre on semilocal Prüfer +schemes (Theorem 9.0.1), the torsors P and rP are trivial. Hence we may assume that d ą 0. Then, +by Lemma 11.3, there are a smooth, affine A-curve C, an A-finite closed subscheme Z Ă C, a section +s P CpAq, a reductive C-group scheme G with s˚G » GA, +- a G -torsor F over C that trivializes over CzZ such that s˚F » P, and +- a G -torsor rF over Cr 1 +rs that trivializes over Cr 1 +rszZr 1 +rs such that ps|Ar 1 +r sq˚p rFq » rP. +By Theorem 10.1 (i), the G-torsor s˚F » P is trivial. By Theorem 10.1 (ii), in the case ps|Ar 1 +r sq˚pG q » +GAr 1 +r s is totally isotropic, the GAr 1 +r s-torsor ps|Ar 1 +r sq˚p r +Fq » rP is trivial. +□ +12. +Torsors under a constant reductive group scheme +In this section we prove the following variant of Theorem 11.1, in which the R-smooth scheme X need +not be proper, but the reductive group scheme G is supposed to descend to the Prüfer ring R. Thus, +we established the Grothendieck–Serre conjecture and a version of Nisnevich conjecture for ‘constant’ +47 + +reductive group schemes. As for the proof, we use a variant of Lindel’s Lemma (Proposition 7.2.1) and +glueing techniques to reduce to the case already settled by Theorem 11.1. +Theorem 12.1. For a semilocal Prüfer domain R, a nonzero element r P R, an irreducible affine R- +smooth scheme X, a finite subset x Ă X, and a reductive R-group scheme G, +(i) any generically trivial G-torsor over A :“ OX,x is trivial, that is, +ker +` +H1pA, Gq Ñ H1pFrac A, Gq +˘ +“ t˚u; +(ii) if GRr 1 +r s is totally isotropic, then any generically trivial G-torsor over Ar 1 +rs is trivial, that is, +ker +` +H1pAr 1 +rs, Gq Ñ H1pFrac A, Gq +˘ +“ t˚u. +Proof. Let P (resp., rP) be a generically trivial G-torsor over A (resp., over Ar 1 +rs). By shrinking X +around x, we may assume that P is defined over the whole X (resp., rP is defined over the whole +Xr 1 +rs :“ X ˆR Rr 1 +rs). Let d be the relative dimension of X over R. As noted by ˇCesnaviˇcius, since it +suffices to argue that P (resp., rP) is trivial Zariski semilocally on X, we may replace X by X ˆR AN +R +for large N to assume that d ą # x: by pulling back along the zero section X Ñ X ˆR AN +R , the Zariski +semilocal triviality of PXˆRAN +R (resp., rPXr 1 +r sˆRAN +R ) on X ˆR AN +R implies that of P (resp., rP) on X. +By specialization, we may assume that each point of x is closed in the corresponding R-fiber of X (but +not necessarily lies in the closed R-fibers of X). Our goal is to show that P|A (resp., rP|Ar 1 +r s) is trivial. +If d “ 0, then A (resp., Ar 1 +rs) is a semilocal Prüfer domain, so, by the Grothendieck–Serre conjecture on +semilocal Prüfer schemes (Theorem 9.0.1), the torsor P|A (resp., rP|Ar 1 +r s) is trivial. Thus we may assume +that d ą 0 for what follows. +Denote by π : X Ñ S :“ SpecpRq the structural morphism. Let y be the set of maximal points of the +R-fibers of X. +Claim 12.1.1. No points of x specializes to any point of y, that is, x X y “ H. +Proof of the claim. By Lemma 3.1.1(iii), for any y P y, OX,y is a valuation ring having the same value +group as OS,πpyq; in particular, the map πy : Spec OX,y Ñ Spec OS,πpyq induced by π is a homeomorphism, +and is thus injective. Assume by contradiction that x P x specializes to y P y, so Spec OXπpxq,x is a +subset of Spec OX,y. Since the image of Spec OXπpxq,x under πy is the singleton tπpxqu, by the injectivity +of πy, we deduce that dim OXπpxq,x “ 0. This contradicts the fact dim OXπpxq,x “ d ą 0 (because by our +assumption x is a closed point in the corresponding π-fiber). +□ +By Lemma 3.1.1(iii) again, the semilocal ring OX,y, and hence also OX,yr 1 +rs, is a Prüfer domain, so, by +the Grothendieck–Serre conjecture on semilocal Prüfer schemes (Theorem 9.0.1), the generically trivial G- +torsor P|OX,y (resp., rP|OX,yr 1 +r s) is actually trivial. Therefore, using the above claim and prime avoidance, +we can find an element a P ΓpX, OXq such that, denoting Y :“ V paq Ă X, then x Ă Y , y X Y “ H, +and the restriction P|XzY (resp., rP|pXzY qr 1 +r s) is trivial. (We just take a “ a1a2, where a1 is an element +such that y X V pa1q “ H and P|XzV pa1q (resp., rP|pXzV pa1qqr 1 +r s) is trivial, and a2 is delivered from prime +avoidance utilizing the fact x X y “ H so that x Ă V pa2q and y X V pa2q “ H.) +Since d ą # x, we may apply Proposition 7.2.1 to obtain an affine open neighbourhood W Ă X of x, +an affine open subscheme U Ă Ad +R, and an étale surjective R-map f : W Ñ U such that the restriction +f|WXY is a closed immersion and f induces a Cartesian square +W X Y +W +W X Y +U. +f +Applying p´q ˆR Rr 1 +rs yields a similar Cartesian square. By glueing Lemma 6.2.2 (ii), +48 + +(i) we may (non-canonically) glue P|W and the trivial G-torsor over UzfpW X Y q to descend P|W +to a G-torsor Q over U that trivializes over UzfpW X Y q. Since U has a smooth, projective +compactification Pd +R, we may apply Theorem 11.1 (i) to deduce that Q|OU,fpxq is trivial, so P|A “ +P|OW,x is trivial, as desired. +(ii) we may (non-canonically) glue rP|Wr 1 +r s and the trivial G-torsor over pUzfpW X Y qqr 1 +rs to descend +rP|Wr 1 +r s to a G-torsor rQ over Ur 1 +rs that trivializes over Ur 1 +rszfpW XY qr 1 +rs. Since U has a smooth, +projective compactification Pd +R, we may apply Theorem 11.1 (ii) to conclude that rQ|OU,fpxqr 1 +r s is +trivial, so rP|Ar 1 +r s “ rP|OW,xr 1 +r s is trivial, as desired. +□ +13. +Torsors under a quasi-split reductive group scheme +In this section we study generically trivial torsors under quasi-split reductive group schemes. The main +result is the following Theorem 13.1, in which (i) is a version of Nisnevich conjecture that is inspired +by the recent preprint of ˇCesnaviˇcius [Čes22c, Theorem 1.3 (2)], who proved it in the case R is a +Dedekind domain, and (ii) is the Grothendieck–Serre conjecture over one-dimensional Prüfer bases. As +for the proof, we will follow the strategy of [Čes22a] (with its earlier version given by Fedorov [Fed22b]), +which goes through because the main tools, such as toral version of purity (Proposition 8.2.5) and the +Grothendieck–Serre conjecture (Proposition 8.3.2(i)) in our context, are available now. +Theorem 13.1. For a semilocal Prüfer domain R with fraction field K, an irreducible, semilocal, and +essentially smooth R-algebra A, and a quasi-split reductive A-group scheme G, +(i) every generically trivial G-torsor over A bR K is trivial, that is, +ker +` +H1pA bR K, Gq Ñ H1pFrac A, Gq +˘ +“ t˚u; +(ii) if R has Krull dimension 1, then every generically trivial G-torsor is trivial, that is, +ker +` +H1pA, Gq Ñ H1pFrac A, Gq +˘ +“ t˚u. +We start with the following consequence of Lemma 7.1.1, which is the key geometric input permitting a +series of reductions that eventually lead to Theorem 13.1. +Lemma 13.2 (cf. [Čes22a, Proposition 4.1]). For +(i) a semilocal Prüfer domain R of Krull dimension 1 with fraction field K; +(ii) a smooth, faithfully flat, R-algebra A of pure relative dimension d ě 1 over R; +(iii) a finite subset x Ă X :“ Spec A; +(iv) a closed subscheme Y Ă X that satisfies +codimpYK, XKq ě 2 +and +codimpYs, Xsq ě 1 for all s P Spec R; +there are an affine open U Ă Spec A containing x, an affine open S Ă Ad´1 +R +, and a smooth R-morphism +π : U Ñ S of relative dimension 1 such that Y X U is S-finite. +Moreover, if in piq R is allowed to be of arbitrary finite Krull dimension, then the same conclusion holds +provided pivq is replaced by the stronger assumption that Y is R-fiberwise of codimension ě 2 in X. +Proof. Choosing an embedding of X into some affine space over R and taking schematic closure in +the corresponding projective space, we get a projective compactification X of X. Since X is flat and +projective over R, by Lemma 3.1.1(i), all its R-fibers have the same dimension d. Denote by Y Ă X the +schematic closure of Y . To apply Lemma 7.1.1 (vii) and conclude, in which X is X here, W is X here, +and Y is Y here, we need to check that the boundary Y zY is R-fiberwise of codimension ě 2 in X. +By [SP, 01R8], set-theoretically we have Y “ Ť +y tyu, where y runs through the generic points of Y . +In the case Y is R-fiberwise of codimension ě 2 in X, the same holds for Y in X; a fortiori, Y zY is +R-fiberwise of codimension ě 2 in X. Indeed, by Lemma 3.1.1(i), X has equal R-fiber dimension d and +49 + +all non-empty R-fibers of tyu have the same dimension, so, if y lies over sy P Spec R, then +codimptyus, Xsq “ codimptyusy, Xsyq ě 2 +for any specialization sy ù s P Spec R. +Next, we assume that R has Krull dimension 1 and Y is of codimension ě 2 (resp., ě 1) in the generic +(resp., closed) R-fiber of X. If y P Yη, then, by Lemma 3.1.1(i) again, we see that +codimptyus, Xsq “ codimptyuη, Xηq ě 2 +for all s P Spec R; +a fortiori, the contribution of such a y to the R-fiber codimension of Y zY in X is ě 2. +Otherwise, y lies over a height 1 prime (i.e., a closed point) s1 P Spec R, then tyus1 “ tyu Ă Ys1; by +assumption codimpYs1, Xs1q “ codimpYs1, Xs1q ě 1, so we have codimptyus1, Xs1q ě 1. But since the +generic point y of tyus1 is not contained in Y zY , we deduce that the contribution of such a y to the +s1-fiber codimension of Y zY in X is again ě 2. +□ +Lemma 13.3 (Lifting the torsor to a smooth relative curve; cf. [Čes22a, Proposition 4.2]). For a semilocal +Prüfer domain R with fraction field K, the semilocalization A of an irreducible, R-smooth algebra A1 at +a finite subset x Ă SpecpA1q, and a quasi-split reductive A-group scheme G with a Borel subgroup B, +(1) given a generically trivial G-torsor PK over AK :“ A bR K, there are +(i) a smooth, affine relative A-curve C with a section s P CpAq; +(ii) an A-finite closed subscheme Z Ă C; +(iii) a quasi-split reductive C-group scheme G with a Borel subgroup B Ă G whose s-pullback is +B Ă G, compatible with the quasi-pinnings; +(iv) a G -torsor PK over CK :“ C ˆR K whose sAK-pullback is PK such that PK reduces to a +radupG q-torsor over CKzZK (here sAK stands for the image of s in CpAKq). +(2) if R has Krull dimension 1, given a generically trivial G-torsor P, then there are +(i) a smooth, affine relative A-curve C with a section s P CpAq; +(ii) an A-finite closed subscheme Z Ă C; +(iii) a quasi-split reductive C-group scheme G with a Borel subgroup B Ă G whose s-pullback is +B Ă G, compatible with the quasi-pinnings; +(iv) a G -torsor P whose s-pullback is P such that P reduces to a radupG q-torsor over CzZ. +Proof. In case (1) we can first use a limit argument involving Lemma 3.1.3 to reduce to the case when +R has finite Krull dimension. +If A1 is of relative dimension 0 over R, then AK “ FracpAq and A is a semilocal Prüfer domain. Thus, +PK is trivial, and, by the Grothendieck–Serre conjecture on semilocal Prüfer schemes (Theorem 9.0.1), +P is also trivial. In this case we simply take C “ A1 +A, s “ 0 P A1 +ApAq, Z “ H, pG , Bq “ pGA1 +A, BA1 +Aq, and +PK “ pPKqA1 +AK (resp., P “ PA1 +A). Thus, for what follows, we can assume that the relative dimension of +A1 over R is d ą 0. +By spreading out and localizing A1, we may assume that our quasi-split G (in particular, the Borel B) +and torsor P all live over A1, and PK live over A1 +K. By [SGA 3III new, Exposé XXVI, Corollaire 3.6 and +Lemme 3.20], the quotient PK{BK (resp., P{B) is representable by a smooth projective scheme over A1 +K +(resp., over A1). Now we treat the cases (1)-(2) separately. +(1) By the generic triviality of PK, applying the valuative criterion of properness to PK{BK Ñ SpecpA1 +Kq +yields a closed subscheme YK Ă SpecpA1 +Kq of codimension ě 2 such that PK{BK Ñ SpecpA1 +Kq has a +section over SpecpA1 +KqzYK that lifts to a generic section of PK. In other words, pPKqSpecpA1 +KqzYK reduces +to a generically trivial BSpecpA1 +KqzYK-torsor P B +K . Consider the A1-torus T :“ B{ radupBq and the induced +T -torsor +P T +K :“ P B +K { radupBqK +over +SpecpA1 +KqzYK. +50 + +Since P T +K is generically trivial, by Corollary 6.3.2, it extends to a T -torsor Ă +P T +K over SpecpA1qzF for a +closed subscheme F Ă SpecpA1q satisfying +codimpFK, SpecpA1qKq ě 2 +and +codimpFs, SpecpA1qsq ě 1 for all s P SpecpRq; +by purity for tori (Theorem 8.2.4), this torsor further extends to the whole SpecpA1q. As Ă +P T +K is generically +trivial, by the Grothendieck–Serre conjecture for tori (Proposition 8.3.2(i)), we may localize A1 around +x to assume that Ă +P T +K, and hence also P T +K, is already trivial. In other words, pPKqSpecpA1 +KqzYK reduces to +a radupBq-torsor over SpecpA1 +KqzYK. +Denote by Y the schematic closure of YK in SpecpA1q; by Lemma 3.1.1(i), it is R-fiberwise of codimension +ě 2 in SpecpA1q. Applying Lemma 13.2 to the R-smooth algebra A1 and the closed subscheme Y Ă +SpecpA1q, we obtain an affine open U Ă SpecpA1q containing x, an affine open S Ă Ad´1 +R +, and a smooth +R-morphism π : U Ñ S of relative dimension 1 such that Y X U is S-finite. +Recall that A is the semilocal ring of U at x. Denote +C :“ U ˆS Spec A +and +Z :“ pY X Uq ˆS Spec A. +Then C is a smooth affine relative A-curve, the diagonal in C induces a section s P CpAq, and the closed +subscheme Z Ă C is A-finite. So (1)(i) and (1)(ii) hold. Let B Ă G be the pullback of BU Ă GU +under the first projection pr1 : C Ñ U, and let PK be the pullback of pPKqUK under the first projection +pr1 : CK Ñ UK. Then, PK is a G -torsor over CK, and, by construction, the s-pullback (resp., sAK- +pullback) of B Ă G (resp., of PK) is B Ă G (resp., PK). Finally, since PK reduces to a radupBq-torsor +over SpecpA1 +KqzYK, PK reduces to a radupBq-torsor over CKzZK. So (1)(iii) and (1)(iv) also hold. +(2) Recall that, by Lemma 3.1.1(iii), the local rings of all maximal points of R-fibers of SpecpA1q are +valuation rings. By the generic triviality of P, applying the valuative criterion of properness to P{B Ñ +SpecpA1q yields a closed subscheme Y Ă SpecpA1q, which avoids all the codimension 1 points of the +generic fiber SpecpA1 +Kq and all the maximal points of R-fibers of SpecpA1q, such that P{B Ñ SpecpA1q +has a section over SpecpA1qzY that lifts to a generic section of P. In other words, Y satisfies +codimpYK, SpecpA1qKq ě 2 +and +codimpYs, SpecpA1qsq ě 1 for all s P SpecpRq. +Therefore, PSpecpA1qzY reduces to a generically trivial BSpecpA1qzY -torsor P B. +Consider the A1-torus +T :“ B{ radupBq and the induced T -torsor +P T :“ P B{ radupBq +over +SpecpA1qzY. +By purity for tori (Theorem 8.2.4), P T extends to a T -torsor Ă +P T . As Ă +P T is generically trivial, by the +Grothendieck–Serre conjecture for tori (Proposition 8.3.2(i)), we may localize A1 around x to assume +that Ă +P T , and hence also P T , is already trivial. In other words, PSpecpA1qzY reduces to a radupBqSpecpA1qzY - +torsor. +Now, applying Lemma 13.2 to the R-smooth algebra A1 and the closed subscheme Y Ă SpecpA1q, we +obtain an affine open U Ă SpecpA1q containing x, an affine open S Ă Ad´1 +R +, and a smooth R-morphism +π : U Ñ S of relative dimension 1 such that Y X U is S-finite. +Recall that A is the semilocal ring of U at x. Denote +C :“ U ˆS Spec A +and +Z :“ pY X Uq ˆS Spec A. +Then C is a smooth affine relative A-curve, the diagonal in C induces a section s P CpAq, and the closed +subscheme Z Ă C is A-finite. So (2)(i) and (2)(ii) hold. Let B Ă G and P be the pullback of BU Ă GU +and PU under the first projection pr1 : C Ñ U, respectively. Then, P is a G -torsor over C, and, by +construction, the s-pullback of B Ă G and P are B Ă G and P, respectively. Finally, since P reduces to +a radupBq-torsor over SpecpA1qzY , P reduces to a radupBq-torsor over CzZ. So (2)(iii) and (2)(iv) also +hold. +□ +Lemma 13.4 ([Čes22a, Lemma 5.2]). For a semilocal ring A whose local rings are geometrically uni- +branch, an ideal I Ă A, reductive A-groups G and G1 that on geometric A-fibers have the same type, +fixed quasi-pinnings of G and G1 extending Borel A-subgroup B Ă G and N 1 Ă G1 and an A{I-group +isomorphism +ι : GA{I +„ +ÝÑ G1 +A{I +respecting the quasi-pinnings; in particular, +ιpBA{Iq “ B1 +A{I, +51 + +there are +(i) a faithfully flat, finite, étale A-algebra rA equipped with an A{I-point a : rA ։ A{I; and +(ii) an rA-group isomorphism rι: G r +A +„ +ÝÑ G1 +r +A respecting the quasi-pinnings such that a˚prιq “ ι. +Notice that the original version [Čes22a, Proposition 5.1] assumed further A to be Noetherian, but the +Noetherianess of A was not used anywhere in the proof. +Lemma 13.5 (Changing the relative curve C to equate G and GC; cf. [Čes22a, Proposition 5.2]). In the +setting of Lemma 13.3, for both cases (1) and (2) we may replace C by an étale neighbourhood of impsq +to achieve further that pG , Bq “ pGC, BCq. +Proof. Consider the semilocalization SpecpDq of C at the closed points of impsq Y Z; since C is normal, +all the local rings of D are geometrically unibranch. The image of the section s : Spec A Ñ SpecpDq +gives rise to a closed subscheme SpecpD{Iq Ă SpecpDq. By the conclusion of Lemma 13.3, the restriction +of BD Ă GD and BD Ă GD to SpecpD{Iq agree with each other in a way compatible with their +quasi-pinnings. +Thus, by Lemma 13.4, there is a faithfully flat, finite, étale D-algebra rD, a point +rs : rD ։ D{I » A lifting s : D ։ D{I » A such that B r +D Ă G r +D is isomorphic to B r +D Ă G r +D +compatibly with the fixed identification of rs-pullbacks. We then spread out the finite étale morphism +Specp rDq Ñ SpecpDq to a finite étale morphism rC Ñ C1 for an open C1 Ă C that contains impsq Y Z, +while preserving an rs P rCpAq, and an isomorphism between B r +C Ă G rC and B r +C Ă G r +C. Now it remains to +replace C, s, Z and PK (resp., P) by rC, rs, Z ˆC rC and pPKq r +CK (resp., P rC). +□ +Lemma 13.6 (Changing the relative smooth curve C for descending to A1 +A; [Čes22a, Proposition 6.5]). +In the setting of Lemma 13.3, for both cases (1) and (2), in addition to pG , Bq “ pGC, BCq, we may +change C to achieve further that there is a flat A-map C Ñ A1 +A that maps Z isomorphically to a closed +subscheme Z1 Ă A1 +A with +Z » Z1 ˆA1 +A C. +Proof. Assume that, in both cases (1) and (2) of Lemma 13.3, we have achieved the conclusion of +Lemma 13.5. We have the data of a smooth affine relative A-curve C, a section s P CpAq, and an +A-finite closed subscheme Z Ă C; replacing Z by Z Y impsq, we may assume that s factors through Z. +However, in general, the A-finite scheme Z may be too large to embed into A1 +A. (For instance, if R “ k +is a finite field, then Z can’t be embedded into A1 +k as soon as 7 Zpkq ą 7 k.) For this, we first apply +Panin’s ‘finite fields tricks’ [Čes22a, Proposition 7.4] to obtain a finite morphism rC Ñ C that is étale +at the points in ˜Z :“ rC ˆC Z such that s lifts to rs P rCpAq, and there are no finite fields obstruction to +embedding rZ into A1 +A in the following sense: for every maximal ideal m Ă A, +7 +! +z P rZκpmq : rκpzq : κpmqs “ d +) +ă 7 +! +z P A1 +κpmq : rκpzq : κpmqs “ d +) +for every +d ě 1. +Then, by [Čes22a, Lemma 6.3], there are an affine open C1 Ă rC containing imprsq, a quasi-finite, flat +A-map C1 Ñ A1 +A that maps Z isomorphically to a closed subscheme Z1 Ă A1 +A with +Z » Z1 ˆA1 +A C1. +It remains to replace C by C1, Z by rZ, s by rs, PK by pPKqC1 +K (resp., P by PC1). +□ +Lemma 13.7 (Descend to A1 +A via patching; cf. [Čes22a, Proposition 7.4]). In the setting of Lemma 13.3, +for both cases (1) and (2), we may achieve further that pG , Bq “ pGC, BCq, C “ A1 +A, and s “ 0 P A1 +ApAq. +Proof. By the reduction given in Lemma 13.6, we have a flat A-curve C, a section s P CpAq, an A-finite +closed subscheme Z Ă C, a quasi-finite, affine, flat A-map C Ñ A1 +A that maps Z isomorphically to a +closed subscheme Z1 Ă A1 +A with Z “ Z1 ˆA1 +A C, and a G-torsor PK over CK whose sAK-pullback is PK +(resp., a G-torsor P over C whose s-pullback is P) and whose restriction to CKzZK (resp., CzZ ) reduces +to a radupBq-torsor. Now, since Z “ Z1 ˆA1 +A C » Z1, [Čes22a, Lemma 7.2] implies the pullback maps +H1pA1 +AzZ1, radupGqq ։ H1pCzZ, radupGqq +52 + +and +H1pA1 +AKzZ1 +K, radupGqq ։ H1pCKzZK, radupGqq +are surjective. Combining these, we see that PK|CKzZK (resp., P|CzZ) descends to a G-torsor QK (resp., +Q) over A1 +AKzZ1 +K (resp., A1 +AzZ1) that reduces to a radupBq-torsor. By the glueing Lemma 6.2.2(ii), we +may (non-canonically) glue PK with QK (resp., P with Q) to descend PK (resp., P) to a G-torsor Ą +PK +(resp., rP) over A1 +AK (resp., over A1 +A) that reduces to a radupBq-torsor over A1 +AKzZ1 +K (resp., over A1 +AzZ1). +It remains to replace C by A1 +A, Z by Z1, s P CpAq by its image in A1 +ApAq, and PK by Ą +PK (resp., P by +rP). Finally, by shifting, we may assume even that s “ 0 P A1 +ApAq. +□ +Proof of Theorem 13.1. Let PK (resp., P) be a generically trivial GAK-torsor (resp., G-torsor). By the +reduction Lemma 13.7, we get an A-finite closed subscheme Z Ă A1 +A, and a GA1 +AK -torsor PK (resp., +GA1 +A-torsor P) whose pullback along the zero section is PK (resp., P) such that pPKq|A1 +AK zZK (resp., +P|A1 +AzZ) reduces to a radupBq-torsor. 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Steklov Mathematical Institute, Fontanka 27, 191023 St. Petersburg, Russia +Email address: guo.ning@eimi.ru +Department of Mathematics, Southern University of Science and Technology, Shenzhen, China +Email address: liufei54@pku.edu.cn +56 + diff --git a/ptFPT4oBgHgl3EQf7zXe/content/tmp_files/load_file.txt b/ptFPT4oBgHgl3EQf7zXe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9827ff2da7e44a75bfee08d38da02cde582669b3 --- /dev/null +++ b/ptFPT4oBgHgl3EQf7zXe/content/tmp_files/load_file.txt @@ -0,0 +1,4282 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf,len=4281 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='13206v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='AG] 29 Jan 2023 PURITY AND TORSORS OVER PRÜFER BASES NING GUO AND FEI LIU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We establish Zariski–Nagata purity theorem concerning finite étale covers on smooth schemes over Prüfer rings by proving Auslander’s flatness criterion in this non-Noetherian context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Inspired by Gabber–Ramero’s upper bound of projective dimensions over Prüfer bases, we present an Auslander– Buchsbaum formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' On the basis of the analysis of reflexive sheaves, we prove various purity theorems for torsors under reductive group algebraic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Specifically, by parafactorial results in [EGA IV4] on smooth schemes over normal bases, we prove the purity for cohomology groups of multiplicative type groups at this level of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Subsequently, we take advantage of aforementioned purity results to give affirmative answer to the Grothendieck–Serre conjecture for torsors on smooth schemes over semilocal Prüfer rings in certain cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Along the way, inspired by the recent preprint of ˇCesnaviˇcius [Čes22c], we also prove several versions of Nisnevich conjecture in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Purity and the Grothendieck–Serre on schemes smooth over Prüfer bases .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 2 Acknowledgements .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Auslander’s flatness criterion on schemes smooth over valuation rings .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Generalities on torsors over algebraic spaces .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 53 Date: February 1, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Primary 14F22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Secondary 14F20, 14G22, 16K50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' purity, Zariski–Nagata, Auslander–Buchsbaum, Grothendieck–Serre, vector bundles, principal bundles, Prüfer rings, torsors, homogeneous spaces, group schemes, valuation rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Purity and the Grothendieck–Serre on schemes smooth over Prüfer bases 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Purity and regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In algebraic geometry, purity refers to a diverse range of phenomena in which certain invariants or categories associated to geometric objects are insensitive to the removal of closed subsets of large codimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In the classical Noetherian world, purities, say, for vector bundles (and even torsors), or for finite étale covers, are intimately related to the regularities measured by lengths of regular sequences of geometric objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a concrete instance, the Auslander–Buchsbaum formula depthR M ` proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='dimRM “ depthR R ([AB57, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7]) controls the projective dimension of the finite type module M over the Noetherian local ring R via depths, leading to the purity for vector bundles on regular local rings of dimension two ([Sam64, Proposition 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Granted this, Colliot-Thélène and Sansuc [CTS79, Théorème 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='13] established the purity for reductive torsors over arbitrary regular local ring R of dimension two by bootstrapping from the vector bundle case: the restriction H1 ´etpSpec R, Gq „ ÝÑ H1 ´etpSpec RztmRu, Gq is bijective for every reductive R-group scheme G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Nevertheless, not only does the term ‘regularity’ make sense for Noetherian rings, its non-Noetherian generalization can still enlighten us to contemplate purity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Regularity of Prüfer rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Originally formulated by Bertin [Ber71], [Ber72, Définition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5] for coherent local rings, we say that a ring R is regular if every finitely generated ideal of R has finite projective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This coincides with the classical notion of regularity when restricting to Noetherian rings by Serre’s homological characterization [Ser56, Théorème 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A typical non-Noetherian example can be sought in Prüfer rings, namely, the rings whose all local rings are valuation rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By definition, an integral domain V is a valuation ring if every x P pFrac V qzV satisfies x´1 P V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Beyond fields, Noetherian valuation rings are exactly discrete valuation rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The regularity of Prüfer rings thus follows from the fact that all finitely generated ideals of valuation rings are principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In addition to the regularity and other nature (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1), the ubiquity of Prüfer rings in the study of nonarchimedean geometry, Zariski–Riemann spaces, among others, motivates us to investigate their algebro-geometric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Basic setup I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The purity part of the present article focuses on a semilocal affine Prüfer scheme S with dim S ą 0 (and with dim S ă 8 if necessary), an S-flat finite type algebraic space X with regular S-fibers, and a closed subset Z Ă X such that j : XzZ ãÑ X is quasi-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a point x P X lying in an open subscheme, the local ring of X at x makes sense and we denote A :“ OX,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' When involving torsors on X, we let G be an X-group algebraic space that étale-locally permits an embedding G ãÑ GLn such that GLn {G is X-affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This condition is fulfilled if G is X-reductive1, or finite and locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Auslander–Buchsbaum over Prüfer bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Gabber–Ramero’s upper bound of projective di- mensions of coherent modules over X unveils a glimpse of the Prüferian Auslander–Buchsbaum formula Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1: if x P X lies over a closed point s P S, then every finitely presented A-module M satisfies proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimA M ` depthAM “ depthA A “ d ` 1, where d “ dim OXs,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Here proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimAp0q “ ´8 and depthA M is the smallest i such that the i-th local cohomology of M be nonzero (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Our proof is significantly different from the classical case [AB57, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Specifically, taking Gabber–Ramero’s boundness [GR18, Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1] as an input, we bypass the interpretation of projective dimensions in terms with Tor functors, which is a crucial ingredient in Auslander–Buchsbaum’s argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In the sequel, we will only use Gabber–Ramero’s part of Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Purity for torsors on smooth relative curves over Prüfer rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Once the projective di- mensions of reflexive sheaves on X are controlled, by imposing codimensional constraints on Z, we may extend vector bundles on XzZ to X, as in Noetherian scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Subsequently, this allows us to obtain the purity Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 for G-torsors: if Z satisfies Zη “ H for each generic point η P S and codimpZs, Xsq ě 1 for all s P S, and X is an S-curve, then restriction induces the following equivalence of categories of G-torsors TorspX´et, Gq „ ÝÑ TorsppXzZq´et, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1By this we mean a smooth affine X-group algebraic space G whose X-geometric fibers are (connected) reductive algebraic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, étale-locally on X, G splits so admits a closed immersion G ãÑ GLn,X for some integer n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' by [Alp14, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1], the reductivity of G implies that the quotient GLn,X{G is X-affine of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 2 In particular, passing to isomorphism classes of objects, we have the following bijection of pointed sets H1 ´etpX, Gq » H1 ´etpXzZ, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Meanwhile, a local version Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 allows us to loose constraints on the relative dimension of X: if either x P Xη with dim OXη,x “ 2, or x P Xs with s ‰ η and dim OXs,x “ 1, then every G-torsor over Spec OX,xztxu extends uniquely to a G-torsor over Spec OX,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This permits us to iteratively extend reductive torsors beyond a closed subset of higher fiberwise codimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Zariski–Nagata over Prüfer bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The Zariski–Nagata purity, known as “purity of branch locus”, states that every finite extension A Ă B of rings with A regular Noetherian and B normal is unramified if and only if so it is in codimension one on Spec B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This purity was settled by Zariski [Zar58] in a geometric context, and more algebraically by Nagata [Nag59] based on Chow’s local Bertini theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In contrast to them, Auslander gave an alternative proof [Aus62, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4] by skillful homological methods leading to a criterion for flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In [SGA 2new, Exposé X, §3], Grothendieck reformulated their results into a purity concerning finite étale covers and proved this purity on Noetherian local rings that is a complete intersection of dimension ě 3 by reducing the assertion to hypersurfaces via several passages involving formal completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Nevertheless, a practical deficiency of the later argument is that, even over a rank-one valuation ring V with pseudo-uniformizer ̟, the coherence of the ̟-adic completion pA of A is unknown to us, not to mention the primary decomposition on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To circumvent this technical obstacle, we revert to Auslander’s argument by establishing a Prüferian counterpart Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 of the criterion for flatness [Aus62, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Granted this, we acquire the Prüferian Zariski–Nagata Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2: the pullback FÉtX „ ÝÑ FÉtXzZ is an equivalence for every closed subset Z Ă X in the basic setup §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3 that satisfies the following condition codimpZη, Xηq ě 2 for each generic point η P S and codimpZs, Xsq ě 1 for all s P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, for every geometric point x: Spec Ω Ñ XzZ with a separably closed field Ω, the map π´et 1 pXzZ, xq Ñ π´et 1 pX, xq is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Grothendieck–Serre on semilocal Prüfer rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The Grothendieck–Serre conjecture predicts that, for a regular local ring R and a reductive R-group scheme G, every generically trivial G-torsor is trivial, that is, the following restriction map of nonabelian cohomology pointed sets has trivial kernel: ker pH1 ´etpR, Gq Ñ H1 ´etpFrac R, Gqq “ t˚u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The conjecture was settled in the equicharacteristic case and in certain unramified mixed characteristic cases, see the histrical summary below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thanks to the purity for cohomology of groups of multiplicative type, we prove the non-Noetherian counterpart of Colliot-Thélène–Sansuc’s result for tori and then obtain a product formula for tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Based on this, the similar argument in [Guo20] leads to a passage from the semilocal case to the local case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, we settle the Grothendieck–Serre on semilocal Prüfer rings in §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Basic setup II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The second half of this article deals mainly with the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer ring R, an irreducible R-smooth scheme X, the semilocalization A :“ OX,x of X at a finite subset x Ă X contained in a single affine open of X, and a reductive A-group scheme G, we study the trivialization behaviour of G-torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Grothendieck–Serre on smooth projective schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This result was proved by the second author and simultaneously by an unpublished work of Panin and the first author in the Noetherian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We show that, when X is R-projective in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8 and G has a reductive model over X, every generically trivial G-torsor on A is trivial, that is, ker pH1pA, Gq Ñ H1pFrac A, Gqq “ t˚u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To prove this, we use crucially our purity Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 after spreading out to extend the domain of the torsor in question to an open subset as large as possible: according to that purity, a generically trivial torsor on OX,x extends to a torsor on an open neighbourhood of x whose complementary closed has codimension ě 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', ě 2) in the generic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', non-generic) R-fibers of X, see Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This codimension bound is sharp enough for us to apply the geometric presentation Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 and glueing techniques to reduce the problem to studying torsors on relative affine lines that we treat in detail in §10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Grothendieck–Serre under constant reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume that G is ‘constant’, namely, it is a pullback from the Prüfer base ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then every generically trivial G-torsor on A is trivial, that is, ker pH1pA, Gq Ñ H1pFrac A, Gqq “ t˚u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For this, we first devise a variant (in some aspect, a stronger form) of Lindel’s lemma (Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1), which states that, for a closed subscheme Y Ă X that avoids all the maximal points of the R-fibers of X, the pair pY, Xq Zariski-locally on X can be presented as an elementary étale neighbourhood of a similar pair pY 1, X1q, where X1 is an open of some projective R-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This allows us to use glueing techniques to reduce to studying generically trivial torsors on opens of projective R-spaces, which is done in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Grothendieck–Serre under quasi-split groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As for the quasi-split case of the Grothendieck– Serre, we will follow a similar strategy of [Čes22a] (with its earlier version given by Fedorov [Fed22b]), where the key input is our toral version of purity Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5 and Grothendieck–Serre type Propo- sition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Precisely, by the valuative criterion of properness, a generically trivial torsor on X, say, reduces to a generically trivial torsor under a Borel B away from a closed subset Z of X that has codimension ě 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', ě 1) in the generic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', non-generic) R-fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Further, utilizing the aforementioned toral purity and Grothendieck–Serre type results, one shows that the above torsor further reduces to a radupBq-torsor on XzZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In conclusion, when G is quasi-split, we prove Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 that ker ` H1pA bR K, Gq Ñ H1pFrac A, Gq ˘ “ t˚u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' if R has Krull dimension 1, then every generically trivial G-torsor is trivial, that is, ker ` H1pA, Gq Ñ H1pFrac A, Gq ˘ “ t˚u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Nisnevich’s purity conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now, we turn to Nisnevich’s purity conjecture, where we require the total isotropicity of group schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A reductive group scheme G defined over a scheme S is totally isotropic at s P S if every Gi in the decomposition [SGA 3III new, Exposé XXIV, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='10 (i)] Gad OS,s – ś i ResAi{OS,spGiq contains a Gm,Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If this holds for all s P S, then G is totally isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proposed by Nisnevich [Nis89, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3] and modified due to the anisotropic counterexamples of Fedorov [Fed22b, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1], the Nisnevich conjecure predicts that, for a regular semilocal ring R, a regular parameter r P R (that is, r P mzm2 for every maximal ideal m Ă R), and a reductive R-group scheme G such that GR{rR is totally isotropic, every generically trivial G-torsor on Rr 1 rs is trivial, that is, the following map H1pRr 1 rs, Gq Ñ H1pFrac R, Gq has trivial kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The case when R is a local ring of a regular affine variety over a field and G “ GLn was settled by Bhatwadekar–Rao in [BR83] and was subsequently extended to arbitrary regular local rings containing fields by Popescu [Pop02, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Nisnevich in [Nis89] proved the conjecture in dimension two, assuming that R is a local ring with infinite residue field and that G is quasi-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the state of the art, the conjecure was settled in equicharacteristic case and in several mixed characteristic case by Česnavičius in [Čes22c, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3] (previously, Fedorov [Fed21] proved the case when R contains an infinite field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Besides, the toral case and some low dimensional cases are known and surveyed in [Čes22b, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 (1)] including Gabber’s result [Gab81, Chapter I, Theorem 1] for the local case dim R ď 3 when G is either GLn or PGLn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In this article, we prove several variants of Nisnevich conjecture over Prüfer bases, see Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (ii) and Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The Grothendieck–Serre conjecture: a history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since proposed by Serre [Ser58, page 31] and Grothendieck [Gro58, pages 26–27, Remarques 3], [Gro68a, Remarques 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='11 a)], the Grothendieck–Serre conjecture has already various known cases beyond the trivial dim R “ 0 case for fields, as listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) The case when G is a torus is proved by Colliot-Thélène and Sansuc in [CTS87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) The case when dim R “ 1, namely, R is a discrete valuation ring, was addressed by Nisnevich in [Nis82] and [Nis84], then is improved and generalized to the semilocal Dedekind case in [Guo22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Several special cases were proved in [Har67], [BB70], [BTIII] over discrete valuation rings, and in [PS16], [BVG14], [BFF17], [BFFH20] for the semilocal Dedekind case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iii) The case when R is Henselian was settled in [BB70] and [CTS79, Assertion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1] by reducing the triviality of G-torsors to residue fields then inducting on dim R to reach Nisnevich’s resolved case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 4 (iv) The equicharacteristic case, namely, when R contains a field k, was established by Fedorov and Panin [FP15] when k is infinite (see also [PSV15], [Pan20b] for crucial techniques) and by Panin [Pan20a] when k is finite, which was later simplified by [Fed22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Before these, several equichar- acteristic subcases were proved in [Oja80],[CTO92], [Rag94], [PS97], [Za˘ı00], [Oja01], [Oja04], [Pan05], [Zai05], [Che10], [PSV15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (v) When R is of mixed characteristic, Česnavičius [Čes22a] settled the case when G is quasi-split and R is unramified (that is, for p :“ charpR{mRq, the ring R{pR is regular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Prior to this, Fedorov [Fed22b] proved the split case under additional assumptions on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Recently, Česnavičius [Čes22c, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3] settled a generalized Nisnevich conjecture under certain conditions, which specializes to the equal and mixed characteristic cases of the Grothendieck–Serre proved in [FP15], [Pan20a], [Čes22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (vi) There are sporadic cases where R or G are speical (with possible mixed characteristic condition), see [Gro68a, Remarque 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='11 a)], [Oja82], [Nis89], [Fed22b], [Fir22], [BFFP22], [Pan21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Notations and conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' All rings in this paper are commutative with units, unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a point s of a scheme (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', for a prime ideal p of a ring), we let κpsq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', κppq) denote its residue field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a global section s of a scheme S, we write Sr 1 ss for the open locus where s does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a ring A, we let Frac A denote its total ring of fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a morphism of algebraic spaces S1 Ñ S, we let p´qS1 denote the base change functor from S to S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' if S “ Spec R and S1 “ Spec R1 are affine schemes, we will also write p´qR1 for p´qS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let S be an algebraic space, and let G be an S-group algebraic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For an S-algebraic space T , by a G-torsor over T we shall mean a GT :“ G ˆR T -torsor (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote by TorspSfppf, Gq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', TorspS´et, Gq) the groupoid of G-torsors on S that are fppf-locally (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', étale-locally) trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' specifically, if G is S-smooth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', G is S-reductive, see below), then every fppf-locally trivial G-torsor is étale-locally trivial, so we have TorspSfppf, Gq “ TorspS´et, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For an algebraic space S, a reductive S-group algebraic space is a smooth affine S-group algebraic space whose geometric S-fibers are (connected) reductive algebraic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a scheme S this coincides with the definition of reductive S-groups schemes given in [SGA 3III new].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The authors would like to thank Kęstutis Česnavičius and Ivan Panin for their constant encouragements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We thank Matthew Morrow and Colliot-Thélène for proposing the Grothendieck– Serre on smooth schemes over semilocal Prüfer rings during the defense of the first author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' On several occasions during the past few months, we talked about some aspects of this article with Kęstutis Čes- navičius, Arnab Kundu, Shang Li, and Ivan Panin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We thank them for these conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We thank Kęstutis Česnavičius for helping us to remove the assumptions on finite residue fields in our original for- mulation of the Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' After an earlier version of this paper was finished, Kęstutis Česnavičius kindly sent to us his note which contained a sketch of a different proof of the Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(i) in the Noetherian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We thank Jiandi Zou for useful suggestions about the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research, the innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 851146), the grant 075-15-2022-289, and the excellent environment for research of the Euler International Mathematical Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Auslander–Buchsbaum formula over valuation rings The goal of this section is to establish Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, the Auslander–Buchsbaum formula over finite rank valuation rings as an analogue of the classical regular case [AB57, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Based on the upper- bound of projective dimensions [GR18, Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1], we induct by using the notion of depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Coherent rings and schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a ring A, a finitely generated A-module M is coherent if its any finitely generated A-submodule is finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A ring A is coherent if it is a coherent A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a scheme X, an OX-module F is coherent if, for every affine open U Ă X, ΓpU, Fq is a coherent ΓpU, OUq-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A scheme X is locally coherent if OX is a coherent OX-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A locally coherent scheme is coherent if it is quasi-compact quasi-separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Noetherian rings and Prüfer rings are coherent rings ([SP, 05CY, 0EWV]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Although Noetherian schemes are coherent, open subschemes of affine integral Prüfer schemes are not coherent 5 in general: there exists a valuation ring V such that Spec V ztmV u has no closed points and is not quasi-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let A be a coherent ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) For every multiplicative subset S Ă A, the localization S´1A is a coherent ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) Any A-module M is coherent if and only if it is finitely presented over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Further, the full subcategory of coherent A-modules is an abelian subcategory of the category of A-modules and is closed under taking extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (i), see [Gla89, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the first assertion of (ii), see [FK18, Chapter 0, Corol- lary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a local ring A and the closed point x P Spec A, consider the following functor Γtxu : A-Mod Ñ A-Mod M ÞÑ ker ´ ΓpSpec A, Ă Mq Ñ ΓpSpec Aztxu, Ă Mq ¯ sending M to its largest A-submodule supported on txu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The functor Γtxu is left exact so gives rise to a right derived functor RΓtxu : D`pA-Modq Ñ D`pA-Modq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The depth of M P D`pA-Modq is depthApMq :“ suptn P Z | RiΓtxuM “ 0 for all i ă nu P Zě0 Y t`8u, For an A-module N supported on txu, we also consider the following closely related quantity τNpMq :“ suptn P Z | Exti ApN, Mq “ 0 for all i ă nu P Zě0 Y t`8u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a local ring A, an A-module M, and an M-regular sequence pf1, ¨ ¨ ¨ , frq in mA, depthApMq “ depthApM{ řr i“1 fiMq ` r and τNpMq “ τNpM{ řr i“1 fiMq ` r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The two equalities are proved similarly, so we only treat the one concerning depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By induction on r, we are reduced to the case when r “ 1 and f1 “ f is a nonzerodivisor of M in mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' From the short exact sequence 0 Ñ M fÝÑ M Ñ M{fM Ñ 0, we derive the following long exact sequence ¨ ¨ ¨ Ñ Ri´1ΓtxuM fÝÑ Ri´1ΓtxuM Ñ Ri´1ΓtxupM{fMq Ñ RiΓtxuM fÝÑ RiΓtxuM Ñ ¨ ¨ ¨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If depthA M “ `8, then M “ 0 so it suffices to assume that depthA M “ d for an integer d ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If d “ 0, then there is a nontrivial A-submodule of M supported on txu, contradicting to the assumption that f P mA is a nonzerodivisor of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, we have d ě 1 and RiΓtxuM “ 0 for every 0 ď i ď d´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The displayed long exact sequence implies that RiΓtxupM{fMq “ 0 for every 0 ď i ď d ´ 2 (if d ´ 2 ă 0 then such i does not exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If Rd´1ΓtxupM{fMq “ 0, then the map RdΓtxuM ãÑ RdΓtxuM induced by multiplication by f is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' However, since the nonzero A-module RdΓtxuM is supported on txu and f P mA, we deduce that RdΓtxuM “ 0, that is, depthAM ą d, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, we have Rd´1ΓtxupM{fMq ‰ 0 and depthApM{fMq “ d ´ 1 “ depthA M ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume that A is Noetherian, and take N “ A{I for an ideal I of A (for instance, N “ A{mA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then for any finitely generated A-module M we have depthA M “ τNpMq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Indeed, utilizing Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5, one verifies easily that both of them equals the length of any maximal M-regular sequence in mA (so the length is independent of all choices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' However, this is false when A is non-Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For instance, we let A :“ V be a non-discrete valuation ring of finite rank and let N :“ V {mV be its residue field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Take M “ V {fV for a nonzero f P mV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then one checks immediately that depthV pV {fV q “ 0, but there are no nonzero elements of V {fV annihilated by mV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus HomV pV {mV , V {fV q “ 0, and so τV {mV pV {fV q ě 1 ą 0 “ depthV pV {fV q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a valuation ring V of finite rank, a V -flat finitely presented scheme X, and a point x P X with image s P Spec V such that OXs,x is regular, (i) we have depthApAq “ d ` 1, where d “ dim OXs,x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) for any A-module N supported on txu, we have Exti ApN, Aq “ 0 for all i ď d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) Since by assumption V has nonzero finite rank, we can pick an element f P mV such that dim V {pfq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let pg1, ¨ ¨ ¨ , gdq be a sequence in mA such that their images in the regular local ring A{mV A forms a regular system of parameters, and hence also forms a regular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the flatness criterion [EGA IV3, Théorème 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8], pg1, ¨ ¨ ¨ , gdq is a regular sequence of A, and the quotient ring A :“ A{pg1, ¨ ¨ ¨ , gdq is V -flat with mV A “ mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, pg1, ¨ ¨ ¨ , gd, fq is a regular sequence of A in mA such that dim A{pfA ` řd i“1 giAq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5 yields depthApAq “ depthApA{pfA ` řd i“1 giAqq ` d ` 1 “ d ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) Repeating the preceding argument involving Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5, we deduce the following inequality τNpAq “ τNpA{pfA ` řd i“1 giAqq ` d ` 1 ě d ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the definition of τNpAq, this is equivalent to the displayed vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a local ring pA, mAq, a nonzero A-module M supported on tmAu, and a matrix H P MmˆnpAq, if the A-linear map HM : M ‘n Ñ M ‘m induced by H (via left multiplication) is injective, then H admits a left inverse, or, equivalently, H exhibits A‘n as a direct summand of A‘m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Recall [SP, 0953] that the assumption on the support of M means that, for any w P M and any finitely generated ideal I Ă A, we have INw “ 0 for large enough N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let H “ phijq, then McCoy’s theorem [Gla89, page 211] implies that the ideal generated by the minors of order n of H does not annihilate a nonzero element of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Indeed, the invertibility of minors already yields a left inverse of H and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Precisely, since M is supported at tmAu, there exist some i, j for which hij P Aˆ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We may assume that h11 P Aˆ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By subtracting suitable multiples of the first row of H to other rows (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' the first column of H to other columns), we may also assume that h1j “ 0 for 1 ă j ď n and hi1 “ 0 for 1 ă i ď m (the assumption and conclusion of the lemma are preserved if we replace H by H1HH2, where H1 P MmˆmpAq and H2 P MnˆnpAq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In other words, we have H “ ph11q ‘ H1, where H1 P Mpm´1qˆpn´1qpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then the map H1 M : M ‘pn´1q Ñ M ‘pm´1q induced by H1 is also injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' So we may assume by induction that H1 admits a left inverse H2 P Mpn´1qˆpm´1qpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then ph´1 11 q ‘ H2 is a left inverse of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Now, we acquire the Prüferian analogy of the Auslander–Buchsbaum formula [AB57, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (Auslander–Buchsbaum formula over valuation rings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a valuation ring V of finite nonzero rank, a V -flat finite type scheme X, a point x P X lying over the closed point s P Spec V such that OXs,x is regular, and the local ring A :“ OX,x, every finitely presented A-module M satisfies proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq ` depthApMq “ depthApAq “ d ` 1, where d “ dim OXs,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (By convention, proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimAp0q “ ´8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let M be a finitely presented nonzero A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We will induct on proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq to verify the formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Note that, by [GR18, Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1], we have proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq ď d`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq “ 0, or, M is A-free, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7 we have depthApMq “ d`1, so proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq`depthApMq “ d`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Next, assume that proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq ě 1, so every partial resolution 0 Ñ M 1 ιÝÑ A‘n Ñ M Ñ 0 is non-split and satisfies proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApM 1q “ proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We exploit the associated long exact sequence ¨ ¨ ¨ Ñ Ri´1ΓtxuM 1 Ñ Ri´1ΓtxuA‘n Ñ Ri´1ΓtxuM Ñ RiΓtxuM 1 Ñ RiΓtxuA‘n Ñ ¨ ¨ ¨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq “ 1, then M 1 » A‘m for some m ě 1, and the map A‘m » M 1 ιÝÑ A‘n is given by an n ˆ m matrix H P MnˆmpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We have known that proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApM 1q “ d ` 1, and so RiΓtxuM 1 “ 0 for all i ď d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It follows from the above long exact sequence that RiΓtxuM “ 0 for all i ď d´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If RdΓtxuM “ 0, then left multiplication by H induces an injection ` Rd`1ΓtxuA ˘‘m “ Rd`1ΓtxupA‘mq » Rd`1ΓtxuM 1 ãÑ Rd`1ΓtxupA‘nq “ ` Rd`1ΓtxuA ˘‘n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since Rd`1ΓtxuA is a nonzero A-module supported on txu, we deduce from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8 that H admits a left inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This implies that ι splits, and so M is A-free, contradicting our assumption proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, depthApMq “ d, and we thus obtain the desired formula proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq ` depthApMq “ d ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 7 If proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq ą 1, then proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApM 1q “ proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Applying the induction hypothesis to M 1, we have depthApM 1q “ d ` 1 ´ pproj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq ´ 1q “ d ` 2 ´ proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq, which is ď d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It follows from the above long exact sequence and the fact depthApA‘nq “ d ` 1 that Ri´1ΓtxuM » RiΓtxuM 1 for all i ď d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Combining this with the bound depthApM 1q ď d, we deduce that depthApMq “ depthApM 1q ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, by induction hypothesis, we have proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApMq ` depthApMq “ ` proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' dimApM 1q ` 1 ˘ ` ` depthApM 1q ´ 1 ˘ “ d ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This finishes the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Geometry of schemes over Prüfer bases In this section, we recollect useful geometric properties on scheme over Prüfer bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Geometric properties and reduction methods Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a valuation ring V with spectrum S, a finite type irreducible S-scheme X, a point x P X and its image s P S, the following assertions hold (i) all nonempty S-fibers have the same dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) if X is S-flat , then X is finitely presented over S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iii) if X is S-flat, then for any maximal point ξ P Xs, the local ring OX,ξ is a valuation ring such that the extension OS,s ãÑ OX,ξ of valuation rings induces an isomorphism of value groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iv) for x1 P X that is distinct with x whose image is denoted by s1, if x P tx1u, then ‚ either htpsq “ htps1q (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', s “ s1) and then dimpOXs1 ,x1q ă dimpOXs,xq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' ‚ or htps1q ă htpsq and then dimpOXs1 ,x1q ď dimpOXs,xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (i), see [EGA IV3, Lemme 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (ii), see [Nag66, Theorem 3’].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (iii), see [MB22, Théorème A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now, to prove (iv), we may assume that X is affine and of some pure relative dimension, say, n, over V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By assumption, we have s P ts1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume that we are not in the first case, then htps1q ă htpsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The schematic closure tx1u is a finite type dominant scheme over ts1u (the spectrum of a valuation ring), so by (i), all its non-empty fibers have the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus, we deduce from tx1u Ą txu that dimptx1us1q “ dimptx1usq ě dimptxusq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, we have dimpOXs1 ,x1q “ n ´ dimptx1us1q ď n ´ dimptxusq “ dimpOXs,xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ The following Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 provides us a passage to the case when there is a section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a valuation ring V , an essentially finitely presented (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', essentially smooth) V - local algebra A, there are an extension of valuation rings V 1{V with trivial extension of value groups, and an essentially finitely presented (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', essentially smooth) V -map V 1 Ñ A with finite residue fields extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume A “ OX,x for an affine scheme X finitely presented over V and a point x P X lying over the closed point s P SpecpV q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let t “ tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='degpκpxq{κpsqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As κpxq is a finite extension of l :“ κpsqpa1, ¨ ¨ ¨ , atq for a transcendence basis paiqt 1 of κpxq{κpsq, we have t “ diml Ω1 l{κpsq ď dimκpxq Ω1 κpxq{κpsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Choose sections b1, ¨ ¨ ¨ , bt P ΓpX, OXq such that db1, ¨ ¨ ¨ , dbt P Ω1 κpxq{κpsq are linearly independent over κpxq, where the bar stands for their images in κpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Define p : X Ñ At V by sending the standard coordinates T1, ¨ ¨ ¨ , Tt of At V to b1, ¨ ¨ ¨ , bt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since db1, ¨ ¨ ¨ , dbt P Ω1 κpxq{κpsq are linearly independent, the image η :“ ppxq is the generic point of At κpsq, so V 1 :“ OAt V ,η is a valuation ring whose value group is ΓV 1 » ΓV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Note that κpxq{κpηq is finite, the map V 1 Ñ A induces a finite residue fields extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' When V Ñ A is essentially smooth, the images of db1, ¨ ¨ ¨ , dbt under the map Ω1 X{V b κpxq Ñ Ω1 κpxq{κpsq are linearly independent, so are their images in Ω1 X{V b κpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, p is essentially smooth at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 8 In the sequel, we will use the following limit argument repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Every semilocal Prüfer domain R is a filtered direct union of its subrings Ri such that: (i) for every i, Ri is a semilocal Prüfer domain of finite Krull dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and (ii) for i large enough, Ri Ñ R induces a bijection on the sets of maximal ideals hence is fpqc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Write FracpRq “ YiKi as the filtered direct union of the subfields of FracpRq that are finitely generated over its prime field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let Ri :“ R X Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then R “ YiRi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to see that every Ri is a semilocal Prüfer domain whose local rings have finite ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let tpju1ďjďn be the set of maximal ideals of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then R “ Ş 1ďjďn Rpj is the intersection of the valuation rings Rpj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus we have Ri “ Ş 1ďjďn ` Ki X Rpj ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since Ki{K has finite transcendence degree, by Abhyankar’s inequality, every Ki X Rpj is a valuation ring of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [BouAC, VI, §7, Proposition 1–2], Ri is a semilocal Prüfer domain, and its local rings at maximal ideals are precisely the minimal elements of the set tKi X Rpju1ďjďn under inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This implies (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (ii), it suffices to show that for i large enough there are no strict inclusion relation between Ki X Rpj1 and Ki X Rpj2 for j1 ‰ j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Indeed, if πj P pjz Ť j1‰j pj1 for 1 ď j ď n, then (ii) holds for any i for which tπju1ďjďn Ă Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Reflexive sheaves on schemes over Prüfer bases with regular fibers 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Reflexive sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume that X is a locally coherent scheme, see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For an OX-module F, its dual is denote by F _ :“ HomOXpF, OXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A coherent OX-module F is reflexive if the canonical map F Ñ F __ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since every coherent OX-module G is Zariski-locally finitely presented O‘m X Ñ O‘n X Ñ G Ñ 0, by taking dual, G _ is finitely copresented as 0 Ñ G _ Ñ O‘n X Ñ O‘m X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, the dual G _ of a coherent OX-module G is also coherent (equivalently, finitely presented).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Moreover, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 shows that for integral X and every coherent OX-module G , the double dual G __ is OX-reflexive, hence G __ is the reflexive hull of G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 (reflexive hull).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a locally coherent integral scheme X and two OX-modules F and G , if F is coherent and G is reflexive, then H :“ HomOXpF, G q is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, the double dual F __ :“ HomOXpHomOXpF, OXq, OXq is a reflexive OX-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the coherence of H , it suffices to take a presentation O‘m X Ñ O‘n X Ñ F Ñ 0 of F and its sheaf homomorphism with G so that H “ kerpG ‘n Ñ G ‘mq which is coherent by [SP, 01BY].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a domain R, a finitely presented R-module M, and an exact sequence 0 Ñ M Ñ M 1 Ñ M 2 of finite R-modules, if M 1 is reflexive and M 2 is torsion-free, then M is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote p´q_ “ HomRp´, Rq and consider the following commutative diagram 0 M M 1 M 2 M __ M 1__ M 2__.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [SP, 0AV0], M 1 is torsion-free, so is M, hence the map M ãÑ M __ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to show that this map is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the map u: M 1_ Ñ M _, consider the exact sequence M 1_ Ñ M _ Ñ cokerpuq Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As M 1 is reflexive, it is finitely presented, so [SP, 0583] applies, yielding the exact sequence HomRpM 1 bR K, Kq Ñ HomRpM bR K, Kq Ñ cokerpuq bR K Ñ 0, where K :“ Frac R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since K is R-flat, the injectivity of M ãÑ M 1 implies that cokerpuqK “ 0, hence cokerpuq is R-torsion and cokerpuq_ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, M __ ãÑ M 1__ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Because M 2 is torsion- free, the map M 2 ãÑ M 2__ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By snake lemma, M ։ M __ is surjective so M is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Since H is coherent, it is finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The desired reflexivity follows from Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ By reflexive hull, reflexive sheaves extend from quasi-compact open (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [GR18, Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8(i)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 9 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a coherent reduced scheme X with an quasi-compact open U Ă X, the restriction OX-Rflx Ñ OU-Rflx is essentially surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It suffices to assume that X is irreducible, so X is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Every reflexive OU-module F, by [GR18, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='24 (ii)], extends to a finitely presented quasi-coherent OX-module Ă F, which is coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2, the reflexive hull Ă F __ is a reflexive extension of F on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a locally coherent integral scheme X and two OX-modules F and G , if F is coher- ent and G is reflexive, then the natural map HomOXpF __, G q „ ÝÑ HomOXpF, G q is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Locally on X the reflexive OX-module G fits into an exact sequence 0 Ñ G Ñ O‘m X Ñ O‘n X , hence we have the following commutative diagram of OX-modules with exact rows 0 HomOXpF __, G q HomOXpF __, O‘m X q HomOXpF __, O‘n X q 0 HomOXpF, G q HomOXpF, O‘m X q HomOXpF, O‘n X q By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2, F _ is reflexive, hence the two rightmost vertical arrows are bijective and so is the leftmost vertical arrow, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let X Ñ S be a finite type morphism with regular fibers between topologically Noetherian schemes, let j : U ãÑ X be a quasi-compact open immersion with complement Z :“ XzU satisfying codimpZs, Xsq ě 1 for every s P S and codimpZη, Xηq ě 2 for every generic point η P S, and let F be a reflexive OX-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume that S is a cofiltered inverse limit of integral schemes pSλqλPΛ with generic point ηλ and surjective transition maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, there is a λ0 P Λ, a finite type morphism Xλ0 Ñ Sλ0 with regular fibers such that Xλ0 ˆSλ0 S » X, a closed subscheme Zλ0 Ă Xλ0 such that Zλ0 ˆSλ0 S » Z, the open immersion jλ0 : Xλ0zZλ0 ãÑ Xλ0 is quasi-compact, and the following codimppZλ0qs, pXλ0qsq ě 1 for every s P Sλ0 and codimppZλ0qη0, pXλ0qη0q ě 2 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Also, there is a reflexive OXλ0 -module Fλ0 whose inverse image on X is F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The condition that X has regular S-fibers descends to Xλ0 by [EGA IV2, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The reflexive OX-module F descends thanks to [EGA IV3, Théorème 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2] and by applying [EGA IV3, Corollaire 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5] to F „ ÝÑ F __.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Because Z is contructible closed, by [EGA IV3, Théorème 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='11], it descends to Zλ such that p´1 λ pZλq “ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For fλ : Xλ Ñ Sλ, by the transversity of fibers and [EGA IV2, Corollaire 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6], Zλ does not contain any irreducible components of f ´1 λ psλq for any sλ P Sλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Finally, the image of the generic point η P S is the generic point ηλ P Sλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [EGA IV2, Corollaire 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4], we have codimppZλqηλ, pXλqηλq “ codimpZη, Xηq ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a valuation ring V with spectrum S and a flat, locally of finite type morphism f : X Ñ S of integral schemes with regular fibers, the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) For every x P X and every coherent OX-module F that is reflexive at x, we have proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='dimOX,xFx ď maxp0, n ´ 1q, where n “ dim Of ´1pfpxqq,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) For a closed subset Z Ă X such that j : XzZ ãÑ X is quasi-compact and satisfies the following codimpZs, Xsq ě 1 for all s P S and codimpZη, Xηq ě 2 for the generic point η P S, the restriction functors induce the following equivalences of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' OX-Rflx „ ÝÑ OXzZ-Rflx Pic X „ ÝÑ Pic XzZ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) In particular, for every X-affine finite type algebraic space Y , we have a bijection of sets Y pXq » Y pXzZq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 10 (iii) For a closed subset Z Ă X such that j : XzZ ãÑ X is quasi-compact and XzZ contains all the associated points of the generic fiber of X and every X-separated algebraic space Y , the map Y pXq ãÑ Y pXzZq is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iv) For a closed subset Z Ă X satisfying the assumption in (ii) and a quasi-compact quasi-separated morphism p: W Ñ XzZ such that p˚OW is a reflexive OXzZ-module, we have the Cartesian square AffXzZW AffXW XzZ X, paff ν j where AffXzZW “ SpecXzZpp˚OW q and AffXW “ SpecXpj˚p˚OW q, such that paff and ν are finite, paff is the relative normalization [SP, 035H] of XzZ in W and ν is the relative normalization of X in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, ν˚pOAffpW{Xqq is a reflexive OX-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (v) For a closed subset Z Ă X satisfying the assumption in (ii) and a finite flat locally finitely presented morphism p: W Ñ XzZ, the morphism ν : AffXW Ñ X is the relative normalization of X in W such that pAffXWqXzZ “ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, ν˚pOAffXW q is a reflexive OX-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The assertion (i) is [GR18, Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (iii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (ii), by Lemmata 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6, we may assume that V has finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since |X| is the finite disjoint union of its S-fibers Xs, which are Noetherian spaces, we know that X is topologically Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, every open subset of X is quasi-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4, the functors (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) are essentially surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the faithfulness, consider two morphisms α, β : F Ñ G between reflexive OX-modules such that α|XzZ “ β|XzZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To show that α “ β, since it is a local problem, it suffices to check that αx “ βx : Fx Ñ Gx for every x P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Take a presentation O‘m X,x Ñ O‘n X,x Ñ Fx Ñ 0 and copresentation 0 Ñ Gx Ñ O‘m1 X,x Ñ O‘n1 X,x, then αx and βx induce two morphisms between these copresentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then we are reduced to the case when Fx and Gx are free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We may assume that Fx “ O‘r X,x and Gx “ O‘s X,x, so the following isomorphisms lead to α “ β HomOX,xpFx, Gxq “ HomOX,xpO‘r X,x, O‘s X,xq » Homj˚OX,xpj˚O‘r X,x, j˚O‘s X,xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) are full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If F and G are two reflexive OX-modules with a morphism φ: j˚F Ñ j˚G , then by [GR18, Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='9], taking j˚p´q induces the following morphism rφ: F » j˚j˚F Ñ j˚j˚G » G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the second assertion of (ii), by the sheaf property, the problem is étale local on X, so we can assume that X is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Choose an embedding Y ãÑ An X for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The assumption implies that XzZ is scheme-theoretically dense in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, for every morphism φ: XzZ Ñ Y , if φ extends uniquely to a morphism rφ: X Ñ An X, then rφ´1pY q is a closed subscheme of X containing XzZ and by [EGA IV4, Lemme 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8], coincides with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In other words, if rφ exists uniquely, then it factorises as X ψÑ Y ãÑ An X such that ψ is the unique extension of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This reduces us to the case Y “ An X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now, by the reflexivity of OX and the full faithfulness of OX-Rflx „ ÝÑ OXzZ-Rflx, we have the desired bijections An XpXq “ HomOXpOX, O‘n X q » HomOXpOXzZ, O‘n XzZq “ An XpXzZq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To prove (iii), we first prove that XzZ Ă X is scheme-theoretically dense in the sense of [SP, 0834].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [SP, 0836], we need to show that OX Ñ j˚OXzZ is injective, which through the terminology of [GR18, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='19], signifies that δpz, OXq ą 0 for all z P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It suffices to take étale coverings of X by schemes and use the depth formula [GR18, Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='46] for flat morphisms to deduce that all z P Z satisfies δpx, OXq ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since j is quasi-compact, by [SP, 0835], the schematic image of XzZ is X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, we apply [SP, 084N] to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The (iv) follows from (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (v), note that p˚OW is OXzZ-reflexive since by [SP, 02KB], p is finite locally free, hence it is a special case of (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Auslander’s flatness criterion on schemes smooth over valuation rings The goal is to establish Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 as a counterpart of Auslander’s flatness criterion [Aus62, Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3] on schemes smooth over valuation rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As expected, our criterion leads to a Zariski–Nagata purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a valuation ring V with spectrum S and closed point s P S, an S-smooth finite type scheme X, a point x P X lying over s with local ring A :“ OX,x, and a reflexive A-module M, EndApMq is isomorphic to a direct sum of copies of M if and only if M is A-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As Auslander’s proof, our strategy relies on an estimate of the length of cohomology groups of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To begin with, we introduce the length function on torsion modules over valuation rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lengths of torsion modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a nontrivial valuation ring V with fraction field K, value group Γ and a valuation map ν : K Ñ Γ, every finitely presented torsion V -module M is of the form M » À i V {aiV for finitely many ai P V zt0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Define the length of M as δpMq “ ř i νpaiq P Γě0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The element δpMq is well defined, and δpMq “ 0 if and only if M “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Every acyclic, bounded complex M ‚ of torsion, finitely presented V -modules satisfies ř jp´1qjδpM jq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a nontrivial valuation ring V , an essentially smooth V -local algebra pA, mAq, and the collection A-Modtor,fp of all finitely presented A-modules M such that SupppMq Ă tmAu, there exist a totally ordered abelian group Γ and a map l: A-Modtor,fp Ñ Γě0 satisfying the following properties: ‚ for A-module M P A-Modtor,fp, we have lpMq “ 0 if and only if M “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' ‚ for every acyclic, bounded complex M ‚ such that M j P A-Modtor,fp for each j, one has ř jp´1qjlpM jq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' First we assume that the structural map V Ñ A admits a section A ։ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In this case we claim that M is finitely presented over V and is V -torsion, so we can simply let Γ be the valuation group of V and set lpMq :“ δpMq, where δ is delivered from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Indeed, it is clear that M is V -torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Any section Spec V Ñ Spec A is a regular immersion [SP, 067R], so there is a finitely generated ideal J Ă A such that V » A{J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, since M P A-Modtor,fp, we see that JnM “ 0 for a large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' On the other hand, the essential smoothness of A over V implies that J{J2 is a free V » A{J-module whose rank equals the rank of the free A-module Ω1 A{V , and there is a natural isomorphism of graded V » A{J-algebras À ně0 Jn{Jn`1 » Sym‚ A{JpJ{J2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, A{Jn is a finite free V -module for every n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, by tensoring a presentation AN Ñ AN Ñ M Ñ 0 of M with A{Jn for a large enough n, we get a desired finite presentation of the V -module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In the general case, we first use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 to reduce to the case when the residue fields extension of V Ñ A is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, if B is the integral closure of V in an algebraic closure of FracpV q, we let V 1 be a valuation ring of FracpBq centered at a maximal ideal of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It’s clear that V 1 is absolutely integral closed, so it is strictly Henselian and there exists a V -map φ : A{mA Ñ V 1{mV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let A1 :“ A bV V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then φ induces a V 1-map φ1 : A1 Ñ V 1{mV 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' let p Ă A1 be its kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then A1 p is essentially smooth over V 1 and φ1 induces a V 1-map A1 p Ñ V 1{mV 1, which, by the Henselianity of V 1, lifts to a V 1-map A1 p Ñ V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the previous paragraph, the lemma is true for A1 p, say, with corresponding map l1 valued in Γ, where Γ is the valuation group of V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since A Ñ A1 p is faithfully flat, it suffices to define lpMq :“ l1pM bA A1 pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a valuation ring V , a V -smooth finite type scheme X, a point x P X that lies over a non-generic point s P SpecpV q, and a map of finitely presented OX,x-modules M Ñ N that induces an isomorphism over SpecpOX,xqztxu, we have an isomorphism Exti OX,xpN, OX,xq „ ÝÑ Exti OX,xpM, OX,xq for every i ă d and a monomorphism Extd OX,xpN, OX,xq ãÑ Extd OX,xpM, OX,xq, where d :“ dim OXs,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let ker, coker, and im be the kernel, cokernel, and image of M Ñ N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By assumption and the coherence of OX,x, ker and coker are coherent, or, equivalently, finitely presented OX,x-modules ([SP, 05CX]) supported at txu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consider the following short exact sequences 0 Ñ ker Ñ M Ñ im Ñ 0, 0 Ñ im Ñ N Ñ coker Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By applying HomOX,xp´, OX,xq, we get two long exact sequences concerning Ext’s, and the lemma follows from the vanishing Exti OX,xpker, OX,xq “ 0 and Exti OX,xpcoker, OX,xq “ 0 for i ď d (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For finitely presented A :“ OX,x-modules M and N, Exti ApM, Nq and TorA i pM, Nq are finitely presented over A for all i ě 0 and are zero for i ą d ` 1, where d “ dim OXs,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [GR18, Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (i)], since A is coherent, the coherent A-module ([SP, 05CX]) M has a resolution by finite free A-modules of length ď d ` 1: F‚ Ñ M, Fi “ 0 for i ą d ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then Exti ApM, Nq “ HipHompF‚, Nqq and TorA i pM, Nq “ HipF‚ b Nq are all coherent, or equivalently, finitely presented A-modules, and are zero for i ą d ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a finitely presented A :“ OX,x-module M, we have a natural isomorphism EndApMq__ „ ÝÑ EndApM __q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' First, we define a natural map EndApMq__ Ñ EndApM __q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Note that M __ is A-reflexive due to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5, where M __ plays the role of G , we get a natural isomorphism HomApM, M __q „ ÐÝ EndApM __q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It suffices to consider the natural maps EndApMq Ñ HomApM, M __q „ ÐÝ EndApM __q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2, the two rightmost modules are reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Taking double dual yields the desired map of reflexive A- modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to check that the map EndApMq__ Ñ EndApM __q is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The equivalence of categories of reflexive modules in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii) reduces us to checking this at x P X that is either a one-codimensional point of the generic V -fiber or a maximal point of a non-generic V -fiber, where, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(iii), A is a valuation ring, so there is an N P Zě0 and finitely many ai P mAzt0u such that M » A‘N À p‘iA{aiAq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, we conclude by the isomorphisms EndApMq__ » EndApA‘Nq » EndApM __q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The proof proceeds as the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Preliminary cases and reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' First, since X is locally of finite presentation over S and M is finitely presented over A, by a standard limit argument involving Lemmata 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6, we are reduced to the case when V is a finite-rank valuation ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Secondly, if V 1 is a valuation ring of an algebraic closure of FracpV q that dominates V and if x1 P X1 :“ X ˆV V 1 is a point lying over x P X, then MA1 :“ M bA A1 is a finitely presented reflexive A1-module and EndA1pMA1q » EndApMq bA A1 is isomorphic to a direct sum of copies of MA1, where A1 :“ OX1,x1 (because A1 is faithfully flat over A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By faithfully flat descent [SP, 08XD, 00NX], the freeness of M over A is equivalent to the freeness of MA1 over A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, by replacing V by V 1, A by A1, and M by MA1, we are reduced to the case when FracpV q is algebraically closed (this assumption will be only used in the very end of the proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Set dx :“ dimpOXs,xq and r :“ rankpV q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The case r “ 0 and dx arbitrary is classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The case r arbitrary and dx “ 0 is trivial, because A is a valuation ring (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(iii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The case r arbitrary and dx “ 1 follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Subsequently, we may assume dx ě 2 in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Case 1: r is arbitrary and dx “ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now, we deal with the crucial case when r arbitrary and dx “ 2 by induction on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The induction hypothesis is that the assertion holds for dx “ 2 and r1 ď r ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Notice that, for any proper generalization x1 P X of x that lies over, say, s1 P SpecpV q, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(iv), we have either s1 “ s and dx1 ă 2, or htps1q ă r and dx1 ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, by induction hypothesis and 13 the preliminary cases above, the assertion holds for OX,x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since Mx1 is a finitely presented reflexive OX,x1-module and EndOX,x1 pMx1q “ EndOX,xpMq bOX,x OX,x1 » p à Mq bOX,x OX,x1 “ à Mx1, the induction hypothesis applies to the OX,x1-module Mx1, implying that Mx1 is OX,x1-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In other words, Ă M is locally free over Spec Aztxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consider the following evaluation map M _ bA M Ñ HomApM, Mq, f b m ÞÑ rm1 ÞÑ fpm1qms, which, by the local freeness of Ă M over Spec Aztxu, is an isomorphism over Spec Aztxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since dx “ 2 ą 1, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4, we apply Ext1 Ap´, Aq to the above map to obtain the following isomorphism Ext1 ApM _ b M, Aq » Ext1 ApEndApMq, Aq » Ext1 ApM, Aq‘rkM (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) of A-modules that are supported on txu by the local freeness of Ă M over Spec Aztxu, where rkM “ dimFrac A M bA Frac A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5, the modules in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) are also finitely presented over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the adjunction HomApM, HomApM _, ´qq » HomApM b M _, ´q, we take their derived functors valued at A, so the E2-page of the associated Grothendieck spectral sequence yields a monomorphism Ext1 ApM, Mq ãÑ Ext1 ApM b M _, Aq p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1q » Ext1 ApM, Aq‘rkM , where we have used M __ » M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' again, by the local freeness of Ă M over Spec Aztxu and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5, they are finitely presented supported on txu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, the map l from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3 applies so we have lpExt1 ApM, Mqq ď rkM ¨ lpExt1 ApM, Aqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2) Since M is reflexive, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(i), we have proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='dimAM ď dx´1 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We prove proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='dimpMq “ 0 by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='dimpMq “ 1, then M has a free resolution 0 Ñ F1 Ñ F0 Ñ M Ñ 0 by finite A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As M is not free, the sequence is nonsplit, corresponding to a nontrivial extension class in Ext1 ApM, F1q » Ext1 ApM, AqrankpF1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, we have C :“ Ext1 ApM, Aq ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Applying HomAp´, Aq to F‚ Ñ M yields an exact sequence 0 Ñ M _ Ñ F _ 0 Ñ F _ 1 Ñ Ext1 ApM, Aq Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Tensoring it with M, we get an exact sequence F _ 0 bA M Ñ F _ 1 bA M Ñ Ext1 ApM, Aq bA M Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since coker pF _ 0 b M Ñ F _ 1 b Mq » coker pHomApF0, Mq Ñ HomApF1, Mqq “ Ext1 ApM, Mq, we deduce that Ext1 ApM, Mq » Ext1 ApM, Aq bA M “ C bA M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By tensoring 0 Ñ F1 Ñ F0 Ñ M Ñ 0 with C “ Ext1 ApM, Aq (which is a nonzero finitely presented A-module supported at txu, by the locally freeness of Ă M over Spec Aztxu), we get an exact sequence 0 Ñ TorA 1 pC, Mq Ñ C bA F1 Ñ C bA F0 Ñ C bA M Ñ 0 of finitely presented A-modules supported on txu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Applying the map l from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, we obtain lpC bA Mq “ lpC bA F0q ´ lpC bA F1q ` lpTorA 1 pC, Mq “ rkM ¨ lpCq ` lpTorA 1 pC, Mqq, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3) where rkM “ rankpF0q ´ rankpF1q ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' On the other hand, since C bA M » Ext1 ApM, Mq, we deduce lpC bA Mq p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2q ď rkM ¨ lpCq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4) The combination of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4) leads to lpTorA 1 pC, Mqq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' So, we have an exact sequence 0 Ñ C bA F1 Ñ C bA F0 Ñ C bA M Ñ 0, which combined with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8 implies that the map F1 Ñ F0 splits, that is, M is A-free, contradicting our assumption that proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='dimApMq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This completes the case when r is arbitrary and dx “ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Case 2: r is arbitrary and dx ą 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We deduce by double induction on the pair pr “ htpsq, dxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By induction hypothesis, the assertion holds for all smooth V -scheme X1 and all points x1 P X1 such that htps1q ď htpsq and dx1 ď dx, where s1 P SpecpV q lies below x1, and at least one of equalities is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(iv), the induction hypothesis applies to OX,x1 for all proper generalization x1 P X of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since Mx1 is a finitely presented reflexive OX,x1-module and EndOX,x1 pMx1q “ EndOX,xpMqx1 » à Mx1, the induction hypothesis gives that Mx1 is OX,x1-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In other words, Ă M is locally free over Spec Aztxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 14 Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5 ([SP, 057F]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume that the residue field extension of V Ñ A is separable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', this holds if κpsq :“ V {mV is perfect), then there exists an a P A such that A :“ A{paq is essentially V -smooth and dimpA{mV Aq “ dx ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since our V has algebraically closed fraction field (by the first paragraph), all of its primes have alge- braically closed residue fields, so we can choose a P A as in the above claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since a is nonzerodivisor in A and M “ HomApM _, Aq, we see that a is M-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Set M :“ M{aM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Applying HomApM, ´q to the short exact sequence 0 Ñ M aÝÑ M Ñ M Ñ 0, we get an exact sequence 0 Ñ HomApM, Mq aÝÑ HomApM, Mq Ñ HomApM, Mq Ñ Ext1 ApM, Mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Substituting our assumption HomApM, Mq – M ‘rkM into it yields an exact sequence of A-modules 0 Ñ M ‘rkM Ñ HomApM, Mq Ñ T Ñ 0, where T Ă Ext1 ApM, Mq is a finitely presented A-submodule (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5), which, by the locally freeness of Ă M over Spec Aztxu, is supported on txu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since dimpA{mV Aq “ dx´1 ě 2, taking dual (as A-modules) of the above short exact sequence and using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4, we see that pM _q‘rkM » HomApM, Mq_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Taking dual further and invoking Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6, we get the following isomorphism pM __q‘rkM » HomApM __, M __q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since the double dual M __ is finitely presented over A and is reflexive (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2), we can apply our induction hypothesis to the A-module M __ and conclude that it is A-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The same lemma also implies that M _ is A-reflexive, so M _ » M ___ is A-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Finally, we show that M is A-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since Ă M is locally free over Spec Aztxu, the natural map M Ñ M __ is an isomorphism over Spec Aztxu, and, since dimpA{mV Aq “ dx ´ 1 ą 1, we may apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 to see that Ext1 ApM, Aq » Ext1 ApM __, Aq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since a is M-regular, we deduce that Ext1 ApM, Aq » Ext1 ApM bL A A, Aq » Ext1 ApM, Aq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Applying HomApM, ´q to the short exact sequence 0 Ñ A aÝÑ A Ñ A Ñ 0 we get an exact sequence 0 Ñ M _ aÝÑ M _ Ñ HomApM, Aq Ñ Ext1 ApM, Aq aÝÑ Ext1 ApM, Aq Ñ Ext1 ApM, Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As all modules are finitely presented over A and Ext1 ApM, Aq “ 0, Nakayama’s lemma gives that Ext1 ApM, Aq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, M _{aM _ » HomApM, Aq “ M _ is A-free (by the previous paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' From this we can deduce that M is A-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Indeed, the A-free module M _{aM _ has projective dimension 1 over A, thus, for any finitely presented A-module N, we can derive from 0 Ñ M _ aÝÑ M _ Ñ M _{aM _ Ñ 0 an exact sequence of finitely presented A-modules Ext1 ApM _, Nq aÝÑ Ext1 ApM _, Nq Ñ Ext2 ApM _{aM _, Nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As Ext2 ApM _{aM _, Nq “ 0, by Nakayama’s lemma, we have Ext1 ApM _, Nq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, for any surjection A‘n ։ M _ with, say, kernel N, the extension class of the short exact sequence 0 Ñ N Ñ A‘n Ñ M _ Ñ 0 is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This implies that M _ is A-free, hence so is M “ M __.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Generalities on torsors over algebraic spaces 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Throughout this section, we let S denote a base scheme, X an algebraic space over S, and G an X-group algebraic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (1) A (right) G-torsor (for the fppf topology) is an X-algebraic space P equipped with a G-action a : P ˆX G Ñ P such that the following conditions hold: (i) the induced morphism P ˆX G „ ÝÑ P ˆX P, pp, gq ÞÑ pp, app, gqq, is an isomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and (ii) there exists a fppf covering tXi Ñ XuiPI of algebraic spaces [SP, 03Y8] such that PpXiq ‰ H for every i P I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 15 (2) For G-torsors P1 and P2, a morphism P1 Ñ P2 is a G-equivariant morphism P1 Ñ P2 of X- algebraic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (3) By a trivialization of a G-torsor P we mean a G-equivariant isomorphism t : G „ ÝÑ P, where G acts on itself via right multiplication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' this amounts to the choice of a section tp1Gq P PpXq (if exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A G-torsor P is trivial if there exists a trivialization, or, equivalently, if PpXq ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Note that every morphism of two G-torsors is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To see this, one may pass to a fppf covering of X to reduce to the case when both torsors are trivial, in this case the assertion is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' One can also define a sheaf torsor for an X-group algebraic space G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It is a sheaf P : pSch{Sqopp fppf Ñ Set equipped with a map P Ñ X of sheaves and a G-action a : P ˆX G Ñ P such that the above two condi- tions (i) and (ii) in (1) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' However, it turns out that such a sheaf torsor is necessarily representable by an algebraic space, so working with sheaf torsors adds no more generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To see this, let tXi Ñ XuiPI be a fppf covering as in (ii) that trivializes P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then every P ˆX Xi » G ˆX Xi is an algebraic space, and the map Ů i P ˆX Xi Ñ P is representable by algebraic spaces and is a fppf covering, because it is the base change of the fppf covering Ů i Xi Ñ X of algebraic spaces via P Ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Here, all coproducts are taken in the category of sheaves on pSch {Sqfppf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It follows from (3) of [SP, 04S6] that P is an algebraic space, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let P1, P2 be two G-torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Define a functor IsomXpP1, P2q : pSch{Xqopp Ñ Set which associates to any scheme T over X the set of GT -equivariant isomorphisms P1,T Ñ P2,T over T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For two G-torsors P1 and P2, IsomXpP1, P2q is an algebraic space over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Further, G Ñ X is quasi-compact (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', étale, smooth, flat, separated, (locally) of finite type, (locally) of finite presentation, quasi-affine, affine, or finite) if and only if IsomXpP1, P2q Ñ X is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since IsomXpP1, P2q is fppf locally on X isomorphic to G, it admits a representable fppf covering by algebraic spaces, hence it is an algebraic space by [SP, 04S6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The list properties of morphisms of algebraic spaces are all stable under base changes and are fppf local on the target, see [SP, 03KG] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', [SP, 03XT, 03ZF, 03MM, 03KM, 040Y, 0410, 03WM, 03WG, 03ZQ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, since IsomXpP1, P2q is fppf locally on X isomorphic to G, the properties of G are inherited by and can be detected from IsomXpP1, P2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Since every G-torsor P Ñ X trivializes over a fppf covering tXi Ñ Xu, one may try to obtain P by glueing the trivial GXi-torsors PXi using the canonical isomorphisms φij : pPXiqXij » PXij » PXjqXji, where Xij “ Xi ˆX Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It turns out that, unlike the case of schemes, this is always possible in the framework of algebraic spaces, see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Note that, by taking U :“ Ů Xi, we may assume that PU is trivial for a fppf covering U Ñ X with U an algebraic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5 (Descent datum for torsors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let S, X and G be as in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let U Ñ X be a fppf covering of algebraic spaces over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For every integer n ě 0, denote by U pnq the n-fold fiber product of U over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The category of descent datum for G-torsors relative to U Ñ X, denoted Tors ´ pU p2q Ñ Uqfppf, G ¯ , has pairs pQ, φq as objects, where ‚ Q Ñ U is a GU-torsor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and 16 ‚ φ : pr˚ 1Q „ ÝÑ pr˚ 2Q is an isomorphism of GUp2q-torsors such that the following diagram commutes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', the cocycle condition holds) pr˚ 12pr˚ 1Q pr˚ 12pr˚ 2Q pr˚ 23pr˚ 1Q pr˚ 23pr˚ 2Q pr˚ 13pr˚ 1Q pr˚ 13pr˚ 2Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' pr˚ 12pφq » » pr˚ 23pφq » pr˚ 13pφq A morphism from a pair pQ, φq to another pair pQ1, φ1q is a morphism θ : Q Ñ Q1 of GU-torsors compatible with φ and φ1, that is, pr˚ 2pθqφ “ φ1pr˚ 1pθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To every G-torsor P one can associate a pair ΨpPq :“ pPU, canq via base changes, where can is the canonical isomorphism pr˚ 1pPUq » PUp2q » pr˚ 2pPUq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus we obtain a functor Ψ : TorspXfppf, Gq Ñ TorsppU p2q Ñ Uqfppf, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6 (Descent G-torsors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Ψ is an equivalence of category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In other words, every descent datum pQ, φq for G-torsors are effective in the sense that there exists a G-torsor P and an isomorphism Q » PU compatible with θ and the canonical descent datum for PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The full faithfulness of Ψ follows from the sheaf property of the functor IsomXpP1, P2q for any G-torsors P1 and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To show that Ψ is essential surjective, we pick a descent datum pQ, φq, and we need to show that there exists a G-torsor P such that pPU, canq » pQ, φq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' When both X and U are schemes, this is proven in [SP, 04U1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The case of algebraic spaces can be proved similarly, and we repeat the argument for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' First we view Q as a sheaf on the site pAS{Uqfppf (by the natural equivalence of the topoi associated to pAS{Uqfppf and pSch {Uqfppf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since descent datums for sheaves on any site are always effective [SP, 04TR], we may find a sheaf P on the site pAS{Xqfppf and an isomorphism of sheaves PU » Q compatible with the descent datums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Further, since maps of sheaves on any site can be glued [SP, 04TQ], the GU-action on Q descent to a G-action on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' All the assumptions (i) and (ii) of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 hold, because they can be checked on the fppf covering U Ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to see that P is representable by an algebraic space over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' However, this follows from (3) of [SP, 04S6], in view of the fact that the map Q Ñ P is representable by algebraic spaces and is a fppf covering (being a base change of the fppf covering U Ñ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ We end this section with the following result, which will be used repeatedly in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let S be a scheme, X an algebraic space over S, and G an X-group algebraic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let f : Y Ñ X be a morphism of algebraic spaces over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume the following conditions hold: (i) for every fppf covering T Ñ X with T a scheme, the pullback functor f ˚ T : TorspTfppf, GT q Ñ TorsppYT qfppf, GYT q is fully faithful, where YT :“ Y ˆX T , and fT :“ f ˆX T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and (ii) for every GY -torsor P, there is a fppf covering T Ñ X with T a scheme such that PYT lies in the essential image of f ˚ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then pullback induces an equivalence f ˚ : TorspXfppf, GT q „ ÝÑ TorspYfppf, GY q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Similarly, if G Ñ X is smooth, then we have an equivalence f ˚ : TorspX´et, GT q „ ÝÑ TorspY´et, GY q, provided that one replaces ‘fppf’ by ‘étale’ everywhere in the above assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We prove the Lemma for fppf torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to check that f ˚ is essentially surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let P be a GY -torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By assumption (ii) there is a fppf covering T Ñ X with T a scheme and a GT -torsor Q such that f ˚ T Q » PYT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Using this isomorphism we can transform the canonical descent datum on PYT to a descent datum θ : pr˚ 1f ˚ T Q „ ÝÑ pr˚ 2f ˚ T Q 17 on f ˚ T Q (relative to the covering YT Ñ Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For every integer n ě 0, denote by T pnq the n-fold fiber product of T over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Using the canonical identifications pr˚ 1f ˚ T Q “ f ˚ T p2qpr˚ 1Q and pr˚ 2f ˚ T Q “ f ˚ T p2qpr˚ 2Q, the full faithfulness of fT p2q implies that there is a unique isomorphism τ : pr˚ 1Q „ ÝÑ pr˚ 2Q of GT p2q-torsors such that f ˚ T p2qpτq “ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since pr˚ 13pθq “ pr˚ 13pf ˚ T p2qpτqq “ f ˚ T p3qpr˚ 13pτq and pr˚ 13pθq “ pr˚ 23pθqpr˚ 12pθq “ pr˚ 23 ` f ˚ T p2qpτq ˘ pr˚ 12 ` f ˚ T p2qpτq ˘ “ f ˚ T p3q ppr˚ 23pτqq f ˚ T p3q ppr˚ 12pτqq “ f ˚ T p3q ppr˚ 23pτqpr˚ 12pτqq , the full faithfulness of f ˚ T p3q implies that pr˚ 13pτq “ pr˚ 23pτqpr˚ 12pτq, that is, τ is a descent datum on Q relative to T Ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6, there is a G-torsor R and an isomorphism pQ, φq » pRT , canq of descent datums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Pulling back to YT , we get an isomorphism of descent datums pPYT , canq » f ˚ T pQ, τq » pRYT , canq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6 again (applied to the covering YT Ñ Y ), we see that f ˚pRq “ RY » P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Purity for torsors and finite étale covers We begin with generalities about linear groups that will be fundamental in multiple types of purities for reductive torsors, where the overall argument is bootstrapped from that for vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, in this process, controlling on the projective dimensions of extended reflexive sheaves leads to relative- dimensional constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, we obtain the purity for reductive torsors on relative curves §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We then present local variants of the acquired purity results §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2, where the constraints on dimensions are more flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By virtue of this, we shrink complements of domains of reductive torsors to a higher- codimensional closed subset §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, laying the groundwork for later proofs of the Grothendieck–Serre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Finally, by our Auslander’s flatness criterion, we present a Prüferian counterpart of the Zariski–Nagata purity in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Coaffine locally linear groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let X be an algebraic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' An X-group algebraic space G is linear if there exists a group monomorphism G ãÑ GLpV q for a locally free OX-module V of finite rank;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' it is fppf (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', étale) locally linear if there exists a fppf covering (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', an étale covering) X1 Ñ X such that GX1 is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A locally linear X-group algebraic space G is coaffine, if it locally has an X-affine coset GLpV q{G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For instance, if a linear group G Ă GLpV q is reductive or finite locally free, then GLpV q{G is X-affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In the sequel, we mainly consider locally linear coaffine X-group algebraic spaces G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Purity for reductive torsors on relative curves Now we study the extension behavior of torsors over relative curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Motivated by [EGA IV4, Proposi- tion 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4] that every invertible sheaf on a curve over a field extend across finitely many closed points, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 concerns relative curves over valuation rings and generalizes [Guo20, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Torsors on relative curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a valuation ring V with spectrum S, a V -flat finite type scheme X with regular one-dimensional V -fibers, and a closed subscheme D Ă X such that (i) D is finite locall free over V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and (ii) D factors through an affine open Spec R Ă X, we consider the completion pR :“ lim ÐÝn R{In, where I Ă R is the ideal determined by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote BD :“ Spec pR as the formal neighborhood of D and UD :“ BDzD for the punctured formal neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 18 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a valuation ring V with spectrum S, an S-flat finite type scheme X with regular one-dimensional S-fibers, an S-finite locally free closed subscheme D Ă X inside an affine open X0 Ă X with complementary open j : XzD ãÑ X, then the restriction functor between the categories VectX Ñ VectXzD is essentially surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, for the formal neighborhood BD :“ p XD, the punctured neighborhood UD :“ BDzD, we have H1 ZarpUD, GLnq “ H1 ´etpUD, GLnq “ t˚u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Every vector bundle E on XzD by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 extends to a reflexive sheaf rE on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(i) implies that rE is a vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now let V be a vector bundle on UD and denote the Henselization of the pair pX0, Dq by pBh D, Dq with U h D :“ Bh DzD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then [BČ22, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='22] descends V to a vector bundle V h on U h D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since Bh D is the limit of elementary étale neighorhoods of D Ă X0, by a limiat argument, V h descends to a vector bundle V 1 on an S-flat finite type scheme X1 with regular one-dimesensional S-fibers and the open X1zD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since VectX1 Ñ VectX1zD is essentially surjective, V 1 extends to a vector bundle r V 1 on X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, there exists a vector bundle r V on BD extending V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since pBD, Dq is a Henselian pair, by [Čes22b, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1], we have an isomorphism VectBD » VectD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Note that D is semilocal and affine, so r V is trivial, in particular, V is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal affine Prüfer scheme S, an S-flat finite type algebraic space X with regular one-dimensional S-fibers, and its closed subset Z such that j : XzZ ãÑ X is quasi-compact and Zη “ H for each generic point η P S and codimpZs, Xsq ě 1 for all s P S, the pushforward j˚p´q and restriction as inverse induce an equivalence between categories of vector bundles VectXzZ „ ÝÑ VectX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We simply verify the assumptions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7 for G “ GLn,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For vector bundles E1 and E2, Y :“ IsomXpE1, E2q is X-affine of finite type (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4), so Y pXzZq “ Y pXq by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The same holds when we base change to every étale X-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (ii), by taking étale atlas, we may assume that X is a scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii), every vector bundle E on XzZ extends to a reflexive OX-module j˚E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To show that the reflexive OX-module j˚E is a vector bundle, it suffices to exploit Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [CTS79, Théorème 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal affine Prüfer scheme S, an S-flat finite type algebraic space X with regular one-dimensional S-fibers, an X-group algebraic space G that is étale- locally linear and coaffine2, and a closed subset Z Ă X such that j : XzZ ãÑ X is quasi-compact and Zη “ H for each generic point η P S and codimpZs, Xsq ě 1 for all s P S, restriction of torsors induces the following equivalence of categories of G-torsors TorspX´et, Gq „ ÝÑ TorsppXzZq´et, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, passing to isomorphism classes of objects, we have an isomorphism H1 ´etpX, Gq » H1 ´etpXzZ, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We simply verify the assumptions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) Since the assumption on the fiber codimension still holds when we base change to every étale scheme over X, it suffices to verify that the restriction functor TorspX´et, Gq Ñ TorsppXzZq´et, Gq is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Indeed, for any G-torsors P1 and P2, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4, Y :“ IsomXpP1, P2q is an X-affine algebraic space of finite type, so Y pXzZq “ Y pXq by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 2A special case is when X is an affine scheme and G is X-reductive, as explained in a footnote of the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 19 (ii) Étale locally on X, every G-torsor on XzZ extends to a G-torsor on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To see this, we may assume that X is affine and G Ă GLn,X, then exploit the commutative diagram with exact rows pGLn,X{GqpXq H1 ´etpX, Gq H1 ´etpX, GLn,Xq pGLn,X{GqpXzZq H1 ´etpXzZ, Gq H1 ´etpXzZ, GLn,Xq, » where the bijectivity of the left vertical arrow follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii) and our assump- tion GLn,X{G being affine over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For every G-torsor P on XzZ, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, we may replace X by an affine open cover to ensure that the induced GLn,XzZ-torsor P ˆGXzZ GLn,XzZ is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A diagram chase implies that there exists a G-torsor Q on X such that Q|XzZ » P, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Local variants of purity results The following is a variant of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a finite-rank valuation ring R with spectrum pS, ηq, an S-flat finite type scheme X with regular fibers, an X-group scheme G that is étale-locally linear and coaffine, and a point x that is (i) either x P Xη with dim OXη,x “ 2, or (ii) x P Xs with s ‰ η and dim OXs,x “ 1, every G-torsor over Spec OX,xztxu extends uniquely to a G-torsor over Spec OX,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The argument of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 reduces us to the case of vector bundles, namely, G “ GLn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then the assertion (i) follows from the classical purity (see for instance, [Gab81, §1, Lemma 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (ii), by the quasi-compactness of SpecpOX,xqztxu and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii), every vector bundle E on Spec OX,xztxu, extends to a reflexive sheaf j˚pE q on Spec OX,x, which, by the assumption dim OXs,x “ 1 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(i), is projective, hence the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For an algebraic space S with a finitely presented closed subspace Z Ă X and an affine morphism of algebraic spaces f : X1 Ñ X, denote Z1 :“ Z ˆX X1, U :“ XzZ, and U 1 :“ U ˆX X1, consider the following Cartesian square U 1 X1 U X, fU j1 f j where j and j1 are open immersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If f is faithfully flat and induces an isomorphism Z » Z1, then (i) The restriction Ψ: FÉtX „ ÝÑ FÉtU ˆFÉtU1 FÉtX1 is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, if FÉtX1 Ñ FÉtU1 is essentially surjective (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', an equivalence), then so is FÉtX Ñ FÉtU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) If X, X1 are schemes, then for a quasi-affine, flat, finitely presented X-group scheme G, the following base change functor is an equivalence of categories of G-torsors TorspXfppf, Gq „ ÝÑ TorspX1 fppf, Gq ˆTorspU1 fppf,Gq TorspUfppf, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) Consider the fibered category AFF over the category of algebraic spaces such that every algebraic space T has the fiber category AFFpT q, the category of T -affine algebraic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [MB96, Théorème 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1], then base change induces the following equivalence of categories ΦAFF : AFFpXq „ ÝÑ AFFpX1q ˆAFFpU1q AFFpUq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence Ψ is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the essential surjectivity, it suffices to patch finite étale covers over U and X1 to an X-affine algebraic space, and conclude by using faithfully flat descent for finite étale properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) See [Čes22a, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 20 Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a local scheme X, the closed point x and punctured spectrum U :“ Xztxu, if for the Henselization Xh of X at x with punctured spectrum U h, FÉtXh „ ÝÑ FÉtUh is an equivalence if and only if so is FÉtX „ ÝÑ FÉtU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let X1 Ñ X be a flat morphism of affine schemes that are smooth over a semilocal Prüfer domain R with spectrum pS, ηq such that there is a closed subscheme Z Ă X satisfies the following (i) codimpZs, Xsq ě 1 for every s P S and codimpZη, Xηq ě 2 for the generic point η P S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and (ii) X1 Ñ X induces an isomorphism between Z and its preimage Z1 :“ Z ˆX X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote U :“ XzZ and U 1 :“ U ˆX X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For an affine, smooth X-group (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', U-group) F with a filtration F “ F0 Ą F1 Ą ¨ ¨ ¨ Ą Fn “ 0 by affine smooth S-normal subgroups (U-normal subgroups) such that every subquotient Fi{Fi`1 is a vector group on X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', such that Fi{Fi`1 is a vector group on S and is central in F{Fi`1), the map H1 ´etpU, Fq Ñ H1 ´etpU 1, Fq has trivial kernel (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', is surjective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' When F is an X-group, since X and X1 are affine, both H1pX, Fq and H1pX1, Fq vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, for every F-torsor P on U that becomes trivial over U 1, we utilize Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 to patch trivial torsors on X1 and U to obtain a trivial F-torsor rP on X such that rP|U » P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, P is trivial and the displayed map has trivial kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now assume that F is a U-group and we induct on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' When n “ 1, then F is associated to a vector bundle F on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let j : U ãÑ X denote the open immersion, then for j˚F we apply [GR18, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='17 (iii)] to deduce that RΓZpX, j˚Fq » RΓZ1pX1, j˚Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, we have HipU, Fq „ ÝÑ HipU 1, Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' When n ą 1, we invoke the nonabelian cohomology sequences [Gir71, Chapitre IV, Remarque 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='10] for a central extension to acquire the following commutative diagram with exact rows H1pU, Fn´1q H1pU, Fq H1pU, F{Fn´1q H2pU, Fn´1q H1pU 1, Fn´1q H1pU 1, Fq H1pU 1, F{Fn´1q H2pU 1, Fn´1q „ „ by a diagram chase up to twist technique [Gir71, Chapitre III, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(i)], we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R with spectrum S and generic point η, an S-smooth algebraic space X, and a point x P X that is not any maximal point of S-fibers of X such that dim OX,x ě 2, then pullback induces an equivalence of categories of finite étale covers FÉtSpec OX,x „ ÝÑ FÉtSpec OX,xztxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Further, for a qc open immersion j : U ãÑ X such that every z P XzU satisfies the condition for x, FÉtX Ñ FÉtU is essentially surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If x R Xη and dim OXs,x “ 1, then the assertion is due to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The remained case is proved below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To show that FÉtX Ñ FÉtU is essentially surjective, let f : rU Ñ U be a finite étale cover and we use Noetherian induction to reduce to showing that the finite étale cover f extends to an open subset of U strictly containing U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Pick a maximal point of XzU so U ˝ :“ U ˆX Spec OX,x “ Spec OX,xztxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Restricting f over U ˝ to f ˝ : rU ˝ Ñ U ˝, the equivalence FÉtSpec OX,x „ ÝÑ FÉtSpec OX,xztxu yields a finite étale cover W Ñ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A spreading out [SP, 0BQ5, 0EY3] provides an open neighborhood x P U 1 Ą U with a finite étale cover W 1 Ñ U 1 extending rU Ñ U, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let S be a semilocal affine geometrically unibranched scheme with total ring of fractions K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For an étale locally constant group scheme E over S of finite type3, the map H1 ´etpS, Eq ãÑ H1 ´etpK, Eq has trivial kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let T be an E-torsor that trivializes over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This signifies that T pKq ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since S is geometrically unibranched, by [SGA 3II, Exposé X, Théorème 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='16] (the Noetherian assumption is 3This means that after a finite étale covering, the constant group is a finite type abelian group, see [SGA 3II, X, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1] 21 removable), E is isotrivial, so there is a finite étale covering S1 Ñ S with total ring of fractions K1 such that ES1 is a constant group in finite type abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, we have the commuative diagram T pSq T pS1q T pS1 ˆ S1q T pKq T pK1q T pK1 ˆ K1q so descent yields the equality T pSq “ T pKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (If S is the spectrum of a Prüfer domain and E is S-finite, then this is simplier by valuative criterion for properness) In particular, we have T pSq ‰ H so T is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a valuation ring V with fraction field K, every reductive K-group scheme G has at most one reductive V -model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To see this, we let G be a reductive V -model of G and consider the commutative diagram with exact rows H0 ´etpV, OutpGqq H1 ´etpV, Gadq H1 ´etpV, AutpGqq H1 ´etpV, OutpGqq H0 ´etpK, OutpGqq H1 ´etpK, Gadq H1 ´etpK, AutpGqq H1 ´etpK, OutpGqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' f0 f1 f2 f3 The map f1 is injective by [Guo20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By diagram chase, f2 has trivial kernel, so we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Extending generically trivial torsors Granted the purity Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, we extend reductive torsors outside a closed subset of higher codi- mension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal affine Prüfer scheme S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' an S-flat finite type scheme X with regular S-fibers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' a closed subset Z Ă X such that XzZ Ă X is quasi-compact and satisfies the following condition codimpZη,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Xηq ě 2 for each generic point η P S and codimpZs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Xsq ě 1 for all s P S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and a reductive X-group scheme G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' there is a closed subset Z1 Ă Z satisfying the following condition codimpZ1 η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Xηq ě 3 for each generic point η P S and codimpZ1 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Xsq ě 2 for all s P S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' such that every G-torsor on XzZ extends to a G-torsor on XzZ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Write R “ colimλPΛRλ as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By a standard limit argument ([SP, 0EY1, 0C0C]), for large enough λ P Λ, the scheme X, the open XzZ Ă X, and the reductive X-group scheme G descend to a quasi-compact quasi-separated Rλ-smooth scheme Xλ, a quasi-compact open pXzZqλ Ă Xλ, and a reductive Xλ-group scheme Gλ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Also, up to enlarging λ, the G-torsor over XzZ in question descends to a Gλ-torsor over pXzZqλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6 that descends the fiberwise codimension of Z, we are reduced to the case when all local rings of R are valuation rings of finite ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let PXzZ be a G-torsor over XzZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since S has finitely many points and each fiber Xs is Noetherian, there are finitely many points x P Z satisfying one of the assumptions (i)-(ii) of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' among these points we pick a maximal one under the generalization, say x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The maximality of x implies that pXzZqXSpecpOX,xq “ SpecpOX,xqztxu, so, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, the G-torsor PXzZ|XzZXSpecpOX,xq extends to a G-torsor Px over SpecpOX,xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As X is topologically Noetherian, we may spread out Px to obtain a G-torsor PUx over an open neighbourhood Ux of x such that PXzZ|pXzZqXUx » PUx|pXzZqXUx as G-torsors over pXzZqXUx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, we may glue PXzZ and PUx to obtain a G-torsor over U1 :“ pXzZqYUx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since Z1 :“ XzU1 contains strictly fewer points x satisfying the assumptions (i) or (ii) of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, we extend P iteratively to find the desired closed subset Z1 Ă X such that PXzZ extends over XzZ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer affine scheme S, an S-flat finite type scheme X with regular S-fibers, finitely many points x Ă X contained in an affine open, a nonzero element r P OX,x, and a reductive X-group scheme G, every generically trivial G-torsor over OX,xr 1 rs extends to a G-torsor over an open neighbourhood U of SpecpOX,xr 1 rsq whose complementary closed Z :“ XzU satisfies the following codimpZη, Xηq ě 3 for each generic point η P S and codimpZs, Xsq ě 2 for all s P S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 22 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As in the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, we may assume that S has finite Krull dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' in particular, X is topologically Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let P be a generically trivial G-torsor over OX,xr 1 rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By spreading out, P extends to a G-torsor PU over U :“ Spec Rr 1 rs for a subring R Ă OX,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to extend U and PU to ensure that Z :“ XzU satisfies the assumptions of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let z P Z be such that either (i) z P Xη and dim OX,z “ 1, in which case SpecpOX,zq X U is a maximal point of X, or (ii) z is a maximal point of Xs with s ‰ η, in which case SpecpOX,zq, and hence also SpecpOX,zq X U, is the spectrum of a valuation ring (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(iii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the Grothendieck–Serre over valuation rings [Guo20], the generically trivial G-torsor PU|SpecpOX,zqXU is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus, as in the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, we can glue PU with the trivial G-torsor over a small enough open neighbourhood of z to extend PU across such a point z P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Note that Z contains finitely many points z satisfying the above assumption (i) or (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, iteratively extend U and PU, we may assume that Z does not contain any point z satisfying (i) or (ii), when Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Purity for finite locally free torsors and the Zariski–Nagata With the purity for reflexive sheaves and Auslander’s flatness criterion Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 in hand, we obtain the purity theorem for finite locally free torsors and establish our non-Noetherian Zariski–Nagata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (Purity for finite locally free groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) For a semilocal affine Prüfer scheme S, an S-smooth algebraic space X, an X-finite locally free group algebraic space G, and a closed subset Z Ă X such that j : XzZ ãÑ X is quasi-compact and codimpZη, Xηq ě 2 for each generic point η P S and codimpZs, Xsq ě 1 for all s P S, the restriction functor induces the following equivalence of categories of G-torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' TorspXfppf, Gq „ ÝÑ TorsppXzZqfppf, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, passing to isomorphism classes of objects, we have the following isomorphism H1 fppfpX, Gq » H1 fppfpXzZ, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) For a finite-rank valuation ring R with spectrum S, an S-smooth scheme X, an X-finite locally free group scheme G, and a point x P X that is not a maximal point of S-fibers of X such that dim OX,x ě 2, the restriction functor induces the following equivalence of category of G-torsors TorsppSpec OX,xqfppf, Gq „ ÝÑ TorsppSpec OX,xztxuqfppf, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, passing to isomorphism classes of objects, we have the following isomorphism H1 fppfpSpec OX,x, Gq » H1 fppfpSpec OX,xztxu, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) We simply verify the assumptions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By considering the space IsomX of isomor- phisms of two torsors (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4), we deduce from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii) that the restriction functor is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The same holds when we base change to every étale X-scheme over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Next, we show that, étale locally on X, any G-torsor on XzZ extends to a G-torsor over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For this, we may assume that X is an affine scheme and S is the spectrum of a valuation ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By a standard limit argument involving Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, we reduce to the case when S has finite Krull dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since every R-fiber of X is Noetherian and S has finitely many points, X is topologically Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let P be a GXzZ-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then j˚OP by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(iv) is a reflexive OX-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' First, we prove the OX-flatness of j˚OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since X is topologically Noetherian, we use Noetherian induction to reduce to the case when X is local and essentially smooth over R and Z “ txu is its closed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, our Auslander’s flatness criterion Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 reduces us to showing that the following is an isomorphism HomOXpj˚OP, j˚OPq » pj˚OPq‘r , where r “ rankOXOG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Note that in such local case, we have OG » O‘r X , consider the following map of reflexive OX-modules HomOXpOG, j˚OPq Ñ HomOXpj˚OP, j˚OPq, f ÞÑ ´ j˚OP j˚ρ ÝÝÑ OG bOX j˚OP pf,idq ÝÝÝÑ j˚OP ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 23 This is an isomorphism: by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii), it suffices to argue over XzZ, then its explicit inverse is g ÞÑ ´ OGXzZ idb1 ÝÝÝÑ OGXzZ bOXzZ OP pρ,idq´1 ÝÝÝÝÝÑ OP bOXzZ OP pg,idq ÝÝÝÑ OP ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, we prove that the G-torsor structure of P extends uniquely to that of SpecXpj˚OPq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As G is finite locally free, by projection formula [SP, 01E8], taking j˚ of the co-action ρ : OP Ñ j˚OG bOXzZ OP yields j˚ρ: j˚OP Ñ OG bOX j˚OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To check that j˚ρ is a co-action, we verify the associativity, the commutativity of the following diagram j˚OP OG bOX j˚OP OG bOX j˚OP OG bOX OG bOX j˚OP, j˚pρq j˚pρq idbj˚pρq µGbid where µG : OG Ñ OG bOX OG is the co-multiplication of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since all sheaves involved are OX-reflexive, the commutativity over XzZ by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii) extends over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Finally, the following map pj˚ρ, 1 b idq: j˚OP bOX j˚OP Ñ OG bOX j˚OP, by the OX-flatness of j˚OP and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii), is an isomorphism since so is its restriction on XzZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) This can be proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For instance, for the essential surjectivity of the restriction functor, the finite rank assumption on V guarantees j : Spec OX,xztxu ãÑ Spec OX,x to be quasi-compact quasi- separated, and so j˚OP is a reflexive OX,x-module (by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii)) for any G-torsor P over Spec OX,xztxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then one uses Auslander’s flatness criterion Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 to show that j˚OP is OX,x-free and inherits the G-torsor structure on P, giving the desired extension of P to Spec OX,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 (Zariski–Nagata: purity for finite étale covers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) For a semilocal affine Prüfer scheme S, an S-smooth algebraic space X, and a closed subset Z Ă X such that XzZ ãÑ X is quasi-compact and satisfies the following condition codimpZη, Xηq ě 2 for each generic point η P S and codimpZs, Xsq ě 1 for all s P S, the pullback functor induces the following equivalence between categories of finite étale covers FÉtX „ ÝÑ FÉtXzZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, for every geometric point x: Spec Ω Ñ XzZ with a separably closed field Ω, the map π´et 1 pXzZ, xq Ñ π´et 1 pX, xq is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) For a finite-rank valuation ring R with spectrum S and generic point η P S, an S-smooth scheme X, and a point which is either x P Xη with dim OXη,x “ 2, or x P Xs with s ‰ η and dim OXs,x “ 1, the pullback functor induces the following equivalence of categories of finite étale covers FÉtSpec OX,x „ ÝÑ FÉtSpec OX,xztxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) Full faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For two finite étale covers πi : Xi Ñ X, i “ 1, 2, consider the X-functor Y :“ HomXpX1, X2q that parameterizes X-morphisms from X1 to X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' it is a subfunctor of HomXpπ2,˚OX2, π1,˚OX1q con- sisting of sections compatible with algebraic structures of π2,˚OX2 and π1,˚OX1, which amount to the commutativity of a certain diagram of OX-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' So Y Ă HomXpπ2,˚OX2, π1,˚OX1q is a closed sub- functor Zariski-locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, Y is an algebraic space finite over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (Using the infinitesimal criterion for formal smoothness, one can check that Y Ñ X is even finite étale, but we will not need this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=') By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii), we have Y pXq » Y pXzZq, thereby proving the full faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Essential surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let V Ñ XzZ be a finite étale cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We need to show that it extends to a finite étale cover of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the full faithfulness, we may use glueing in the étale topology to reduce to the case that X is an affine scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the S-smoothness of X, X and also V is normal, so, by breaking X and V into connected components, we may assume that both X and V are integral schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let rV Ñ XzZ be 24 a connected finite étale Galois cover dominating V Ñ XzZ, say with Galois group G :“ GalprV {pXzZq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let H :“ GalprV {V q Ă G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(i), the G-Galois cover rV Ñ XzZ extends (uniquely) to a G- Galois cover ˜W Ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Grothendieck–Galois correspondence, the subcover Ă W{H Ñ X corresponding to the subgroup H Ă G is a finite étale cover that extends V Ñ XzZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) This is proved in the same way as (i), using Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(ii) in place of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Geometric lemmata for the Grothendieck–Serre 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Geometric presentation lemma over Prüfer bases In both of the works of Fedorov and ˇCesnaviˇcius on mixed charateristic Grothendieck–Serre, a certain type geometric results in the style of Gabber-Quillen play a prominent role, see [Fed22b, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='18] and [Čes22a, Variant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This is also true in our context, and we begin with an analog of [Čes22a, Variant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let R be a semilocal Prüfer ring, X a projective, flat R-scheme with fibers of pure dimension d ą 0, OXp1q a R-ample line bundle on X, W Ă Xsm an open, x Ă W finitely many points, and Y Ă X a closed subscheme that is R-fiberwise of codimension ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Upon replacing OXp1q by any large power, there exists nonzero h0 P ΓpX, OXp1qq, h1 P ΓpX, OXpw1qq, ¨ ¨ ¨ , hd´1 P ΓpX, OXpwd´1qq with w1, ¨ ¨ ¨ , wd´1 ą 0, such that (i) the hypersurface H0 :“ V ph0q Ă X is disjoint from x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) the hypersurfaces Hi :“ V phiq Ă X satisfy Y X H0 X ¨ ¨ ¨ X Hd´1 “ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iii) in the following commutative diagram with vertical maps determined by the h0, ¨ ¨ ¨ , hd´1: XzH0 XzpH0 X ¨ ¨ ¨ X Hd´1q X :“ BlXph0, ¨ ¨ ¨ , hd´1q Ad´1 R PRp1, w1, ¨ ¨ ¨ , wd´1q PRp1, w1, ¨ ¨ ¨ , wd´1q, π π π the map π is smooth of relative dimension 1 at x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iv) we have Y X H0 X π´1pπpxqq “ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (v) if Y zXsm is R-fiberwise of codimension ě 2 in X, then π is smooth at Y X π´1pπpxqq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (vi) if Y zW is R-fiberwise of codimension ě 2 in X, then pY zWq X π´1pπpxqq “ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (vii) if Y zW is R-fiberwise of codimension ě 2 in X, then there are affine opens S Ă Ad´1 R and x Ă U Ă W X π´1pSq Ă XzH0 such that π : U Ñ S is smooth of relative dimension 1 and Y X U “ Y X π´1pSq is S-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This can be proved similarly as [Čes22a, Variant 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A variant of Lindel’s lemma According to a lemma of Lindel [Lin81, Proposition 1 et seq Lemma], an étale extension of local rings A Ñ B with trivial extension of residue fields automatically induces isomorphisms A{rnA „ ÝÑ B{rnB, where n ě 1, for a well-chosen non-unit r P A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In our context in which the prescribed B is essentially smooth over a valuation ring, we will prove the following variant of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' by allowing to fix the r P B in advance, at the cost of that A is a carefully-chosen local ring of an affine space over that valuation ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This result will be the key geometric input for dealing with torsors under a reductive group scheme that descends to the Prüfer base ring, and, as the cited work of Lindel on the Bass–Quillen conjecture for vector bundles, it reduces us to studying torsors on opens of affine spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 25 Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let Λ be a semilocal Prüfer domain, X an irreducible, Λ-smooth affine scheme of pure relative dimension d ą 0, Y Ă X a finitely presented closed subscheme that avoids all the maximal points of the Λ-fibers of X, and x Ă X a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume that for every maximal ideal m Ă Λ with finite residue field, there are at most maxp# κpmq, dq ´ 1 points of x lying over m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' There are an affine open neighbourhood W Ă X of x, an affine open subscheme U Ă Ad Λ, and an étale surjective Λ-morphism f : W Ñ U such that the restriction f|WXY : W X Y Ñ U is a closed immersion and f induces a Cartesian square: W X Y W W X Y U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' f Moreover, if Y is a Cartier divisor on X, then W X Y is a Cartier divisor on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The assumption on the cardinality of x holds, for instance, either if x is a singleton or if d ą # x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The latter will be critical to settle the general semilocal case of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' On the other hand, the following finite field obstruction shows a certain assumption on #x is necessary: if d “ 1 and Λ “ k is a finite field, then the map f delivered from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 gives a closed immersion x ãÑ A1 k, which is impossible as soon as # x ą # k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To prove Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 we begin with the following reduction: Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 reduces to the case when x consists of closed points of the closed Λ-fibers of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As an initial step, by a standard limit argument involving Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, we can reduce to the case when SpecpΛq has a finite underlying space (which we will assume from now on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If for each x P x the closure txu contains a closed point x1 of the closed Λ-fibers of X and if the new collection tx1 : x P xu satisfies the same cardinality assumption on x, we can simply replace each x by x1 to complete the reduction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' However, it may happen that txu does not contain any point of the closed Λ-fibers of X, and even if it does, the new collection tx1 : x P xu may destroy the cardinality assumption on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To overcome this difficulty, we will use a trick by adding auxiliary primes to SpecpΛq (and adding the corresponding fibers to X and Y ) so that txu contains closed points of the closed Λ-fibers of X for all x P x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' More precisely, we will show that there are a semilocal Prüfer domain Λ1, an open embedding SpecpΛq Ă SpecpΛ1q, an irreducible, affine, Λ1-smooth scheme X1 of pure relative dimension d, a closed Λ1-subscheme Y 1 Ă X1 that avoids all the maximal points of the Λ1-fibers of X1, and a Λ-isomorphism X1 Λ » X that identifies Y 1 Λ with Y such that the assumptions of the first sentence of this paragraph hold for our new X1 and Y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To construct the desired Λ1 (and X1, Y 1), we can first use the specialization technique to reduce to the case when all points of x are closed in the corresponding Λ-fibers of X, that is, if x P x lies over p Ă Λ, then x is κppq-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the rest of proof we will assume, without lose of generality, that there is exactly one point of x, say x, that lies over some non-maximal prime of Λ, say p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Write Λp “ Ť A as a filtered union of its finitely generated Z-subalgebras A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By a standard limit argument ([SP, 0EY1, 0C0C]), for large enough A, (a) XΛp descends to an irreducible, affine, A-smooth scheme X of pure relative dimension d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (b) the finitely presented closed subscheme YΛp Ă XΛp descends to a closed A-subscheme Y Ă X which, upon enlarging A, avoids all the maximal points of the A-fibers of X: by [EGA IV3, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1], the subset ts P SpecpA : dim Ys “ du Ă SpecpA is constructible, and its pullback to SpecpΛpq “ limA SpecpA is empty, hence after enlarging A we can assume that it is already empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (c) the κppq-finite point x descends to a A{pA-finite closed subscheme rx Ă XA{pA, where pA :“ AXp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For any prime Λ Ą q Ą p with htpqq “ htppq ` 1, choose an element aq P qzp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We assume that (d) a´1 q P A for all such q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (This guarantees the equality A ¨ Λm “ Λp for every maximal ideal m Ă Λ containing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=') 26 Since a maximal ideal m Ă Λ containing p gives rise to a non-trivial valuation ring Λm{pΛm of κppq, the field κppq is not finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As κppq “ Ť A A{pA, by enlarging A we may assume that A{pA is also not a finite field, and therefore we can find a nonzero prime p1 Ă A{pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (We have used the following fact: for a finite type Z-algebra, a prime ideal is maximal if and only if its residue field is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=') Choose a valuation ring of κppAq with center p1 in A{pA, and then extend it to a valuation ring Vp1 of κppq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let V be the composite of Λp and Vp1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' explicitly, V is the preimage of Vp1 under the reduction map Λp ։ κppq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then V is a valuation ring of FracpΛq, and, by the above assumption (d), the equality V ¨ Λm “ Λp holds for any maximal ideal m Ă Λ containing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, by [BouAC, VI, §7, Proposition 1-2], Λ1 :“ Λ X V is a semilocal Prüfer domain whose spectrum is obtained by glueing SpecpΛq with SpecpV q along their common open SpecpΛpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, we may glue X with XV along XΛp to extend X to an irreducible, affine, Λ1-smooth scheme X1 of pure relative dimension d, with a closed Λ1-subscheme Y 1 Ă X1 obtained by glueing Y with YV along YΛp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' by construction, Y 1 avoids all the maximal points of the Λ1-fibers of X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since the closed subscheme rxV Ă XV is V -finite, we may specialize x to a point of rxV Ă X1 that lies over the closed point of SpecpV q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, by replacing Λ by Λ1, X by X1 and Y by Y 1, we can reduce to the already treated case when all points of x specialize to closed points of the closed Λ-fibers of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Henceforth, we may assume that x consists of closed points of the closed Λ-fibers of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, since the relative dimension of X{Λ is d ą 0, the closed subset x Ť Y does not contain any maximal points of the R-fibers of X, and so, by prime avoidance, there is an a P ΓpX, OXq such that a vanishes on x Ť Y but does not vanish at any maximal points of Λ-fibers of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since for the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 we are free to enlarge Y to a closed subscheme of X that still avoids all the maximal points of the Λ-fibers of X, by replacing Y by V paq Ă X, we reduce to the case ‚ x consists of closed points of the closed Λ-fibers of X, and ‚ x Ă Y “ V paq for some a P ΓpX, OXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the rest of the proof we will assume this throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a field k, an affine k-variety X, a closed subscheme Y Ă X of pure dimension e ą 0, a finite subset of closed points x Ă Y X Xsm, and an arbitrary element ptpxqq P ś xPx κpxq, there is a morphism h : X Ñ A1 k that is smooth at x such that h|Y has fiber dimension e ´ 1 and such that hpxq “ tpxq for every x P x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Choose a finite subset of closed points y Ă Y that is disjoint from x and that contains precisely 1 point of every irreducible component of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For every integer n ą 0 denote by xpnq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', ypnq) the nth infinitesimal neighbourhood of x (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', y) in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let hx P H0pxp1q, Oxp1qq be such that hxpxq “ tpxq and dhxpxq ‰ 0 P mx{m2 x for every x P x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) By prime avoidance, there exists a hy P H0pX, OXq whose restriction to every irreducible component of Yred is not identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the faithfully flatness of OYred,y “ ź yPy OYred,y Ñ ź yPy { OYred,y “ lim n H0pypnq X Yred, OypnqXYredq, we see that for large enough n, the restriction of hy to every component of ypnq X Yred is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let h P H0pX, OXq be any element whose restriction to xp1q is hx and whose restriction to ypnq is congruent to hy for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since X is smooth at x, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) implies that the morphism h : X Ñ A1 k (obtained by sending the standard coordinate of A1 k to h) is smooth at x and hpxq “ tpxq for every x P x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since the restriction of h to every irreducible component of ypnq X Yred and hence also to Yred is nonzero, the morphism h is non-constant on every irreducible component of Y , so h|Y has fiber dimension e ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' There exists a Λ-morphism g : X Ñ Ad´1 Λ such that (i) it smooth of relative dimension 1 at x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) the restriction g|Y is quasi-finite at x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and (iii) for x P x lying over m, one has κpmq “ κpgpxqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 27 In addition, if d ą #px X Xκpmqq for every maximal ideal m Ă Λ with finite residue field, then we may find such a g under which all points of x have pairwise distinct images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We first reduce the lemma to the case when Λ “ k is a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume that for every maximal ideal m Ă Λ there exists a κpmq-morphism gm : Xκpmq Ñ Ad´1 κpmq that is smooth at x X Xκpmq such that the restriction gm|Yκpmq is quasi-finite at x X Xκpmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We then use Chinese remainder theorem to lift the maps tgmum simultaneously to obtain a Λ-morphism g : X Ñ Ad´1 Λ which would verify the first assertion of the lemma: only the flatness of g at x need to be checked, but this follows from the fibral criterion of flatness [EGA IV3, Théorème 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In addition, if all points of x X Xκpmq have pairwise distinct images under gm, then the resulting morphism g verifies the second assertion of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In case Λ “ k being a field, our assumptions become that X is a k-smooth affine variety of pure dimension d ą 0 and Y “ V paq is a closed k-subvariety of pure codimension 1 that contains x, and, for the second assertion, our assumption becomes that d ą # x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a collection of maps t1, ¨ ¨ ¨ , td´1 : x Ñ k, taking products yields maps pt1, ¨ ¨ ¨ , tiq : x Ñ Ai kpkq “ ki for 1 ď i ď d ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We now apply Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 inductively to show: Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For 1 ď i ď d ´ 1, there exists a k-morphism gi : X Ñ Ai k such that ‚ gi is smooth at x with gi|x “ pt1, ¨ ¨ ¨ , tiq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and ‚ every irreducible component of gi|´1 Y pgipxqq intersecting x has dimension d ´ 1 ´ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume the morphism gi´1 has been constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We apply Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4, with k being the ring k1 of global sections of gi´1pxq here, X being g´1 i´1pgi´1pxqq here, Y being the union Y 1 of all the irreducible components of gi´1|´1 Y pgi´1pxqq meeting x here, and t being ti|k1, to obtain a k1- morphism h : g´1 i´1pgi´1pxqq Ñ A1 k1 that is smooth at x such that h|Y 1 has fiber dimension d ´ 1 ´ i and such that h|x “ ti|k1, where ti|k1 : x ti ÝÑ k Ñ k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to take gi :“ pgi´1,rhq : X Ñ Ai k “ Ai´1 k ˆk A1 k for any lifting rh P H0pX, OXq of h P H0 ´ g´1 i´1pgi´1pxqq, Og´1 i´1pgi´1pxqq ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Starting from a map pt1, ¨ ¨ ¨ , td´1q : x Ñ kd´1, the map g :“ gd´1 of the Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 immediately settles the first assertion of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the second assertion, it suffices to note that, under the stated assumption, there always exists an injection x ãÑ kd´1: for an infinite field k, kd´1 is infinite, and, for a finite field k, # kd´1 ě d ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Consider the map pg, aq : X Ñ Ad Λ “ Ad´1 Λ ˆΛ A1 Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By construction, it is quasi-finite at x, and, by the openness of the quasi-finite locus of a finite type morphism, we may shrinking X if needed to assume that it is already quasi-finite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' since the generic Λ-fibers of its domain and codomain are irreducible varieties of the same dimension d, it is also dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, by Zariski’s main theorem [EGA IV4, Corollaire 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='13], pg, aq factors as X jÝÑ X h1 ÝÑ Ad Λ, where X is an integral affine scheme, j is an open immersion, and h1 is finite, dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (Unless Λ is a DVR, ΓpX, OXq is, in general, only a finite type Λ-subalgebra of the integral closure of Λrt1, ¨ ¨ ¨ , tds in the function field of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=') Denote g :“ pr1 ˝ h1, where pr1 : Ad Λ Ñ Ad´1 Λ is the projection onto the first pd ´ 1q-coordinates, and let a P ΓpX, OXq be the pullback of the last standard coordinate of Ad Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then h1 “ pg, aq, and g (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', a) restricts to g (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', a) on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In what follows, we shall identify the points of jpxq with the corresponding points of x via j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Write S Ă Spec Λ for the union of the closed points of Spec Λ (with the reduced structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' There exists an element b P ΓpX, OXq such that the morphism h2 :“ pg, bq : X Ñ Ad Λ “ Ad´1 Λ ˆΛ A1 Λ has the following properties: 28 (i) set-theoretically we have h´1 1 ph1pxqq X h´1 2 ph2pxqq “ x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) h2 is étale around x and induces a bijection x „ ÝÑ h2pxq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and (iii) h2 induces an isomorphism of residue fields κph2pxqq „ ÝÑ κpxq for every x P x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since h1 is finite, surjective, g´1pgpxqq is an S-curve that contains g´1pgpxqq as an open subcurve, so it is S-smooth around x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a point x P x lying over a maximal ideal m Ă Λ, its first infinitesimal neighbourhood in g´1pgpxqq is isomorphic to Specpκpxqruxs{pu2 xqq, where ux is an uniformizer of g´1pgpxqq at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Recall the fact that the residue field of a point on a smooth curve over a field is a simple extension of that field, see [Čes22a, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It follows that, for x P x lying over m, there exists a closed κpmq-immersion xp1q ãÑ A1 κpmq “ A1 gpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a maximal ideal m Ă Λ with finite residue field, under our assumption that #px XXκpmqq ă maxp# κpmq, dq, either x contains at most # κpmq´1 points lying over m or the fiber of gκpmq contains at most 1 point of x (Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, we may arrange the above immersions so that they jointly give a closed immersion over Ad´1 Λ : ğ xPx xp1q ãÑ A1 gpxq Ă A1 Ad´1 Λ “ Ad Λ, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) where we regard gpxq Ă Ad´1 Λ as a closed subscheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Note that the complement of the image of the morphism (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) in Ad Λ has at least 1 rational point Ad´1 Λ fiberwisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus, by sending any y P ph´1 1 ph1pxqqzxq to a suitable rational point of A1 gpyq, we may further extend (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) to a Ad´1 Λ morphism u: Z :“ `Ů xPx xp1q˘ Ů ´Ů yPh´1 1 ph1pxqqzx y ¯ Ñ Ad Λ such that upxq X uph´1 1 ph1pxqqzxq “ H, or, what amounts to the same, h´1 1 ph1pxqq X u´1pupxqq “ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2) As Z is a closed subscheme of the affine scheme X, we can lift u˚ptq P ΓpZ, OZq to obtain an element b P ΓpX, OXq, where t is the standard coordinate on A1 Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consider the morphism h2 :“ pg, bq : X Ñ Ad Λ “ Ad´1 Λ ˆΛ A1 Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Viewing X as a Ad´1 Λ scheme via g, the base change of h2 to gpxq Ă Ad´1 Λ restricts to u on Z, so h2 is unramified at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now (i) follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2), (iii) is a consequence of our choice of the morphism (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (ii), it suffices to argue that h2 is flat at x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' however, since the domain and the codomain of h2 are Λ-flat of finite presentation, the fibral criterion of flatness [EGA IV3, Théorème 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='10] reduces us to checking the flatness of the Λ-fibers of h2 at x, while the latter follows from the flatness criterion [EGA IV2, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Let Λrh˚ 1pt1q, ¨ ¨ ¨ , h˚ 1ptd´1q, a, bs Ă ΓpX, OXq be the Λ-subalgebra generated by a, b and h˚ 2ptiqp“ h˚ 1ptiq “ g˚ptiqq for 1 ď i ď d ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We introduce the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' ‚ Let V :“ SpecpΛrh˚ 1pt1q, ¨ ¨ ¨ , h˚ 1ptd´1q, a, bsq, and let h3 : X Ñ V be the morphism induced by the inclusion Λrh˚ 1pt1q, ¨ ¨ ¨ , h˚ 1ptd´1q, a, bs Ă ΓpX, OXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' ‚ Let v1 : V Ñ Ad Λ be the map such that v˚ 1 ptiq “ h˚ 1ptiq for 1 ď i ď d ´ 1 and v˚ 1 ptdq “ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' ‚ Let v2 : V Ñ Ad Λ be the map such that v˚ 2 ptiq “ h˚ 1ptiq for 1 ď i ď d ´ 1 and v˚ 2 ptdq “ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Note that there is a natural surjection Λrh˚ 1pt1q, ¨ ¨ ¨ , h˚ 1ptd´1q, bs ։ Λrh˚ 1pt1q, ¨ ¨ ¨ , h˚ 1ptd´1q, a, bs{paq “ ΓpV, OV q{paq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' this implies that v2 induces a closed immersion v2 : SpecpΓpV, OV q{paqq ãÑ V v2 ÝÑ Ad Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 29 We have the following commutative diagram of morphisms of affine schemes: X X V Ad Λ Ad Λ j h2 h1 h3 v2 v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The map h3 induces a bijection x „ ÝÑ h3pxq with h´1 3 ph3pxqq “ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Further, h3 induces an isomorphism of semilocal rings OV,h3pxq » OX,x “ OX,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6(ii)-(iii), we see that h3 induces a bijection x „ ÝÑ h3pxq and an isomorphism of residue fields κph3pxqq „ ÝÑ κpxq for every x P x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Chasing the above diagram we see that h´1 3 ph3pxqq Ă h´1 1 ph1pxqq X h´1 2 ph2pxqq “ x, where the last equality is Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As h3 is finite, surjective, we see that h´1 3 ph3pxqq “ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6(ii), h3 is unramified at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It follows that the base change of h3 to Spec OV,h3pxq is Spec OX,x Ñ Spec OV,h3pxq, and it is actually an isomorphism: letting J be the Jacobson radical of the semilocal ring OV,h3pxq, since the natural map ź xPx κph3pxqq » OV,h3pxq{J h˚ 3 ÝÝÑ OX,x{JOX,x » ź xPx κpxq is an isomorphism (in particular, surjective), an application of Nakayama lemma shows h˚ 3 : OV,h3pxq » OX,x “ OX,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ End of the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Define f :“ h2 ˝ j : X Ñ Ad Λ, which we may assume to be étale upon replacing X by an affine open neighbourhood of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7, there exists an affine open neighbourhood W 1 0 Ă V of h3pxq such that W0 :“ h´1 3 pW0q Ă jpXq and h3|W0 : W0 „ ÝÑ W 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We shall identify W0 as an open subscheme of X via j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As noted above, v2 induces a closed immersion v2 : Y 1 :“ SpecpΓpV, OV q{paqq ãÑ Ad Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, the topology of Y 1 is induced from that of Ad Λ via v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Note also that, since a vanishes on x, h3pxq Ă Y 1 Ă V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, there exists an affine open neighbourhood U Ă Ad Λ of fpxq “ v2ph3pxqq such that v´1 2 pUq Ă W 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, f induces a closed immersion of affine schemes YU :“ f ´1pUq X Y “ ph3 ˝ jq´1pv´1 2 pUq X Y 1q “ ph3 ˝ jq´1pv´1 2 pUqq » v´1 2 pUq ãÑ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since f is separated and étale, any section of f ˆAd Λ,f YU, such as the one induced by the inclusion YU ãÑ X, is an inclusion of a clopen, so X ˆAd Λ,f YU “ rY1 \\ rY2 with rY1 „ ÝÑ YU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let W Ă f ´1pUq be an affine open whose preimage in X ˆAd Λ,f YU is rY1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then f|W : W Ñ U is an étale morphism such that f|WXY : W XY ãÑ U is a closed immersion and such that W ˆU,f pW XY q „ ÝÑ W XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As any étale map is open, we may shrink U around fpxq to ensure that f|W : W Ñ U is also surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This proves the first assertion of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The second assertion follows from descent theory, because the ideal sheaf of W X Y on U pulls back to that of W X Y on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Cohomology of groups of multiplicative type Inspired by the purity results in [ČS21, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8], we investigate the parafactoriality over Prüfer bases and then present the purity for cohomology of group schemes of multiplicative type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 30 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Geometrically parafactorial pairs 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Parafactorial pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let pX, OXq be a ringed space with a closed subspace Z Ă X and open immersion j : XzZ ãÑ X, if for every open subspace U Ă X the restriction Pic X „ ÝÑ Pic XzZ, L ÞÑ L |UXpXzZq is an equivalence of categories, then the pair pX, Zq is parafactorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, we have L » j˚j˚L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A local ring A is parafactorial if the pair pSpec A, Spec A{mAq is parafactorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We list several parafactorial pairs pX, Zq and local rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) when A is a Noetherian factorial local ring, by [EGA IV4, Exemples 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='9 (ii)], it is parafactorial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) by [EGA IV4, Proposition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8], a local ring A is parafactorial if and only if Pic pSpec Aztxuq “ 0 and A » ΓpSpec Aztxu, rAq for the closed point x P Spec A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iii) when X is a locally Noetherian and locally complete intersection and Z satisfies codimpZ, Xq ě 4, by [SGA 2new, Exposé XI, Théorème 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='13 (ii)], the pair pX, Zq is parafactorial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iv) for a normal scheme S, an S-smooth scheme X and a closed subset Z Ă X satisfying codimpZη, Xηq ě 2 for each generic point η P S and codimpZs, Xsq ě 1 for every s P S, by [EGA IV4, Proposition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3], the pair pX, Zq is parafactorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now we assume that X is a scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A parafactorial pair pX, Zq is geometrically parafactorial, if for every X-étale X1 with the base change Z1 :“ ZˆX X1, the pair pX1, Z1q is parafactorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a local ring A, if its strict Henselization Ash is parafactorial, then A is geometrically parafactorial (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [ČS21, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a topologically locally Noetherian scheme X and a closed subscheme Z Ă X, (i) the pair pX, Zq is parafactorial if and only if OX,z is parafactorial for every z P Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) the pair pX, Zq is geometrically parafactorial if and only if Osh X,z is parafactorial for every z P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The assertion (ii) follows the same argument of (i), except viewing Osh X,z as the inverse limit of étale neighborhoods of z P X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume that pX, Zq is parafactorial and for each z P Z, denote Uz :“ Spec OX,z and U ˝ z :“ Uzztzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To show that OX,z is parafactorial, we prove that every invertible OUz-module L0 is isomorphic to OU˝ z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then by [EGA IV3, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='13] and [EGA I, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2], U ˝ z is the inverse limit of B˝ :“ BzpB X tzuq where B ranges over all open affine neighborhoods of z P X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since every B˝ is topologically Noetherian and separated, by a limit argument [SP, 0B8W], there exists an open affine neighborhood B of z P X and an invertible OB˝-module LB˝ such that L0 » LB˝|U˝ z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By assumption and [EGA IV4, Corollaire 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6 (i)(ii)], the pair pB, B Xtzuq is parafactorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, there exists an invertible OB-module Ă LB such that Ă LB|B˝ » LB˝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Shrinking B if necessary, we have Ă LB » OB hence L0 » OU˝ z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the other side, assume that OX,z are parafactorial for all z P Z, which, combined with [EGA IV4, Proposition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5], reduces us to showing that for every invertible OXzZ-module L , the pushforward j˚L is an invertible OX-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For this, we use Noetherian induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Namely, consider the subset Ω :“ tx P X | j˚L is invertible on an open neighborhood of xu Then Ω Ă X is a non-empty open whose complementary closed is XzΩ “: Y Ă Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [EGA IV2, Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1], the quasi-compact quasi-separated morphism j guarantees that the formation of j˚L commutes with arbitrary flat base changes (in particular, localizations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Pick a maximal point y P Y Ă Z so OX,y is parafactorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The maximality of y P Y implies that Ω X Uy “ U ˝ y, so L0 :“ pj˚L q|U˝ y is an invertible OU˝ y -module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The parafactoriality of OX,y yields an extension of L0 to an invertible OUy- module Ă L0, which, by the limit argument [SP, 0B8W] again, descends to an invertible OW -module Ă LW for an open neighborhood W of y P X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Shrinking W if necessary, we may assume that the restrictions of j˚L and Ă LW on Ω X W are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' With this gluing datum, let Ω1 :“ Ω Y W, so there is an invertible Ω1-module L 1 such that L 1|W “ Ă LW and L 1|Ω “ pj˚L q|Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since XzZ Ă Ω1 and L 1|XzZ “ L , hence OX » j˚OXzZ and pj˚L q|Ω1 » L 1, which leads to a desired contradiction with the definition of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a normal scheme S and an S-scheme X satisfying one of the following 31 (i) either X Ñ S is a smooth morphism of topologically locally Noetherian schemes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' or (ii) S is semilocal Prüfer of finite dimension and X is S-flat locally of finite type with regular S-fibers then every x P X that does not contain any maximal point of S-fibers of X and dim OX,x ě 2 satisfies OX,x is geometrically parafactorial, namely, Osh X,x is parafactorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The parafactoriality of Osh X,x is that of pSpec Osh X,x, txuq, which by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2(ii), is equivalent to the parafactoriality of pSpec OX1,x1, tx1uq for all X-étale X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since all X1 and x1 satisfy the conditions in the statement above ([BS15, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='10 (3)]), we are reduced to showing that OX,x is parafacto- rial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For the Zariski closure Z :“ txu, by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 again, we are reduced to finding a small open neighborhood U of x P X such that pU, Z X Uq is a parafactorial pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now, take an arbitrary open neighborhood U of x P X, by [EGA IV3, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3] applied to Z Ă X, shrinking U, we may assume that U X Z does not contain any irreducible components of S-fibers of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If a z P Z lies over a maximal point η P S, since x specializes to z, then we have dim OXη,z “ dim OX,z ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, we have codimpXη X Z, Xηq ě 2 and by §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(iv) and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii), the desired parafactoriality of pU, Z X Uq follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Purity for groups of multiplicative type Now we study purity for groups of multiplicative type in the situation of higher relative dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We start with the following generalization of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 when G “ M is a X-group algebraic space of multiplicative type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For an algebraic space X with a closed subspace Z Ă X such that for every geometric point z Ñ Z, the strict local ring OX,z is parafactorial, the open immersion j : XzZ ãÑ X and a finite type multiplicative type X-group algebraic space M , the following map between fppf sheaves on X M „ ÝÑ j˚j˚M is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, we have H0 ZpX, M q “ H1 ZpX, M q “ 0 and ΓpX, Pq » ΓpU, Pq for every M -torsor P on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For an M -torsor P, to show that ΓpX, Pq » ΓpU, Pq, it suffices to prove that P » j˚j˚P, which can be checked fppf locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, it suffices to prove the first assertion in the case when X is a scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [SGA 3II, Exposé X, Corollaire 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5], M is quasi-isotrivial, namely, there is an étale surjective morphism r X Ñ X such that M ˆX r X splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We need to show that the morphism M Ñ j˚j˚M is an isomorphism fppf locally at all z P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Suppose f : X1 Ñ X is a flat morphism inducing g: X1zZ1 Ñ XzZ, where Z1 :“ Z ˆX X1 with the open immersion j1 : X1zZ1 ãÑ X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Taking inverse image of M Ñ j˚j˚M , we obtain f ˚M Ñ f ˚j˚j˚M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [EGA IV2, Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1], the formation of j˚p´q commutes with flat base change, hence f ˚j˚j˚M » j1 ˚g˚j˚M “ j1 ˚pj1q˚f ˚M and the inverse image of M Ñ j˚j˚M is f ˚M Ñ j1 ˚pj1q˚f ˚M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We may assume that X1 “ Spec Osh X,z and Z1 “ tzu, so the desired isomorphism is reduced to an isomorphism M „ ÝÑ j1 ˚pj1q˚M for a split finite type multiplicative group sheaf M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For an X1-group µn, we have the following short exact sequence 0 Ñ µn Ñ Gm ˆn Ñ Gm Ñ 0, hence j1 ˚pj1q˚µn “ kerpj1 ˚pj1q˚Gm ˆn Ñ j1 ˚pj1q˚Gmq, reducing us to the case when MX1 “ Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since pX1, Z1q is parafactorial, we have Oˆ X1 „ ÝÑ j1 ˚pj1q˚Oˆ X1, so the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a finite-rank valuation ring R with spectrum S and generic point η P S, an S-flat finite type scheme X with regular S-fibers, a point x P X, and an OX,x-torus T , (1) if either x P Xη with dim OXη,x ě 2, or x P Xs with s ‰ η and dim OXs,x ě 1, then we have Hi txupOX,x, T q “ 0 for 0 ď i ď 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (2) otherwise, OX,x is a valuation ring, then if T is flasque we have H2 txupOX,x, T q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 32 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (1) Notice that the finite-rank assumption on R guarantees X being topologically locally Noe- therian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the local-to-global E2 spectral sequence [SGA 4II, Exposé V, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4], Hp ´etpOX,x, Hq txupT qq ñ Hp`q txu pOX,x, T q, where Hq txupT q is the sheafification of the étale presheaf ´ U hÝÑ SpecpOX,xq ¯ ÞÑ Hq h´1pxqpU, T q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, it suffices to prove the vanishing of the sheaves Hq txupT q for 0 ď q ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We calculate their stalks at a geometric point x lying over x: Hq txupT qx “ Hq txupOsh X,x, T q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now, since TOsh X,x » Gdim T m,Osh X,x, and, since by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3 Osh X,x is parafactorial, we have Hq ´etpSpecpOsh X,xq, T q » Hq ´etpSpecpOsh X,xqztxu, T q for 0 ď q ď 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' as Osh X,x is strictly Henselian, we have H2 ´etpSpecpOsh X,xq, T q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Looking at the local cohomology exact sequence for the pair pSpecpOsh X,xq, ¯xq and T , we see that Hq t¯xupOsh X,x, T q “ 0 for 0 ď q ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This implies Hq txupT q “ 0 for 0 ď q ď 2, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (2) In this case, either x P Xη with dim OXη,x ď 1, then OX,x is a discrete valuation ring, or x is a maximal point of some fiber of X Ñ S, then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(iii), OX,x is a valuation ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The desired vanishing is proven in [Guo20, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [ČS21, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For an algebraic space X, an open subspace U Ă X with complement i : Z :“ XzU ãÑ X, and an abelian sheaf F on pSch{Xqfppf, if for any integer q ě 0, Hq ZpFq denotes the étale-sheafification of the presheaf X1 ÞÑ Hq Z1pX1, Mq where Z1 :“ Z ˆX X1, then we have a convergent spectral sequence Epq 2 “ Hp ´etpX, Hq ZpFqq ñ Hp`q Z pX, Mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [ČS21, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8 (a)]) For an algebraic space X, a quasi-compact open immersion j : U ãÑ X with complement Z :“ XzU, and an X-group algebraic space M of multiplicative type, if for every geometric point z Ñ Z, the strict local ring OX,z is parafactorial, then restric- tion functor TorspXfppf, Mq „ ÝÑ TorspUfppf, Mq induces an equivalence of categories of M-torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, passing to isomorphism classes of objects, we have the following isomorphisms Hi fppfpX, Mq „ ÝÑ Hi fppfpU, Mq for i ď 1 and H2 fppfpX, Mq ãÑ H2 fppfpU, Mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) For a semilocal Prüfer domain R with spectrum S, a quasi-compact quasi-separated S-smooth scheme X, a quasi-compact open U Ă X with complement Z :“ XzU, and an X-torus T such that TOX,z is flasque for every z P Z for which OX,z is a valuation ring, then we have H1 ´etpX, T q ։ H1 ´etpU, T q and H2 ´etpX, T q ãÑ H2 ´etpU, T q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) By the local cohomology exact sequence for the pair pX, Zq and the sheaf M, everything reduces to show the vanishings Hq ZpX, Mq “ 0 for 0 ď q ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the spectral sequence in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, it suffices to show the vanishings of Hq ZpMq, the étale-sheafification of the presheaf X1 ÞÑ Hq Z1pX1, Mq where Z1 :“ Z ˆX X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Further, the quasi-compactness of j allows us to identify the stalk of Hq ZpMq at a geometric point z Ñ Z as Hq tzupOX,z, Mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence we may assume that M split as µn or Gm, and since µn “ kerpGm ˆn Ñ Gmq, it suffices to show that Hq tzupOX,z, Gmq “ 0 for 0 ď q ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since OX,z is parafactorial, we have HqpSpecpOX,zq, Gmq » HqpSpecpOX,zqztzu, Gmq for 0 ď q ď 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 33 as OX,z is strictly Henselian, we have H2pSpecpOX,zq, Gmq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Looking at the local cohomology exact sequence for the pair pSpecpOX,zq, zq and T , we deduce the desired vanishings Hq tzupOX,z, Gmq “ 0 for 0 ď q ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) By the local cohomology exact sequence ¨ ¨ ¨ Ñ H1pX, T q Ñ H1pU, T q Ñ H1 ZpX, T q Ñ H2pX, T q Ñ H2pU, T q Ñ ¨ ¨ ¨ , the assertion is equivalent to the vanishing H2 ZpX, T q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since X is quasi-compact quasi-separated and U Ă X is a quasi-compact open, by a limit argument involving Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, we reduce to the case R having finite Krull dimension, so X is topologically Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Recall the coniveau spectral sequence [Gro68b, §10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1] Epq 2 “ à zPZppq Hp`q tzu pT q ñ Hp`q Z pX, T q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' the topological Noetherianness of X allows us to identify Hp`q tzu pT q :“ colim Hp`q tzuXUpU, T q as Hp`q tzu pOX,z, T q, where U runs over the open neighbourhoods of z in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, it is enough to show H2 tzupOX,z, T q “ 0, which has been solved by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a normal scheme S and an S-algebraic space X satisfying one of the following (i) either X Ñ S is a smooth morphism of topologically Noetherian algebraic spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' or (ii) S is semilocal Prüfer of finite dimension and X is S-flat locally of finite type with regular S-fibers, a quasi-compact open U Ă X with complementary closed Z :“ XzU satisfying the following condition codimpZη, Xηq ě 2 for every generic point η P S and codimpZs, Xsq ě 1 for all s P S, and a finite type X-group algebraic space M of multiplicative type, the following restriction functor TorspXfppf, Mq „ ÝÑ TorspUfppf, Mq induces an equivalence of categories of M-torsors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, passing to isomorphism classes of objects, we have the following isomorphisms H0pX, Mq » H0pU, Mq, H1 fppfpX, Mq » H1 fppfpU, Mq, H2 fppfpX, Mq ãÑ H2 fppfpU, Mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We simply verify the assumptions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' First, the restriction functor is fully faithful, because M is X-affine so is Y :“ IsomXpP1, P2q for arbitrary M-torsors P1 and P1 on X (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4), which implies that Y pXq » Y pUq (note that Y is an AutGpP1q » M-torsor, so we have Y pXq » Y pUq by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The same holds when we base change to every scheme étale over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Next, we show that, fppf locally on X, every M-torsor on U extends on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For this we may assume that X is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since X is normal, M is isotrivial, so there is an X-torus T and a finite X-group µ of multiplicative type fitting into the short exact sequence 1 Ñ T Ñ M Ñ µ Ñ 1, From which we leverage the following commutative diagram with exact rows µpXq H1 fppfpX, T q H1 fppfpX, Mq H1 fppfpX, µq µpUq H1 fppfpU, T q H1 fppfpU, Mq H1 fppfpU, µq, where µpXq » µpUq follows from the X-affineness of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A diagram chase reduces us to showing that H1 fppfpX, T q „ ÝÑ H1 fppfpU, T q and H1 fppfpX, µq „ ÝÑ H1 fppfpU, µq are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since the extension problem is fppf local, we may assume that M splits, without loss of generalities, say M » Gm or M “ µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7(ii), the X-affineness of M implies that MpX1q » MpU 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 34 When M “ Gm, we have Pic X „ ÝÑ Pic U because pX, Zq is a parafactorial pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to prove that H1 ´etpX, µnq » H1 ´etpU, µnq, for which we consider the commutative diagram 0 OpXqˆ{OpXqˆn H1 ´etpX, µnq n PicpXq 0 0 OpUqˆ{OpUqˆn H1 ´etpU, µnq n PicpUq 0 A diagram chase leads to the desired isomorphism H1 ´etpX, µnq „ ÝÑ H1 ´etpU, µnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Finally, all fppf local extension data glue together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence we obtain the desired essential surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Grothendieck–Serre type results for groups of multiplicative type Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let φ : X Ñ Y be a morphism of schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let L be an invertible OX-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If (1) Y is quasi-compact quasi-separated, integral, and normal, (2) there exist a smooth projective morphism φ : X Ñ Y , with geometrically integral fibers, and a quasi-compact open immersion X ãÑ X over Y , and (3) L is trivial when restricted to the generic fiber of φ, then L » φ˚N for some invertible OY -module N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' When Y is Noetherian, this follows from a much more general result [SP, 0BD6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' for instance, (2) can be replaced by the assumption that X Ñ Y is faithfully flat of finite presentation, with integral fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The general case can be deduced from this via Noetherian approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' More precisely, we first use [SP, 01ZA] to write Y “ limi Yi for a filtered inverse system tYiu of finite type integral Z-schemes with affine transition morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since the normalization of a finite type integral Z-scheme is finite, we may assume that each Yi is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Next, by [SP, 01ZM, 0C0C], for some i0 there exist a finite type smooth morphism φi0 : Xi0 Ñ Yi0 such that X » Xi0 ˆYi0 Y as Y -schemes, an open subscheme Xi0 Ă Xi0 whose pullback to X identifies with X, and, by [SP, 0B8W], there is an invertible OXi0 -module Li0 whose pullback to X is isomorphic to L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For any i ě i0 denote by φi : Xi :“ Xi0 ˆYi0 Yi Ñ Yi the base change of φi0|Xi0 to Yi, and denote by Li the pullback of Li0 to Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [SP, 01ZM, 01ZP], any projective embedding of X over Y descends to a projective embedding of Xi over Yi for large enough i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' in particular, φi is projective for large enough i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since Y is normal, the assumption (3) implies that the Stein factorization of φ is itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' in particular, OY „ ÝÑ φ˚OX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This implies that the finite extension OYi0 ãÑ φi0,˚OXi0 is an isomorphism, because its base change to the function field of Y is so and Yi0 is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, by Zariski’s main theorem, φi0 has connected geometric fibers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' as it is also smooth, all its fibers are even geometrically integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By limit formalism, for large enough i, Li is trivial when restricted to the generic fiber of φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, for large enough i, the morphism φi : Xi Ñ Yi and the invertible OXi-module Li satisfy all the assumptions of the Lemma, so Li » φ˚ i Ni for some invertible OYi-module Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then L » φ˚N where N is the pullback of Ni to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [CTS87, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a Prüfer domain R with spectrum pS, ηq, an irreducible scheme X essentially smooth over S with function field KpXq, an X-group scheme M of multiplicative type, and a connected finite étale Galois covering X1 Ñ X splitting M 4, the restriction maps H1 fppfpX, Mq Ñ H1 fppfpKpXq, Mq and H2 fppfpX, Mq Ñ H2 fppfpKpXq, Mq are injective in each of the following cases: (i) X “ SpecpAq and A is a semilocal ring essentially smooth over R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) For some essentially smooth semilocal R-algebra A, there exists a quasi-compact open immersion X ãÑ X, where X is a smooth projective A-scheme, with geometrically integral fibers, such that PicpXLq “ 0 for any finite separable fields extension L{FracpAq, and M “ NX for N an A-group of multiplicative type (for instance, X could be any quasi-compact open subscheme of PN A );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 4Such a covering always exists, because X is normal and so M is isotrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 35 (iii) any subcovering X2 Ñ X of X1 Ñ X satisfies PicpX2q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Further, if M is a flasque X-torus, then in all cases piq-piiiq the restriction map H1 ´etpX, Mq „ ÝÑ H1 ´etpKpXq, Mq is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It is clear that (i) is a particular case of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let us show that (ii) is a particular case of (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let A Ñ B be a connected finite étale Galois covering that splits N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Take X1 :“ X ˆA B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the normality of A and the smoothness of X Ñ SpecpAq, X is also normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, since X Ñ SpecpAq has geometrically integral generic fiber, the natural map π´et 1 pXq Ñ π´et 1 pSpec Aq is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This implies that any subcovering X2 Ñ X of X1 Ñ X is of the form X2 “ X ˆA C for some subcovering A Ñ C of A Ñ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By assumption, PicpXFracpCqq “ 0, so we may apply Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 to the morphism X ˆA C Ñ SpecpCq to deduce that the pullback map PicpSpecpCqq Ñ PicpX ˆA Cq is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since C is semilocal, we conclude that PicpSpecpCqq “ 0 “ PicpX ˆA Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It is thus enough to prove all assertions only for (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume first that M “ T is an X-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Take a flasque resolution 1 Ñ F Ñ P Ñ T Ñ 1, where F is a flasque X-torus and P is a quasitrivial X-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This yields a commutative diagram H1 ´etpX, Pq H1 ´etpX, T q H2 ´etpX, Fq H1 ´etpKpXq, T q H2 ´etpKpXq, Fq ρ1 ρ2 with exact rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now the quasitrivial torus P is isomorphic to a finite direct product of tori ResX2{XGm,X2 for finite étale subcoverings X2 Ñ X of X1 Ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, assumption (iii) implies that H1 ´etpX, Pq “ 0, and so the injectivity of ρ1 reduces to that of ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To prove that ρ2 is injective we pick a P H2 ´etpX, Fq for which a|KpXq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By spreading out, we may assume that X is a localization of an irreducible, smooth, affine R-scheme r X, F “ rFX for a flasque r X-torus rF, and a “ ra|X for some class ra P H2p r X, rFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since ra|KpXq “ 0, for a large enough hypersurface Z Ă r X, ra|Ă XzZ “ 0 P H2 ´etp r XzZ, rFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4(ii), ra “ 0, so a “ ra|X “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This proves the injectivity of ρ2 and hence also of ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now let M be an arbitrary X-group of multiplicative type, then there is an X-subtorus T Ă M such that µ :“ M{T is X-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, for any generically trivial M-torsor P, the µ-torsor P{T is finite over X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' as X is normal, this implies pP{T qpXq “ pP{T qpKpXqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, P{T Ñ X has a section that lifts to a generic section of P Ñ X, that is, P reduces to a generically trivial T -torsor PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the injectivity of ρ1, PT and hence also P is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This proves the injectivity of H1 ´etpX, Mq Ñ H1 ´etpKpXq, Mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' On the other hand, there is a short exact sequence 1 Ñ M Ñ F Ñ P Ñ 1 of X-groups of multiplicative type, where F is flasque and P is quasitrivial, both split after base change by X1 Ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This yields the following commutative diagram with exact rows H1 fppfpX, Pq H2 fppfpX, Mq H2 fppfpX, Fq H2 fppfpKpXq, Mq H2 fppfpKpXq, Fq ρ3 ρ2 Since we have already shown that H1 fppfpX, Pq “ 0 and ρ2 is injective, the injectivity of ρ3 follows from a diagram chase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Finally, if M is a flasque X-torus, the bijectivity of H1 fppfpX, Mq Ñ H1 fppfpKpXq, Mq will follow if one proves its surjectivity, but the latter follows from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4(ii) via a limit argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 36 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Grothendieck–Serre on a semilocal Prüfer domain The main result of this section is the following mild generalization of [Guo20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R with fraction field K, and a reductive R-group scheme G, the following restriction map has trivial kernel: ker ` H1 ´etpR, Gq Ñ H1 ´etpK, Gq ˘ “ t˚u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We fix the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R of finite Krull dimension, all the maximal ideals pmiqr i“1 of R, the local rings Oi :“ Rmi, an element a P R such that V paq “ tmiur i“1, let pR (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', p Oi) denote the a-adic completion of R (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', of Oi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then p Oi is an a-adic complete valuation ring of rank 1, and we have pR » śr i“1 p Oi, compatibly with the topologizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote pKi :“ Frac pOi “ p Oir 1 as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Topologize Rr 1 as by declaring timpanR Ñ Rr 1 asquně1 to be a fundamental system of open neighbourhood of 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' the associated completion is Rr 1 as Ñ pRr 1 as » śr i“1 pOir 1 as “ śr i“1 pKi, where each pKi is a complete valued field, with pseudo-uniformizer (the image of) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, for an R-scheme X, we have a map ΦX : XpRr 1 asq Ñ śr i“1 Xp pKiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If X is locally of finite type over R, we endow the right hand side with the product topology where each Xp pKiq, by, for example, Conrad, has a natural topology induced from that of pKi, which we will call the a-adic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If moreover X is affine, we can canonically topologize XpRr 1 asq by choosing a closed embedding X ãÑ AN R and endowing XpRr 1 asq ãÑ Rr 1 asN with the subspace topology (this is independent of the choices of the embeddings), then ΦX is a continuous map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lifting maximal tori of reductive group schemes over semilocal rings Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a scheme S, an S-smooth finitely presented group scheme G whose S-fibers are connected and affine, and a finite subset I Ă S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If I satisfies the following conditions (i) I is contained in an affine open subset of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) for each residue field κi of S at i P I, the fiber Gκi is a κi-reductive group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and (iii) 7κi ě dimpGκi{Ziq for the center Zi Ă Gκi, then there is an open neighborhood U of I such that the following map is surjective TorpGqpUq ։ ś iPI TorpGqpκiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [SGA 3II, Exposé XVI, Théorème 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2], there is an open neighborhood U of I such that G|U is a U-reductive group scheme, so we may replace S by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [SGA 3II, Exposé XII, Théorème 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7 c)], G has a reductive center Z and we have Zi “ pZqκi for every i P I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since TorpGq » TorpG{Zq, we may replace G by G{Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [SGA 3II, Exposé XIV, Théorème 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='18], the maximal tori of G are exactly the subgroups of type (C), which are bijectively assigned by D ÞÑ LiepDq to the Cartan subalgebras of g :“ LiepGq ([SGA 3II, Exposé XIV, Théorème 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It suffices to lift a Cartan subalgebra c0 Ă ś iPI gκi to that of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote ci :“ pc0qκi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since for each i P I, we have 7κi ě dimpG{Zq “ dimpGq, by [Bar67, Theorem 1], ci is of the form Nilpaiq :“ Ť n kerpadpan i qq for some ai P ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence [SGA 3II, Exposé XIII, Corollaire 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7] implies that each ai P ci is a regular element of gκi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We take a section a of g passing through all ai and claim that V :“ ts P Spec R such that as P gs is regularu is an open subset of Spec R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We may assume that R is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since the nilpotent rank of g is locally constant, there is an open neighborhood U of I such that the nilpotent rank of g is constant on each connected component Uα of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' On each Uα, the Killing polynomial of g at every s P Uα is uniformly Pα,gsptq “ trαptn´rα ` pc1qstn´rα´1 ` ¨ ¨ ¨ ` pcn´rαqsq such that pcn´rαqs is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus, the regular locus in g is the principle open subset Ş αtcn´rα ‰ 0u Ă Wpgq so V is nonempty and open, hence shrinking U if necessary, we have V “ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, the regular elements pai P ciqiPI are lifted to a quasi-regular section a P g, which by [SGA 3III new, Exposé XIV, Corollaire 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7], is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By definition of regular sections, there is a Cartan subalgebra of g containing a and is the desired lifting of c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 37 Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R of finite Krull dimension, we use the notations in the setup §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a reductive R-group scheme G, the scheme TorpGq of maximal tori of G, and the a-adic topology on TorpGqp p Kiq, the image of the following map is dense: TorpGqpRr 1 asq Ñ śr i“1 TorpGqp p Kiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The proof proceeds in the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The ring A :“ lim ÝÑkě0 CauchyěkpRr 1 asq is a semilocal ring with residue fields Frac pOi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let I be the kernel of the surjection A ։ śr i“1 Frac pOi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since A{I is a product of fields, it suffices to show that 1 ` I Ă Aˆ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a sequence pbNqN P I, its tail lies in impakR Ñ Rr 1 asq for all k ą 0, so the tail of p1 ` bNqN is invertible in Rˆ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since Rr 1 as is semilocal, the tail of p1 ` bNqN is termwise invertible in Rr 1 as and the inverses form a Cauchy sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We combine the Step 1 and Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 to obtain the following surjective map lim ÝÑmě0 ´ TorpGq ` CauchyěmpRr 1 asq ˘¯ » TorpGq ´ lim ÝÑmě0 ` CauchyěmpRr 1 asq ˘¯ ։ śr i“1 TorpGqpFrac pOiq, which signifies that every Cauchy sequence in the image of TorpGqpRr 1 asq converges in śr i“1 TorpGqFrac pOi, hence the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Harder’s weak approximation Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R of finite Krull dimension, we use the setup §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a Rr 1 as-torus T , let Li{ pKi be minimal Galois field extensions splitting Tx Ki and consider the norm map Ni : T pLiq Ñ T p pKiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, the image U of śr i“1 Ni is a-adically open and is contained in impT pRr 1 asq Ñ śr i“1 T p pKiqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The proof proceeds as the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The image U is a-adically open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For each i, there is a short exact sequence of tori 1 Ñ Ti Ñ ResLi{x Ki TLi Ñ Tx Ki Ñ 1 and the norm map Ni : ResLi{x Ki TLip pKiq Ñ pResLi{x Ki TLi{Tiqp p Kiq, which by [Čes15, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3 (a) and §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8 (2)] is a-adically open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As a product of open subsets, U is open in śr i“1 T p pKiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We prove that U is contained in the closure of impT pRr 1 asqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Equivalently, we show that every u P U and every a-adically open neighborhood Bu Ă U satisfy that Bu X impT pRr 1 asqq ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let rR{Rr 1 as be a minimal Galois cover splitting T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consider the following commutative diagram T p ˜Rq śr i“1 T pLiq T pRr 1 asq śr i“1 T p pKiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' NĂ R{Rr 1 a s śr i“1 Ni Take a preimage v P pśr i“1 Niq´1puq Ă śr i“1 T pLiq and let Bv Ă śr i“1 T pLiq be the preimage of Bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since T rR splits, the image of T p rRq in śr i“1 T pLiq is a-adically dense, hence T p rRq ˆśr i“1 T pLiq Bv ‰ H, namely, there is r P T p rRq whose image is in Bv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let s :“ N rR{Rr 1 a sprq P T pRr 1 asq, then the image of s under the map T pRr 1 asq Ñ śr i“1 T p pKiq is contained in Bu, so the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R of finite Krull dimension, we use the setup §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a reductive R-group scheme G and for each i a fixed maximal torus Ti Ă Gx Ki with minimal Galois field extension Li{ pKi splitting Ti, consider the following norm map Ni : T pLiq Ñ T p pKiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 38 Then the image U of the map śr i“1 Ni is an a-adically open subgroup of śr i“1 T p pKiq and is contained in the closure of impGpRr 1 asq Ñ śr i“1 Gp pKiqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the same arguement in Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, the image U is a-adically open in ś i“1 T p pKiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to show that U Ă impGpRr 1 asqq, which proceeds as the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The map φi : Gp pKiq Ñ TorpGqp p Kiq defined by g ÞÑ gT g´1 is a-adically open for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since the image of T pRr 1 asq Ñ śr i“1 T p pKiq is a-adically dense, for every open neighborhood W Ă śr i“1 Gp pKiq of id, we have ppśr i“1 φiqpWqq X ImpTorpGqpRr 1 asq Ñ śr i“1 TorpGqp p Kiqq ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, there exist a torus T 1 P TorpGqpRr 1 asq and a pgiqr i“1 P W such that giTig´1 i “ T 1 x Ki for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For any u P U, consider the map śr i“1 Gp pKiq Ñ śr i“1 Gp pKiq defined by g ÞÑ g´1ug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, we apply the Step 1 to the preimage W of U under this map: there is a γ “ pγiqr i“1 P W and a torus T 1 P TorpGqpRr 1 asq such that γ´1 i Tiγi “ T 1 x Ki for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, u P γUγ´1 “ γppśr i“1 NiqpTipLiqqqγ´1, which by transport of structure, is pśr i“1 NiqpT 1 x KipLiqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, the last term is contained in the closure of impT 1pRr 1 asq Ñ śr i“1 T p pKiqq, so is contained in impGpRr 1 asqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R of finite Krull dimension, we use the setup §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a reductive group scheme G over R, the closure GpRr 1 asq of the image of GpRr 1 asq Ñ śr i“1p pKiq, GpRr 1 asq contains an open normal subgroup N of śr i“1 Gp pKiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The proof proceeds in the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) For each i, we fix a maximal torus Ti Ă Gx Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 provides the open subgroup U Ă śr i“1 Tip pKiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since each component of the norm map defining U is the image of the pKi- points of ResLi{x KipTiqLi Ñ Ti, and ResLi{x KipTiqLi is a Zariski dense open subset of an affine space over pKi, we have U X śr i“1 T reg i p pKiq ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) Fix an element τ P U X śr i“1 T reg i p pKiq, by [SGA 3II, Exposé XIII, Corollaire 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2], for each i, fi : Gx Ki ˆ Ti Ñ Gx Ki, pg, tq ÞÑ gtg´1 is smooth at pid, τq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, there is a Zariski open neighborhood B of pid, τq such that pśr i“1 fiq|B : B Ñ śr i“1 Gx Ki is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [GGMB14, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4], the map Bpśr i“1 pKiq Ñ śr i“1 Gp p Kiq is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then the image of W :“ Bpśr i“1 pKiq X pśr i“1 Gp p Kiq ˆ Uq under f “ śr i“1 fi is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Subsequently, all śr i“1 Gp pKiq translations of W have open images, so there is an open subset U0 Ă U such that E :“ fpśr i“1 Gp pKiq ˆ U0q is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now we define N as the subgroup of śr i“1 Gp pKiq generated by E, then E is an open subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Further, by construction, E is stable under conjugations by śr i“1 Gp pKiq, thus N is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iii) We prove that N is contained in the closure of impGpRr 1 asq Ñ śr i“1 Gp pKiqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since E is the union of all conjugates of U0, which are contained in GpRr 1 asq by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2, so E is in this closure, and so is N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R of finite Krull dimension, we use the setup §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a reductive group scheme G over R, a maximal torus Ti Ă G p Oi for each i, and any a-adically open neighborhood W of id P śr i“1 Gp pKiq such that W Ă GpRr 1 asq X śr i“1 Gp p Oiq, there exist g “ pgiqi P W and a maximal torus T P TorpGqpRq such that for every i, we have Tx Ki “ gipTiqx Kig´1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, GpRr 1 asqXśr i“1 Gp p Oiq is an a-adically open neighborhood of id P śr i“1 Gp pKiq, so it makes sense to take its subset W such that W is a neighborhood of id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now consider the a-adically open map φ: śr i“1 Gp pKiq Ñ śr i“1 TorpGqp p Kiq defined by gi ÞÑ gipTiqx Kig´1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then φpWq is an a-adically open neighborhood of pTiqi P śr i“1 TorpGqp p Kiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since śr i“1 TorpGqp p Oiq Ă śr i“1 TorpGqp p Kiq is also an 39 a-adically open neighborhood of pTiqi, we have an open intersection φpWq X śr i“1 TorpGqp p Oiq ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then the density of the image of TorpGqpRr 1 asq Ñ śr i“1 TorpGqp pKiq provided by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 yields an element T P TorpGqpRq „ ÝÑ TorpGqpRr 1 asq ˆśr i“1 TorpGqpx Kiq śr i“1 TorpGqp p Oiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, T is a maximal torus of G over R satisfying the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' With the notations in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, we have GpRr 1 asq ¨ śr i“1 Gp p Oiq “ impGpRr 1 asq Ñ śr i“1 Gp pKiqq ¨ śr i“1 Gp p Oiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Product formula over semilocal Prüfer domains, passage to the local case Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R of finite Krull dimension, we use the notations in the setup §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For an R-torus T , we have the following product formula śr i“1 T p pKiq “ impT pRr 1 asq Ñ śr i“1 T p pKiqq ¨ śr i“1 T p pOiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let Rh denote the Henselization of the pair pR, aRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then we have the commutative digram 0 T pRq T pRr 1 asq H1 ta“0upR, T q H1pR, T q H1pRr 1 as, T q 0 T pRhq T pRhr 1 asq H1 ta“0upRh, T q H1pRh, T q H1pRhr 1 as, T q, whose exact rows are the local cohomology exact sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since the case of tori for Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 is proved, the two horizontal arrows of the rightmost squares are injective, hence the coset T pRhr 1 asq{T pRhq » H1 ta“0upRh, T q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By excision [Mil80, III, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='28], we have an isomorphism H1 ta“0upR, T q – H1 ta“0upRh, T q, which leads to a surjection T pRr 1 asq ։ H1 ta“0upRh, T q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, we obtain the product formula T pRhr 1 asq “ impT pRr 1 asq Ñ T pRhr 1 asqq ¨ T pRhq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) On the other hand, by [BČ22, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='17], the image of T pRhr 1 asq Ñ śr i“1 T p pKiq is dense in śr i“1 T p pKiq with respect to the topology fixed in §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since each T p pOiq Ă T p pKiq is an open subgroup, we have im ` T pRhr 1 asq Ñ śr i“1 T p pKiq ˘ ¨ śr i“1 T p pOiq “ śr i“1 T p pKiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2) Consequently, the combination of (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2) leads to the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R of finite Krull dimension, we use the notations in the setup §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a reductive R-group scheme G, we have śr i“1 Gp pKiq “ im ´ GpRr 1 asq Ñ śr i“1 Gp pKiq ¯ ¨ śr i“1 Gp p Oiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We will proceed verbatim as in [Guo20, §4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We choose a minimal parabolic pOi-subgroup Pi for each Gi :“ G ˆR pOi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote Ui :“ radupPiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) for the maximal split torus Ti Ă Pi, we have śr i“1 Tip pKiq Ă impGpRr 1 asq Ñ śr i“1 Gp pKiqq ¨ śr i“1 Gp p Oiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [SGA 3III new, Exposé XXVI, Corollaire 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='11], there is a maximal torus rTi of Gi containing Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, p rTiqx Ki is a maximal torus of Gx Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then we apply Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 to all rTi: there are a g “ pgiqi P GpRr 1 asq ¨ śr i“1 Gp p Oiq and a maximal torus T0 Ă G such that pT0qx Ki “ gip rTiqx Kig´1 i for every i, which combined with the product formula Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 for T0 yields śr i“1 rTp pKiq “ śr i“1 g´1 i T0p pKiqgi Ă śr i“1 impGpRr 1 asqq ¨ Gp p Oiqgi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since g P impGpRr 1 asqq X śr i“1 Gp p Oiq, the inclusion displayed above implies that śr i“1 rTip p Kiq Ă impGpRr 1 asqq ¨ śr i“1 Gp p Oiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, we obtain the following desired inclusion śr i“1 Tip pKiq Ă śr i“1 rTip pKiq Ă impGpRr 1 asqq ¨ śr i“1 Gp p Oiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 40 (ii) we have śr i“1 Uip pKiq Ă impGpRr 1 asq Ñ śr i“1 Gp p Kiqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consider the Ti-action on Gi defined by Ti ˆ Gi Ñ Gi, pt, gq ÞÑ tgt´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Recall the open normal subgroup N Ă śr i“1 Gp p Kiq constructed in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, then each N XUip pKiq is open in Uip pKiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The dynamic argument in [Guo20] shows that Uip pKiq “ N XUip pKiq, hence Uip pKiq Ă N for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, we have śr i“1 Uip pKiq Ă impGpRr 1 asq Ñ śr i“1 Gp pKiqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iii) we have śr i“1 Pip p Kiq Ă impGpRr 1 asq Ñ śr i“1 Gp p Kiqq ¨ śr i“1 Gp p Oiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The quotient Hi :“ Li{Ti is anisotropic, therefore we have Hip pKiq “ Hip p Oiq for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consider the commutative diagram 0 Tip p Oiq Lip p Oiq Hip pOiq H1p p Oi, Tiq “ 0 0 Tip pKiq Lip p Kiq Hip pKiq H1p pKi, Tiq “ 0 with exact rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By diagram chase, we have Lip pKiq “ Tip pKiq ¨ Lip p Oiq for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Subsequently, the combination of (i) and (ii) yields the inclusion śr i“1 Pip pKiq Ă impGpRr 1 asq Ñ śr i“1 Gp pKiqq ¨ śr i“1 Gp pOiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iv) Recall [SGA 3III new, Exposé XXVI, Théorème 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 and Corollaire 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2] that for each Pi, there is a parabolic subgroup Qi of Gi such that Pi X Qi “ Li fitting into the following surjection radupPiqp pKiq ¨ radupQiqp pKiq ։ Gp pKiq{Pip pKiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This surjection, combined with the result of (ii) gives an inclusion śr i“1 Gp pKiq Ă impGpRr 1 asq Ñ śr i“1 Gp pKiqq ¨ śr i“1 Pip pKiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now we further use the result of (iii) to obtain śr i“1 Gp pKiq Ă impGpRr 1 asq Ñ śr i“1 Gp p Kiqq ¨ śr i“1 Gp p Oiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, we have the following product formula śr i“1 Gp pKiq “ im ´ GpRr 1 asq Ñ śr i“1 Gp pKiq ¯ ¨ śr i“1 Gp p Oiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Torsors on a smooth affine relative curve In this section we prove the following result concerning triviality of torsors on a smooth affine relative curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The idea of the proof ultimately depends on the geometry of affine Grassmannians developed by Fedorov, who proved Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (i) for C “ A1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A similar result can also be found in the recent preprint [Čes22c, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (Section theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let R be a semilocal domain whose local rings at primes are geomet- rically unibranch5, C a smooth, affine, relative R-curve, and G a reductive C-group scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let A be a R-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let P be a G-torsor over CA :“ C ˆR A that trivializes over CAzZA for some R-finite closed subscheme Z Ă C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a section s P CpRq, if either (i) A is semilocal, or (ii) s˚ ApGq is totally isotropic, then the pullback s˚ ApPq is trivial as an s˚ ApGq-torsor, where sA stands for the image of s in CApAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To prove Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, we first use Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 to reduce to the case when G is the base change of a reductive R-group scheme, and then to the case when C “ A1 R, see Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As for the latter, one can approach it via the geometry of affine Grassmannians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We start with the following result concerning equating reductive group schemes, which was already known to experts, see also [Čes22c, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 5According to [SP, 0BPZ], a local ring A is geometrically unibranch if its reduction Ared :“ A{ a p0q is a domain, and if the integral closure of Ared in its fraction field is a local ring whose residue field is purely inseparable over that of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [SP, 06DM], A is geometrically unibranch iff its strict Henselization Ash has a unique minimal prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 41 Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 (Equating reductive group schemes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let B be a semilocal ring whose local rings are geometrically unibranch, and let G1, G2 be two reductive B-group schemes whose geometric B-fibers are of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let T1 Ă G1, T2 Ă G2 be maximal B-tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume that, for some ideal I Ă B, there is an isomorphism of B{I-group schemes ι : pG1qB{I » pG2qB{I such that ιppT1qB{Iq “ pT2qB{I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' There are a faithfully flat, finite, étale B-algebra B1, a section s : B1 ։ B{I, and an isomorphism of B1-groups ι1 : pG1qB1 » pG2qB1 such that ιppT1qB1q “ pT2qB1 and whose s-pullback is ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' According to [SGA 3III new, Exposé XXIV, Corollaire 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2], the condition on the geometric B-fibers ensures that the functor X :“ IsomBppG1, T1q, pG2, T2qq parameterizing the isomorphisms of the pairs pG1, T1q and pG2, T2q is representable by a B-scheme and is a H :“ AutBppG1, T1qq-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We need to show that, for any ι P XpB{Iq, there are a faithfully flat, finite, étale B-algebra B1, an ι1 P XpB1q, and a section s : B1 ։ B{I such that spι1q “ ι P XpB{Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', H is an extension of an étale locally constant B-group scheme by T ad 1 , the quotient of T1 by the scheme-theoretic center of G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' According to [SGA 3III new, Exposé XXIV, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6], T ad 1 acts freely on X and the quotient X :“ X{T ad 1 is represented by a faithfully flat B-scheme that is étale locally constant on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As B is geometrically unibranch, by [SGA 3III new, Exposé X, Corollaire 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='14], every connected component of X is finite, étale over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As the image of ι : SpecpB{Iq Ñ X Ñ X intersects only finitely many connected components of X, the union of these components is the spectrum of a finite étale B-algebra A, and there are an ι P XpAq and a section t : A ։ B{I such that tpιq “ ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By adding more connected components of X into SpecpAq if needed, we may assume that A is faithfully flat over B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let Y :“ X ˆX,ι SpecpAq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' it is a T ad A -torsor equipped with a point ι P Y pA{Jq Ă XpA{Jq, where J :“ ker pA ։ B{Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [Čes22b, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2], there are a faithfully flat, finite, étale A-algebra B1, a section s1 : B1 ։ A{J » B{I, and an ι1 P Y pB1q Ă XpB1q such that s1pι1q “ ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The proof of the Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 reduces to the case when C “ A1 R and G is the base change of a reductive R-group scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let B be the semilocal ring of C at the closed points of impsqYZ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' its local rings are geometrically unibranch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By abuse of notation, we may view s : B ։ R as a section of the R-algebra B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As B is semilocal, by [SGA 3II, Exposé XIV, Corollaire 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='20], GB admits a maximal B-torus TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since the pullbacks of the paris pGB, T q and pps˚pGqqB, ps˚pT qqBq along s are the same, by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2, there are a faithfully flat, finite, étale B-algebra B1, a section s1 : B1 ։ R that lifts s, and a B1-isomorphism ι : pGB1, TB1q » pps˚pGqqB1, ps˚pT qB1q whose s-pullback is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We may spread out SpecpB1q Ñ SpecpBq to obtain a finite étale covering C1 Ñ U of a small enough affine open neighbourhood U of impsq Y Z in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By shrinking U if necessary, we may assume that the isomorphism ι is defined over C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In both cases of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 we may replace C by C1, Z by C1 ˆC Z, s by s1, and P by P|C1 A to reduce to the case when G is the base change of the reductive R-group scheme s˚pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Next, in order to apply glueing Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2(ii) to achieve that C “ A1 R, we need to modify C so that Z embeds into A1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For this, we first replace Z by Z Y impsq to assume that s factors through Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then we apply Panin’s ‘finite field tricks’ [Čes22a, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4] to obtain a finite morphism rC Ñ C that is étale at the points in rZ :“ rC ˆC Z such that s lifts to rs P rCpRq, and there are no finite fields obstruction to embedding rZ into A1 R in the following sense: for every maximal ideal m Ă R, 7 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' z P rZκpmq : rκpzq : κpmqs “ d ) ă 7 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' z P A1 κpmq : rκpzq : κpmqs “ d ) for every d ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 42 Then, by [Čes22a, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3], there are an affine open C2 Ă rC containing imprsq, a quasi-finite, flat R-map C2 Ñ A1 R that maps Z isomorphically to a closed subscheme Z1 Ă A1 R with Z » Z1 ˆA1 R C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (Actually, by shrinking C2 around imprsq, one can show that C2 Ñ A1 R is étale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=') For both cases of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, since P|C2 A is a G-torsors that trivializes over C2 Az rZA, we may use Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2(ii) to glue PC2 A with the trivial G-torsor over A1 A to obtain a G-torsor P1 over A1 A that trivializes over A1 AzZ1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let s1 P A1 RpRq be the image of rs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' then s1˚pP1q » s˚pPq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to replace C by A1 R, Z by Z1, s by s1, and P by P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ The analysis of torsors on A1 R ultimately depends on the geometry of affine Grassmannians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A nice summary of and complement on the relevant techniques can be found in [Čes22b, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, we will use the following result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' it is a slight variant of [Čes22b, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6], which in turn is a mild generalization of [Fed22b, Theorem 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal ring R with connected spectrum and a reductive R-group scheme G, let Gad » ź i ResRi{RpGiq be the canonical decomposition of the adjoint quotient Gad [SGA 3III new, Exposé XXIV, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='10], where Gi is an adjoint simple Ri-group scheme, and Ri is a finite, étale R-algebra with connected spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let Y Ă A1 R be a R-finite, étale, closed subscheme with the following properties: (i) for every i, there is a clopen Yi Ă Y ˆR Ri such that pGiqYi contains a copy of Gm,Yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) for every maximal ideal m Ă Ri such that pGiqκpmq is isotropic, the line bundle OP1 κpmqp1q is trivial over P1 κpmqzpYiqκpmq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iii) the line bundle OP1 Rp1q is trivial over P1 RzY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let P be a G-torsor over P1 R that trivializes over P1 RzZ for some R-finite closed subscheme Z Ă A1 RzY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume that for every maximal ideal m Ă R the Gad-torsor over P1 κpmq induced by P lifts to a generically trivial pGadqsc-torsor over P1 κpmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then the restriction P|P1 RzY is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Recall that, by [SGA 3III new, Exposé XXVI, Corollaire 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='12], (i) is equivalent to that the base change of pGiqYi to every connected component of Yi contains a proper parabolic subgroup scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For instance, if G is quasi-split, we can just take Yi “ Y ˆR Ri to ensure (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In practice, we achieve (i) by guaranteeing base change of pGiqYi to connected components of Yi contain proper parabolics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (ii), we can take Yi so that Yipκpmqq ‰ H for every maximal ideal κpmq Ă Ri with pGiqκpmq isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (iii), we just need to choose Y so that it contains finite étale R-schemes of degrees d and d ` 1 for some d ě 1, because Opdq and Opn ` 1q are both trivial on P1 RzY , and so is Op1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We will deduce Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 from (the proof of) a particular case of [Čes22b, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (We remind that the assumption (ii) of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' should read as ‘pGiqYi contains a copy of Gm,Yi’, as its proof shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=') The R-finite étale Y is the vanishing locus of a monic polynomial t in the standard coordinate of A1 R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' namely, t is the characteristic polynomial of this standard coordinate acting on rR :“ ΓpY, OY q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The formal completion of P1 R along Y has coordinate ring rRrrtss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Recall that, by formal glueing, a G-torsor over P1 R can be viewed as the glueing of its restriction to P1 RzY and to rRrrtss along the ‘intersection’ rRpptqq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' since our torsor P is trivial over an open neighbourhood U Ă P1 R of Y , both of the restriction P|UzY and P| rRrrtss are trivial, and once a trivialization of the former was chosen, all such glueings are parameterized by elements of Gp rRpptqqq{Gp rRrrtssq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In particular, since Gp rRpptqqq acts on Gp rRpptqqq{Gp rRrrtssq (via left multiplication), an element of Gp rRpptqqq yields a modification of P along Y : it is the G-torsor over P1 R whose restriction to P1 RzY and to rRrrtss are the same as P, but their corresponding glueings, viewed as elements of Gp rRpptqqq{Gp rRrrtssq, differ by a left translation by the element of rRpptqq we choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote by Pad the Gad-torsor over P1 R induced by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since the formation of H1pP1 R, ´q commutes with taking products, Pad corresponds to a collection pPad i q, where Pad i is a ResRi{RpGiq-torsor over P1 R 43 satisfying the analogous assumptions (i)-(iii) of the Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since R Ñ Ri is finite étale and Gi is Ri-smooth, we have R1f˚Gi “ 1 for the map f : SpecpRiq Ñ SpecpRq induced by R Ñ Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the exact sequence from [Gir71, Chapitre V, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3], 1 Ñ H1pP1 R, ResRi{RpGiqq Ñ H1pP1 Ri, Giq Ñ H1pP1 R, R1f˚Giq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus Q ÞÑ ResRi{RpQq defines a bijection of pointed sets H1pP1 Ri, Giq „ ÝÑ H1pP1 R, ResRi{RpGiqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In par- ticular, each Pad i corresponds to a Gi-torsor Qi over P1 Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As one can see immediately, the assumptions (i)-(iii) of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 for the ResRi{RpGiq-torsor Pad i translate into the assumptions [Čes22b, Propo- sition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6] (i)-(iv) for the Gi-torsor Qi over P1 Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the proof of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', for some element αi P im ´ Gsc i pp rR bR Riqpptqqq Ñ Gipp rR bR Riqpptqqq ¯ , the corresponding modification of Qi along Y ˆR Ri is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We can view α :“ pαiq P im ´ pGadqscp rRpptqqq Ñ Gadp rRpptqqq ¯ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' as pGadqsc Ñ Gad factors through pGadqsc Ñ G, α lifts to rα P Gp rRpptqqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote by Q the modification of P along Y using rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By our construction, the Gad-torsor Qad over P1 R induced by Q corresponds to the collection of modifications of the Pad i “ ResRi{RpQiq along Y using αi P Gipp rR bR Riqpptqqq “ ResRi{Rp rRpptqqq, which is trivial, so that Qad is trivial, to the effect that Q reduces to a torsor over P1 R under the center ZG of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now, as the last paragraph of the proof of [Čes22b, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6] shows, any ZG-torsor over P1 R is the sum of a constant torsor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', the pullback of a ZG-torsor over R) and λ˚pOp1qq for a unique cocharacter λ of ZG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, by our assumption (iii), Q is a constant torsor, and, by checking along the infinity section, it is even trivial, so is P|P1 RzY “ Q|P1 RzY , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ The following result will help us to construct the desired R-finite, étale schemes Yi and Y from the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let R be a semilocal ring with connected spectrum, let R1 be a finite, étale R-algebra with connected spectrum, let W Ă A1 R be a R-finite closed scheme, and let G1 be a simple R1-group scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' There is a R1-finite, étale scheme Y1, and a closed immersion Y1 Ă A1 RzW over R such that pG1qY1 contains a copy of Gm,Y1, and, for every maximal ideal m Ă R1 with pG1qκpmq isotropic, the line bundle OP1 κpmqp1q is trivial over P1 κpmqzpY1qκpmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (Notice that Y1 is a clopen of Y1 ˆR R1, thus naturally embeds into A1 R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=') In addition, there is a R1-finite, étale scheme Y 1 and a closed immersion Y 1 ãÑ A1 RzW over R such that the line bundle OP1 Rp1q is trivial over P1 RzY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let Par1 Ñ SpecpR1q be the scheme parameterizing proper parabolic subgroup schemes of the reductive R1-group scheme G1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' it is smooth projective over R1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [SGA 3III new, Exposé XXVI, Corol- laire 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Fix an embedding Par1 ãÑ PN R1 over R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Write Par1 “ Ůt i“1 Pt as a disjoint union of its connected components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' every Pt has a constant relative dimension dt over R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For every maximal ideal m Ă R1 with pG1qκpmq isotropic, a proper parabolic subgroup of pG1qκpmq gives a point bm P Par1pκpmqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Fix an i “ 1, ¨ ¨ ¨ , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For every maximal ideal m Ă R1, by Bertini theorem (including Poonen’s version over finite fields), one can find a hypersurface in PN κpmq of large enough degree such that it passes through all points bm that lies in Pi and it intersects pPiqκpmq transversally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We may assume that the above hypersurfaces have the same degree for all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the Chinese Remainder theorem, one can lift these simultaneously to get a hypersurfaces H Ă PN R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then H X Pi is a smooth projective R1-scheme of pure relative dimension di ´ 1, and bm P H X Pi whenever bm P Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The same argument can be applied to the hypersurface section H X Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Continuing in this way, we finally arrive at a R1-finite, étale, closed subscheme Yi Ă Pi such that bm P Yi whenever bm P Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote Y 1 1 :“ Ůt i“1 Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Unfortunately, Y 1 1 may not embed into A1 RzW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' So let’s first modify Y 1 1 using Panin’s ‘finite field tricks’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let d ą 0 be a large enough integer such that, for every maximal ideal n Ă R, (1) we have d ą dimκpnq ΓpWκpnq, OWκpmqq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (2) for every maximal ideal n1 Ă ΓpY 1 1, OY 1 1q lying over n and every n ě d, there are at least degpY 1 1{Rq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', at least one) closed point(s) on A1 κpnq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', on A1 κpn1q) of exact degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 44 For every maximal ideal n1 Ă ΓpY 1 1, OY 1 1q we choose a monic polynomial hn1 P κpn1qrus of degree 2d ` 1 such that: (i) if κpn1q is finite, hn1 is a product of two irreducible polynomials of degrees d and d`1, respectively (which is possible by (2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) if κpn1q is infinite, hn1 is a separable polynomial and has at least one root in κpn1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let h P ΓpY 1 1, OY 1 1qrus be a common monic lifting of hn1 for all n1 Ă ΓpY 1 1, OY 1 1q, and define Y1 “ Spec ˆΓpY 1 1, OY 1 1qrus phq ˙ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' it is finite, étale over Y 1 1, and hence also over R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By (1)-(2), there is a closed immersion ğ nĂR pY1qκpnq ãÑ A1 RzW over R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' by Nakayama’s lemma, any of its lifting Y1 ãÑ A1 RzW over R (which exists by Chinese Remainder theorem) is also a closed immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By construction, the restriction of pG1qY 1 1 to every connected component of Y 1 1 contains a proper parabolic subgroup scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus, by [SGA 3III new, Exposé XXVI, Corollaire 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='12], pG1qY 1 1 contains Gm,Y 1 1, and so pG1qY1 contains Gm,Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By (i)-(ii), for m Ă R1 with pG1qκpmq isotropic, the line bundle OP1 κpmqp1q is trivial over P1 κpmqzpY1qκpmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To construct Y 1, it suffices to produce, for a large enough d, a R-finite, étale, closed subschemes Y2 Ă A1 R of R-degrees d and d ` 1 which are disjoint from W, and then take Y 1 :“ Y1 Ů Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To achieve this, one just need to imitate the above procedure for constructing Y1 from Y 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Details are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the reduction Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, we may assume throughout that C “ A1 R and G is a reductive R-group scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Up to shifting we may assume that s “ 0R P A1 RpRq is the zero section, and base changing to A reduces us further to the case A “ R at the cost that R need not be a domain or geometrically unibranch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus, in case (i), our R is semilocal, and, in case (ii), our G is totally isotropic (but R need not be semilocal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By decomposing SpecpRq into connected components, we can assume that R has connected spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For both cases (i)-(ii), by glueing P with the trivial G-torsor over P1 RzZ we extend P to a G-torsor Q over P1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [Fed22b, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3] or [Čes22b, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5], up to replacing Q and Z by their pullbacks by P1 R Ñ P1 R, t ÞÑ td, where d is divisible by the R-fibral degree of the simply-connected central cover pGadqsc Ñ Gad, we may assume that for every maximal ideal m Ă R the Gad-torsor over P1 κpmq induced by Q lifts to a generically trivial pGadqsc-torsor over P1 κpmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In both cases (i)-(ii), assume that R is semilocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For any R-finite closed subscheme W0 Ă A1 R, there exists a R-finite, étale, closed subscheme Y Ă A1 RzW0 such that Q|P1 RzY is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We write the canonical decomposition of Gad as in Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Replacing W0 by W0 Y Z, we may assume that Z Ă W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Applying Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5 separately to each simple Ri-group scheme Gi (with appropriate choices of W’s), we get Ri-finite, étale schemes Yi such that pGiqYi is totally isotropic, a closed immersion Ů i Yi ãÑ A1 RzW0 over R such that for every maximal ideal m Ă Ri with pGiqκpmq isotropic, the line bundle OP1 κpmqp1q is trivial over P1 κpmqzpYiqκpmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Applying the second part of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5 to W :“ p\\iYiq Ů W0, we get a R-finite, étale, closed subscheme Y 1 Ă A1 Rz ´ p\\iYiq ğ W0 ¯ such that OP1 Rp1q is trivial over P1 RzY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let Y :“ Y 1 Ůp\\iYiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then all the assumptions (i)-(iii) of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 are verified, so we conclude that Q|P1 RzY is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ For (i), we take W0 “ Z Y 0R, then the above Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 gives a R-finite, étale, closed subscheme Y Ă A1 RzW0 such that Q|P1 RzY is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since Y X 0R “ H, we deduce that the pullback of Q along s “ 0R is also trivial, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (ii), we will follow [Čes22c, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3] to show that both P “ Q|A1 R and Q|P1 Rz0R descend to G- torsors over R, and then we are done: both of these descendants agree with the restriction of Q along 45 1R P A1 RpRq, so they agree with the restriction of Q along 8R, which is trivial, and hence they must be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Quillen patching [Čes22b, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5 (b)], for the descent claim we may replace R by its localizations at maximal ideals to assume that R is local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now, since R is local, we may apply the above Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 to W0 “ 0R to find a R-finite, étale, closed subscheme Z1 Ă A1 Rz0R such that Q|P1 RzZ1 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to apply Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 twice, with Y “ 0R and Y “ 8R respectively, to show that both Q|P1 Rz0R and Q|P1 Rz8R are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Torsors under a reductive group scheme over a smooth projective base The main result of this section is the following: Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R, an r P Rzt0u, an irreducible, smooth, projective R-scheme X, a finite subset x Ă X with semilocal ring A :“ OX,x, and a reductive X-group scheme G, (i) any generically trivial G-torsor over A is trivial, that is, ker pH1pA, Gq Ñ H1pFrac A, Gqq “ t˚u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) if GAr 1 r s is totally isotropic, then any generically trivial G-torsor over Ar 1 rs is trivial, that is, ker pH1pAr 1 rs, Gq Ñ H1pFrac A, Gqq “ t˚u The case (i) is a version of the Grothendieck–Serre conjecture in the case the relevant reductive group scheme GA has a reductive model over some smooth projective compactification of SpecpA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The case (ii) provides a version of Nisnevich conjecture for such ‘nice’ reductive groups satisfying the total isotropicity assumption: if R is a discrete valuation ring with uniformizer r and if R Ñ A is a local homomorphism of local rings, then r P mAzm2 A, and (ii) says that any generically trivial G-torsor over Ar 1 rs is trivial (the isotropicity assumption on GA is essential, see, for instance, [Fed21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' An inspection of the proof below shows that, if Xns Ă X denotes the loci where a finitely presented morphism X Ñ SpecpRq is non-smooth, then Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 still holds provided that X is only a flat projective R-scheme such that Xns is R-fiberwise of codimension ě 2 in X, x X Xns “ H, and G is a reductive XzXns-group scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To prove Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, we first derive from Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 the following key result, which reduces the proof of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 to studying torsors on a smooth affine relative curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R of finite Krull dimension, an irreducible, smooth, pro- jective R-scheme X of pure relative dimension d ą 0, a finite subset x Ă X, and a reductive X-group scheme G, the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) Given a generically trivial G-torsor P over A :“ OX,x, there are a smooth, affine A-curve C, an A-finite closed subscheme Z Ă C, and a section s P CpAq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' a reductive C-group scheme G satisfying s˚G » GA and a G -torsor F such that F|CzZ is trivial and s˚F » P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) Given an r P Rzt0u and a generically trivial G-torsor rP over Ar 1 rs, there are a smooth, affine A-curve C, an A-finite closed subscheme Z Ă C, and a section s P CpAq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' a reductive C-group scheme G such that s˚G » GA, a G -torsor rF over Cr 1 rs :“ C ˆA Ar 1 rs such that rF|Cr 1 r szZr 1 r s is trivial and ps|Ar 1 r sq˚p r Fq » rP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2, P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rP) extends to a G-torsor P0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', Ă P0) over an open neighbourhood W Ă X of x (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', an open neighbourhood Ă W Ă X of SpecpAr 1 rsq) such that codimppXzWqK, XKq ě 3 and codimppXzWqs, Xsq ě 2 for all s P SpecpRq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and codimppXzĂ WqK, XKq ě 3 and codimppXzĂ Wqs, Xsq ě 2 for all s P SpecpRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Here, K is the fraction field of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let z Ă X be the set of maximal points of the R-fibers of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' the above codimension bounds implies z Ă W (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', z Ă Ă W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(iii), the semilocal ring OX,z, 46 and hence also OX,zr 1 rs, is a Prüfer domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the Grothendieck–Serre on semilocal Prüfer schemes (Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1), the generically trivial G-torsor pP0q|OX,z (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', pĂ P0q|OX,zr 1 r s) is actually trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus there exists a closed subscheme Y Ă X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rY Ă X) that avoids all the maximal points of R-fibers of X such that the restriction pP0q|XzY (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', pĂ P0q|pXz rY qr 1 r s) is trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' such a Y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rY ) is R-fiberwise of codimension ą 0 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now, we treat the two cases (i)–(ii) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (i) By the above, XzW is R-fiberwise of codimension ě 2 in X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' a fortiori, the same codimension bound holds for Y zW in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, we can apply Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (vii) to obtain an affine open S Ă Ad´1 R , an affine open neighbourhood U Ă W of x, and a smooth morphism π: U Ñ S of pure relative dimension 1 such that U X Y is S-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let τ : C :“ U ˆS Spec A Ñ Spec A be the base change of π to Spec A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let Z and F be the pullbacks of U X Y and pP0q|U under pr1 : C Ñ U, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, via τ, C is a smooth affine A-curve, Z Ă C is a A-finite closed subscheme, and F is a G :“ pr˚ 1pGUq-torsor that trivializes over CzZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Finally, the diagonal in C induces a section s P CpAq with s˚F » P (as s˚G “ GA-torsors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) Since SpecpAr 1 rsq consists of points of Xr 1 rs :“ X ˆR Rr 1 rs that specializes to some point of x, we deduce from the inclusion SpecpAr 1 rsq Ă Ă W that no points of pXzĂ Wqr 1 rs “ Xr 1 rszĂ Wr 1 rs specializes to any points of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence, the closure pXzĂ Wqr 1 rs (in X) is disjoint from x, so Ă W 1 :“ XzpXzĂ Wqr 1 rs is an open neighbourhood of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Notice that X is topological Noetherian, because its R-fibers are projective varieties over fields, and by our assumption SpecpRq has a finite underlying space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since by the above pXzĂ Wqr 1 rs is Rr 1 rs-fiberwise of codimension ě 2 in Xr 1 rs, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(i) applied to the closures of the (finitely many) maximal points of pXzĂ Wqr 1 rs, the closure pXzĂ Wqr 1 rs “ XzĂ W 1 is R-fiberwise of codimension ě 2 in X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' a fortiori, the same holds for rY zĂ W 1 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consequently, we can apply Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (vii) to obtain an affine open rS Ă Ad´1 R , an affine open neighbourhood rU Ă Ă W 1 of x, and a smooth morphism rπ : rU Ñ rS of pure relative dimension 1 such that rU X rY is rS-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Notice that rUr 1 rs Ă Ă W 1r 1 rs “ Ă Wr 1 rs, so we have the restriction pĂ P0q| rUr 1 r s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let τ : C :“ rU ˆ rS SpecpA Ñ SpecpA be the base change of rπ to SpecpA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let Z be the pullback of rU X rY under pr1 : C Ñ rU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let rF be the pullback of pĂ P0q| rUr 1 r s under pr1 : Cr 1 rs Ñ rUr 1 rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, via τ, C is a smooth affine A-curve, Z Ă C is a A-finite closed subscheme, and r F is a G :“ pr˚ 1pG rUq-torsor over Cr 1 rs that trivializes over Cr 1 rszZr 1 rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Finally, the diagonal in C induces a section s P CpAq with s˚ Ar 1 r sp r Fq » rP, and s˚G “ GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Proof of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By a standard limit argument involving Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, one easily reduces to the case when R has finite Krull dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now, let P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rP) be a generically trivial G-torsor over A :“ OX,x (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', over Ar 1 rs) which we want to trivialize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let d be the relative dimension of X over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If d “ 0, then A and Ar 1 rs are semilocal Prüfer domains, so, by the Grothendieck–Serre on semilocal Prüfer schemes (Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1), the torsors P and rP are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hence we may assume that d ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, by Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, there are a smooth, affine A-curve C, an A-finite closed subscheme Z Ă C, a section s P CpAq, a reductive C-group scheme G with s˚G » GA, a G -torsor F over C that trivializes over CzZ such that s˚F » P, and a G -torsor rF over Cr 1 rs that trivializes over Cr 1 rszZr 1 rs such that ps|Ar 1 r sq˚p rFq » rP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (i), the G-torsor s˚F » P is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (ii), in the case ps|Ar 1 r sq˚pG q » GAr 1 r s is totally isotropic, the GAr 1 r s-torsor ps|Ar 1 r sq˚p r Fq » rP is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Torsors under a constant reductive group scheme In this section we prove the following variant of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, in which the R-smooth scheme X need not be proper, but the reductive group scheme G is supposed to descend to the Prüfer ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus, we established the Grothendieck–Serre conjecture and a version of Nisnevich conjecture for ‘constant’ 47 reductive group schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As for the proof, we use a variant of Lindel’s Lemma (Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1) and glueing techniques to reduce to the case already settled by Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R, a nonzero element r P R, an irreducible affine R- smooth scheme X, a finite subset x Ă X, and a reductive R-group scheme G, (i) any generically trivial G-torsor over A :“ OX,x is trivial, that is, ker ` H1pA, Gq Ñ H1pFrac A, Gq ˘ “ t˚u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) if GRr 1 r s is totally isotropic, then any generically trivial G-torsor over Ar 1 rs is trivial, that is, ker ` H1pAr 1 rs, Gq Ñ H1pFrac A, Gq ˘ “ t˚u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rP) be a generically trivial G-torsor over A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', over Ar 1 rs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By shrinking X around x, we may assume that P is defined over the whole X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rP is defined over the whole Xr 1 rs :“ X ˆR Rr 1 rs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let d be the relative dimension of X over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As noted by ˇCesnaviˇcius, since it suffices to argue that P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rP) is trivial Zariski semilocally on X, we may replace X by X ˆR AN R for large N to assume that d ą # x: by pulling back along the zero section X Ñ X ˆR AN R , the Zariski semilocal triviality of PXˆRAN R (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rPXr 1 r sˆRAN R ) on X ˆR AN R implies that of P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rP) on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By specialization, we may assume that each point of x is closed in the corresponding R-fiber of X (but not necessarily lies in the closed R-fibers of X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Our goal is to show that P|A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rP|Ar 1 r s) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If d “ 0, then A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', Ar 1 rs) is a semilocal Prüfer domain, so, by the Grothendieck–Serre conjecture on semilocal Prüfer schemes (Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1), the torsor P|A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rP|Ar 1 r s) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus we may assume that d ą 0 for what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote by π : X Ñ S :“ SpecpRq the structural morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let y be the set of maximal points of the R-fibers of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' No points of x specializes to any point of y, that is, x X y “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(iii), for any y P y, OX,y is a valuation ring having the same value group as OS,πpyq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' in particular, the map πy : Spec OX,y Ñ Spec OS,πpyq induced by π is a homeomorphism, and is thus injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume by contradiction that x P x specializes to y P y, so Spec OXπpxq,x is a subset of Spec OX,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since the image of Spec OXπpxq,x under πy is the singleton tπpxqu, by the injectivity of πy, we deduce that dim OXπpxq,x “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' This contradicts the fact dim OXπpxq,x “ d ą 0 (because by our assumption x is a closed point in the corresponding π-fiber).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(iii) again, the semilocal ring OX,y, and hence also OX,yr 1 rs, is a Prüfer domain, so, by the Grothendieck–Serre conjecture on semilocal Prüfer schemes (Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1), the generically trivial G- torsor P|OX,y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rP|OX,yr 1 r s) is actually trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, using the above claim and prime avoidance, we can find an element a P ΓpX, OXq such that, denoting Y :“ V paq Ă X, then x Ă Y , y X Y “ H, and the restriction P|XzY (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rP|pXzY qr 1 r s) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (We just take a “ a1a2, where a1 is an element such that y X V pa1q “ H and P|XzV pa1q (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rP|pXzV pa1qqr 1 r s) is trivial, and a2 is delivered from prime avoidance utilizing the fact x X y “ H so that x Ă V pa2q and y X V pa2q “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=') Since d ą # x, we may apply Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 to obtain an affine open neighbourhood W Ă X of x, an affine open subscheme U Ă Ad R, and an étale surjective R-map f : W Ñ U such that the restriction f|WXY is a closed immersion and f induces a Cartesian square W X Y W W X Y U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' f Applying p´q ˆR Rr 1 rs yields a similar Cartesian square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By glueing Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 (ii), 48 (i) we may (non-canonically) glue P|W and the trivial G-torsor over UzfpW X Y q to descend P|W to a G-torsor Q over U that trivializes over UzfpW X Y q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since U has a smooth, projective compactification Pd R, we may apply Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (i) to deduce that Q|OU,fpxq is trivial, so P|A “ P|OW,x is trivial, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) we may (non-canonically) glue rP|Wr 1 r s and the trivial G-torsor over pUzfpW X Y qqr 1 rs to descend rP|Wr 1 r s to a G-torsor rQ over Ur 1 rs that trivializes over Ur 1 rszfpW XY qr 1 rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since U has a smooth, projective compactification Pd R, we may apply Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (ii) to conclude that rQ|OU,fpxqr 1 r s is trivial, so rP|Ar 1 r s “ rP|OW,xr 1 r s is trivial, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Torsors under a quasi-split reductive group scheme In this section we study generically trivial torsors under quasi-split reductive group schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The main result is the following Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, in which (i) is a version of Nisnevich conjecture that is inspired by the recent preprint of ˇCesnaviˇcius [Čes22c, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3 (2)], who proved it in the case R is a Dedekind domain, and (ii) is the Grothendieck–Serre conjecture over one-dimensional Prüfer bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As for the proof, we will follow the strategy of [Čes22a] (with its earlier version given by Fedorov [Fed22b]), which goes through because the main tools, such as toral version of purity (Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5) and the Grothendieck–Serre conjecture (Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2(i)) in our context, are available now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R with fraction field K, an irreducible, semilocal, and essentially smooth R-algebra A, and a quasi-split reductive A-group scheme G, (i) every generically trivial G-torsor over A bR K is trivial, that is, ker ` H1pA bR K, Gq Ñ H1pFrac A, Gq ˘ “ t˚u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) if R has Krull dimension 1, then every generically trivial G-torsor is trivial, that is, ker ` H1pA, Gq Ñ H1pFrac A, Gq ˘ “ t˚u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We start with the following consequence of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, which is the key geometric input permitting a series of reductions that eventually lead to Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [Čes22a, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For (i) a semilocal Prüfer domain R of Krull dimension 1 with fraction field K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) a smooth, faithfully flat, R-algebra A of pure relative dimension d ě 1 over R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iii) a finite subset x Ă X :“ Spec A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iv) a closed subscheme Y Ă X that satisfies codimpYK, XKq ě 2 and codimpYs, Xsq ě 1 for all s P Spec R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' there are an affine open U Ă Spec A containing x, an affine open S Ă Ad´1 R , and a smooth R-morphism π : U Ñ S of relative dimension 1 such that Y X U is S-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Moreover, if in piq R is allowed to be of arbitrary finite Krull dimension, then the same conclusion holds provided pivq is replaced by the stronger assumption that Y is R-fiberwise of codimension ě 2 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Choosing an embedding of X into some affine space over R and taking schematic closure in the corresponding projective space, we get a projective compactification X of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since X is flat and projective over R, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(i), all its R-fibers have the same dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote by Y Ă X the schematic closure of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' To apply Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1 (vii) and conclude, in which X is X here, W is X here, and Y is Y here, we need to check that the boundary Y zY is R-fiberwise of codimension ě 2 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [SP, 01R8], set-theoretically we have Y “ Ť y tyu, where y runs through the generic points of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In the case Y is R-fiberwise of codimension ě 2 in X, the same holds for Y in X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' a fortiori, Y zY is R-fiberwise of codimension ě 2 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Indeed, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(i), X has equal R-fiber dimension d and 49 all non-empty R-fibers of tyu have the same dimension, so, if y lies over sy P Spec R, then codimptyus, Xsq “ codimptyusy, Xsyq ě 2 for any specialization sy ù s P Spec R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Next, we assume that R has Krull dimension 1 and Y is of codimension ě 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', ě 1) in the generic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', closed) R-fiber of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If y P Yη, then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(i) again, we see that codimptyus, Xsq “ codimptyuη, Xηq ě 2 for all s P Spec R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' a fortiori, the contribution of such a y to the R-fiber codimension of Y zY in X is ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Otherwise, y lies over a height 1 prime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', a closed point) s1 P Spec R, then tyus1 “ tyu Ă Ys1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' by assumption codimpYs1, Xs1q “ codimpYs1, Xs1q ě 1, so we have codimptyus1, Xs1q ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' But since the generic point y of tyus1 is not contained in Y zY , we deduce that the contribution of such a y to the s1-fiber codimension of Y zY in X is again ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3 (Lifting the torsor to a smooth relative curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [Čes22a, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal Prüfer domain R with fraction field K, the semilocalization A of an irreducible, R-smooth algebra A1 at a finite subset x Ă SpecpA1q, and a quasi-split reductive A-group scheme G with a Borel subgroup B, (1) given a generically trivial G-torsor PK over AK :“ A bR K, there are (i) a smooth, affine relative A-curve C with a section s P CpAq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) an A-finite closed subscheme Z Ă C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iii) a quasi-split reductive C-group scheme G with a Borel subgroup B Ă G whose s-pullback is B Ă G, compatible with the quasi-pinnings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iv) a G -torsor PK over CK :“ C ˆR K whose sAK-pullback is PK such that PK reduces to a radupG q-torsor over CKzZK (here sAK stands for the image of s in CpAKq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (2) if R has Krull dimension 1, given a generically trivial G-torsor P, then there are (i) a smooth, affine relative A-curve C with a section s P CpAq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (ii) an A-finite closed subscheme Z Ă C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iii) a quasi-split reductive C-group scheme G with a Borel subgroup B Ă G whose s-pullback is B Ă G, compatible with the quasi-pinnings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (iv) a G -torsor P whose s-pullback is P such that P reduces to a radupG q-torsor over CzZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In case (1) we can first use a limit argument involving Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3 to reduce to the case when R has finite Krull dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' If A1 is of relative dimension 0 over R, then AK “ FracpAq and A is a semilocal Prüfer domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus, PK is trivial, and, by the Grothendieck–Serre conjecture on semilocal Prüfer schemes (Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1), P is also trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In this case we simply take C “ A1 A, s “ 0 P A1 ApAq, Z “ H, pG , Bq “ pGA1 A, BA1 Aq, and PK “ pPKqA1 AK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P “ PA1 A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus, for what follows, we can assume that the relative dimension of A1 over R is d ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By spreading out and localizing A1, we may assume that our quasi-split G (in particular, the Borel B) and torsor P all live over A1, and PK live over A1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By [SGA 3III new, Exposé XXVI, Corollaire 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6 and Lemme 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='20], the quotient PK{BK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P{B) is representable by a smooth projective scheme over A1 K (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', over A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now we treat the cases (1)-(2) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (1) By the generic triviality of PK, applying the valuative criterion of properness to PK{BK Ñ SpecpA1 Kq yields a closed subscheme YK Ă SpecpA1 Kq of codimension ě 2 such that PK{BK Ñ SpecpA1 Kq has a section over SpecpA1 KqzYK that lifts to a generic section of PK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In other words, pPKqSpecpA1 KqzYK reduces to a generically trivial BSpecpA1 KqzYK-torsor P B K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consider the A1-torus T :“ B{ radupBq and the induced T -torsor P T K :“ P B K { radupBqK over SpecpA1 KqzYK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 50 Since P T K is generically trivial, by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2, it extends to a T -torsor Ă P T K over SpecpA1qzF for a closed subscheme F Ă SpecpA1q satisfying codimpFK, SpecpA1qKq ě 2 and codimpFs, SpecpA1qsq ě 1 for all s P SpecpRq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' by purity for tori (Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4), this torsor further extends to the whole SpecpA1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As Ă P T K is generically trivial, by the Grothendieck–Serre conjecture for tori (Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2(i)), we may localize A1 around x to assume that Ă P T K, and hence also P T K, is already trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In other words, pPKqSpecpA1 KqzYK reduces to a radupBq-torsor over SpecpA1 KqzYK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote by Y the schematic closure of YK in SpecpA1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(i), it is R-fiberwise of codimension ě 2 in SpecpA1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Applying Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 to the R-smooth algebra A1 and the closed subscheme Y Ă SpecpA1q, we obtain an affine open U Ă SpecpA1q containing x, an affine open S Ă Ad´1 R , and a smooth R-morphism π : U Ñ S of relative dimension 1 such that Y X U is S-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Recall that A is the semilocal ring of U at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote C :“ U ˆS Spec A and Z :“ pY X Uq ˆS Spec A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then C is a smooth affine relative A-curve, the diagonal in C induces a section s P CpAq, and the closed subscheme Z Ă C is A-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' So (1)(i) and (1)(ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let B Ă G be the pullback of BU Ă GU under the first projection pr1 : C Ñ U, and let PK be the pullback of pPKqUK under the first projection pr1 : CK Ñ UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, PK is a G -torsor over CK, and, by construction, the s-pullback (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', sAK- pullback) of B Ă G (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', of PK) is B Ă G (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', PK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Finally, since PK reduces to a radupBq-torsor over SpecpA1 KqzYK, PK reduces to a radupBq-torsor over CKzZK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' So (1)(iii) and (1)(iv) also hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (2) Recall that, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1(iii), the local rings of all maximal points of R-fibers of SpecpA1q are valuation rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the generic triviality of P, applying the valuative criterion of properness to P{B Ñ SpecpA1q yields a closed subscheme Y Ă SpecpA1q, which avoids all the codimension 1 points of the generic fiber SpecpA1 Kq and all the maximal points of R-fibers of SpecpA1q, such that P{B Ñ SpecpA1q has a section over SpecpA1qzY that lifts to a generic section of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In other words, Y satisfies codimpYK, SpecpA1qKq ě 2 and codimpYs, SpecpA1qsq ě 1 for all s P SpecpRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Therefore, PSpecpA1qzY reduces to a generically trivial BSpecpA1qzY -torsor P B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consider the A1-torus T :“ B{ radupBq and the induced T -torsor P T :“ P B{ radupBq over SpecpA1qzY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By purity for tori (Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4), P T extends to a T -torsor Ă P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' As Ă P T is generically trivial, by the Grothendieck–Serre conjecture for tori (Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2(i)), we may localize A1 around x to assume that Ă P T , and hence also P T , is already trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In other words, PSpecpA1qzY reduces to a radupBqSpecpA1qzY - torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now, applying Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2 to the R-smooth algebra A1 and the closed subscheme Y Ă SpecpA1q, we obtain an affine open U Ă SpecpA1q containing x, an affine open S Ă Ad´1 R , and a smooth R-morphism π : U Ñ S of relative dimension 1 such that Y X U is S-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Recall that A is the semilocal ring of U at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Denote C :“ U ˆS Spec A and Z :“ pY X Uq ˆS Spec A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then C is a smooth affine relative A-curve, the diagonal in C induces a section s P CpAq, and the closed subscheme Z Ă C is A-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' So (2)(i) and (2)(ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let B Ă G and P be the pullback of BU Ă GU and PU under the first projection pr1 : C Ñ U, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, P is a G -torsor over C, and, by construction, the s-pullback of B Ă G and P are B Ă G and P, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Finally, since P reduces to a radupBq-torsor over SpecpA1qzY , P reduces to a radupBq-torsor over CzZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' So (2)(iii) and (2)(iv) also hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4 ([Čes22a, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' For a semilocal ring A whose local rings are geometrically uni- branch, an ideal I Ă A, reductive A-groups G and G1 that on geometric A-fibers have the same type, fixed quasi-pinnings of G and G1 extending Borel A-subgroup B Ă G and N 1 Ă G1 and an A{I-group isomorphism ι : GA{I „ ÝÑ G1 A{I respecting the quasi-pinnings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' in particular, ιpBA{Iq “ B1 A{I, 51 there are (i) a faithfully flat, finite, étale A-algebra rA equipped with an A{I-point a : rA ։ A{I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' and (ii) an rA-group isomorphism rι: G r A „ ÝÑ G1 r A respecting the quasi-pinnings such that a˚prιq “ ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Notice that the original version [Čes22a, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1] assumed further A to be Noetherian, but the Noetherianess of A was not used anywhere in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5 (Changing the relative curve C to equate G and GC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [Čes22a, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In the setting of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, for both cases (1) and (2) we may replace C by an étale neighbourhood of impsq to achieve further that pG , Bq “ pGC, BCq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Consider the semilocalization SpecpDq of C at the closed points of impsq Y Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' since C is normal, all the local rings of D are geometrically unibranch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' The image of the section s : Spec A Ñ SpecpDq gives rise to a closed subscheme SpecpD{Iq Ă SpecpDq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the conclusion of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, the restriction of BD Ă GD and BD Ă GD to SpecpD{Iq agree with each other in a way compatible with their quasi-pinnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Thus, by Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4, there is a faithfully flat, finite, étale D-algebra rD, a point rs : rD ։ D{I » A lifting s : D ։ D{I » A such that B r D Ă G r D is isomorphic to B r D Ă G r D compatibly with the fixed identification of rs-pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We then spread out the finite étale morphism Specp rDq Ñ SpecpDq to a finite étale morphism rC Ñ C1 for an open C1 Ă C that contains impsq Y Z, while preserving an rs P rCpAq, and an isomorphism between B r C Ă G rC and B r C Ă G r C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now it remains to replace C, s, Z and PK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P) by rC, rs, Z ˆC rC and pPKq r CK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P rC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6 (Changing the relative smooth curve C for descending to A1 A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [Čes22a, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In the setting of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, for both cases (1) and (2), in addition to pG , Bq “ pGC, BCq, we may change C to achieve further that there is a flat A-map C Ñ A1 A that maps Z isomorphically to a closed subscheme Z1 Ă A1 A with Z » Z1 ˆA1 A C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Assume that, in both cases (1) and (2) of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, we have achieved the conclusion of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' We have the data of a smooth affine relative A-curve C, a section s P CpAq, and an A-finite closed subscheme Z Ă C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' replacing Z by Z Y impsq, we may assume that s factors through Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' However, in general, the A-finite scheme Z may be too large to embed into A1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' (For instance, if R “ k is a finite field, then Z can’t be embedded into A1 k as soon as 7 Zpkq ą 7 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=') For this, we first apply Panin’s ‘finite fields tricks’ [Čes22a, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4] to obtain a finite morphism rC Ñ C that is étale at the points in ˜Z :“ rC ˆC Z such that s lifts to rs P rCpAq, and there are no finite fields obstruction to embedding rZ into A1 A in the following sense: for every maximal ideal m Ă A, 7 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' z P rZκpmq : rκpzq : κpmqs “ d ) ă 7 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' z P A1 κpmq : rκpzq : κpmqs “ d ) for every d ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Then, by [Čes22a, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3], there are an affine open C1 Ă rC containing imprsq, a quasi-finite, flat A-map C1 Ñ A1 A that maps Z isomorphically to a closed subscheme Z1 Ă A1 A with Z » Z1 ˆA1 A C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to replace C by C1, Z by rZ, s by rs, PK by pPKqC1 K (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P by PC1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7 (Descend to A1 A via patching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [Čes22a, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' In the setting of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='3, for both cases (1) and (2), we may achieve further that pG , Bq “ pGC, BCq, C “ A1 A, and s “ 0 P A1 ApAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the reduction given in Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='6, we have a flat A-curve C, a section s P CpAq, an A-finite closed subscheme Z Ă C, a quasi-finite, affine, flat A-map C Ñ A1 A that maps Z isomorphically to a closed subscheme Z1 Ă A1 A with Z “ Z1 ˆA1 A C, and a G-torsor PK over CK whose sAK-pullback is PK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', a G-torsor P over C whose s-pullback is P) and whose restriction to CKzZK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', CzZ ) reduces to a radupBq-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Now, since Z “ Z1 ˆA1 A C » Z1, [Čes22a, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2] implies the pullback maps H1pA1 AzZ1, radupGqq ։ H1pCzZ, radupGqq 52 and H1pA1 AKzZ1 K, radupGqq ։ H1pCKzZK, radupGqq are surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Combining these, we see that PK|CKzZK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P|CzZ) descends to a G-torsor QK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', Q) over A1 AKzZ1 K (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', A1 AzZ1) that reduces to a radupBq-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the glueing Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='2(ii), we may (non-canonically) glue PK with QK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P with Q) to descend PK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P) to a G-torsor Ą PK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', rP) over A1 AK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', over A1 A) that reduces to a radupBq-torsor over A1 AKzZ1 K (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', over A1 AzZ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' It remains to replace C by A1 A, Z by Z1, s P CpAq by its image in A1 ApAq, and PK by Ą PK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P by rP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Finally, by shifting, we may assume even that s “ 0 P A1 ApAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ Proof of Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Let PK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P) be a generically trivial GAK-torsor (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', G-torsor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By the reduction Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='7, we get an A-finite closed subscheme Z Ă A1 A, and a GA1 AK -torsor PK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', GA1 A-torsor P) whose pullback along the zero section is PK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P) such that pPKq|A1 AK zZK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P|A1 AzZ) reduces to a radupBq-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Since any A-finite closed subscheme of A1 A is contained in tf “ 0u for some monic polynomial f, we may enlarge Z to assume that A1 AzZ is affine, to the effect that any radupBq-torsor over A1 AKzZK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', over A1 AzZ), such as pPKq|A1 AK zZK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P|A1 AzZ), is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' By section Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1, the pullback of PK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', of P) along the section s P A1 ApAq is trivial, that is, PK (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=', P) is trivial, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' □ References [BouAC] Nicolas Bourbaki, Commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Translated from the French, Reprint of the 1989 English translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [EGA I] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Grothendieck and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Dieudonné, Éléments de géométrie algébrique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Le langage des schémas, Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hautes Études Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 4 (1960), 228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' MR0217083 (36 #177a) [EGA IV2] Alexander Grothendieck and Jean Dieudonné, Éléments de géométrie algébrique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Étude locale des schémas et des morphismes de schémas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' II, Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hautes Études Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 24 (1965), 231 (French).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' MR0199181 (33 #7330) [EGA IV3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Grothendieck and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Dieudonné, Éléments de géométrie algébrique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Étude locale des schémas et des morphismes de schémas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' III, Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hautes Études Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 28 (1966), 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' MR0217086 (36 #178) [EGA IV4] Alexander Grothendieck and Jean Alexandre Eugène Dieudonné, Éléments de géométrie algébrique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Étude locale des schémas et des morphismes de schémas IV, Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Hautes Études Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 32 (1967), 361 (French).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' MR0238860 (39 #220) [SGA 2new] Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 4, Société Mathématique de France, Paris, 2005 (French).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Séminaire de Géométrie Algébrique du Bois Marie, 1962;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Augmenté d’un exposé de Michèle Raynaud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [With an exposé by Michèle Raynaud];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' With a preface and edited by Yves Laszlo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Revised reprint of the 1968 French original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' MR2171939 [SGA 3II] Schémas en groupes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Sémi- naire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Dirigé par M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Demazure et A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Grothendieck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Lecture Notes in Mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 152, Springer-Verlag, Berlin-New York, 1970 (French).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' MR0274459 (43 #223b) [SGA 3III new] Philippe Gille and Patrick Polo (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' ), Schémas en groupes (SGA 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Tome III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Structure des schémas en groupes réductifs, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 8, Société Mathé- matique de France, Paris, 2011 (French).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Séminaire de Géométrie Algébrique du Bois Marie 1962–64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [Alge- braic Geometry Seminar of Bois Marie 1962–64];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A seminar directed by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Demazure and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Grothendieck with the collaboration of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Artin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Bertin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Gabriel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Raynaud and J-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Serre;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Revised and annotated edition of the 1970 French original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' MR2867622 [SGA 4II] Théorie des topos et cohomologie étale des schémas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Tome 2, Lecture Notes in Mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 270, Springer-Verlag, Berlin, 1972 (French).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Dirigé par M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Artin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Grothendieck et J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' L.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1371, Berlin etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' : Springer-Verlag, 1989 (English).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [GGMB14] Ofer Gabber, Philippe Gille, and Laurent Moret-Bailly, Fibrés principaux sur les corps valués henséliens, Algebr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 1 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 5, 573–612 (French, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Théorie cohomologique, Dix exposés sur la cohomologie des schémas, 1968, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 67–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' [Gro68b] , Le groupe de Brauer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Exemples et compléments, Dix Exposés sur la Cohomologie des Schémas, North-Holland, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' USA 44 (1958), 791–796, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='791 (English).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' 55 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Petersburg branch of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Steklov Mathematical Institute, Fontanka 27, 191023 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content=' Petersburg, Russia Email address: guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='ning@eimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='ru Department of Mathematics, Southern University of Science and Technology, Shenzhen, China Email address: liufei54@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} +page_content='cn 56' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptFPT4oBgHgl3EQf7zXe/content/2301.13206v1.pdf'} diff --git a/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf b/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..962da7ebedab4f61c1f126b2f66932eabbb2bc03 --- /dev/null +++ b/qNE0T4oBgHgl3EQfaQBn/content/2301.02332v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ef3d44271e83c8306fbf42dcb9b9e0f9446b47927854dc729f9325fdc73abe78 +size 806267 diff --git a/qNE0T4oBgHgl3EQfaQBn/vector_store/index.faiss b/qNE0T4oBgHgl3EQfaQBn/vector_store/index.faiss new file 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/dev/null +++ b/qNFKT4oBgHgl3EQfzS7F/content/tmp_files/2301.11911v1.pdf.txt @@ -0,0 +1,1304 @@ +Multi-dimensional concept discovery (MCD): A unifying +framework with completeness guarantees +Johanna Vielhaben +johanna.vielhaben@hhi.fraunhofer.de +AI department +Fraunhofer Heinrich-Hertz-Institut +Stefan Blücher +bluecher@tu-berlin.de +Machine Learning Group +TU Berlin +Nils Strodthoff +nils.strodthoff@uol.de +Division AI4Health +Oldenburg University +Abstract +The completeness axiom renders the explanation of a post-hoc XAI method only locally +faithful to the model, i.e. for a single decision. For the trustworthy application of XAI, in +particular for high-stake decisions, a more global model understanding is required. Recently, +concept-based methods have been proposed, which are however not guaranteed to be bound +to the actual model reasoning. To circumvent this problem, we propose Multi-dimensional +Concept Discovery (MCD) as an extension of previous approaches that fulfills a completeness +relation on the level of concepts. Our method starts from general linear subspaces as con- +cepts and does neither require reinforcing concept interpretability nor re-training of model +parts. We propose sparse subspace clustering to discover improved concepts and fully lever- +age the potential of multi-dimensional subspaces. MCD offers two complementary analysis +tools for concepts in input space: (1) concept activation maps, that show where a concept is +expressed within a sample, allowing for concept characterization through prototypical sam- +ples, and (2) concept relevance heatmaps, that decompose the model decision into concept +contributions. Both tools together enable a detailed understanding of the model reasoning, +which is guaranteed to relate to the model via a completeness relation. This paves the way +towards more trustworthy concept-based XAI. We empirically demonstrate the superiority +of MCD against more constrained concept definitions. +1 +Introduction +Explainable AI (XAI) allows to peek insight the black box of inherently complex deep learning models. Local +interpretability methods are particular valuable, as they measure attributions for an individual instance, +which are easily comprehensible for any kind of end-users, see (Covert et al., 2021; Lundberg & Lee, 2017; +Montavon et al., 2018; Samek et al., 2021) for reviews. +For example, local methods make a prediction +interpretable on the level of single images or individual bank customers for an image or credit risk classifier, +respectively. Importantly, the commonly employed completeness axiom (attributions sum up to the model +prediction) ensures a meaningful interpretation of attributions (Lundberg & Lee, 2017; Sundararajan et al., +2017). However, to actually comprehend the model reasoning we require a global model understanding, +which reliably explains the model behavior across multiple instances (e.g. a group of female vs. male bank +customers). We stress that it is not viable to require an end-user to aggregate local attributions into common +model features (concepts). Such a procedure is prone to human confirmation bias and it is not clear how +the imagined concepts align with the actual model reasoning. This urges for novel local and concept-based +interpretability methods, which allow to understand shared model structures (used across multiple samples) +1 +arXiv:2301.11911v1 [cs.LG] 27 Jan 2023 + +D3: arbitrary directions +D4: linear subspaces of +arbitrary dimensionality +D2: orthogonal directions +allow rotation +allow non-orthogonality +allow multi-dimensionality +D1: single neurons +Figure 1: We strive for the most general decomposition of the hidden feature space, spanned by the neurons +c1 +c1 +c1,c2 +c2 +c2,c3 +c3 +c3, into linear structures that form the concepts Ci. The most constrained approach is to identify +concepts with single neurons (D1), i.e. directions in feature space aligned with the neuron axes. If one +allows for an arbitrary rotations of the concept directions, one arrives at D2. Leaving aside the orthogonality +constraint, D3 allows concepts to form arbitrary directions is feature space. Finally, allowing concepts to +form multi-dimensional subspaces, we arrive at the most general approach D4. +Previous concept-based +methods are based on D1, D2 and D3. We choose the most general approach D3, to discover concepts that +are true-to-the-model. +for an individual instance. This idea was first formalized by Kim et al. (2018) and further developed by +ACE (Ghorbani et al., 2019) and its successors (Yeh et al., 2020; Zhang et al., 2021). Crucially, our work (1) +re-introduces completeness within the context of concept-based explanations. Thereby, concepts obtained +within our multi-dimensional concept discovery (MCD) scheme are locally and globally interpretable in terms +of a well-defined completeness decomposition. We outline the benefits of MCD in the following paragraphs. +Novel concepts as multi-dimensional subspaces +Indisputably, concept discovery in neural networks is inherently linked to structures in intermediate feature +layers. In Figure 1, we illustrate different approaches to decompose the hidden feature space into meaningful +concepts, which are mathematically formalized as linear structures. The most constrained definition (left +most panel, D1) is to directly identify concepts with a neural directions (Bau et al., 2017). This means that +a concept is a one-dimensional subspace which aligns with the unit axes in feature space. A slightly more +general definition is to use an arbitrary rotated orthogonal one-dimensional decomposition of the feature +space (D2). Such a concept decomposition can be obtained via a principal component analysis (PCA) of the +feature space (Zhang et al., 2021). Going one step further, we disregard the orthogonality constraint and +allow arbitrary directions in feature space (D3) (Ghorbani et al., 2019; Kim et al., 2018; Yeh et al., 2020; +Zhang et al., 2021). Thereby, we can characterize related concepts which are linearly independent but not +orthogonal (for example different parts of an animal). Allowing for arbitrary multi-dimensional subspaces +unfolds the most general definition of a linear decomposition. Thus, this general approach enables true-to- +the-model concepts as it allows to capture any meaningful linear structure within the hidden feature layer +(benefit 1). +Multi-dimensionality ensures concise explanations Concepts strive to organize the information about +the model reasoning in a concise and accessible manner. As previously outlined, aggregating many different +explanations into a comprehensive model understanding is a challenge for humans. Thus, it is desirable +to grasp the actual model reasoning with a limited number of concepts. Phrased differently, we want to +cover the relevant feature space with only a few concepts and avoid fragmentation into a large number of +low/one-dimensional subspaces. We formalize the relevant feature subspace based on its impact on the model +prediction. Via this intuition we can define a concept completeness score in Section 2.3, which measures the +fraction of model prediction jointly covered by all concepts. Intuitively, multi-dimensional concept subspaces +reach a certain level of completeness with a smaller number of concepts than one-dimensional concepts and +thus deliver more concise explanations (benefit 2). +Sparse subspace clustering for better concept discovery The principles of MCD do not rely on any +particular (clustering) algorithm for concept discovery. We argue in favor of clustering approaches that place +2 + +the fewest restrictions on the discovery process, in order to fully realize the promise of multi-dimensional +concepts. A clustering algorithm that fits the above objective extremely well is sparse subspace clustering +(SSC) (Elhamifar & Vidal, 2013), as it is tailored to discover otherwise unconstrained linear subspaces. Thus, +there is no need for additional measures to reinforce the interpretability of concepts. In contrast, previous +methods use techniques like superpixels in input space (Ghorbani et al., 2019) or regularizers to enforce +concept dissimilarity (Yeh et al., 2020) for this purpose. This potentially breaks the connection between +discovered concepts and the actual reasoning structures in feature space. +Concept decomposition To present the discovered concepts in a human-comprehensible form, it is custom +to visualize regions in input space in which the concept is activated. We assume that the activations in a +hidden layer form a spatially resolved map of feature vectors (given e.g. for convolutional or transformer +models with skip-connections). We uniquely decompose each hidden feature vector into its concept parts +and measure the length of these parts to assess whether a particular concept is activated. +Upsampling +these concept activations from hidden to input layer finally creates comprehensible concept activation maps +in input space. We now restrict to a high-level feature layer which is only succeeded by linear operations +(e.g. a linear classification head with global pooling). This enables to also uniquely decompose the model +prediction into concept parts and thereby define a concept relevance heatmap. Then, the relevances follow +a completeness relation (benefit 3), i.e. summing up the relevance from all concepts is guaranteed to equal +the final prediction. +Concept relevance heatmaps show a spatially resolved decomposition of the model +prediction into concepts and show how indicative a particular concept instantiation is for the predicted class. +We stress, that the concept relevance heatmaps follow directly from the decomposition into concept parts +and do not invoke any additional XAI method nor retraining model parts. Thus, MCD can completely +capture the model reasoning solely in terms of linear operations on concept subspaces. +In summary, MCD is a consistent framework to discover true-to-the-model concepts, which are guaranteed to +rely on the actual model reasoning via the completeness relation. Visualizing activation maps and relevance +heatmaps for prototype samples offers the possibility to characterize concepts more closely. We see the main +utility of MCD in the domain of model understanding and certification. +Concepts provide insights into +model behavior that generalize across samples and are therefore a valuable tool for systematic investigations +of spurious correlations (model biases) (Lapuschkin et al., 2019; Palatnik de Sousa et al., 2021; Weber et al., +2021), as well as for scientific discovery (Blücher et al., 2020; Hägele et al., 2020; McGrath et al., 2022; +Šarčević et al., 2022), where the model serves as proxy for the unknown relationships in the data. This work +builds on an earlier manuscript (Vielhaben et al., 2022) and crucially extends it. +2 +Multi-dimensional Concept Discovery (MCD) +We organize this methodological section into three parts: First, we introduce our novel concept definition. +Second, we describe practical concept discovery procedures that align with this definition. Most prominently, +we introduce SSC but also elaborate on possible alternatives. Third, we introduce a concept decomposition +and discuss how to construct local and global concept importance that fulfill a concept completeness relation. +Figure 2 presents a schematic summary of our MCD framework. +2.1 +Concept definition +Concepts are inherently tied to the hidden representations of intermediate feature layers. In Figure 1, we +illustrate the structures that concepts could possibly form in hidden feature space: from single directions +(D1-D3) to linear subspaces (D4). MCD opts for the most general structure, i.e., arbitrarily orientated +multi-dimensional linear subspaces. Note, that exploring even more general structures, such as concepts as +sub-manifolds in feature space, is an interesting idea. However, these do not allow for a decomposition of +the feature vector and hence do not lead to a completeness property, which is central to the definition of +concept relevance maps (see Section 2.3). +We start out with a user-specified set of samples S for which we aim to discover concepts. The sample +selection S is unrestricted: the user can decide for class-specific samples/concepts or use all training samples +to obtain completely class-unspecific concepts. Next, we split the model f into two parts, f = g ◦ h, where +3 + +feature +vector in +input sample +representation +mapping + +police +car +prediction +mapping +concept +layer +width +height +features +Model +Multi-dimensional concept discovery (MCD) +Basis +PCA +samples +Clustering +Determine subspace bases +set of features   +Decompose feature vector + into concept bases +pooling +prediction +concept relevance +heatmap +Testing +Training +project into + +and rescale to input +rescale +to input size +concept +completeness +concept +activation map +Figure 2: Schematic illustration of the MCD framework for concept discovery. The lower left panel illus- +trates how the model is split into a representation and prediction mapping. Feature vectors are extracted +from the representation mapping of a sample. The upper left panel illustrates concept discovery method- +ology of MCD (Section 2.2). First, randomly choose and cluster a set of feature vectors {φφφ} from a selection +of samples (using any clustering algorithm). Second, construct subspace bases for all clusters Cl via PCA +(intrinsic dimension dl). The upper right panel corresponds to the construction of concept activation +maps and the lower right panel shows the construction of concept relevance heatmaps, both laid out in +Section 2.3. +h is the mapping to a hidden feature layer, which is mapped to the prediction by g. Our definition then +relies on all hidden representations h(α) ∈ RH×W ×F of the input samples α ∈ S (height H, width W and +number of features F, see upper left panel in Figure 2). We spatially deconstruct the feature maps h(α) +and obtain a feature vector1 φφφα +xy ∈ RF for each location (x, y) ∈ {1, . . . , H} × {1, . . . , W}. We now strive to +identify concepts as (linear) structures in this F-dimensional feature space and pose no additional restrictions +(one-dimensionality and/or orthogonality) on the structure to the subspaces. +Definition 1. We define a concept Cl as a dl dimensional linear subspace in the F-dimensional feature +space, spanned by the basis vectors cccl +j, +Cl = span +�� +cccl +j|j = 1, . . . , dl�� +. +(1) +In particular, the dimensionality dl can vary among the concepts l = 1, . . . , nc. We denote the number of +concepts as nc and assume without loss of generality that their subspaces are pairwise disjoint.2 In particular, +our concept definition does not require orthogonal subspaces. Further, we do not require the nc concepts +Cl to cover the whole feature space. However, for the decomposition in Section 2.3, we need a set of all cccl +js +that spans the entire feature space. For this purpose, we define Cnc+1 to be the orthogonal complement of +the subspace spanned by all concepts, i.e., Cnc+1 = span(C1, . . . , Cnc)⊥. +2.2 +Concept Discovery +Typically, concept discovery, i.e., obtaining concepts as defined by Equation (1), can be subdivided into two +steps: First, cluster a user-defined set of feature vectors {φφφα +x,y} (usually sourced from the initial samples S) +and second, identify a representative basis for each concept cluster (lower left panel in Figure 2). +1Vectors are denoted lower-case bold (φφφ ∈ RF ). +2This assumption was never violated in our experiments, but it could be enforced by removing the intersection between the +subspaces from both and considering it as a separate concept. +4 + +Clustering feature vectors In principle, any clustering method can be considered to discover concept +clusters in feature space. This includes well-established baselines such as k-means clustering or PCA. Both +have previously been proposed in (Zhang et al., 2021) to identify one-dimensional subspaces. However, k- +means does not incorporate any information about the final objective to identify linear subspaces as opposed +to general clusters and PCA is restricted to orthogonal, one-dimensional subspaces. +We therefore propose a dedicated approach for this particular purpose and draw on the rich body of literature +on sparse subspace clustering (SSC) (You et al., 2016a; Soltanolkotabi & Candes, 2012; You et al., 2016b; +Elhamifar & Vidal, 2013). As nicely laid out in (Elhamifar & Vidal, 2013), SSC is ideally suited to identify +clusters of linear subspaces and provides a number of advantages over standard clustering algorithms, which +are directly applied to the data: SSC does not take advantage of the spatial proximity of the data, it can be +implemented robustly against noise and outliers and does not require specifying the cluster dimensionalities +in advance. +The various clustering algorithm mentioned above give rise to different MCD flavors: +• MCD-SSC For SSC, the concept discovery can be divided into two phases: Identifying a concept- +determining self-representation and applying spectral clustering. We provide technical details on +the particular subspace algorithm in Appendix A. +• MCD-kmeans As a simple baseline, we consider k-means clustering directly applied to the features. +Like SSC, it leads to multi-dimensional and in general non-orthogonal subspaces. +However, the +clustering algorithm does not include any information about the linear subspaces as desired clustering +target. +• MCD-PCA Finally, we consider PCA applied to the features directly. +This corresponds to the +concept discovery algorithm considered by ICE (Zhang et al., 2021). +Note, that this approach +already encompasses the basis identification step and directly leads to one-dimensional, orthogonal +subspaces by construction. +Constructing concept bases Irrespective of the chosen clustering algorithm, we have now identified clus- +ters C1, . . . , Cnc, which contain all feature vectors φφφα +x,y from the training set. Next, we strive to characterize +each concept via a subspace basis rather than its cluster members. To this end, we aim to identify a basis +Cl that robustly covers all samples in Cl. Here, we apply principal component analysis (PCA) and deter- +mine the intrinsic dimension dl of the subspace using a heuristic proposed by Fukunaga & Olsen (1971) and +implemented by Bac et al. (2021). The PCA components up to the intrinsic dimension dl then serve as a +basis vectors cccl +j for the subspace Cl. +Even though this leads to a slightly simpler interpretation, we will not assume that two different subspaces Cl +and Cm are orthogonal, as general subspace clustering algorithms do not enforce this. This could be enforced +through the use of dedicated orthogonal subspace clustering methods (Rahmani & Atia, 2017a), however, +at the potential cost of slightly sub-optimal subspace clusters (Rahmani & Atia, 2017b). Alternatively, this +could be implemented by sequentially rotating each identified subspace into the orthogonal complement of +its predecessors. The latter leads to the last MCD flavor: +• MCD-SSC-orth Here, we devise a post-hoc adaptation of the MCD-SSC approach to explore the +impact of orthogonal subspaces. We construct these subspaces in an iterative fashion. Starting +with an empty set, we explore the effect of adding one of the subspaces discovered by MCD-SSC +by considering adding the subspace rotated into the orthogonal complement of the span of the +subspaces in the set so far. Then, we select the candidate subspace that leads to the largest increase +in completeness, as defined in the following paragraph. +2.3 +Concept decomposition +Now, we discuss how new features vectors {φφφα +x,y} (obtained from a test set sample α) and the weights of the +final linear classifier layer can be analyzed via a decomposition in terms of previously discovered concepts Cl. +5 + +To this end, we propose concept activation maps, concept relevance heatmaps and a global concept relevance +score. +Concept activation maps quantify the activation of a chosen concept at a certain spatial location in the +input space of a sample α. +For this purpose, we decompose the feature vectors {φφφα +x,y} into its concept contributions. +Since the union of all concepts (including the orthogonal complement) forms a basis for the entire feature +space, we can uniquely decompose any feature vector φφφ as +φφφ = +nc+1 +� +l=1 +dl +� +i=1 +φl +icccl +i ≡ +nc+1 +� +l=1 +φφφl . +(2) +For a fixed sample α, we normalize φφφ such that the maximum length across all elements of the feature layer +is 1, i.e., maxx,y|φφφα +x,y|. Now, one can interpret |φφφl| as a measure for the extent to which a certain concept is +expressed in the given feature vector. Performing this step for every feature vector φφφα +x,y within a sample α +leads to a concept activation map whose spatial dimensions match those of the feature layer. For CNNs, we +follow the example of Selvaraju et al. (2020) and compute the corresponding concept activation map in input +space by bilinear upsampling in the spatial dimensions. Our concept activation maps extend the concept +visualization of Zhang et al. (2021) to multi-dimensional concepts (upper right panel in Figure 2). For the +final explanation, we also use them to characterize a concept in terms of prototypical examples. Here, we +sort test set samples by maxx,y|φφφα +x,y| and choose the top-k samples as concept prototypes. +We stress, that our methodology is applicable beyond CNNs. In particular, one can decompose feature +representations of any model based on MCD. +However, the prerequisite for showing concept (activation) maps in input space is the locality of the trained +model, i.e., the ability to associate locations in feature and input space. Whereas this locality is built in +as an inductive bias into convolutional architectures, it also emerges for vision transformer models during +training, as manifested for example in localized attention maps (Caron et al., 2021). +To the best of our knowledge, we show the first concept-based explanations for a vision transformer model +in Section 4, where we modify the upsampling to account for the model’s tokenization procedure. +Concept relevance heatmaps and completeness relation As a general requirement, any concept-based +XAI method should quantify the relevance of a concept in terms of its impact on the classification decision. +To this end, we specialize to the last hidden layer, which is only followed by linear operations (e.g. mean +pooling and a linear classification head). We discuss the broad class of models to which this applies in the +last paragraph of this section and empirically in Section 4. Now, for a given class, the weight vector www ∈ RF +linearly connects the final F-dimensional feature space with the scalar class-prediction. As before, φφφ ∈ RF +corresponds to a (potentially spatially pooled) feature vector in this very layer (see Figure 2 lower right +panel). +First, we are interested local (per-sample) concept relevance. For this, we can decompose the class logit +under consideration, φφφ · www + b, up to the bias term b, as +φφφ · www = +nc+1 +� +l=1 +φφφl · www ≡ +nc+1 +� +l=1 +rl . +(3) +We start by discussing the case where the class logit for sample α is obtained as +1 +W H +� +x,y φφφα +xy · www, i.e. after +global average pooling. First, we use the feature vector φφφα ∈ RF after pooling. Then, the decomposition +above defines a local concept relevance rl = φφφl · www. Aggregating relevances rl from all concepts recovers the +class logit prediction (up to the bias term), and thus, Equation (3) defines a completeness relation. 3 +3In the special case of one-dimensional concepts, rl reduces to the local concept relevance in (Zhang et al., 2021). +6 + +4 Second, we apply Equation (3) to the feature vectors φφφα +xy before pooling. This leads to a relevance heatmap +rl +xy that has the same spatial dimension as the feature layer. Importantly, rl +xy reduces to rl after spatial +pooling. +As for the concept activation maps, we use spatial upsampling to map rl +xy back to the input +space and obtain concept relevance heatmaps. Since upsampling preserves the completeness relation, these +decompose the local relevance maps rx,z = +1 +W Hφφφα +xy · www used by Zhou et al. (2016) (commonly referred to as +class activation maps (CAMs)) into concept contributions. +Global relevance and completeness score Next, we establish a global (model-wide) concept relevance +score. Recall, that all cccl +j defined above represent a basis for the feature space RF . Hence, we can directly +decompose the weight vector www into (analogously to Equation (2)) +www = +nc+1 +� +l=1 +dl +� +i=1 +wl +icccl +i ≡ +nc+1 +� +l=1 +wwwl , +(4) +where wwwl = �dl +i=1 wl +icccl +i and by construction, wwwl · wwwnc+1 = 0 for l = 1, . . . nc. In this case, we have +|www|2 = |wwwnc+1|2 + | +nc +� +l=1 +wwwl|2 = +nc+1 +� +l=1 +|wwwl|2 + +nc +� +l,k=1,l̸=k +|wwwl||wwwk| cos(∠(wwwl,wwwk)) +(5) +The first equality allows us to define +η({Cl}) = 1 − |wwwnc+1|2/|www|2 +(6) +as a completeness score (fraction of www which is explained by all concepts {C1, . . . , Cnc}) with respect to +a given class. To the best of our knowledge, we are the first to introduce a concept completeness score +directly based on model parameters. Previous work (Yeh et al., 2020) defined a related measure based on +model accuracy. Note, that for an orthonormal basis the second term in Equation (5) (cosine) disappears. +Then |wwwl|/|www| can be directly interpreted as (global) concept relevances, which sum up to the previous +completeness score over all concepts. Further, the angles in Equation (5) are lower- and upper-bounded +by the corresponding minimal or maximal principal angles5 between the two corresponding subspaces, i.e., +θkl +min ≡ minmθkl +m ≤ ∠(wwwk,wwwl) ≤ maxmθkl +m ≡ θkl +max. This means we can lower- and upper-bound |www|2 by +nc+1 +� +l +|wwwl|2 + +nc +� +l,k=1,l̸=k +|wwwl||wwwk| cos(θlk +max) ≤ |www|2 ≤ +nc+1 +� +l +|wwwl|2 + +nc +� +l,k=1,l̸=k +|wwwl||wwwk| cos(θlk +min) . +(7) +Obviously, lower and upper bound coincide in the case of orthogonal subspaces. This implies, that the |wwwl| +are also informative in the non-orthogonal case, provided the principal angles between the different subspaces +are given. This highlights the intricate connection between (global) relevances and the geometry in feature +space, i.e., the relative orientation of the concept spaces (specified via principal angles between pairs). +Finally, we briefly comment on the applicability of our approach for local and global concept relevances via +Equation (3) and Equation (4). In the form described above it can be used for any model with a linear layer +as final layer, potentially preceded by a global pooling layer, if one aims to spatially resolve the relevances +instead of considering only pooled feature vectors. This latter category covers a broad range of modern CNN +architectures such as ResNets, Inception-based model but also vision transformers, that do not base their +prediction on a CLS token, such as Swin transformers (Liu et al., 2021). We envision, that our approach is +even applicable, in approximate form, to other feature layers apart from the final hidden layer if one locally +approximates the remainder of the model by a linear model, similarly as it is done by Ribeiro et al. (2016) +or by Selvaraju et al. (2020) to generalize (Zhou et al., 2016). +4We briefly comment on the remaining commonly desired Shapley axioms Lundberg & Lee (2017). +The local concept +relevance trivially fulfills them, since it is built on a linear additive model. Formally, the hidden activation φα of a given sample +α are segmented into concept contributions/unique features φl +α. +Thus, the value function corresponding to the underlying +Shapley values is given by vα(S) = � +l∈S φl +α · w (linear in φl) for S ⊆ {1, . . . , nc + 1}. +5A formal definition of principal angles is given in Appendix B. +7 + +3 +Related Work +ACE (Ghorbani et al., 2019) uses a superpixel segmentation algorithm and k-means clustering to identify +class-specific concept candidates for TCAV (Kim et al., 2018). +The concept discovery scheme of ACE +has several shortcomings: The segmentation into candidate concept patches is model-independent and thus, +segments are not necessarily meaningful as perceived by the model. To enable clustering of intermediate CNN +activations, segments are resized and mean padded to the original input shape. This leads to artificial, off- +manifold samples with potentially distorted aspect ratios and discards the overall scale information. Finally, +ACE relies on multiple heuristics to discard segments/clusters both before and after k-means clustering. In +contrast, MCD is coherently based on hidden model representations without relying on additional pre- or +post-processing. Similar limitations apply to methods that rely on ACE-discovered labeled concepts, like +(Li et al., 2021), which uses Shapley values for concept importance, and (Wu et al., 2020), which occludes +particular neurons for neuron-wise relevances and transforms them into concept importances via concept +classification. Recently, Crabbé & van der Schaar (2022) proposed a generalization of TCAV by invoking the +kernel trick, which generalizes the concept definition towards non-linear structures. However, unlike MCD, +it does not allow quantifying the relevance of a concept towards the model prediction and can only verify +predefined concepts instead of discovering them. +ICE (Zhang et al., 2021) defines concepts as directions in feature space. Technically, this is achieved via +dimensionality reduction techniques applied to concatenated flattened feature maps. ICE measures the im- +portance of its class-wise concepts using TCAV. Interestingly, ICE introduces the notion of a concept weight, +which is analogous to our concept relevances on the logit layer. However, they do not consider spatially re- +solved concept relevance heatmaps and only address the special case of single-dimensional subspaces. Given +these restrictions, ICE can be seen as a special realization of the MCD framework, which uses dimensionality +reduction methods like PCA as clustering algorithms. Other methods learn concept vectors and a mapping +to feature space either for all classes simultaneously (ConceptSHAP (Yeh et al., 2020)) or for each class sep- +arately (MACE (Kumar et al., 2021), PACE (Kamakshi et al., 2021)). ConceptShap, MACE and PACE all +use additional regularizers to enforce concept dissimilarity. In contrast, MCD restricts the concept discovery +process as little as possible. Importantly, each method above defines a custom measure for concept impor- +tance, which is based on approximations of the original model. In contrast, the local and global concept +relevance within MCD are solely based on the original model parameters. +Other approaches (Chormai et al., 2022) use a concept definition similar to ours, but use information from +external attribution methods as well as orthogonality constraints to restrict the discovered concepts, whereas +MCD works without such restrictions. +There is a complementary line of work of frameworks that try to identify concepts associated with particular +neurons in hidden CNN representations, in conjunction with (Bau et al., 2017) or without (Achtibat et al., +2022) special concept-annotated datasets. Network Dissection (Bau et al., 2017) investigates the alignment of +human-understandable concepts and particular single hidden features (neurons). Net2vec (Fong & Vedaldi, +2018) extended this by allowing concepts to be represented by combinations of neurons. +Lastly, there is a line of research that constructs inherently interpretable concept models by design with +(Koh et al., 2020; Radenovic et al., 2022) or without relying on concept annotations (Chen et al., 2019). +Our approach is best comparable with (Chen et al., 2019), as both can be reduced to a linear model operating +on concepts that can be characterized via prototypes. We stress the essential difference, that our approach +does not require retraining (with special training objectives) but is an interpretable reformulation of the +original model. +4 +Results +We carry out our experiments on ImageNet (Deng et al., 2009). As model architectures, we consider ResNet +models (He et al., 2016) using original weights as provided by torchvision and updated weights as provided by +timm (Wightman, 2019) with an improved training procedure (Wightman et al., 2021). We also present re- +sults for a swin vision transformer (Liu et al., 2021), again using weights provided by timm (SwinS3base224). +8 + +In the following, we will refer to these models as ResNet50, ResNet50v2 and, Swin-T, respectively. We base +most of our experiments on images from a diverse selection of ten ImageNet classes, which roughly align +with CIFAR10 classes6 +4.1 +Completeness arithmetic +First, we provide a concrete example for an MCD explanation and showcase its completeness relation intro- +duced in Section 2.3 (benefit 3). To this end, Figure 3 shows an MCD-SSC explanation of a ResNet50v2 +prediction for a sample of the police van class in ImageNet. The number of concepts was chosen such that the +completeness measure in Equation (6) reaches η = 0.5. The three information components of the explanation +all provide complementary information: +(1) Concept relevance heatmaps decompose the local relevance maps into a sum of concept-specific relevance +heatmaps according to the completeness relation Equation (3). They show the alignment of a feature vector +component φφφl associated with concept Cl and the weight vector of a specific class. Roughly speaking, this +alignment indicates how typical the network perceives the particular instantiation of the concept for the class +under consideration. Applying mean pooling leads to a corresponding decomposition of the class logit under +consideration (up to the bias term) into contributions corresponding to different concepts. The completeness +relation on the level of concept relevance heatmaps as well on the level of logits represents a unique feature of +the MCD framework. Interestingly, for the explanation in Figure 3, only the orthogonal complement concept +contributes negatively to the class logit. The contributions of the first two concepts clearly dominate the +class logit. +(2) Concept activation maps stem from a decomposition of the feature vectors into a sum of vectors coming +from different, distinct concept subspaces, see Equation (2). Their non-negative score show how much a +particular feature vector aligns with a specific concept subspace. These maps help to identify input regions +where the concept is highly expressed. We color-code concept activation maps as a transparent overlay +over the image where transparent regions indicate high activation. +To guide the eye, we also include a +yellow(white) contour line at a threshold value of 0.5(0.4). +As a sanity check, we compute the Pearson correlation coefficient between the positive part of each concept +relevance map and each concept activation map for MCD-SSC and ResNet50v2. Among all test set samples, +we find a mean correlation of 0.45 for the concepts of the CIFAR10 classes. This confirms that concept +relevance is high in sample areas where the respective concept is strongly activated. +(3) Concept prototypes allow characterizing a concept subspace through examples. Here, we display the +concept activation maps of three test set samples that show the highest activation with the given concept. +In many cases, an intuitive meaning of a concept can be inferred most easily from these samples and numer- +ous previous approaches present concepts in this way (Zhang et al., 2021; Achtibat et al., 2022; Yeh et al., +2020). In case of the explanation in Figure 3, this could be windows/livery, livery, blue lights, building, tires +(and the orthogonal complement covering mainly the background). In addition, we also indicate the global +concept relevances for the different concepts according to Equation (4). +In summary, the sample in Figure 3, is classified as a police van mainly due to its windows/livery, which are +perceived as typical for the class by the network and are also the most relevant concept for the class globally. +Further, all other concepts are expressed in the sample and contribute positively, except for the orthogonal +complement. +4.2 +Empirical evaluation +We compare the the MCD flavors and previous methods listed in Table 1 in terms of (1) true-to-the-modelness +(benefit 1) and (2) conciseness (benefit 2) of the explanations. We base our evaluation on the CIFAR10 classes +mentioned above, and work with the ResNet50v2 model, for which we extract concepts from the last hidden +layer. For all methods within the MCD framework, we fix the number of concepts in a class-dependent way +such that we reach a completeness score of η = 0.5. +9 + +7.77 +3.14 +0.52 +0.94 +(-1.63) ++ ++ ++ ++ +class logit - bias +10.74 += +prototypes +concept +C1 +C2 +C3 +C4 +C5 +local relevance +concept space +complement += +mean +0.52 +0.42 +0.35 +0.22 +global relevance += ++ +0.86 +activation +mean +mean +mean +mean +mean +LOCAL +GLOBAL ++ ++ ++ +Figure 3: Completeness relation for the police van class in ImageNet. Concepts are discovered via MCD-SSC +for ResNet50v2. The number of concepts is chosen such that the completeness score reaches η = 0.5. We +distinguish between local (sample-specific) and global properties (characterizing a set of samples). Locally, +we consider concept relevance maps, which quantify the spatially resolved contribution of a concept to the +prediction. These satisfy a completeness relation, as explicitly shown in the first line. Concept activation +maps provide complementary information and indicate how much a concept is activated depending on the +spatial location in the sample. Globally, the overall relevance of a particular concept is quantified by the +global relevance scores. Finally, we also present concept prototypes (concept activation maps of the most +strongly activated samples) to characterize a particular concept. +Table 1: Summary of concept discovery methods considered in this work. +Method +Multi-dim. +Arbitrary +orientation +MCD-SSC + + +MCD-SSC-ortho + + +MCD-kmeans + + +ICE/MCD-PCA (Zhang et al., 2021) + + +ACE (Ghorbani et al., 2019) + + +4.2.1 +Comparing true-to-the-modelness via concept flipping +In order to compare the methods in Table 1 in terms of true-to-the-modelness, we invoke the Smallest +Destroying Concepts (SDC) benchmark as proposed in (Ghorbani et al., 2019) and (Wu et al., 2020). For +concepts that reflect the model’s actual reasoning structure in feature space and true-to-the-model concept +relevance scores, SDC should show a sharp decline of the model accuracy with the number of flipped concepts. +Here, we already mention the trivial solution for the sharpest decline, which is assigning the whole object to +a single concept and provides little insight into the model behavior. +We obtain concept masks, i.e. hard concept assignments, in input space by taking the argmax of the corre- +sponding concept activation maps over all concepts including the orthogonal complement. After the argmax +operation, we disregard the orthogonal complement, i.e., we do not remove it during the SDC experiments. +6namely (airliner, beach wagon, hummingbird, siamese cat, ox, golden retriever, tailed frog, zebra, container ship, police +van) +10 + +1305700.0 +0.2 +0.4 +0.6 +0.8 +percentage of deleted pixel +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +prediction accuracy +MCD-SSC +MCD-SSC-ortho +MCD-kmeans +ICE +ACE +sample +MCD-SSC +MCD-SSC-ortho +MCD-kmeans +ICE +ACE +Figure 4: Left: Concepts are flipped one at a time in descending order of local concept importance/TCAV +score, respectively. We measure the decline in model accuracy and show the mean accuracy across CIFAR10 +classes against the fraction of deleted pixels. Meaningful concept discovery and quantification methods are +supposed to show a sharp decline in this figure, but the decline should not happen after flipping only a single +concept (i.e. the whole object). Right: Qualitative comparison between hard concept assignments. +For each concept mask we obtain local relevance scores by pooling the corresponding concept relevance +heatmaps over the respective regions. This provides concept masks in input space which are ordered ac- +cording to their importance. ACE does not provide a measure of per-sample concept relevance. Therefore, +we revert to the order of their (global) TCAV scores after discarding concepts where statistical testing in +comparison to random input samples fails to stay below p = 0.05. To provide a qualitative impression of +the concept relevance heatmaps across methods, we show them together with concept activation maps for +selected samples in Figures 6 to 8. +To evaluate SDC, we subsequently remove concepts, as represented by the corresponding segments, in order +of their sample-wise (local) relevance starting from high to low. To inpaint the removed segments, we use a +classical imputation algorithm (Bertalmio et al., 2001), which leads to comparably realistic imputed images. +Thus, the model is evaluated on-manifold in contrast to imputing with gray patches as often done in the +literature. For similar reasons, we avoid the Smallest Sufficient Concepts (SSC) benchmark, which would +require high-quality imputation algorithms to avoid evaluating the model far from the data manifold. +In the left panel of Figure 4, we show the results of the SDC experiment. In contrast to previous studies +(Ghorbani et al., 2019; Wu et al., 2020), we report the model performance depending on the fraction of oc- +cluded pixels, which is essential for comparability since the segment size varies between different approaches. +In order to show a meaningful average of the samples across all classes we flip only as many concepts as are +present for the class with minimum number of concepts nc. As mentioned above, a meaningful concept dis- +covery and quantification method should show a sharp decline in this figure, but the full decline of the score +should not happen after flipping only a single concept (covering the whole object). We observe the latter for +ICE/MCD-PCA and MCD-SCC-ortho, both of which rely on orthogonal concepts. We hypothesize that this +behavior relates to the greedy way these orthogonal subspaces are constructed. In particular, ICE/MCD- +PCA only detects a single relevant and expressive concept, as the accuracy curve stagnates after flipping +the first concept. Among the remaining algorithms, MCD-SSC shows the strongest decline as compared to +MCD-kmeans and ACE. In Figure 4 (right panel), we also show hard concept assignments for an example +image of the golden retriever class, which form the basis of the concept flipping experiment described above. +These visually support the findings of the concept flipping experiment. Most approaches only discover a +single concept for the dog (apart from a potential genuine background concept). MCD-SCC shows the most +finegrained decomposition. This trend is also supported by the average subspace dimension dl as stated in +Table 2. +11 + +To summarize the results of the concept flipping experiment, our general MCD definition leads to concepts +that are most true-to-the-model, as the two unconstrained MCD flavors (MCD-SSC and MCD-kmeans), +show the steepest descent among all methods without reverting to the non-informative solution of a single +relevant concept. +4.2.2 +Conciseness of explanations +In Section 1, we describe why it is desirable, to explain the model reasoning with as few meaningful concepts +as completely as possible, i.e. to deliver concise concept explanations. Similarly to the above section, there +is a trivial solution to extremely concise concepts for the ImageNet classification task, which is to relate +a major part of feature space to a single concept of high dimensionality. Therefore, we characterize the +conciseness of concept explanations not only by the number of concepts nc that is required to reach a certain +completeness score, but also by average subspace dimension dl. Additionally, we evaluate the mean (scaled) +Grassmann distance ∆kl +c , as defined in Equation (10), between all concept pairs (k, l) within one class c to +quantify how dissimilar two concepts are.7 In summary, we argue that concepts should be concise (small nc), +but dissect the feature space into meaningful building blocks of model reasoning. While the latter is difficult +to quantify, we argue that there is a trade-off between (1) covering feature space with very few concepts of +high dimensionality and potentially small distance vs. (2) dissecting it into a high number of concepts with +small dimensionality (extreme case: one-dimensional). To support this reasoning, we also inspect the visual +impression of concepts for a selection of classes. +We list the number of concepts nc that is required to reach a completeness score of η = 0.5 for ResNet50v2 on +the CIFAR10 classes and dl in Table 2. To provide a visual comparison of the concepts discovered by these +methods, we show concept activation maps of prototypes for basketball, golden retriever and airliner class +in ImageNet in Figures 9 to 11 and judge how broad they appear in input space. MCD-kmeans discovers +the smallest number of concepts with the highest mean concept dimensionality of 74.7 and the smallest +inter-concept distance (mean(∆kl +c ) = 0.83) among all methods. We argue that this is reflected in the visual +appearance of the concept prototypes, which are visually broad and difficult to distinguish. +MCD-SSC +discovers on average 4.8 concepts with a smaller mean concept dimensionality of 44.2. Visually, concepts +seem medium broad and are easier to distinguish in input space than for MCD-kmeans, which is reflected in +a higher inter-concept distance. When requiring orthogonality mean(∆kl +c ) = π/2 = 1.57, like for MCD-SSC- +ortho and ICE, we see that only one concept appears medium broad in input space while all others are almost +not activated. We argue, that the orthogonality constraint hinders the concepts to reflect a natural similarity +between certain concepts. This aligns with the conclusions drawn from the SDC benchmark. Most notably, +to achieve a comparable model faithfulness (completeness score of η = 0.5) 30 times more one-dimensional +ICE concepts than multi-dimensional MCD concepts are required, meaning this method delivers concept +explanations that are not concise. Intuitively, a single concept is split up into several concepts, which is +also reflected in their weak activation on test set samples. Lastly, the visual impression of ACE concepts +is fixed by the choice of the superpixel algorithm. While ACE concepts are all one-dimensional, they do +not provide a mechanism to quantify how complete they are, thus we cannot quantify nc required to reach +a completeness of 50%. As an overall summary, MCD-SSC is superior in dissecting the feature space into +enclosed and meaningful concepts. +4.3 +Use case: MCD concepts reveal differences in classification strategies between model +architectures and training procedures +Finally, we showcase how MCD can unravel different classification strategies depending on the model archi- +tecture (ResNet50 vs. Swin-T) and the training strategies (ResNet50 vs. ResNet50v2). The test accuracies +for the subset of CIFAR10-classes are 0.80 (ResNet50), 0.84 (ResNet50v2) and 0.86 (Swin-T). Here, we +focus on MCD-SCC and, as before, restrict ourselves to concepts in the last hidden feature layer. First, +we compare the discovered concepts between the models by the activation maps of concept prototypes for +the beach wagon class of ImageNet in Figure 5. We fix the number of concepts to nc = 5. For Swin-T, +7We use a scaled version of the original Grassmann distance that aggregates the principle angles (in radian) between two +subspaces, for which 0 ≤ ∆kl +c +≤ π. Two special cases are ∆kl +c += 0 meaning subspace bases vectors are perfectly aligned, and +∆kl +c = π/2, meaning they are orthogonal. +12 + +Table 2: Summary of concept discovery methods considered in this work in comparison to prior work from +Zhang et al. (2021) (ICE/MCD-PCA) and Ghorbani et al. (2019) (ACE). We measure average subspace +dimension dl and the number of concepts nc that is required to reach a completeness score of η = 0.5 +for ResNet50v2 on the CIFAR10 classes. A small number of relevant concepts nc is desirable, since this +summarizes the complete model into an accessible and meaningful format. Here, multi-dimensional concepts +have an advantage. Additionally, we evaluate the mean (scaled) Grassmann distance ∆kl +c , see Equation (10), +between all concept pairs (k, l) within one class c to quantify the distinctness between concepts. The visual +inspection is based on prototypes of the basketball, golden retriever and airliner class concepts in Figures 9 +to 11. Medium broad and distinct concepts are the most informative. +Method +mean(dl) +mean(nc) +mean(∆kl +c ) +Visual inspection +MCD-SSC +44.2 +4.8 +1.19 +medium broad +MCD-SSC-ortho +44.2 +4.8 +1.57 +only one broad (rest narrow) +MCD-kmeans +74.7 +2.7 +0.83 +very broad +ICE/MCD-PCA +1 +146.7 +1.57 +only one broad (rest narrow) +ACE +1 +n.a. +n.a. +medium broad +we only apply a spatial upsampling of the concept activation maps from the feature to the input space to +14 × 14 in order to account for the 16 × 16 patch tokenization. We find that ResNet50 concepts, which could +roughly be identified as (car body, windows, car roof, wheels, street), are more narrow than the expression +of Swin-T and ResNet50v2 concepts. The latter are related to broader views of the car, such as concepts (1, +2, 4) for ResNet50v2 and concepts (1, 3) for Swin-T. Interestingly, ResNet50v2 concepts reach a much lower +completeness score of η = 0.49 than ResNet50 (η = 0.89) and Swin-T (η = 0.84) for fixed nc = 5. In Figure 5 +we show the relation between the total concept space dimensionality, the number of concepts nc and the +completeness score η across the CIFAR10-classes. Even for nc = 30, the ResNet50v2 concepts have a lower η +than those of the ResNet50 for nc = 3, although the former covers already a much larger part of the concept +space. These observations support the statement that feature space of the ResNet50v2 exhibits comparably +richer structure than Resnet50. This is an interesting difference in the character of the feature space as a +consequence of two different training procedures for the same architecture, revealed by MCD-SSC concepts. +Interestingly, the dependence of η on nc for the concepts between two models with different architectures, +ResNet50 and Swin-T, is quite similar. This also aligns with the visual appearance of the concepts. +To summarize, Swin-T and ResNet50 build on broader and more versatile concepts. +In comparison, +ResNet50v2 builds more narrow and thus specific concepts for its classification strategy. These broad con- +cepts are not unexpected for a transformer architecture like Swin-T with coarse self-attention windows, but +a rather surprising finding for ResNet50 in comparison to ResNet50v2. +5 +Summary and Discussion +In this work, we put forward MCD, a general framework for concept discovery based on the hidden repre- +sentation of a trained deep neural network. Unlike prior work in the field, we propose a general concept +definition (incorporating previous approaches) as multi-dimensional linear subspaces without restricting to +single directions or enforcing orthogonality between subspaces. We use concept activation maps to visualize +concepts in input space. Considering the final hidden layer representation, we can reformulate the original +model as a linear classifier acting on linear concept subspaces without the need to retrain with a special +objective. This leads to a completeness relation, i.e., a natural decomposition of class logits into contribu- +tions corresponding to specific concepts and allows to resolve their spatial importance in terms of concept +relevance heatmaps. As a particularly suited realization of our framework, we put forward MCD-SCC, which +relies on sparse subspace clustering for concept discovery. Based on qualitative and quantitative insights, we +show the superiority of MCD-SCC over other MCD flavors that build on traditional clustering algorithms. +13 + +0 +200 +400 +600 +800 +1000 +1200 +conceptspace dimensionality +l +d l +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +completeness +ResNet50v2 +ResNet50 +Swin-T +concept space +complement +Swin-T +Resnet50v2 +ResNet50 +Figure 5: Left: Mean concept space completeness score ν for the CIFAR10 classes across architectures +against the dimensionality of the union of all concept subspaces � +l dl. The number of concepts can be +inferred from the points on the line where the first point on each line corresponds to nc = 3 and the last +one to nc = 30. ResNet50v2 shows a much lower completeness score at roughly the same nc and � +l dl as +ResNet50. The feature space dimensionality is F = 2048 for ResNet50(v2) and F = 768 for Swin-T. Right: +We show MCD-SSC concept activation maps for concept prototypes for ResNet50, ResNet50v2 and Swin-T +and the beach wagon class in ImageNet. We fixed the number of concepts to nc = 5. In this way, ResNet50v2 +reaches η = 0.49, ResNet50 η = 0.89 and Swin-T η = 0.84. Each row shows a single concept and is titled by +its global concept importance. The last row shows the orthogonal complement of the concept space. +We showcase the ability of MCD via discriminating between hidden representations obtained from different +model architectures and training strategies. This paves the way towards further novel use-cases for MCD +concepts such as gaining insights in the natural sciences, e.g., identifying sub-classes of cancerous cells in +histopathology or summarizing model behavior beyond single examples and thereby systematically discover +model biases. 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More specifically, +using the notation from (Elhamifar & Vidal, 2013), given the feature vectors Φ = [φφφ1, . . . ,φφφn] ∈ RF ×n, we +identify a sparse coefficient matrix RRR = [rrr1, . . . ,rrrn] ∈ Rn×n such that +φφφj = Φrrrj where rii = 0. +(8) +The particular kind of sparsity constraints that are imposed on Equation (8) and how it is optimized depends +on the chosen SSC algorithm. Here, we use elastic net subspace clustering (You et al., 2016a), which is robust +against noise and scales well for large sample sizes. In all our experiments, we fix the hyperparameter γ, +which balances sparsity vs. +robustness, to γ = 10. +We confirmed that the results are not sensitive to +variation of this parameter over a range of values from 5 to 50. As computation time for SSC is dependent +on this parameter, we chose γ such that this is minimized. +We remove outliers based on the ℓ1-norm as in (Soltanolkotabi & Candes, 2012), where we empirically fix +the percentile threshold to 0.75 and re-fit the sparse self-representation for the remaining elements. +Another scalable alternative to the elastic net clustering is orthogonal matching pursuit (OMP)(You et al., +2016b), which is, however, not robust against noise and does not allow for outlier removal via thresholding. +Finally, the original sparse subspace clustering method from (Elhamifar & Vidal, 2013) is robust against +noise and outliers but does not scale to large datasets. The particular robustness and scalability properties +make elastic net subspace clustering (with thresholding) an ideal choice for the first step of our concept +discovery method. +Spectral clustering In a second step, we perform spectral clustering with the affinity matrix W = |R|+|RT |, +which encodes the similarity of two feature vectors according to their self-representations. We determine the +17 + +number of clusters nc either via the largest gap in the spectrum of the Laplacian (Von Luxburg, 2007) or +use a predetermined value. This step assigns every input feature φφφi to a particular cluster C1, . . . , Cnc or to +the set of outliers. +B +Characterizing relation between subspaces by principal angles +In this section, we briefly review the definition of principal angles, which can be used to characterize the rela- +tion between two linear subspaces. The principal angles θAB +i +(Jordan, 1875) (i = 1, . . . , min(dim A, dim B)) +between two linear subspaces A, B, are defined recursively via +cos θAB +i += +max +aaa∈A,bbb∈B +aaaTbbb +|aaa||bbb| =: aaaT +i bbbi +|aaa||bbbi| , +(9) +where the maximum is taken subject to the orthogonality constraints aaaTaaaj = 0 and bbbTbbbj = 0 for j = +1, . . . , i − 1. +To quantify the similarity between two subspaces A and B, we use a scaled version of their Grassmann +distance Hamm (2008), which is defined as, +∆AB = 1/ +� +min(dim A, dim B) +� +(θAB +1 +)2 + . . . + (θAB +min(dim A,dim B))2 . +(10) +This allows comparing the similarity of concepts within a given class or across classes regardless of the +concept subspaces’ dimensionality. +C +Qualitative results +For a qualitative comparison between of the concept activation maps and relevance heatmaps between the +methods in Section 4.2, we provide results for selected samples in Figures 6 to 8. In Figures 9 to 11 we show +the respective concept prototypes for all concept discovery approaches. +18 + ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ +MCD-SSC +MCD-SSC-ortho ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ } += += +MCD-kmeans +ICE +ACE +4.78 +concept space +complement +Figure 6: Concept heatmaps and activation maps for ResNet50v2 and a randomly chosen sample from the +basketball class in ImageNet. The number of concepts is chosen such that the completeness score reaches +η = 0.5. Concepts are ordered from left to right according to global concept relevance. Concept heatmaps +are titled by the pooled local concept relevance that sums to the prediction logit minus the bias. For ICE, +we only show the first six out of 105 and for ACE the first six out of 25 concepts. +19 + ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ +MCD-SSC +MCD-SSC-ortho ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ } += += +6.28 +concept space +complement +MCD-kmeans +ICE +ACE +Figure 7: Concept heatmaps and activation maps for ResNet50v2 and a randomly chosen sample from the +golden retriever class in ImageNet. The number of concepts is chosen such that the completeness score +reaches η = 0.5. Concepts are ordered from left to right according to global concept relevance. Concept +heatmaps are titled by the pooled local concept relevance that sums to the prediction logit minus the bias. +For ICE, we only show the first six out of 142 and for ACE the first six out of 25 concepts. +20 + +MCD-SSC +MCD-SSC-ortho +} +concept space +complement +MCD-kmeans +ICE +ACE ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ += += +8.66 +Figure 8: Concept heatmaps and activation maps for ResNet50v2 and a randomly chosen sample from the +airliner class in ImageNet. +The number of concepts is chosen such that the completeness score reaches +η = 0.5. Concepts are ordered from left to right according to global concept relevance. Concept heatmaps +are titled by the pooled local concept relevance that sums to the prediction logit minus the bias. For ICE, +we only show the first seven out of 141 and for ACE the first seven out of 25 concepts. +21 + +MCD-SSC +MCD-SSC-ortho +MCD-kmeans +ICE +ACE +Figure 9: Concept activation maps for concept prototypes for basketball class of ImageNet. The last row +shows prototype for the complement, except for ACE, where no complement exists. For ICE, we only show +the first six out of 105 and for ACE the first six out of 25 concepts. +22 + +1042PLERAANCLRU33MCD-SSC +MCD-SSC-ortho +MCD-kmeans +ICE +ACE +Figure 10: Concept activation maps for concept prototypes for golden retriever class of ImageNet. The last +row shows prototype for the complement, except for ACE, where no complement exists. For ICE, we only +show the first six out of 142 and for ACE the first six out of 25 concepts. +23 + +UNASYDMCD-SSC +MCD-SSC-ortho +MCD-kmeans +ICE +ACE +Figure 11: Concept activation maps for concept prototypes for airliner class of ImageNet. The last row +shows prototype for the complement, except for ACE, where no complement exists. For ICE, we only show +the first seven out of 141 and for ACE the first seven out of 25 concepts. +24 + +FFIRDSTAIRWorldwide Services +conizing theworld ol cornmetceeaworldethopian.HollandExel.dleglantairtransatWorldwide Services +Fonizingtheworld ol cotmmetceDRAGOEhoplaaairtransatWorldwideServices +ronizingtheworldofcommerceWWoridwidesoplan..easEthoplan.airtransat \ No newline at end of file diff --git a/qNFKT4oBgHgl3EQfzS7F/content/tmp_files/load_file.txt b/qNFKT4oBgHgl3EQfzS7F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f7a6ef3ed5be199f04fa5f45ff7719d2a98ba42c --- /dev/null +++ b/qNFKT4oBgHgl3EQfzS7F/content/tmp_files/load_file.txt @@ -0,0 +1,920 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf,len=919 +page_content='Multi-dimensional concept discovery (MCD): A unifying framework with completeness guarantees Johanna Vielhaben johanna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='vielhaben@hhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='fraunhofer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='de AI department Fraunhofer Heinrich-Hertz-Institut Stefan Blücher bluecher@tu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='de Machine Learning Group TU Berlin Nils Strodthoff nils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='strodthoff@uol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='de Division AI4Health Oldenburg University Abstract The completeness axiom renders the explanation of a post-hoc XAI method only locally faithful to the model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' for a single decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For the trustworthy application of XAI, in particular for high-stake decisions, a more global model understanding is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Recently, concept-based methods have been proposed, which are however not guaranteed to be bound to the actual model reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To circumvent this problem, we propose Multi-dimensional Concept Discovery (MCD) as an extension of previous approaches that fulfills a completeness relation on the level of concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Our method starts from general linear subspaces as con- cepts and does neither require reinforcing concept interpretability nor re-training of model parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We propose sparse subspace clustering to discover improved concepts and fully lever- age the potential of multi-dimensional subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' MCD offers two complementary analysis tools for concepts in input space: (1) concept activation maps, that show where a concept is expressed within a sample, allowing for concept characterization through prototypical sam- ples, and (2) concept relevance heatmaps, that decompose the model decision into concept contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Both tools together enable a detailed understanding of the model reasoning, which is guaranteed to relate to the model via a completeness relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This paves the way towards more trustworthy concept-based XAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We empirically demonstrate the superiority of MCD against more constrained concept definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 1 Introduction Explainable AI (XAI) allows to peek insight the black box of inherently complex deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Local interpretability methods are particular valuable, as they measure attributions for an individual instance, which are easily comprehensible for any kind of end-users, see (Covert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Lundberg & Lee, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Montavon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Samek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021) for reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For example, local methods make a prediction interpretable on the level of single images or individual bank customers for an image or credit risk classifier, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Importantly, the commonly employed completeness axiom (attributions sum up to the model prediction) ensures a meaningful interpretation of attributions (Lundberg & Lee, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Sundararajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' However, to actually comprehend the model reasoning we require a global model understanding, which reliably explains the model behavior across multiple instances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' a group of female vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' male bank customers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We stress that it is not viable to require an end-user to aggregate local attributions into common model features (concepts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Such a procedure is prone to human confirmation bias and it is not clear how the imagined concepts align with the actual model reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This urges for novel local and concept-based interpretability methods, which allow to understand shared model structures (used across multiple samples) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='11911v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='LG] 27 Jan 2023 D3: arbitrary directions D4: linear subspaces of arbitrary dimensionality D2: orthogonal directions allow rotation allow non-orthogonality allow multi-dimensionality D1: single neurons Figure 1: We strive for the most general decomposition of the hidden feature space, spanned by the neurons c1 c1 c1,c2 c2 c2,c3 c3 c3, into linear structures that form the concepts Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The most constrained approach is to identify concepts with single neurons (D1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' directions in feature space aligned with the neuron axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' If one allows for an arbitrary rotations of the concept directions, one arrives at D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Leaving aside the orthogonality constraint, D3 allows concepts to form arbitrary directions is feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Finally, allowing concepts to form multi-dimensional subspaces, we arrive at the most general approach D4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Previous concept-based methods are based on D1, D2 and D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We choose the most general approach D3, to discover concepts that are true-to-the-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' for an individual instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This idea was first formalized by Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (2018) and further developed by ACE (Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2019) and its successors (Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Crucially, our work (1) re-introduces completeness within the context of concept-based explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Thereby, concepts obtained within our multi-dimensional concept discovery (MCD) scheme are locally and globally interpretable in terms of a well-defined completeness decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We outline the benefits of MCD in the following paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Novel concepts as multi-dimensional subspaces Indisputably, concept discovery in neural networks is inherently linked to structures in intermediate feature layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In Figure 1, we illustrate different approaches to decompose the hidden feature space into meaningful concepts, which are mathematically formalized as linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The most constrained definition (left most panel, D1) is to directly identify concepts with a neural directions (Bau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This means that a concept is a one-dimensional subspace which aligns with the unit axes in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' A slightly more general definition is to use an arbitrary rotated orthogonal one-dimensional decomposition of the feature space (D2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Such a concept decomposition can be obtained via a principal component analysis (PCA) of the feature space (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Going one step further, we disregard the orthogonality constraint and allow arbitrary directions in feature space (D3) (Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Thereby, we can characterize related concepts which are linearly independent but not orthogonal (for example different parts of an animal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Allowing for arbitrary multi-dimensional subspaces unfolds the most general definition of a linear decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Thus, this general approach enables true-to- the-model concepts as it allows to capture any meaningful linear structure within the hidden feature layer (benefit 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Multi-dimensionality ensures concise explanations Concepts strive to organize the information about the model reasoning in a concise and accessible manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' As previously outlined, aggregating many different explanations into a comprehensive model understanding is a challenge for humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Thus, it is desirable to grasp the actual model reasoning with a limited number of concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Phrased differently, we want to cover the relevant feature space with only a few concepts and avoid fragmentation into a large number of low/one-dimensional subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We formalize the relevant feature subspace based on its impact on the model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Via this intuition we can define a concept completeness score in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='3, which measures the fraction of model prediction jointly covered by all concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Intuitively, multi-dimensional concept subspaces reach a certain level of completeness with a smaller number of concepts than one-dimensional concepts and thus deliver more concise explanations (benefit 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Sparse subspace clustering for better concept discovery The principles of MCD do not rely on any particular (clustering) algorithm for concept discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We argue in favor of clustering approaches that place 2 the fewest restrictions on the discovery process, in order to fully realize the promise of multi-dimensional concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' A clustering algorithm that fits the above objective extremely well is sparse subspace clustering (SSC) (Elhamifar & Vidal, 2013), as it is tailored to discover otherwise unconstrained linear subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Thus, there is no need for additional measures to reinforce the interpretability of concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In contrast, previous methods use techniques like superpixels in input space (Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2019) or regularizers to enforce concept dissimilarity (Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2020) for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This potentially breaks the connection between discovered concepts and the actual reasoning structures in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concept decomposition To present the discovered concepts in a human-comprehensible form, it is custom to visualize regions in input space in which the concept is activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We assume that the activations in a hidden layer form a spatially resolved map of feature vectors (given e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' for convolutional or transformer models with skip-connections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We uniquely decompose each hidden feature vector into its concept parts and measure the length of these parts to assess whether a particular concept is activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Upsampling these concept activations from hidden to input layer finally creates comprehensible concept activation maps in input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We now restrict to a high-level feature layer which is only succeeded by linear operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' a linear classification head with global pooling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This enables to also uniquely decompose the model prediction into concept parts and thereby define a concept relevance heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Then, the relevances follow a completeness relation (benefit 3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' summing up the relevance from all concepts is guaranteed to equal the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concept relevance heatmaps show a spatially resolved decomposition of the model prediction into concepts and show how indicative a particular concept instantiation is for the predicted class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We stress, that the concept relevance heatmaps follow directly from the decomposition into concept parts and do not invoke any additional XAI method nor retraining model parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Thus, MCD can completely capture the model reasoning solely in terms of linear operations on concept subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In summary, MCD is a consistent framework to discover true-to-the-model concepts, which are guaranteed to rely on the actual model reasoning via the completeness relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Visualizing activation maps and relevance heatmaps for prototype samples offers the possibility to characterize concepts more closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We see the main utility of MCD in the domain of model understanding and certification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concepts provide insights into model behavior that generalize across samples and are therefore a valuable tool for systematic investigations of spurious correlations (model biases) (Lapuschkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Palatnik de Sousa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021), as well as for scientific discovery (Blücher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Hägele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' McGrath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Šarčević et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2022), where the model serves as proxy for the unknown relationships in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This work builds on an earlier manuscript (Vielhaben et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2022) and crucially extends it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 2 Multi-dimensional Concept Discovery (MCD) We organize this methodological section into three parts: First, we introduce our novel concept definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Second, we describe practical concept discovery procedures that align with this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Most prominently, we introduce SSC but also elaborate on possible alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Third, we introduce a concept decomposition and discuss how to construct local and global concept importance that fulfill a concept completeness relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Figure 2 presents a schematic summary of our MCD framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='1 Concept definition Concepts are inherently tied to the hidden representations of intermediate feature layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In Figure 1, we illustrate the structures that concepts could possibly form in hidden feature space: from single directions (D1-D3) to linear subspaces (D4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' MCD opts for the most general structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', arbitrarily orientated multi-dimensional linear subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Note, that exploring even more general structures, such as concepts as sub-manifolds in feature space, is an interesting idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' However, these do not allow for a decomposition of the feature vector and hence do not lead to a completeness property, which is central to the definition of concept relevance maps (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We start out with a user-specified set of samples S for which we aim to discover concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The sample selection S is unrestricted: the user can decide for class-specific samples/concepts or use all training samples to obtain completely class-unspecific concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' we split the model f into two parts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' f = g ◦ h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='vector in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='input sample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='mapping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='police ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='car ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='mapping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='concept ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='width ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='height ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='Multi-dimensional concept discovery (MCD) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='Basis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='PCA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='samples ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='Clustering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='Determine subspace bases ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='set of features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='Decompose feature vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='into concept bases ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='concept relevance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='heatmap ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='Testing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='project ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='into ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='and rescale to input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='rescale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='to input size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='concept ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='completeness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='concept ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='activation map ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='Figure 2: Schematic illustration of the MCD framework for concept discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The lower left panel illus- trates how the model is split into a representation and prediction mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Feature vectors are extracted from the representation mapping of a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The upper left panel illustrates concept discovery method- ology of MCD (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' First, randomly choose and cluster a set of feature vectors {φφφ} from a selection of samples (using any clustering algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Second, construct subspace bases for all clusters Cl via PCA (intrinsic dimension dl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The upper right panel corresponds to the construction of concept activation maps and the lower right panel shows the construction of concept relevance heatmaps, both laid out in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' h is the mapping to a hidden feature layer, which is mapped to the prediction by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Our definition then relies on all hidden representations h(α) ∈ RH×W ×F of the input samples α ∈ S (height H, width W and number of features F, see upper left panel in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We spatially deconstruct the feature maps h(α) and obtain a feature vector1 φφφα xy ∈ RF for each location (x, y) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' , H} × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' , W}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We now strive to identify concepts as (linear) structures in this F-dimensional feature space and pose no additional restrictions (one-dimensionality and/or orthogonality) on the structure to the subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We define a concept Cl as a dl dimensional linear subspace in the F-dimensional feature space, spanned by the basis vectors cccl j, Cl = span �� cccl j|j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' , dl�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (1) In particular, the dimensionality dl can vary among the concepts l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' , nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We denote the number of concepts as nc and assume without loss of generality that their subspaces are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2 In particular, our concept definition does not require orthogonal subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Further, we do not require the nc concepts Cl to cover the whole feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' However, for the decomposition in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='3, we need a set of all cccl js that spans the entire feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For this purpose, we define Cnc+1 to be the orthogonal complement of the subspace spanned by all concepts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', Cnc+1 = span(C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' , Cnc)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2 Concept Discovery Typically, concept discovery, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', obtaining concepts as defined by Equation (1), can be subdivided into two steps: First, cluster a user-defined set of feature vectors {φφφα x,y} (usually sourced from the initial samples S) and second, identify a representative basis for each concept cluster (lower left panel in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 1Vectors are denoted lower-case bold (φφφ ∈ RF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 2This assumption was never violated in our experiments, but it could be enforced by removing the intersection between the subspaces from both and considering it as a separate concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 4 Clustering feature vectors In principle, any clustering method can be considered to discover concept clusters in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This includes well-established baselines such as k-means clustering or PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Both have previously been proposed in (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021) to identify one-dimensional subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' However, k- means does not incorporate any information about the final objective to identify linear subspaces as opposed to general clusters and PCA is restricted to orthogonal, one-dimensional subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We therefore propose a dedicated approach for this particular purpose and draw on the rich body of literature on sparse subspace clustering (SSC) (You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Soltanolkotabi & Candes, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Elhamifar & Vidal, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' As nicely laid out in (Elhamifar & Vidal, 2013), SSC is ideally suited to identify clusters of linear subspaces and provides a number of advantages over standard clustering algorithms, which are directly applied to the data: SSC does not take advantage of the spatial proximity of the data, it can be implemented robustly against noise and outliers and does not require specifying the cluster dimensionalities in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The various clustering algorithm mentioned above give rise to different MCD flavors: MCD-SSC For SSC, the concept discovery can be divided into two phases: Identifying a concept- determining self-representation and applying spectral clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We provide technical details on the particular subspace algorithm in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' MCD-kmeans As a simple baseline, we consider k-means clustering directly applied to the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Like SSC, it leads to multi-dimensional and in general non-orthogonal subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' However, the clustering algorithm does not include any information about the linear subspaces as desired clustering target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' MCD-PCA Finally, we consider PCA applied to the features directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This corresponds to the concept discovery algorithm considered by ICE (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Note, that this approach already encompasses the basis identification step and directly leads to one-dimensional, orthogonal subspaces by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Constructing concept bases Irrespective of the chosen clustering algorithm, we have now identified clus- ters C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' , Cnc, which contain all feature vectors φφφα x,y from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Next, we strive to characterize each concept via a subspace basis rather than its cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To this end, we aim to identify a basis Cl that robustly covers all samples in Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Here, we apply principal component analysis (PCA) and deter- mine the intrinsic dimension dl of the subspace using a heuristic proposed by Fukunaga & Olsen (1971) and implemented by Bac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The PCA components up to the intrinsic dimension dl then serve as a basis vectors cccl j for the subspace Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Even though this leads to a slightly simpler interpretation, we will not assume that two different subspaces Cl and Cm are orthogonal, as general subspace clustering algorithms do not enforce this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This could be enforced through the use of dedicated orthogonal subspace clustering methods (Rahmani & Atia, 2017a), however, at the potential cost of slightly sub-optimal subspace clusters (Rahmani & Atia, 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Alternatively, this could be implemented by sequentially rotating each identified subspace into the orthogonal complement of its predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The latter leads to the last MCD flavor: MCD-SSC-orth Here, we devise a post-hoc adaptation of the MCD-SSC approach to explore the impact of orthogonal subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We construct these subspaces in an iterative fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Starting with an empty set, we explore the effect of adding one of the subspaces discovered by MCD-SSC by considering adding the subspace rotated into the orthogonal complement of the span of the subspaces in the set so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Then, we select the candidate subspace that leads to the largest increase in completeness, as defined in the following paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='3 Concept decomposition Now, we discuss how new features vectors {φφφα x,y} (obtained from a test set sample α) and the weights of the final linear classifier layer can be analyzed via a decomposition in terms of previously discovered concepts Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 5 To this end, we propose concept activation maps, concept relevance heatmaps and a global concept relevance score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concept activation maps quantify the activation of a chosen concept at a certain spatial location in the input space of a sample α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For this purpose, we decompose the feature vectors {φφφα x,y} into its concept contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Since the union of all concepts (including the orthogonal complement) forms a basis for the entire feature space, we can uniquely decompose any feature vector φφφ as φφφ = nc+1 � l=1 dl � i=1 φl icccl i ≡ nc+1 � l=1 φφφl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (2) For a fixed sample α, we normalize φφφ such that the maximum length across all elements of the feature layer is 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', maxx,y|φφφα x,y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Now, one can interpret |φφφl| as a measure for the extent to which a certain concept is expressed in the given feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Performing this step for every feature vector φφφα x,y within a sample α leads to a concept activation map whose spatial dimensions match those of the feature layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For CNNs, we follow the example of Selvaraju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (2020) and compute the corresponding concept activation map in input space by bilinear upsampling in the spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Our concept activation maps extend the concept visualization of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (2021) to multi-dimensional concepts (upper right panel in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For the final explanation, we also use them to characterize a concept in terms of prototypical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Here, we sort test set samples by maxx,y|φφφα x,y| and choose the top-k samples as concept prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We stress, that our methodology is applicable beyond CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In particular, one can decompose feature representations of any model based on MCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' However, the prerequisite for showing concept (activation) maps in input space is the locality of the trained model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', the ability to associate locations in feature and input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Whereas this locality is built in as an inductive bias into convolutional architectures, it also emerges for vision transformer models during training, as manifested for example in localized attention maps (Caron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To the best of our knowledge, we show the first concept-based explanations for a vision transformer model in Section 4, where we modify the upsampling to account for the model’s tokenization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concept relevance heatmaps and completeness relation As a general requirement, any concept-based XAI method should quantify the relevance of a concept in terms of its impact on the classification decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To this end, we specialize to the last hidden layer, which is only followed by linear operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' mean pooling and a linear classification head).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We discuss the broad class of models to which this applies in the last paragraph of this section and empirically in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Now, for a given class, the weight vector www ∈ RF linearly connects the final F-dimensional feature space with the scalar class-prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' As before, φφφ ∈ RF corresponds to a (potentially spatially pooled) feature vector in this very layer (see Figure 2 lower right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' First, we are interested local (per-sample) concept relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For this, we can decompose the class logit under consideration, φφφ · www + b, up to the bias term b, as φφφ · www = nc+1 � l=1 φφφl · www ≡ nc+1 � l=1 rl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (3) We start by discussing the case where the class logit for sample α is obtained as 1 W H � x,y φφφα xy · www, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' after global average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' First, we use the feature vector φφφα ∈ RF after pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Then, the decomposition above defines a local concept relevance rl = φφφl · www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Aggregating relevances rl from all concepts recovers the class logit prediction (up to the bias term), and thus, Equation (3) defines a completeness relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 3 3In the special case of one-dimensional concepts, rl reduces to the local concept relevance in (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 6 4 Second, we apply Equation (3) to the feature vectors φφφα xy before pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This leads to a relevance heatmap rl xy that has the same spatial dimension as the feature layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Importantly, rl xy reduces to rl after spatial pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' As for the concept activation maps, we use spatial upsampling to map rl xy back to the input space and obtain concept relevance heatmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Since upsampling preserves the completeness relation, these decompose the local relevance maps rx,z = 1 W Hφφφα xy · www used by Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (2016) (commonly referred to as class activation maps (CAMs)) into concept contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Global relevance and completeness score Next, we establish a global (model-wide) concept relevance score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Recall, that all cccl j defined above represent a basis for the feature space RF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Hence, we can directly decompose the weight vector www into (analogously to Equation (2)) www = nc+1 � l=1 dl � i=1 wl icccl i ≡ nc+1 � l=1 wwwl , (4) where wwwl = �dl i=1 wl icccl i and by construction, wwwl · wwwnc+1 = 0 for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In this case, we have |www|2 = |wwwnc+1|2 + | nc � l=1 wwwl|2 = nc+1 � l=1 |wwwl|2 + nc � l,k=1,l̸=k |wwwl||wwwk| cos(∠(wwwl,wwwk)) (5) The first equality allows us to define η({Cl}) = 1 − |wwwnc+1|2/|www|2 (6) as a completeness score (fraction of www which is explained by all concepts {C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' , Cnc}) with respect to a given class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To the best of our knowledge, we are the first to introduce a concept completeness score directly based on model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Previous work (Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2020) defined a related measure based on model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Note, that for an orthonormal basis the second term in Equation (5) (cosine) disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Then |wwwl|/|www| can be directly interpreted as (global) concept relevances, which sum up to the previous completeness score over all concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Further, the angles in Equation (5) are lower- and upper-bounded by the corresponding minimal or maximal principal angles5 between the two corresponding subspaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', θkl min ≡ minmθkl m ≤ ∠(wwwk,wwwl) ≤ maxmθkl m ≡ θkl max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This means we can lower- and upper-bound |www|2 by nc+1 � l |wwwl|2 + nc � l,k=1,l̸=k |wwwl||wwwk| cos(θlk max) ≤ |www|2 ≤ nc+1 � l |wwwl|2 + nc � l,k=1,l̸=k |wwwl||wwwk| cos(θlk min) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (7) Obviously, lower and upper bound coincide in the case of orthogonal subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This implies, that the |wwwl| are also informative in the non-orthogonal case, provided the principal angles between the different subspaces are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This highlights the intricate connection between (global) relevances and the geometry in feature space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', the relative orientation of the concept spaces (specified via principal angles between pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Finally, we briefly comment on the applicability of our approach for local and global concept relevances via Equation (3) and Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In the form described above it can be used for any model with a linear layer as final layer, potentially preceded by a global pooling layer, if one aims to spatially resolve the relevances instead of considering only pooled feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This latter category covers a broad range of modern CNN architectures such as ResNets, Inception-based model but also vision transformers, that do not base their prediction on a CLS token, such as Swin transformers (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We envision, that our approach is even applicable, in approximate form, to other feature layers apart from the final hidden layer if one locally approximates the remainder of the model by a linear model, similarly as it is done by Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (2016) or by Selvaraju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (2020) to generalize (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 4We briefly comment on the remaining commonly desired Shapley axioms Lundberg & Lee (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The local concept relevance trivially fulfills them, since it is built on a linear additive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Formally, the hidden activation φα of a given sample α are segmented into concept contributions/unique features φl α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Thus, the value function corresponding to the underlying Shapley values is given by vα(S) = � l∈S φl α · w (linear in φl) for S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' , nc + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 5A formal definition of principal angles is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 7 3 Related Work ACE (Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2019) uses a superpixel segmentation algorithm and k-means clustering to identify class-specific concept candidates for TCAV (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The concept discovery scheme of ACE has several shortcomings: The segmentation into candidate concept patches is model-independent and thus, segments are not necessarily meaningful as perceived by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To enable clustering of intermediate CNN activations, segments are resized and mean padded to the original input shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This leads to artificial, off- manifold samples with potentially distorted aspect ratios and discards the overall scale information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Finally, ACE relies on multiple heuristics to discard segments/clusters both before and after k-means clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In contrast, MCD is coherently based on hidden model representations without relying on additional pre- or post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Similar limitations apply to methods that rely on ACE-discovered labeled concepts, like (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021), which uses Shapley values for concept importance, and (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2020), which occludes particular neurons for neuron-wise relevances and transforms them into concept importances via concept classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Recently, Crabbé & van der Schaar (2022) proposed a generalization of TCAV by invoking the kernel trick, which generalizes the concept definition towards non-linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' However, unlike MCD, it does not allow quantifying the relevance of a concept towards the model prediction and can only verify predefined concepts instead of discovering them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' ICE (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021) defines concepts as directions in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Technically, this is achieved via dimensionality reduction techniques applied to concatenated flattened feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' ICE measures the im- portance of its class-wise concepts using TCAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Interestingly, ICE introduces the notion of a concept weight, which is analogous to our concept relevances on the logit layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' However, they do not consider spatially re- solved concept relevance heatmaps and only address the special case of single-dimensional subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Given these restrictions, ICE can be seen as a special realization of the MCD framework, which uses dimensionality reduction methods like PCA as clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Other methods learn concept vectors and a mapping to feature space either for all classes simultaneously (ConceptSHAP (Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2020)) or for each class sep- arately (MACE (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021), PACE (Kamakshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' ConceptShap, MACE and PACE all use additional regularizers to enforce concept dissimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In contrast, MCD restricts the concept discovery process as little as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Importantly, each method above defines a custom measure for concept impor- tance, which is based on approximations of the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In contrast, the local and global concept relevance within MCD are solely based on the original model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Other approaches (Chormai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2022) use a concept definition similar to ours, but use information from external attribution methods as well as orthogonality constraints to restrict the discovered concepts, whereas MCD works without such restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' There is a complementary line of work of frameworks that try to identify concepts associated with particular neurons in hidden CNN representations, in conjunction with (Bau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2017) or without (Achtibat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2022) special concept-annotated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Network Dissection (Bau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2017) investigates the alignment of human-understandable concepts and particular single hidden features (neurons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Net2vec (Fong & Vedaldi, 2018) extended this by allowing concepts to be represented by combinations of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Lastly, there is a line of research that constructs inherently interpretable concept models by design with (Koh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Radenovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2022) or without relying on concept annotations (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Our approach is best comparable with (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2019), as both can be reduced to a linear model operating on concepts that can be characterized via prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We stress the essential difference, that our approach does not require retraining (with special training objectives) but is an interpretable reformulation of the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 4 Results We carry out our experiments on ImageNet (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' As model architectures, we consider ResNet models (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2016) using original weights as provided by torchvision and updated weights as provided by timm (Wightman, 2019) with an improved training procedure (Wightman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We also present re- sults for a swin vision transformer (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021), again using weights provided by timm (SwinS3base224).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 8 In the following, we will refer to these models as ResNet50, ResNet50v2 and, Swin-T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We base most of our experiments on images from a diverse selection of ten ImageNet classes, which roughly align with CIFAR10 classes6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='1 Completeness arithmetic First, we provide a concrete example for an MCD explanation and showcase its completeness relation intro- duced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='3 (benefit 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To this end, Figure 3 shows an MCD-SSC explanation of a ResNet50v2 prediction for a sample of the police van class in ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The number of concepts was chosen such that the completeness measure in Equation (6) reaches η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The three information components of the explanation all provide complementary information: (1) Concept relevance heatmaps decompose the local relevance maps into a sum of concept-specific relevance heatmaps according to the completeness relation Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' They show the alignment of a feature vector component φφφl associated with concept Cl and the weight vector of a specific class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Roughly speaking, this alignment indicates how typical the network perceives the particular instantiation of the concept for the class under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Applying mean pooling leads to a corresponding decomposition of the class logit under consideration (up to the bias term) into contributions corresponding to different concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The completeness relation on the level of concept relevance heatmaps as well on the level of logits represents a unique feature of the MCD framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Interestingly, for the explanation in Figure 3, only the orthogonal complement concept contributes negatively to the class logit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The contributions of the first two concepts clearly dominate the class logit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (2) Concept activation maps stem from a decomposition of the feature vectors into a sum of vectors coming from different, distinct concept subspaces, see Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Their non-negative score show how much a particular feature vector aligns with a specific concept subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' These maps help to identify input regions where the concept is highly expressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We color-code concept activation maps as a transparent overlay over the image where transparent regions indicate high activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To guide the eye, we also include a yellow(white) contour line at a threshold value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='5(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' As a sanity check, we compute the Pearson correlation coefficient between the positive part of each concept relevance map and each concept activation map for MCD-SSC and ResNet50v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Among all test set samples, we find a mean correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='45 for the concepts of the CIFAR10 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This confirms that concept relevance is high in sample areas where the respective concept is strongly activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (3) Concept prototypes allow characterizing a concept subspace through examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Here, we display the concept activation maps of three test set samples that show the highest activation with the given concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In many cases, an intuitive meaning of a concept can be inferred most easily from these samples and numer- ous previous approaches present concepts in this way (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Achtibat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Yeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In case of the explanation in Figure 3, this could be windows/livery, livery, blue lights, building, tires (and the orthogonal complement covering mainly the background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In addition, we also indicate the global concept relevances for the different concepts according to Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In summary, the sample in Figure 3, is classified as a police van mainly due to its windows/livery, which are perceived as typical for the class by the network and are also the most relevant concept for the class globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Further, all other concepts are expressed in the sample and contribute positively, except for the orthogonal complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2 Empirical evaluation We compare the the MCD flavors and previous methods listed in Table 1 in terms of (1) true-to-the-modelness (benefit 1) and (2) conciseness (benefit 2) of the explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We base our evaluation on the CIFAR10 classes mentioned above, and work with the ResNet50v2 model, for which we extract concepts from the last hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For all methods within the MCD framework, we fix the number of concepts in a class-dependent way such that we reach a completeness score of η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='77 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='94 (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='63) + + + + class logit - bias 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='74 = prototypes concept C1 C2 C3 C4 C5 local relevance concept space complement = mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='22 global relevance = + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='86 activation mean mean mean mean mean LOCAL GLOBAL + + + Figure 3: Completeness relation for the police van class in ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concepts are discovered via MCD-SSC for ResNet50v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The number of concepts is chosen such that the completeness score reaches η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We distinguish between local (sample-specific) and global properties (characterizing a set of samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Locally, we consider concept relevance maps, which quantify the spatially resolved contribution of a concept to the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' These satisfy a completeness relation, as explicitly shown in the first line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concept activation maps provide complementary information and indicate how much a concept is activated depending on the spatial location in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Globally, the overall relevance of a particular concept is quantified by the global relevance scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Finally, we also present concept prototypes (concept activation maps of the most strongly activated samples) to characterize a particular concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Table 1: Summary of concept discovery methods considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Method Multi-dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Arbitrary orientation MCD-SSC \x13 \x13 MCD-SSC-ortho \x13 \x17 MCD-kmeans \x13 \x13 ICE/MCD-PCA (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2021) \x17 \x17 ACE (Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2019) \x17 \x13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='1 Comparing true-to-the-modelness via concept flipping In order to compare the methods in Table 1 in terms of true-to-the-modelness, we invoke the Smallest Destroying Concepts (SDC) benchmark as proposed in (Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2019) and (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For concepts that reflect the model’s actual reasoning structure in feature space and true-to-the-model concept relevance scores, SDC should show a sharp decline of the model accuracy with the number of flipped concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Here, we already mention the trivial solution for the sharpest decline, which is assigning the whole object to a single concept and provides little insight into the model behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We obtain concept masks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' hard concept assignments, in input space by taking the argmax of the corre- sponding concept activation maps over all concepts including the orthogonal complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' After the argmax operation, we disregard the orthogonal complement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', we do not remove it during the SDC experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 6namely (airliner, beach wagon, hummingbird, siamese cat, ox, golden retriever, tailed frog, zebra, container ship, police van) 10 1305700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='8 percentage of deleted pixel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='8 prediction accuracy MCD-SSC MCD-SSC-ortho MCD-kmeans ICE ACE sample MCD-SSC MCD-SSC-ortho MCD-kmeans ICE ACE Figure 4: Left: Concepts are flipped one at a time in descending order of local concept importance/TCAV score, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We measure the decline in model accuracy and show the mean accuracy across CIFAR10 classes against the fraction of deleted pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Meaningful concept discovery and quantification methods are supposed to show a sharp decline in this figure, but the decline should not happen after flipping only a single concept (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' the whole object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Right: Qualitative comparison between hard concept assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For each concept mask we obtain local relevance scores by pooling the corresponding concept relevance heatmaps over the respective regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This provides concept masks in input space which are ordered ac- cording to their importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' ACE does not provide a measure of per-sample concept relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Therefore, we revert to the order of their (global) TCAV scores after discarding concepts where statistical testing in comparison to random input samples fails to stay below p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To provide a qualitative impression of the concept relevance heatmaps across methods, we show them together with concept activation maps for selected samples in Figures 6 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To evaluate SDC, we subsequently remove concepts, as represented by the corresponding segments, in order of their sample-wise (local) relevance starting from high to low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To inpaint the removed segments, we use a classical imputation algorithm (Bertalmio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2001), which leads to comparably realistic imputed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Thus, the model is evaluated on-manifold in contrast to imputing with gray patches as often done in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For similar reasons, we avoid the Smallest Sufficient Concepts (SSC) benchmark, which would require high-quality imputation algorithms to avoid evaluating the model far from the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In the left panel of Figure 4, we show the results of the SDC experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In contrast to previous studies (Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2020), we report the model performance depending on the fraction of oc- cluded pixels, which is essential for comparability since the segment size varies between different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In order to show a meaningful average of the samples across all classes we flip only as many concepts as are present for the class with minimum number of concepts nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' As mentioned above, a meaningful concept dis- covery and quantification method should show a sharp decline in this figure, but the full decline of the score should not happen after flipping only a single concept (covering the whole object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We observe the latter for ICE/MCD-PCA and MCD-SCC-ortho, both of which rely on orthogonal concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We hypothesize that this behavior relates to the greedy way these orthogonal subspaces are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In particular, ICE/MCD- PCA only detects a single relevant and expressive concept, as the accuracy curve stagnates after flipping the first concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Among the remaining algorithms, MCD-SSC shows the strongest decline as compared to MCD-kmeans and ACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In Figure 4 (right panel), we also show hard concept assignments for an example image of the golden retriever class, which form the basis of the concept flipping experiment described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' These visually support the findings of the concept flipping experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Most approaches only discover a single concept for the dog (apart from a potential genuine background concept).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' MCD-SCC shows the most finegrained decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This trend is also supported by the average subspace dimension dl as stated in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 11 To summarize the results of the concept flipping experiment, our general MCD definition leads to concepts that are most true-to-the-model, as the two unconstrained MCD flavors (MCD-SSC and MCD-kmeans), show the steepest descent among all methods without reverting to the non-informative solution of a single relevant concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2 Conciseness of explanations In Section 1, we describe why it is desirable, to explain the model reasoning with as few meaningful concepts as completely as possible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' to deliver concise concept explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Similarly to the above section, there is a trivial solution to extremely concise concepts for the ImageNet classification task, which is to relate a major part of feature space to a single concept of high dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Therefore, we characterize the conciseness of concept explanations not only by the number of concepts nc that is required to reach a certain completeness score, but also by average subspace dimension dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Additionally, we evaluate the mean (scaled) Grassmann distance ∆kl c , as defined in Equation (10), between all concept pairs (k, l) within one class c to quantify how dissimilar two concepts are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='7 In summary, we argue that concepts should be concise (small nc), but dissect the feature space into meaningful building blocks of model reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' While the latter is difficult to quantify, we argue that there is a trade-off between (1) covering feature space with very few concepts of high dimensionality and potentially small distance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (2) dissecting it into a high number of concepts with small dimensionality (extreme case: one-dimensional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To support this reasoning, we also inspect the visual impression of concepts for a selection of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We list the number of concepts nc that is required to reach a completeness score of η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='5 for ResNet50v2 on the CIFAR10 classes and dl in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To provide a visual comparison of the concepts discovered by these methods, we show concept activation maps of prototypes for basketball, golden retriever and airliner class in ImageNet in Figures 9 to 11 and judge how broad they appear in input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' MCD-kmeans discovers the smallest number of concepts with the highest mean concept dimensionality of 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='7 and the smallest inter-concept distance (mean(∆kl c ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='83) among all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We argue that this is reflected in the visual appearance of the concept prototypes, which are visually broad and difficult to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' MCD-SSC discovers on average 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='8 concepts with a smaller mean concept dimensionality of 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Visually, concepts seem medium broad and are easier to distinguish in input space than for MCD-kmeans, which is reflected in a higher inter-concept distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' When requiring orthogonality mean(∆kl c ) = π/2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='57, like for MCD-SSC- ortho and ICE, we see that only one concept appears medium broad in input space while all others are almost not activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We argue, that the orthogonality constraint hinders the concepts to reflect a natural similarity between certain concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This aligns with the conclusions drawn from the SDC benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Most notably, to achieve a comparable model faithfulness (completeness score of η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='5) 30 times more one-dimensional ICE concepts than multi-dimensional MCD concepts are required, meaning this method delivers concept explanations that are not concise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Intuitively, a single concept is split up into several concepts, which is also reflected in their weak activation on test set samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Lastly, the visual impression of ACE concepts is fixed by the choice of the superpixel algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' While ACE concepts are all one-dimensional, they do not provide a mechanism to quantify how complete they are, thus we cannot quantify nc required to reach a completeness of 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' As an overall summary, MCD-SSC is superior in dissecting the feature space into enclosed and meaningful concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='3 Use case: MCD concepts reveal differences in classification strategies between model architectures and training procedures Finally, we showcase how MCD can unravel different classification strategies depending on the model archi- tecture (ResNet50 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Swin-T) and the training strategies (ResNet50 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' ResNet50v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The test accuracies for the subset of CIFAR10-classes are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='80 (ResNet50), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='84 (ResNet50v2) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='86 (Swin-T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Here, we focus on MCD-SCC and, as before, restrict ourselves to concepts in the last hidden feature layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' First, we compare the discovered concepts between the models by the activation maps of concept prototypes for the beach wagon class of ImageNet in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We fix the number of concepts to nc = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For Swin-T, 7We use a scaled version of the original Grassmann distance that aggregates the principle angles (in radian) between two subspaces, for which 0 ≤ ∆kl c ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Two special cases are ∆kl c = 0 meaning subspace bases vectors are perfectly aligned, and ∆kl c = π/2, meaning they are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 12 Table 2: Summary of concept discovery methods considered in this work in comparison to prior work from Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (2021) (ICE/MCD-PCA) and Ghorbani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (2019) (ACE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We measure average subspace dimension dl and the number of concepts nc that is required to reach a completeness score of η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='5 for ResNet50v2 on the CIFAR10 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' A small number of relevant concepts nc is desirable, since this summarizes the complete model into an accessible and meaningful format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Here, multi-dimensional concepts have an advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Additionally, we evaluate the mean (scaled) Grassmann distance ∆kl c , see Equation (10), between all concept pairs (k, l) within one class c to quantify the distinctness between concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The visual inspection is based on prototypes of the basketball, golden retriever and airliner class concepts in Figures 9 to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Medium broad and distinct concepts are the most informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Method mean(dl) mean(nc) mean(∆kl c ) Visual inspection MCD-SSC 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='19 medium broad MCD-SSC-ortho 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='57 only one broad (rest narrow) MCD-kmeans 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='83 very broad ICE/MCD-PCA 1 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='57 only one broad (rest narrow) ACE 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' medium broad we only apply a spatial upsampling of the concept activation maps from the feature to the input space to 14 × 14 in order to account for the 16 × 16 patch tokenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We find that ResNet50 concepts, which could roughly be identified as (car body, windows, car roof, wheels, street), are more narrow than the expression of Swin-T and ResNet50v2 concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The latter are related to broader views of the car, such as concepts (1, 2, 4) for ResNet50v2 and concepts (1, 3) for Swin-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Interestingly, ResNet50v2 concepts reach a much lower completeness score of η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='49 than ResNet50 (η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='89) and Swin-T (η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='84) for fixed nc = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In Figure 5 we show the relation between the total concept space dimensionality, the number of concepts nc and the completeness score η across the CIFAR10-classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Even for nc = 30, the ResNet50v2 concepts have a lower η than those of the ResNet50 for nc = 3, although the former covers already a much larger part of the concept space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' These observations support the statement that feature space of the ResNet50v2 exhibits comparably richer structure than Resnet50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This is an interesting difference in the character of the feature space as a consequence of two different training procedures for the same architecture, revealed by MCD-SSC concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Interestingly, the dependence of η on nc for the concepts between two models with different architectures, ResNet50 and Swin-T, is quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This also aligns with the visual appearance of the concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To summarize, Swin-T and ResNet50 build on broader and more versatile concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In comparison, ResNet50v2 builds more narrow and thus specific concepts for its classification strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' These broad con- cepts are not unexpected for a transformer architecture like Swin-T with coarse self-attention windows, but a rather surprising finding for ResNet50 in comparison to ResNet50v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 5 Summary and Discussion In this work, we put forward MCD, a general framework for concept discovery based on the hidden repre- sentation of a trained deep neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Unlike prior work in the field, we propose a general concept definition (incorporating previous approaches) as multi-dimensional linear subspaces without restricting to single directions or enforcing orthogonality between subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We use concept activation maps to visualize concepts in input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Considering the final hidden layer representation, we can reformulate the original model as a linear classifier acting on linear concept subspaces without the need to retrain with a special objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This leads to a completeness relation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', a natural decomposition of class logits into contribu- tions corresponding to specific concepts and allows to resolve their spatial importance in terms of concept relevance heatmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' As a particularly suited realization of our framework, we put forward MCD-SCC, which relies on sparse subspace clustering for concept discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Based on qualitative and quantitative insights, we show the superiority of MCD-SCC over other MCD flavors that build on traditional clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 13 0 200 400 600 800 1000 1200 conceptspace dimensionality l d l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='0 completeness ResNet50v2 ResNet50 Swin-T concept space complement Swin-T Resnet50v2 ResNet50 Figure 5: Left: Mean concept space completeness score ν for the CIFAR10 classes across architectures against the dimensionality of the union of all concept subspaces � l dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The number of concepts can be inferred from the points on the line where the first point on each line corresponds to nc = 3 and the last one to nc = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' ResNet50v2 shows a much lower completeness score at roughly the same nc and � l dl as ResNet50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The feature space dimensionality is F = 2048 for ResNet50(v2) and F = 768 for Swin-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Right: We show MCD-SSC concept activation maps for concept prototypes for ResNet50, ResNet50v2 and Swin-T and the beach wagon class in ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We fixed the number of concepts to nc = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In this way, ResNet50v2 reaches η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='49, ResNet50 η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='89 and Swin-T η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Each row shows a single concept and is titled by its global concept importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The last row shows the orthogonal complement of the concept space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We showcase the ability of MCD via discriminating between hidden representations obtained from different model architectures and training strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This paves the way towards further novel use-cases for MCD concepts such as gaining insights in the natural sciences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', identifying sub-classes of cancerous cells in histopathology or summarizing model behavior beyond single examples and thereby systematically discover model biases.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' A tutorial on spectral clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Statistics and computing, 17(4):395–416, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Manuel Weber, David Kersting, Lale Umutlu, Michael Schäfers, Christoph Rischpler, Wolfgang P Fendler, Irène Buvat, Ken Herrmann, and Robert Seifert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Just another “clever hans”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' neural networks and fdg pet-ct to predict the outcome of patients with breast cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' European journal of nuclear medicine and molecular imaging, 48(10):3141–3150, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 16 Ross Wightman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Pytorch image models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='com/rwightman/pytorch-image-models, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Ross Wightman, Hugo Touvron, and Hervé Jégou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Resnet strikes back: An improved training procedure in timm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' arXiv preprint 2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='00476, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Weibin Wu, Yuxin Su, Xixian Chen, Shenglin Zhao, Irwin King, Michael R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Lyu, and Yu-Wing Tai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Towards global explanations of convolutional neural networks with concept attribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In IEEE Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 8649–8658, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Chih-Kuan Yeh, Been Kim, Sercan Arik, Chun-Liang Li, Tomas Pfister, and Pradeep Ravikumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' On completeness-aware concept-based explanations in deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Chong You, Chun-Guang Li, Daniel P Robinson, and René Vidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Oracle based active set algorithm for scalable elastic net subspace clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In IEEE Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 3928–3937, 2016a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Chong You, Daniel Robinson, and René Vidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Scalable sparse subspace clustering by orthogonal matching pursuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In IEEE Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 3918–3927, 2016b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Ruihan Zhang, Prashan Madumal, Tim Miller, Krista A Ehinger, and Benjamin IP Rubinstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Invert- ible concept-based explanations for cnn models with non-negative concept activation vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In AAAI Conference on Artificial Intelligence, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 11682–11690, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Learning deep features for discriminative localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In IEEE Conference on Computer Cision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 2921–2929, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' A SSC Algorithmic Details Concept-determining self-representation We compute sparse self-representations R for a random sub- collection of n ≤ N ·H ·W feature vectors {φφφα xy} sampled from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Here, the term self-representation refers to a coefficient matrix that expresses each sample as a linear combination of all other samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' More specifically, using the notation from (Elhamifar & Vidal, 2013), given the feature vectors Φ = [φφφ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' ,φφφn] ∈ RF ×n, we identify a sparse coefficient matrix RRR = [rrr1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' ,rrrn] ∈ Rn×n such that φφφj = Φrrrj where rii = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (8) The particular kind of sparsity constraints that are imposed on Equation (8) and how it is optimized depends on the chosen SSC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Here, we use elastic net subspace clustering (You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2016a), which is robust against noise and scales well for large sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In all our experiments, we fix the hyperparameter γ, which balances sparsity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' robustness, to γ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We confirmed that the results are not sensitive to variation of this parameter over a range of values from 5 to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' As computation time for SSC is dependent on this parameter, we chose γ such that this is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We remove outliers based on the ℓ1-norm as in (Soltanolkotabi & Candes, 2012), where we empirically fix the percentile threshold to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='75 and re-fit the sparse self-representation for the remaining elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Another scalable alternative to the elastic net clustering is orthogonal matching pursuit (OMP)(You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=', 2016b), which is, however, not robust against noise and does not allow for outlier removal via thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Finally, the original sparse subspace clustering method from (Elhamifar & Vidal, 2013) is robust against noise and outliers but does not scale to large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The particular robustness and scalability properties make elastic net subspace clustering (with thresholding) an ideal choice for the first step of our concept discovery method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Spectral clustering In a second step, we perform spectral clustering with the affinity matrix W = |R|+|RT |, which encodes the similarity of two feature vectors according to their self-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' We determine the 17 number of clusters nc either via the largest gap in the spectrum of the Laplacian (Von Luxburg, 2007) or use a predetermined value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' This step assigns every input feature φφφi to a particular cluster C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' , Cnc or to the set of outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' B Characterizing relation between subspaces by principal angles In this section, we briefly review the definition of principal angles, which can be used to characterize the rela- tion between two linear subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The principal angles θAB i (Jordan, 1875) (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' , min(dim A, dim B)) between two linear subspaces A, B, are defined recursively via cos θAB i = max aaa∈A,bbb∈B aaaTbbb |aaa||bbb| =: aaaT i bbbi |aaa||bbbi| , (9) where the maximum is taken subject to the orthogonality constraints aaaTaaaj = 0 and bbbTbbbj = 0 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' , i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' To quantify the similarity between two subspaces A and B, we use a scaled version of their Grassmann distance Hamm (2008), which is defined as, ∆AB = 1/ � min(dim A, dim B) � (θAB 1 )2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' + (θAB min(dim A,dim B))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' (10) This allows comparing the similarity of concepts within a given class or across classes regardless of the concept subspaces’ dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' C Qualitative results For a qualitative comparison between of the concept activation maps and relevance heatmaps between the methods in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='2, we provide results for selected samples in Figures 6 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' In Figures 9 to 11 we show the respective concept prototypes for all concept discovery approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 18 + + + + + + + + + + + + + + + + + MCD-SSC MCD-SSC-ortho + + + + + + + + + + + + + + + + + } = = MCD-kmeans ICE ACE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='78 concept space complement Figure 6: Concept heatmaps and activation maps for ResNet50v2 and a randomly chosen sample from the basketball class in ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The number of concepts is chosen such that the completeness score reaches η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concepts are ordered from left to right according to global concept relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concept heatmaps are titled by the pooled local concept relevance that sums to the prediction logit minus the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For ICE, we only show the first six out of 105 and for ACE the first six out of 25 concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 19 + + + + + + + + + + + + + + + + + MCD-SSC MCD-SSC-ortho + + + + + + + + + + + + + + + + + } = = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='28 concept space complement MCD-kmeans ICE ACE Figure 7: Concept heatmaps and activation maps for ResNet50v2 and a randomly chosen sample from the golden retriever class in ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The number of concepts is chosen such that the completeness score reaches η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concepts are ordered from left to right according to global concept relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concept heatmaps are titled by the pooled local concept relevance that sums to the prediction logit minus the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For ICE, we only show the first six out of 142 and for ACE the first six out of 25 concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 20 MCD-SSC MCD-SSC-ortho } concept space complement MCD-kmeans ICE ACE + + + + + + + + + + + + + + + + + + + + + + + + = = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='66 Figure 8: Concept heatmaps and activation maps for ResNet50v2 and a randomly chosen sample from the airliner class in ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The number of concepts is chosen such that the completeness score reaches η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concepts are ordered from left to right according to global concept relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' Concept heatmaps are titled by the pooled local concept relevance that sums to the prediction logit minus the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For ICE, we only show the first seven out of 141 and for ACE the first seven out of 25 concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 21 MCD-SSC MCD-SSC-ortho MCD-kmeans ICE ACE Figure 9: Concept activation maps for concept prototypes for basketball class of ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The last row shows prototype for the complement, except for ACE, where no complement exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For ICE, we only show the first six out of 105 and for ACE the first six out of 25 concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 22 1042PLERAANCLRU33MCD-SSC MCD-SSC-ortho MCD-kmeans ICE ACE Figure 10: Concept activation maps for concept prototypes for golden retriever class of ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The last row shows prototype for the complement, except for ACE, where no complement exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For ICE, we only show the first six out of 142 and for ACE the first six out of 25 concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 23 UNASYDMCD-SSC MCD-SSC-ortho MCD-kmeans ICE ACE Figure 11: Concept activation maps for concept prototypes for airliner class of ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' The last row shows prototype for the complement, except for ACE, where no complement exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' For ICE, we only show the first seven out of 141 and for ACE the first seven out of 25 concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content=' 24 FFIRDSTAIRWorldwide Services conizing theworld ol cornmetceeaworldethopian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='HollandExel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='dleglantairtransatWorldwide Services Fonizingtheworld ol cotmmetceDRAGOEhoplaaairtransatWorldwideServices ronizingtheworldofcommerceWWoridwidesoplan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='.easEthoplan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} +page_content='airtransat' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNFKT4oBgHgl3EQfzS7F/content/2301.11911v1.pdf'} diff --git a/rdE1T4oBgHgl3EQfjAQ8/vector_store/index.pkl b/rdE1T4oBgHgl3EQfjAQ8/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..7c0febf5c0466447a0b5949b2ce4f7acacd2e1e4 --- /dev/null +++ b/rdE1T4oBgHgl3EQfjAQ8/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d47705d8fad22667dda61cd309bfcb2186dcf7f5a285d7def24d4d7f91d57614 +size 226468 diff --git a/tdAzT4oBgHgl3EQf6v4U/content/tmp_files/load_file.txt b/tdAzT4oBgHgl3EQf6v4U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3af284dcff7e329eb5f4fe90fbfcc3e8f9a2b3c --- /dev/null +++ b/tdAzT4oBgHgl3EQf6v4U/content/tmp_files/load_file.txt @@ -0,0 +1,627 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf,len=626 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='01878v1 [nucl-th] 5 Jan 2023 Cluster-shell competition and effect of adding hyperons Naoyuki Itagaki1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 2 and Emiko Hiyama3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 4 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Osaka Metropolitan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Osaka 558-8585,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Japan 2Nambu Institute for Theoretical and Experimental Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Osaka Metropolitan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Osaka 558-8585,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Japan 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Graduate School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Tohoku University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Sendai 980-8578,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Japan 4RIKEN Nishina Center for Accelerator-Based Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Wako 351-0198,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Japan (Dated: January 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 2023) Background: The fundamental question is how the hyperon plays a role in the nuclear structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' It is of particular importance, especially in the light mass region, to verify the structure change when Λ particle(s) is added to normal nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Purpose: The ground state of 8Be has been know to have a well-developed α–α cluster structure, whereas 12C has a mixed structure of three α clusters and jj-coupling shell model, where α clusters are partially broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Adding Λ particle(s) could induce the structure change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' We compare the Be and C cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Methods: Using the antisymmetrized quasi-cluster model (AQCM), the α-cluster states and jj-coupling shell- model states of 8Be and 12C are prepared on the same footing, and we add Λ particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The cluster-shell competition in the ground state can be well described with this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Using AQCM, we calculate 8Be, 9 ΛBe, 10 ΛΛBe, 12C, 13 Λ C, and 14 ΛΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Results: By adding one or two Λ particle(s), the ground state of 12C approaches the jj-coupling shell model side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' On the other hand, in the Be case, although the Λ particle(s) shrinks the α–α distance, the breaking effect of the cluster structure is rather limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Conclusions: The spin-orbit interaction is the driving force of breaking the α clusters, and whether the glue-like effect of Λ particle(s) attracts the cluster inside the range of this interaction is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' In 14 ΛΛC, the breaking of α clusters in 12C is much enhanced by the addition of the Λ particles than the case of free 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' We also found that breaking α clusters in the ground state of 14 ΛΛC affects the excited state with the pure cluster structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' INTRODUCTION One of the most intriguing phenomena of nuclear struc- ture physics is the competition of the shell and cluster structures [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This is attributed to the effect of the spin- orbit interaction, which strengthens the symmetry of the jj-coupling shell model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' It is well known that this inter- action is vital in explaining the observed magic numbers of 28, 50, 82, and 126 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The spin-orbit interaction also has the effect of breaking clusters [1], where some of the strongly correlated nucleons are spatially localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Nevertheless, the α cluster structure is known to be important in the light mass region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The Be isotopes are known to have the α–α cluster structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 8Be decays into two α clusters, and the molecular-orbital structure of valence neutrons appears in the neutron-rich Be iso- topes [3–5], which is confirmed by the recent ab initio shell-model calculation [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This persistence of the α–α cluster structure is owing to the α–α distance, which is about 3–4 fm and large enough compared with the range of the spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' In light nuclei, it is considered that these two differ- ent pictures (shell and cluster) coexist, and they com- pete with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Although the α–α cluster structure may persist in 8Be, when one more α cluster is added, in 12C, the interaction among α clusters gets stronger, and the system has a shorter α–α distance [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' In this case, the α clusters are trapped in the interaction range of the spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Although the traditional α cluster model (Brink model) [9] is incapable of treating the spin-orbit interaction, its effect is significant if we al- low the breaking of the α clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The ground state of 12C is found to have a mixed nature of shell and clus- ter components [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' On the other hand, the second 0+ state of 12C is well known α clustering state called the Hoyle state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Since this state is nearby the three-α breakup threshold, the wave function is dilute, and this state has a well-developed α clustering structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' It is interesting to investigate how clustering structure is changed when a hyperon such as a Λ particle is injected into 8Be and 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Here it should be noted that there is no Pauli principle between nucleons and a Λ, and the ΛN interaction is attractive, but weaker than NN in- teraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Using this property, some authors studied the structure of 9 ΛBe and 13 Λ C from the viewpoint of dynam- ical change of the core nuclei, 8Be and 12C, due to the addition of Λ particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' For instance, Motoba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' [13], pointed out that the α–α distance in 9 ΛBe was shrunk by about 20 % in comparison with that in the 8Be core nucleus by Λ injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' In the Carbon isotope, one of the present authors (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=') pointed out that dynamical change due to the addition of a Λ particle is dependent on the states in the core nucleus of 12C within the framework of 3α and 3α + Λ three- and four-body OCM (orthogo- nal condition model) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The ground state of 12C, 0+ 1 , is a mixture of shell and cluster structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' the α–α dis- tance does not change due to the addition of a Λ particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' On the other hand, the α–α distance is dramatically con- tracted in the Hoyle state of 13 Λ C, which is well-developed clustering state [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' However, it should be noted that 2 this calculation was done without taking into account the breaking effect of α clusters in 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' In addition, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' [15], they discussed the similarity and difference in several states of 12C and 13 Λ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' In this way, there are some discussions on the change of the α–α distance w/o the Λ particle and the change of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' However, there remain never discussed effects of the clustering in such Be and C isotopes due to addition of Λ particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The question is how the clustering is broken when Λ particles shrinks the α–α distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The traditional cluster model is incapable of describing such breaking situation and we must extend the model space to incorporate the spin- orbit contribution, which is the driving force of breaking clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Thus, in this work, we focus on how the clustering is changed and broken due to the addition of a Λ particle(s) in 8Be, 9 ΛBe, 10 ΛΛBe, 12C, 13 Λ C, and 14 ΛΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' In the case of Be isotopes, as mentioned, the Λ particle(s) shrinks the α– α relative distance [14, 16], but the resultant distance might still be outside the range of the spin-orbit inter- action, and the α cluster structure could persist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' On the contrary, when Λ particle(s) is added to 12C, the distances between clusters get even shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Since the spin-orbit interaction works in the inner regions of the nu- clear systems, the breaking of α clusters is expected to be enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Therefore, the ground state would approach more jj-coupling shell-model side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Indeed, as shown in the study of antisymmetrized molecular dynamics [17], the slightly deformed ground state of 12C is changed into a spherical shape in 13 Λ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' It is worthwhile to check this point in terms of the cluster-shell competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' In most of the conventional α cluster models, the con- tribution of the non-central interactions (spin-orbit and tensor interactions) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' To include the spin-orbit effect, we have developed the antisymmetrized quasi- cluster model (AQCM) [10, 18–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This method allows us to smoothly transform α-cluster model wave func- tions to jj-coupling shell model ones, and we call the clusters that feel the effect of the spin-orbit interaction quasi-clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' We have previously introduced AQCM to 12C and discussed the competition between the clus- ter states and jj-coupling shell model state [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The consistent description of 12C and 16O, which has been a long-standing problem of microscopic cluster models, has been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Also, not only the competition between the cluster states and the lowest shell-model configura- tion, the effect of single-particle excitation was further included in the description of the ground state [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The framework is described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The results are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The conclusions are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' FRAMEWORK The wave function is fully antisymmetrized, and dif- ferent basis states are superposed based on the genera- tor coordinate method (GCM) after the angular momen- tum projection, and the amplitude for each basis state is determined by diagonalizing the norm and Hamiltonian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Single-particle wave function In our framework, every single particle is described in a Gaussian form as in many traditional cluster models, including the Brink model [9], φτ,σ (r) = �2ν π � 3 4 exp � −ν (r − ζ)2� χτ,σ, (1) where the Gaussian center parameter ζ is related to the expectation value of the position of the nucleon, and χτ,σ is the spin-isospin part of the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The α clus- ter is expressed by four nucleons with different spin and isospin sharing the same ζ value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' For the size parameter ν, here we use ν = 1/2b2 and b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='46 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The Slater de- terminant is constructed from these single-particle wave functions by antisymmetrizing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The Λ particle is represented by the same local Gaussian-type wave func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This traditional α cluster wave function cannot take into account the effect of non-central interactions includ- ing the spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' We can extend the model based on the AQCM, by which the contribution of the spin-orbit interaction due to the breaking of α clusters is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Here the ζ values in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' (1) are changed to complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' When the original value of the Gaus- sian center parameter ζ is R, which is real and related to the spatial position of this nucleon, it is transformed by adding the imaginary part as ζ = R + iλespin × R, (2) where espin is a unit vector for the intrinsic-spin orienta- tion of this nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The control parameter λ is associ- ated with the breaking of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' After this transfor- mation, the α clusters are called quasi-clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The two nucleons in the same quasi-cluster with opposite spin ori- entation have ζ values that are complex conjugate to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This situation corresponds to the time-reversal motion of two nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' In our previous analysis on 12C [10], we have intro- duced two parameters representing the distances between quasi-clusters and their breaking (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The subclosure configuration of � s1/2 �4 � p3/2 �8 of the jj-coupling shell model can be obtained at the limit of small relative dis- tances and λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Angular momentum projection and GCM Each AQCM Slater determinant is projected to the eigenstates of parity and angular momentum by using 3 the projection operator P K Jπ, P K Jπ = P π 2J + 1 8π2 � dΩ DJ MK ∗R (Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' (3) Here DJ MK is the Wigner D-function and R (Ω) is the rotation operator for the spatial and spin parts of the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This integration over the Euler angle Ω is numerically performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The operator P π is for the parity projection (P π = (1 + P r) / √ 2 for the positive- parity states, where P r is the parity-inversion operator), which is also performed numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The AQCM basis states with different distances be- tween quasi-clusters and λ values are superposed based on GCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' We also generate Gaussian centers for the Λ particles using random numbers, and the basis states with different positions are superposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The coefficients � cK i � for the linear combination of the Slater determi- nants are obtained together with the energy eigenvalue E when we diagonalize the norm and Hamiltonian ma- trices, namely by solving the Hill-Wheeler equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' � j (< Φi|(P K Jπ)†HP K Jπ|Φj > −E < Φi|(P K Jπ)†P K Jπ|Φj >)cK j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' (4) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Hamiltonian The Hamiltonian consists of kinetic energy and poten- tial energy terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' For the potential part, the interaction consists of the central, spin-orbit, and Coulomb terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The nucleon-nucleon interaction is Volkov No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2 [32] with the Majorana exchange parameter of M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='6, which has been known to reproduce the scattering phase shift of 4He–4He [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' For the spin-orbit part, we use the spin- orbit term of the G3RS interaction [34], which is a re- alistic interaction originally developed to reproduce the nucleon-nucleon scattering phase shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The strength of the spin-orbit interactions [10] is set to V 1 ls = V 2 ls = 1450 MeV, which reproduces the binding energy of 12C from the three-α threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' For the nucleon-Λ interac- tion, we employ only the central part;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' YNG-ND interac- tion [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The kF value for 9 ΛBe and 10 ΛΛBe is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='962 fm−1 as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' [14] and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='17 fm−1 for 13 Λ C and 14 ΛΛC as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' For the Λ-Λ interaction, we adopt the one called “NS” in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' [35], which allows the reproduction of the binding energy of 6 ΛΛHe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Ground states of 8Be, 9 ΛBe, and 10 ΛΛBe We start the discussion with 8Be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Our Hamiltonian gives the energy of −27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='57 MeV for the α cluster, and thus, −55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='1 MeV is the two-α threshold energy (experi- mentally −56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='6 MeV, to which our theoretical value does 0 2 4 −70 −60 −50 −40 Energy (MeV) 8Be 0+ Cluster−cluster distance (fm) (a) 0 2 4 −70 −60 −50 −40 Energy (MeV) 9Be 1/2+ Cluster−cluster distance (fm) (b) ΛΛΛΛ 0 2 4 −70 −60 −50 −40 Energy (MeV) 10Be 0+ Cluster−cluster distance (fm) (c) ΛΛ ΛΛ ΛΛ ΛΛ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' (a): Energy curves of 0+ state of 8Be as a function of the distance between two 4He clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Solid line is for λ = 0 (pure two α’s) and dotted and dashed lines are for two quasi-clusters with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' (b): Same as (a) but for the 1/2+ state of 9 ΛBe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' (c): Same as (a) but for the 0+ state of 10 ΛΛBe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' not contradict).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Figure 1 (a) shows the energy curves of the 0+ state of 8Be as a function of the distance between two 4He clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The solid line is for λ = 0 (pure two α’s), and the dotted and dashed line are for two quasi- clusters with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The energy minimum point appears around the relative distance of ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='5 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This distance is quite large, and this is out- side of the interaction range of the spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Therefore, the λ value that gives the minimum energy is zero (solid line), which means that the α clusters are 4 not broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The α breaking effect can be seen in more inner regions, where the energies of dotted and dashed lines are lower than the solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The α clusters are surely broken there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' However, at short relative distances, the energy itself is high enough, and the spin-orbit inter- action only plays a role in reducing the increase of the excitation energy to some extent when two clusters get closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The situation is slightly different in Figure 1 (b), which is for the 1/2+ of 9 ΛBe, where one Λ particle is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' We superpose 50 Slater determinants with different posi- tions for the Λ particle and diagonalize the Hamiltonian based on the GCM for each cluster-cluster distance and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Owing to the Λ particle added, the attractive effect is increased, and the optimal distance between the two 4He nuclei (lowest energy point) is around 3 fm, slightly shorter than the 8Be case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Here, the solid line (λ = 0) and the dotted line (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='1) almost degenerate, and thus, the α clusters are slightly broken due to the spin- orbit effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The tendency is a bit enhanced in 10 ΛΛBe shown in Fig 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The optimal cluster-cluster distance is less than 3 fm, where the dotted line (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='1) is slightly lower than the solid line (λ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The number of Slater determinants with different positions for the Λ particles is increased to 100 for each 4He–4He distance and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' In this way, since the 4He–4He distances are large in 9 ΛBe and 10 ΛΛBe, we find that the α-cluster braking ef- fect is rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Ground states of 12C, 13 Λ C, and 14 ΛΛC Next we discuss 12C and 13 Λ C, and 14 ΛΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The three-α threshold energy is −82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='7 MeV in our calculation com- pared with the experimental value of −84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='9 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Fig- ure 2 (a) shows the energy curves of 0+ state of 12C with an equilateral triangular configuration as a function of the distance between two 4He clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The solid line is for λ = 0 (pure three α’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Since one 4He is added to 8Be, the energy minimum point appears around the relative distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='5–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='0 fm, shorter by 1 fm than the previ- ous 8Be case before allowing the breaking of α clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Therefore, it is considered that the three α clusters step in the interaction range of the spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The dotted line (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='1) and dashed line (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2) almost degenerate at the region of the lowest energy (the relative cluster-cluster distance shrinks to 2 fm there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This tendency is enhanced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 2 (b), which is for the 1/2+ of 13 Λ C, where one Λ particle is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Owing to the Λ particle added, the attractive effect is increased, and the optimal distance between the 4He nuclei is around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='5 fm (solid line) before breaking the α clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' When we allow the breaking, the energy curves become almost flat inside the relative 4He–4He distance of 2 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The energy minimum points of the dotted (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='1) and dashed (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2) lines are lower than that of the solid line (λ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The attractive effect of the Λ particles is much more en- hanced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 2 (c), which is for the 0+ state of 14 ΛΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The optimal distance between the 4He nuclei (energy mini- mum point) is around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2 fm before breaking the α clus- ters (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' When we allow the breaking, the energy minimum point appears at the relative cluster–cluster distance of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='4 fm, where the dashed line (λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2) gives the lowest energy, and α clusters are significantly broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' We can confirm that the optimal cluster distance gets shorter, and the breaking of α clusters becomes larger with the increasing number of Λ particles added to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Superposition of states with different 4He−4He distance and breaking parameter λ To demonstrate the relation between the effect of α breaking and spin-orbit interaction, we calculate the ground state energies of 8Be, 9 ΛBe, 10 ΛΛBe (Table I) and those of 12C, 13 Λ C, 14 ΛΛC (Table II) with two models: “AQCM”’ which explicitly takes account of the breaking effect of α, and “Brink model” which does not involve the α breaking effect (λ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' We superpose Slater de- terminants with different positions of the Λ particle(s), 4He–4He cluster distances, and α-breaking parameter λ and diagonalize the Hamiltonian based on the GCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' For the Be case (Table I), the energy difference between Brink and AQCM is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2 MeV in 8Be, which means that the spin-orbit interaction does not break the α clusters since they are separated by a certain distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The situation is basically the same when Λ particle(s) is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The difference is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='6 MeV in 9 ΛBe and 10 ΛΛBe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Concerning the ground state energy of 10 ΛΛBe, the binding energy (BΛΛ) of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='4 MeV from 8Be has been reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' [36], which has been revised to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='4 MeV in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' [37] (see the discussions in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' [38, 39]), and the present result (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='23 MeV) is almost consistent with the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' For the C case (Table II), the energy difference be- tween Brink and AQCM is about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='3 MeV in 12C, and this is much enhanced with the increasing number of the Λ particles added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The difference increases to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2 MeV in 14 ΛΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This is because the spin-orbit interaction works in the inner region of the nuclear systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' the glue-like effect of Λ particles shrinks the system and induces more contribution of the spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' To clarify the mixing of the jj-coupling shell model components in each state, we utilize the expectation value of the one-body spin-orbit operator, ˆOLS = � i li · si/ℏ2, (5) where li and si are the orbital angular momentum and the spin operators for the ith nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The sum runs over the nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The expectation value is zero for the pure α cluster state owing to the antisymmetrization effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Also, the li ·si/ℏ2 value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='5 for one nucleon in the p3/2 5 0 2 4 −120 −100 −80 −60 Energy (MeV) 12C 0+ Cluster−cluster distance (fm) (a) 0 2 4 −120 −100 −80 −60 Energy (MeV) 13C 1/2+ Cluster−cluster distance (fm) (b) ΛΛΛΛ 0 2 4 −120 −100 −80 −60 Energy (MeV) 14C 0+ Cluster−cluster distance (fm) (c) ΛΛ ΛΛ ΛΛ ΛΛ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' (a): Energy curves of 0+ state of 12C as a function of the distance between three 4He clusters with equilateral triangular configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Solid line is for λ = 0 (pure three α’s) and dotted and dashed lines are for two quasi-clusters with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' (b): Same as (a) but for the 1/2+ state of 13 Λ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' (c) Same as (a) but for the 0+ state of 14 ΛΛC orbit, and the eigen value is 4 for the subclosure configu- ration of the jj-coupling shell model ( � s1/2 �4 � p3/2 �8) in 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The expectation values of the one-body spin-orbit op- erator for the ground states of 8Be, 9 ΛBe, and 10 ΛΛBe are listed in the column “one-body LS” in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Although the value increases with the number of Λ particles added, it is rather small and cluster structure is considered to be not broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' However, this is completely different in TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Ground state energies of 8Be, 9 ΛBe, and 10 ΛΛBe (“en- ergy (Jπ)”) after performing the GCM calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' “Brink” is for the Brink model (λ = 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' two-α clusters without the breaking, and “AQCM” is for the AQCM calculation, where different λ states are mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' “one-body LS” is for the expec- tation values of the one-body spin-orbit operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The values in the parenthesis show the experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' BΛ, BΛΛ are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' All energies are in MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 8Be energy (0+) one-body LS Brink −54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='00 AQCM −54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='94 (−56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='50) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='12 9 ΛBe energy (1/2+) BΛ one-body LS Brink −60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='00 AQCM −61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='53 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='59 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='71 [17]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='29 10 ΛΛBe energy (0+) BΛΛ one-body LS Brink −69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='00 AQCM −70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='17 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='23 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='4 [37]) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='44 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Ground state energies of 12C, 13 Λ C, and 14 ΛΛC (“en- ergy (Jπ)”) after performing the GCM calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' “Brink” is for the Brink model (λ = 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' three-α clusters with equilat- eral triangular shapes without the breaking, and “AQCM” is for the AQCM calculation, where different λ states are mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' “one-body LS” is for the expectation values of the one-body spin-orbit operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The values in the parenthesis show the experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' All energies are in MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 12C energy (0+) one-body LS Brink −86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='00 AQCM −90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='12 (−92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='16) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='55 13 Λ C energy (1/2+) BΛ one-body LS Brink −97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='00 AQCM −102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='00 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='88 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='69 [17]) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='86 14 ΛΛC energy (0+) BΛΛ one-body LS Brink −110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='00 AQCM −115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='74 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='05 the C case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The expectation values of the one-body spin- orbit operator for the ground states of 12C, 13 Λ C, and 14 ΛΛC are listed in the column “one-body LS” in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The value is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='55 for 12C, and we can reconfirm that the ground state has mixed configurations of shell and cluster aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' As the number of the Λ particles added increases, we can see that the ground states approach the jj-coupling shell model side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The values for 13 Λ C and 14 ΛΛC are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='86 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='05, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' pure α cluster state orthogonal to the ground state We have discussed that the ground states shift to the jj-coupling shell model side by adding Λ particles, and the final question is where the “pure” three-α cluster state appears in 14 ΛΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' We can discuss it by preparing the pure three-α cluster states and orthogonalizing them to the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The shift of the ground state to the jj-coupling shell-model-side after allowing the breaking of α clusters is found to play a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 6 0 2 4 −120 −100 −80 −60 0+ energy (MeV) Cluster−cluster distance (fm) 14C ΛΛ ΛΛ ΛΛ ΛΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2 λλλλ λλλλ = 0 (a) 0 2 4 −120 −100 −80 −60 0+ energy (MeV) Cluster−cluster distance (fm) 14C ΛΛ ΛΛ ΛΛ ΛΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='0 λλλλ λλλλ = 0 (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Excited 0+ state comprised of pure three α clusters in 14 ΛΛC as a function of distances between α–α (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Ground state is represented by the AQCM basis state with the 4He–4He distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='4 fm and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2 (a) and 4He–4He distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2 fm and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='0 (b), which are shown by the solid circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 3 (a) shows the excited 0+ state with equilateral triangular configurations of pure three- α clusters as a function of the relative distances between the α clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' At each α–α distance, the wave function is orthogonalized to the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Here the ground state is represented by the optimal AQCM basis state (4He–4He distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='4 fm and Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2) shown by the solid circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Therefore, the two-by-two matrix is diago- nalized at every point on the horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' It is found that the pure cluster state appears around the excitation energy of Ex = 15 MeV with the relative α–α distance of ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='5 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' To simplify the discussion, the positions for the Gaussian center parameters for the Λ particles are set to origin only in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 3 (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This situation is quite different if the α cluster is as- sumed to be not broken due to the spin-orbit interaction in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This is an artificial calculation, but we can clearly see the influence of the cluster-shell com- petition in the excited state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 3 (b) shows the result when the ground state is represented by the Brink model, which is prepared by changing the λ value to zero and the 4He–4He distance to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The excited 0+ state is quite influenced by this change of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The energy is pushed up by more than 10 MeV, and the optimal α–α distance is increased to ∼3 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This is because if the ground state is a pure three-α cluster state, the excited states need to be more clusterized to satisfy the orthogonal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' On the other hand, if the ground state has different components other than the cluster structure, it is easier for the pure cluster state to be orthogonal to the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This effect has been known in 12C and called the “shrink effect” of the sec- ond 0+ state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' when the α breaking component is mixed in the ground state, the second 0+ state orthogonal to the ground state shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' We found that this shrinking effect is much more enhanced in 14 ΛΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' CONCLUSIONS The effect of adding hyperon(s) in nuclear systems is a fundamental problem in nuclear structure physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' We analyzed this effect in the context of cluster-shell competition and discussed the difference between Be and C cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The antisymmetrized quasi-cluster model (AQCM) is a useful tool to treat the cluster states and shell-model states on the same footing, and we added Λ particle(s) to 8Be and 12C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The cluster breaking effect is negligibly small in 8Be, where α–α cluster structure keeps enough distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' they stay out of the interaction range of the spin-orbit inter- action, which breaks the α clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The situation holds even after Λ particle(s) is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The glue-like effect of Λ particles surely shrinks the cluster-cluster distance, but clusters are not yet broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The situation is completely different in the C case since the additional α cluster shrinks the cluster-cluster dis- tance, and clusters are in the interaction range of the spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The ground state of 12C contains the component of the jj-coupling shell model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The en- ergy difference between the traditional Brink model and AQCM is about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='3 MeV in 12C, and this is much en- hanced with the increasing number of the Λ particles added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The energy difference is about 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content='2 MeV in 14 ΛΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' This is because the spin-orbit interaction works in the inner region of the nuclear systems, and the glue-like ef- fect of Λ particles shrinks the system and induces more contribution of the spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' In 14 ΛΛC, the breaking of α clusters in 12C is much enhanced by the addition of the Λ particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The energy and structure of the excited 0+ state with a pure cluster structure are found to be drastically affected by the transition of the ground state to the jj-coupling shell model side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI Grant Number 19J20543, 22K03618, and JP18H05407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' The nu- merical calculations have been performed using the com- puter facility of Yukawa Institute for Theoretical Physics, Kyoto University (Yukawa-21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' 7 [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Itagaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAzT4oBgHgl3EQf6v4U/content/2301.01878v1.pdf'} +page_content=' Aoyama, S.' metadata={'source': 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a/v9E3T4oBgHgl3EQf-wsu/content/tmp_files/2301.04828v1.pdf.txt b/v9E3T4oBgHgl3EQf-wsu/content/tmp_files/2301.04828v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8c11efd7f8752e8cf26ee3d9591381f834dce751 --- /dev/null +++ b/v9E3T4oBgHgl3EQf-wsu/content/tmp_files/2301.04828v1.pdf.txt @@ -0,0 +1,1541 @@ +Localized covariance estimation: A Bayesian justification∗ +Robert J. Webber† and Matthias Morzfeld‡ +Abstract. A major problem in numerical weather prediction (NWP) is the estimation of high-dimensional co- +variance matrices from a small number of samples. Maximum likelihood estimators cannot provide +reliable estimates when the overall dimension is much larger than the number of samples. +For- +tunately, NWP practitioners have found ingenious ways to boost the accuracy of their covariance +estimators by leveraging the assumption that the correlations decay with spatial distance. In this +work, Bayesian statistics is used to provide a new justification and analysis of the practical NWP +covariance estimators. The Bayesian framework involves manipulating distributions over symmetric +positive definite matrices, and it leads to two main findings: (i) the commonly used “hybrid estima- +tor” for the covariance matrix has a naturally Bayesian interpretation; (ii) the very commonly used +“Schur product estimator” is not Bayesian, but it can be studied and understood within the Bayesian +framework. As practical implications, the Bayesian framework shows how to reduce the amount of +tuning required for covariance estimation, and it suggests that efficient covariance estimation should +be rooted in understanding and penalizing conditional correlations, rather than correlations. +Key words. Covariance estimation, Bayesian statistics, numerical weather prediction +AMS subject classifications. 62H10, 65C20, 86-10 +1. Introduction. In this work, we provide new insights into the estimation of high- +dimensional covariance matrices from a small number of samples. Our work is motivated +by numerical weather prediction (NWP), a setting in which covariance estimation arises nat- +urally and at a vast scale. The goal in NWP is to generate a set of weather forecasts based +on global weather models and real-world observations of the Earth’s atmosphere [2]. All the +standard NWP techniques require estimating the covariance matrix for the near-term weather +forecasts [19,28], but the dimensionality of the covariance estimation problem in NWP is im- +mense. A typical global weather model has billions of unknowns, which are updated by tens +of millions of observations within less than six hours of computing time. The large cost of +running the weather model necessitates that the ensemble size (number of model integrations) +is small compared to the number of unknown weather variables. A typical ensemble size is +≤ 100 and therefore six orders of magnitude smaller than the number of unknowns. +The sample covariance is not an accurate covariance estimator unless the ensemble size is +larger than the number of unknowns [3, 4]. Therefore, covariance estimation at the extreme +scale of NWP can only be accomplished by using additional information and tricks, commonly +referred to as “covariance localization”. The basic idea is that covariances should decay with +spatial distance. On six hour time scales, the weather in La Jolla, California, is uncorrelated +∗Submitted to the editors DATE. +Funding: RJW is supported by the Office of Naval Research through BRC award N00014-18-1-2363 and the +National Science Foundation through FRG award 1952777, under the aegis of Joel A. Tropp. MM is supported by +the US Office of Naval Research (ONR) grant N00014-21-1-2309. +†Computing & Mathematical Sciences, California Institute of Technology, Pasadena, CA (rwebber@caltech.edu). +‡Institute of Geophysics and Planetary Physics, Sripps Institution of Oceanography, University of California, San +Diego, San Diego, CA (matti@ucsd.edu). +1 +arXiv:2301.04828v1 [stat.ME] 12 Jan 2023 + +2 +R.J. WEBBER AND M. MORZFELD +with the weather in Chicago, Illinois. +Localization, in its simplest form, means damping +estimates of long-range of covariances because large magnitudes are caused by sampling error, +not by the existence of long-range covariances [16,17]. Localization started off as an ad hoc +procedure that perhaps grew out of desperation to make NWP work – early NWP attempts +using the ensemble Kalman filter [9], for example, led to useless forecasts because of the large +errors in the forecast covariances. By now, however, localization has been widely accepted as +a necessary ingredient within the NWP community [15]. +Not surprisingly, localization has been studied extensively. In the NWP community, the- +oretical work has focused on adaptive localization methods [1] and theories for optimal lo- +calization [10, 20, 23], but implementing these techniques can be data-intensive. In practice, +localization is often implemented using a Schur product estimator [16, 17, 25], a hybrid co- +variance estimator [5,22,30] (also called a “shrinkage” estimator [24,27]), or a combination of +the two estimators [22], but there is little to no mathematical justification. Meanwhile, the +statistical community has introduced localized estimators with rigorous guarantees [3,4,6,12]. +However, these estimators are only guaranteed to work in the asymptotic limit as the ensem- +ble size and the state dimension jointly grow to infinity. It remains unclear which localized +estimators work best with finite ensemble size and finite state dimension. +To our surprise, localization is not typically understood from the Bayesian perspective, +even though localization is a naturally Bayesian procedure: We estimate an unknown (the +covariance matrix) based on limited data (the ensemble/forecast states) and enforce prior +information about the problem structure (the spatial decay of covariances). This paper is +about describing the Bayesian perspective and explaining why this interpretation of covariance +localization may be useful in practice. Put simply, we ask and answer the following question: +“Are there any Bayesian prior distributions that lead to existing localization methods?” This +question is equivalent to asking, “Which existing localization methods are rooted in a Bayesian +framework?” +To answer this question, we consider two different Bayesian prior distributions. First, we +consider the inverse Wishart distribution [29, Sec. 5.2], a classical distribution over covariance +matrices that leads to the hybrid covariance estimator [5,16,17,22,25,30] as a maximum a pos- +teriori estimator. Second, we consider a new “quadratically constrained” (QC) distribution, +which forces the off-diagonal entries of the precision matrix (inverse covariance matrix) to be +small. We introduce the QC prior to study localization via Schur products [5, 22, 24, 27, 30], +which is a common method for covariance estimation but is surprisingly not Bayesian. We +show that the QC covariance estimator converges to the Schur product estimator as the lo- +calization strength parameter tends toward infinity. In summary, our work provides a new +Bayesian justification for two commonly used localization estimators in NWP. +The Bayesian framework is useful for several reasons. First, the framework is designed for +finite ensemble size and, thus, more practically applicable than statistical techniques that are +largely asymptotic (e.g., [3,4,6,12]). +Second, the Bayesian framework suggests how to adjust the parameters in the hybrid esti- +mator and Schur product estimator as the ensemble size changes. Typically in an operational +setting, this adjustment is done implicitly, since the localization is re-tuned as the ensem- +ble grows larger or smaller. Our Bayesian theory helps with reducing the amount of tuning +required. + +LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION +3 +Last, the Bayesian theory may help to construct new localization estimators for the future. +Every year, Earth models are becoming increasingly complex, e.g., coupled atmosphere, ocean +and sea ice models, or seasonal to sub-seasonal forecast models, and data assimilation is also +being extended to geomagnetic models [11,14]. For all these models, traditional localization +based on a single length scale parameter may no longer be appropriate. +The theoretical +foundations laid here are not limited to a single length scale parameter and are more generally +applicable. As a central feature, our theory emphasizes building the localization scheme via +the precision matrix that describes the conditional correlations between variables, not via the +covariance matrix directly. +The rest of this paper is organized as follows. Section 2 reviews covariance estimation +from the NWP perspective. +Section 3 analyzes covariance estimation from the Bayesian +perspective. Section 4 numerically tests the predictions of the Bayesian theory. Section 5 +proves mathematical theorems to support our Bayesian analysis. Section 6 offers a summary +and some conclusions. +Throughout this paper, we use bold lower case letters to refer to vectors and bold capital +letters to refer to matrices. The ij entry of the matrix A is written Aij. The determinant of a +matrix A is written |A| and the Schur (element-wise) product of compatible matrices A and B +is written A ◦ B. Last, the Frobenius norm of a matrix is written ∥A∥F = +��d +i,j=1 |Aij|2�1/2. +2. A rapid review of NWP covariance estimation. We start by briefly describing how +covariance estimation is accomplished in NWP. To keep things simple, we assume that inde- +pendent samples x1, . . . , xn ∈ Rd are drawn from a mean-zero Gaussian distribution, +xi ∼ N(0, Σ). +We assume that ensemble size n is much smaller than the dimension d. Our goal is to estimate +the d × d positive semidefinite covariance matrix Σ from n ≪ d samples. +A classical estimator for Σ is the maximum likelihood (ML) estimator +ˆΣ = arg max +Σ +n +� +i=1 +p(xi|Σ). +Under the mean zero assumption, the ML estimator is just the “sample covariance” or “em- +pirical covariance” +ˆΣsamp = 1 +n +n +� +i=1 +xixT +i . +The ML estimator is unbiased (mean Σ), and as the number of data points n approaches +infinity, the ML estimator converges to the true covariance Σ at the optimal rate [32, Ch. 8] +(2.1) +√n( ˆΣsamp − Σ) D→ N(0, I(Σ)−1), +where I(Σ)−1 denotes the inverse Fisher information tensor. This means that the ML estima- +tor achieves the optimal O(1/√n) error scaling, and the limiting distribution of √n( ˆΣsamp−Σ) +is as tightly concentrated as possible. However, since we work in a framework where n ≪ p, +the sample covariance is known to be inaccurate [3,4]. + +4 +R.J. WEBBER AND M. MORZFELD +2.1. Schur product estimators. Covariance localization is an approach for increasing the +accuracy of the sample covariance when the ensemble size is small. +The basic idea is to +damp long-range correlations based on the assumption that correlations decay with distance. +Localization can be implemented via a Schur product with a symmetric positive definite +localization matrix L: +(2.2) +ˆΣSchur = ˆΣsamp ◦ L. +A simple example of a localization matrix is based on the Gaussian kernel and has elements +Lij = exp +� +−(dij/ℓ)2� +, +where dij is the distance between grid points i and j, and where ℓ > 0 is a length scale +parameter. Alternately, one can replace the Gaussian kernel function with a Laplacian kernel +function (exponential decay), or with a Gaspari-Cohn kernel function [13]. The latter is zero +at large distances and therefore promotes sparsity in the covariance matrix estimate. +Compared to the sample covariance, the Schur product estimator creates an element-wise +bias of +E[ ˆΣSchur +ij +] − Σij = LijE[ ˆΣsamp +ij +] − Σij = (Lij − 1)Σij, +and changes the element-wise variance by a factor of +Var[ ˆΣSchur +ij +] +Var[ ˆΣsamp +ij +] += +L2 +ijVar[ ˆΣsamp +ij +] +Var[ ˆΣsamp +ij +] += L2 +ij. +Since Li,j is small at large spatial separations, the effect of localization is clear: it introduces +a small bias while drastically reducing the variance of the estimator. +2.2. Hybrid estimators. A second important and widely used covariance estimator is +the “hybrid” estimator ˆΣhyb, which is defined as a convex combination between the sample +covariance ˆΣsamp and a prior covariance estimate Σprior: +(2.3) +ˆΣhyb = αΣprior + (1 − α) ˆΣsamp, +for some α ∈ (0, 1). Compared to the unbiased estimator ˆΣsamp, the hybrid estimator creates +a bias of size +E[ ˆΣhyb] − Σ = αΣprior + (1 − α)E[ ˆΣsamp] − Σ = α(Σprior − Σ) +and changes the variance by a factor of +Var[ ˆΣhyb] +Var[ ˆΣsamp] += (1 − α)2Var[ ˆΣsamp] +Var[ ˆΣsamp] += (1 − α)2. +Thus, the hybrid estimator typically adds a small bias while slightly reducing the variance. + +LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION +5 +In NWP, the hybrid estimator is often presented in an equivalent form +(2.4) +ˆΣhyb, NWP = w1Σclim + w2 ˆΣsamp, +where Σclim is a climatological covariance matrix, derived from a long model run that re- +veals covariance structure inherent to the physical process. The NWP version of the hybrid +estimator in (2.4) and the version we presented in (2.3) are equivalent if we set +α = 1 − w2, +Σprior = +w1 +1 − w2 +ˆΣclim. +In subsection 3.1, we will show that NWP researchers are using a principled Bayesian approach +when applying the hybrid estimator, but the Bayesian ideas are somewhat hidden within the +notation. If we use the right symbols and notation, we can frame the practical NWP estimators +within a rigorous Bayesian perspective. +2.3. Tuning of covariance estimators. The accuracy of the Schur product and hybrid +estimators depends on the various parameters that go into the construction. For the hybrid +estimator in (2.3), one needs to specify the prior covariance matrix Σprior and the interpolation +factor α. +For the Schur product estimator in (2.2), one needs to specify the parameters +that define the localization matrix L. If we use Gaussian or Laplacian kernels to define the +localization matrix, this means that one needs to determine an appropriate length scale ℓ for +localization. +The parameters that define the covariance estimator are usually determined via parameter +tuning, or, using more modern language, a “training” phase. +The idea is to simply try +a few parameters and then determine which parameter combination gives the most useful +results. For example, one can run an ensemble data assimilation algorithm on a set of training +observations and compute the forecast error that results from each choice of parameters. One +then selects the parameters that lead to the smallest forecast errors. +This tuning is expensive, computationally and otherwise. In practice, localization and +hybrid estimators are often combined [22], which means that a relatively large number of +parameters needs to be tuned, which is even more costly. +Even worse, this entire tuning +process must be repeated whenever the underlying model is modified, or if the ensemble size +is increased because more computational power is available. We will see in section 3 that the +Bayesian perspective on covariance localization gives insights that can reduce the efforts that +go into tuning covariance estimators. +3. The Bayesian perspective on covariance estimation. The main goal of Bayesian +statistics is to combine prior information and data to estimate parameters in a model. Bayesian +statistics has three main components: the prior distribution, the likelihood, and the poste- +rior distribution. The Bayesian prior distribution encodes all information before any data are +collected and the likelihood function infuses information from data into the posterior estimate. +Here, we apply Bayesian statistics to the problem of estimating a positive definite covari- +ance matrix Σ ∈ Rd×d from a set of n samples xi, i = 1, . . . , n. We assemble the n samples +into a p × n data matrix X = +� +x1 +· · · +xn +� +, and we express the posterior density function + +6 +R.J. WEBBER AND M. MORZFELD +as +(3.1) +p(Σ|X) +� +�� +� +posterior +∝ p(Σ) +� �� � +prior +p(X|Σ) +� +�� +� +likelihood +. +The symbol ∝ indicates that the left- and right-hand sides are proportional over all choices +of Σ, but the proportionality constant is typically not needed for computing the covariance +estimate. In (3.1), the prior density function p(Σ) is chosen to account for any structural +knowledge of Σ, e.g., the decay of correlations with spatial distance. The likelihood function +p(X|Σ) accounts for information from the data, which in our case are the n samples assembled +in the data matrix X, and the likelihood function takes the form +(3.2) +p(X|Σ) = +��� 1 +2πΣ−1��� +n/2 +exp +� +−1 +2 +n +� +i=1 +xT +i Σ−1xi +� +. +Last, the posterior density function p(Σ|X) gives a distribution of possible covariance ma- +trices. +Using the posterior density, we can calculate the “maximum a posterior” (MAP) +estimator +(3.3) +ˆΣMAP = arg max +Σ +p(Σ|X). +The MAP estimator can be regarded as the single most likely value for the covariance under +the posterior distribution. Other estimators (e.g., mean, median) are equally valid, but are +harder to compute or analyze. +With the uniform prior distribution p(Σ) = Const., the MAP estimator is the same as the +ML estimator +(3.4) +ˆΣMAP = arg max +Σ +p(X|Σ) = 1 +n +n +� +i=1 +xixT +i . +However, more generally, the choice of prior distribution has a non-trivial effect on the +Bayesian posterior — the whole purpose of imposing a prior is to fill in the gaps that the +data leave open. The rest of this paper is about choices of non-uniform priors p(Σ) that +promote structure in the covariance estimate ˆΣMAP, providing justification for existing but +largely empirical covariance estimators in NWP. +3.1. The inverse Wishart prior and hybrid estimators. The inverse Wishart distribution +[29, Sec. 5.2] is a classical distribution defined over symmetric positive definite matrices Σ ∈ +Rd×d by the density +(3.5) +p(Σ) ∝ +��Σ−1��m/2 exp +� +−m +2 tr +� +ΣpriorΣ−1�� +. +There are two parameters in the inverse Wishart distribution: Σprior is the mode (most likely +value) of the distribution, and m is the “sample size” parameter that controls the width of +the distribution around the mode: a large m leads to tightly concentrated distribution. + +LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION +7 +Given an inverse Wishart prior and a data matrix X = +� +x1 +· · · +xn +� +, the Bayesian +posterior is also an inverse Wishart distribution, because the inverse Wishart distribution is +the “conjugate prior” [8] to the mean-zero multivariate Gaussian likelihood: +p(Σ|X) ∝ +��Σ−1��(m+n)/2 exp +� +−m +2 tr +� +ΣpriorΣ−1� +− 1 +2 +n +� +i=1 +xT +i Σ−1xi +� +∝ +��Σ−1��(m+n)/2 exp +� +−m + n +2 +tr +� ˆΣIWΣ−1�� +, +where +(3.6) +ˆΣIW = +m +m + nΣprior + +n +m + n +ˆΣsamp. +In the posterior distribution, the two inverse Wishart parameters are updated in response to +the data: the sample size parameter increases from m to m + n, and the mode changes from +Σprior to ˆΣIW (3.6). +It is now clear that the inverse Wishart prior leads to a covariance estimator (3.6) that is +identical to the hybrid estimator (2.3) with the parameter choice +(3.7) +α = +m +m + n. +In other words, the hybrid estimator is the same estimator that would result from selecting +an inverse Wishart prior and systematically applying a Bayesian analysis. This perspective +provides a Bayesian justification for the hybrid estimator, assuming that the parameters m +and Σprior represent reasonable prior knowledge about the covariance structure. +A major benefit of Bayesian statistics is that it leads to a covariance estimator (3.6) valid +for any sample size n, whereas standard NWP covariance estimators require tuning parameters +whenever the sample size changes (subsection 2.3). When n is large, the Bayesian formula +tells us to adjust our estimator according to +(3.8) +ˆΣIW = ˆΣsamp + O(n−1), +and this scaling with n ensures that ˆΣIW converges to the true covariance at the optimal +asymptotic rate (2.1) as n → ∞. In NWP applications, we anticipate consistent accuracy in +covariance estimation when the localized covariance estimate is adjusted according to (3.6) +and (3.8). We will revisit this idea in the numerical examples in section 4. +3.2. The QC prior and Schur product estimators. We now introduce a new “quadrat- +ically contrained” (QC) distribution to study localization via Schur products. Surprisingly, +localization via Schur products cannot result directly from a Bayesian prior (Proposition 5.1). +However, the QC prior allows us to study Schur product localization in a rigorous and mean- +ingful way, in the asymptotic limit of increasing penalization strength. +The QC distribution is defined over symmetric positive definite matrices by the density +function +p(Σ) ∝ exp +� +−1 +4tr +� +Σ−1� +Θ ◦ Σ−1��� +. + +8 +R.J. WEBBER AND M. MORZFELD +The only parameter in the QC distribution is a symmetric nonnegative-valued matrix Θ. The +QC prior can be “improper” [7], i.e., the density can integrate to infinity for some Θ. However, +the corresponding Bayesian posterior distribution +(3.9) +p(Σ|X) ∝ +��Σ−1��n/2 exp +� +−n +2 tr +� ˆΣsampΣ−1� +− 1 +4tr +� +Σ−1� +Θ ◦ Σ−1��� +. +is well-defined for every Θ. We further show in Proposition 5.3 that this posterior distribution +has a unique positive definite global maximizer given a large localization strength, which +justifies the use of the MAP as a covariance estimator. +3.2.1. Motivation for the QC prior. The QC distribution can be derived as the maximum +entropy or most “random” [18] distribution that constrains the square entries of the precision +matrix Σ−1 to be small (Proposition 5.2). +More specifically, with entropy defined as the +amount of “randomness” in a density p via +H[p] := − +� +p(Σ) log p(Σ) dΣ, +the QC density solves the maximization problem +max +p +� +H[p] − 1 +4 +d +� +i,j=1 +Θij +� +p(Σ)|Σ−1 +ij |2dΣ +� +. +Here, Θ is the parameter that penalizes off-diagonal elements in Σ−1. +At first, it is perhaps strange that we define the QC prior to target off-diagonal elements in +the precision matrix Σ−1, while we aim to explain the Schur product estimator that constrains +elements in the covariance matrix Σ. However, there is a systematic Bayesian explanation for +why targeting the precision matrix is the right approach, based on the conditional correlation +structure. +The conditional correlation between two variables xi and xj measures the degree of asso- +ciation with the effects of all other components of x removed. In many NWP applications, +we expect that conditional correlations, even more so than correlations, should be confined +to small neighborhoods. For example, we expect the weather in La Jolla, California is condi- +tionally uncorrelated with the weather in Chicago, Illinois, after accounting for the weather +in all the in-between locations. The fast decay of conditional correlations has been observed +in many geophysical applications and has been described as the “screening effect” [31]. +In a Gaussian model, the conditional correlations are described explicitly by +corr +� +xi, xj | (xk)k/∈{i,j} +� += − +Σ−1 +ij +(Σ−1 +ii Σ−1 +jj )1/2 . +The magnitude of the conditional correlations is thus proportional to the magnitude of the +Σ−1 elements. The QC prior can be interpreted as enforcing prior knowledge of the screening +effect, by targeting the off-diagonal entries of Σ−1. + +LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION +9 +As an example of the screening effect, we consider a Gaussian process with covariances +defined by the Laplacian kernel +k(x, y) = exp +� +−|x − y| +ℓ +� +, +on a 1D spatial domain (not periodic). When the data is generated from a uniform grid with +mesh size ∆, the corresponding covariance matrix is +Σ = +� +� +� +� +� +� +� +1 +e−∆ +· · · +e−(d−2)∆ +e−(d−1)∆ +e−∆ +1 +· · · +e−(d−3)∆ +e−(d−2)∆ +... +... +... +... +e−(d−2)∆ +e−(d−3)∆ +· · · +1 +e−∆ +e−(d−1)∆ +e−(d−2)∆ +· · · +e−∆ +1 +� +� +� +� +� +� +� +, +and the precision matrix is +Σ−1 = +2 +e∆ − e−∆ +� +� +� +� +� +� +� +� +� +e∆ +−1 +−1 +e∆ + e−∆ +... +... +... +... +... +e∆ + e−∆ +−1 +−1 +e∆ +� +� +� +� +� +� +� +� +� +. +The precision matrix Σ−1 is tridiagonal and, hence, has a faster off-diagonal decay than the +covariance matrix (which has exponential decay). This example thus supports the strategy of +constraining off-diagonal entries in Σ−1, rather than in Σ. +3.2.2. QC covariance estimator. Next, we study the MAP estimator corresponding to +the QC prior. We do so by maximizing the logarithm of the posterior distribution (3.9) +ℓ(Σ) = n +2 log +��Σ−1�� − n +2 tr +� ˆΣsampΣ−1� +− 1 +4tr +� +Σ−1� +Θ ◦ Σ−1�� +. +To find the unique global maximizer of ℓ we set its gradient +∇ℓ = n +2 Σ−1� +−Σ + ˆΣsamp + 1 +nΣ−1 ◦ Θ +� +Σ−1 +equal to zero, and we obtain an implicit equation for the QC estimator +(3.10) +ΣQC = ˆΣsamp + 1 +n(ΣQC)−1 ◦ Θ. +In high dimensions, solving (3.10) is a challenge. Nonetheless, we can extract useful asymptotic +information from (3.10) and make the connection to Schur product estimators. +First, we note that (3.10) implies the QC estimator is the same as the sample covariance +ˆΣQC +ij += ˆΣsamp +ij +. + +10 +R.J. WEBBER AND M. MORZFELD +for any (i, j) entries such that Θij = 0. In other words, the QC estimator trusts the sample +covariance completely if we do not penalize the conditional correlation between xi and xj via +Θi,j > 0. In NWP, it is unusual to penalize variances, so we assume for the rest of this section +that Θij > 0 if and only if i ̸= j, which implies that +ˆΣQC +ii += ˆΣsamp +ii +, +1 ≤ i ≤ p. +We now consider the asymptotic behavior of the QC estimator when we set Θ = sΘref +and raise the penalization strength parameter s → ∞. In this limit, we may write +(3.11) +ˆΣQC = D + s−1∆, +where D is a diagonal matrix with elements Dii = ˆΣQC +ii += ˆΣsamp +ii +and s−1∆ is the matrix +containing all off-diagonal elements of ˆΣQC. As s → ∞, the inverse of the QC estimator is +given by the Taylor series expansion +(3.12) +( ˆΣQC)−1 = D−1 − s−1D−1∆D−1 + O(s−2). +Substituting (3.11) and (3.12) into (3.10), and solving for s−1∆, we find +ˆΣQC +ij += ˆΣsamp +ij +� +1 + +Θij +n ˆΣsamp +ii +ˆΣsamp +jj +�−1 ++ O(s−2). +Therefore, the QC covariance estimator is asymptotically a Schur product estimator that +damps the off-diagonal entries of the covariance matrix. +The Bayesian framework thus begins to provide a theoretical foundation for the Schur +product estimator used in NWP. The QC estimator converges to a Schur product estimator +with localization matrix +(3.13) +Lij = +� +1 + +Θij +n ˆΣsamp +ii +ˆΣsamp +jj +�−1 +. +This approximation becomes increasingly accurate in the limit of a large penalization strength. +3.2.3. Adjusting the length scale. When we use a localization matrix L with a single +length scale parameter, the Bayesian perspective suggests how best to adjust the length scale +with the ensemble size n. As n → ∞, the Schur product formula (3.13) satisfies +(3.14) +Lij = 1 + O(n−1). +For example, if Lij = exp(−dij/ℓ), we need to adjust the length scale as +(3.15) +exp(−dij/ℓ) = 1 + O(n−1) =⇒ 1 +ℓ = O(n−1), +to ensure that (3.14) is satisfied. Thus, we need to increase the length scale at a rate ℓ ∼ n +or faster. If Lij = exp(−d2 +ij/ℓ2), we need to adjust the length scale as +(3.16) +exp +� +−(dij/ℓ)2� += 1 + O(n−1) =⇒ +1 +ℓ2 = O(n−1). + +LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION +11 +This means we need to increase the length scale at a rate ℓ ∼ n1/2 or faster. +The scaling of the length scale with the ensemble size is practically important in NWP. +Typically when the ensemble size changes, the localization length scale(s) are re-tuned, which +is costly, both computationally and otherwise (subsection 2.3). +The Bayesian perspective +naturally provides a scaling of the localization length scale with ensemble size, thus reducing +the amount of tuning necessary when the data assimilation system undergoes an upgrade. +4. Numerical illustration. In this section, we numerically test the scalings for the hybrid +estimator and Schur product estimator that are predicted by our Bayesian theory. We perform +numerical tests with a variety of covariance matrices and a variety of ensemble sizes to check +that the scalings are useful in practice and to reiterate that the theory holds at finite ensemble +size, not only asymptotically (for n, p → ∞). +4.1. Covariance matrices. We estimate covariance matrices for five different Gaussian +models, each with mean zero and a (spatial) dimension of d = 200. Following [20], the five +covariance matrices are defined as follows. +1. Single-scale, Laplacian kernel. The first covariance matrix has elements +Σℓ +ij = exp(−dij/ℓ), +where ℓ is the length scale, which we set to ℓ = 5, and dij is the distance between grid +points i and j, reflecting the periodic domain. +2. Single-scale, Gaussian kernel. The second covariance matrix is similar to the first, but +the decay of covariance is faster. Namely, the matrix has elements +Σℓ +ij = exp(−(dij/ℓ)2) +where ℓ and dij are the length scale and the distance between grid points on the +periodic domain. Here too, we set ℓ = 5. +3. Multiscale covariance. We define a multiscale covariance matrix as the sum of two +single-scale covariances +Σms = 1 +2 +� +Σℓ1 + Σℓ2� +, +and we set ℓ1 = 2 and ℓ2 = 20. We use single-scale covariance matrices defined by a +Gaussian kernel, but similar results can be obtained with a Laplacian kernel. +4. Nonstationary covariance. The next covariance matrix is “nonstationary” [26] and its +elements are +Σns +i,j = (4ℓiℓj)1/4 +(ℓi + ℓj)1/2 exp +� +−2|i − j|2 +ℓi + ℓj +� +, +where √ℓi can be thought of as a local length scale. Here, we consider the case where +ℓi increases linearly from 2.1 to 22 over the domain (not periodic). +5. Pressure-wind covariance. The final covariance matrix models two spatially extended + +12 +R.J. WEBBER AND M. MORZFELD +Single-scale +Multiscale +Nonstationary +Pressure-wind +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +(a) +(b) +(c) +(d) +Figure 1. Four of the five covariance matrices used in numerical experiments. +fields, pressure and wind, that co-vary with each other according to +(4.1) +w = du +dx, +where u is pressure and w is wind. +The pressure has a single-scale covariance Σℓ +(ℓ = 5, Gaussian kernel). Equation (4.1) then implies that the covariance matrix of +both variables (pressure first and then wind) is +Σpw = +� +� +Σℓ +DΣℓ +ΣℓDT +DΣℓDT +� +� +where D ∈ R200×200 is a periodic, centered, second-order discretization of the first +derivative operator. Note that pressure and wind both have dimension 200 so that the +overall dimension for this problem is 400. +Figure 1 shows four of the five covariance matrices used in our numerical experiments. The +figure does not include a plot of the single-scale covariance with a Laplacian kernel because +it looks similar to the single-scale covariance matrix with a Gaussian kernel. +4.2. Benefits of localized covariance estimators. To illustrate the benefit of localization, +we consider estimating a single-scale covariance matrix (Gaussian kernel) of dimension 200 × + +LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION +13 +Covariance matrix +Sample estimate +Hybrid estimate +Localization estimate +-1 +-0.8 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +(a) +(b) +(c) +(d) +Figure 2. The true covariance matrix that we want to estimate from n = 30 samples is shown in the top +left. The sample covariance, shown in the top right, contains large errors because the ensemble size is small +compared to the dimension n = 30 ≪ d = 200. The hybrid (bottom left) and Schur product (bottom right) +localized covariance estimates are more accurate. +200 from n = 30 samples. Figure 2 shows the true covariance matrix, along with the sample +covariance and the hybrid and Schur product estimates. The parameter tuning for the hybrid +and Schur product estimators is described in subsections 4.3 and 4.4. +The figure makes it intuitively clear that hybrid and Schur product estimates are more +accurate than the sample covariance. The reason is that localized estimates take into account +additional information about the covariance structure. In section 3, we explained how this +additional information can be understood as a Bayesian prior distribution that is strongly +influencing the covariance estimates. +4.3. Experiments with hybrid estimators. We now turn our attention to the hybrid +estimator +ˆΣhyb = αΣprior + (1 − α) ˆΣsamp, +and evaluate whether or not it is appropriate to scale the interpolation factor α as α ∼ n−1, +which is suggested by the Bayesian theory (3.7). +We use the following procedure. For each covariance matrix on our list, we apply interpo- + +14 +R.J. WEBBER AND M. MORZFELD +0 +50 +100 +n +0 +0.5 +1 +Opt. +0 +50 +100 +n +0 +0.5 +1 +0 +50 +100 +n +0 +0.5 +1 +0 +50 +100 +n +0 +0.5 +1 +0 +50 +100 +n +0 +0.5 +1 +Shrinkage +(a) +(b) +(c) +(d) +(e) +Laplacian +Gaussian +Multiscale +Nonstationary +Pressure-wind +Figure 3. +Optimal interpolation factor α as a function of ensemble size n, with a comparison to the +theoretical α ∼ n−1 scaling. +lation factors α ranging from 0 to 1, in steps of 0.05. Then, we compute the error +Error(α) := ∥ ˆΣhyb, α − Σ∥F +∥Σ∥F +, +where ˆΣhyb, α is the hybrid estimate of the covariance matrix Σ using the interpolation factor +α. We repeat this procedure 103 times and average the error over independent data sets. The +“optimal” interpolation factor is the one that leads to the smallest averaged error. Repeating +this procedure for various sample sizes allows us to describe how the optimal interpolation +factor α depends on the ensemble size n. +We define the prior covariance matrix Σprior using a single-scale, Gaussian kernel model, +but we vary the length scale for each set of experiments. For the single-scale experiments we +choose ℓ = 1 (Laplacian kernel) and ℓ = 16 (Gaussian kernel); for the multiscale covariance we +choose ℓ = 4; for the nonstationary experiments, we choose ℓ = 8. Finally, for the pressure- +wind experiments we define a prior covariance by +Σprior = +�Σℓ +Σℓ +Σℓ +Σℓ +� +, +where Σℓ is a single-scale Gaussian kernel model with ℓ = 5. We also performed experiments +with different length scales for the various prior covariance matrices and obtained qualitatively +similar results, which is not surprising, since the scaling of α in (3.7) is largely independent +of the choice of the prior covariance matrix. +Results of our experiments are summarized in Figure 3. The figure shows the optimal +interpolation factor as a function of the ensemble size n, along with a n−1 least squares +fit to these data. +In all five cases, the fit of a n−1 polynomial is quite good, confirming +the predictions from the Bayesian theory. +Remarkably, this scaling is independent of the +underlying covariance structure. + +LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION +15 +50 +100 +n +10 +15 +20 +25 +Opt. length scale +50 +100 +n +10 +15 +20 +50 +100 +n +30 +40 +50 +60 +50 +100 +n +30 +40 +50 +60 +50 +100 +n +10 +15 +20 +Laplacian +loc. matrix +(a) +(b) +(c) +(d) +(e) +Laplacian +Gaussian +Multiscale +Nonstationary +Pressure-wind +Gaussian +loc. matrix +50 +100 +n +8 +10 +12 +14 +16 +18 +Opt. length scale +50 +100 +n +6 +8 +10 +12 +50 +100 +n +20 +25 +30 +35 +50 +100 +n +20 +25 +30 +35 +50 +100 +n +6 +8 +10 +12 +(f) +(g) +(h) +(i) +(k) +Figure 4. Optimal length scale as a function of ensemble size, with theoretical ℓ ∼ n (top) and ℓ ∼ n1/2 +(bottom) fits. +4.4. Experiments with Schur product estimators. We now consider the Schur product +estimator +ˆΣSchur = ˆΣsamp ◦ L. +where the localization matrix is defined by the Laplacian kernel Lℓ +ij = exp +� +−dij/ℓ +� +or the +Gaussian kernel Lℓ +ij = exp +� +−(dij/ℓ)2� +, involving a single length-scale parameter ℓ. For the +pressure-wind experiments, we define the overall localization matrix by +L = +�Σℓ +Σℓ +Σℓ +Σℓ +� +, +where Σℓ is a 200×200 single-scale covariance matrix. We evaluate whether or not the scalings +ℓ ∼ n and ℓ ∼ n1/2 are appropriate, as predicted by the Bayesian theory (3.15) and (3.16). +We follow the same protocol as in the experiments with the hybrid estimator. For each +ensemble size n and length scale ℓ, we perform 103 independent experiments and compute the +average of the error defined by +Error(ℓ) = ∥ ˆΣSchur, ℓ +ℓ +− Σ∥F +∥Σ∥F +, +where ˆΣSchur, ℓ +ℓ +is the Schur product estimate using the length scale ℓ. We obtain an optimal +length scale by minimizing this error over different length scales ℓ. Varying the ensemble size +n then allows us to describe how the optimal length scale varies with ensemble size. +Results of our experiments are summarized in Figure 4. The numerical experiments con- +firm the theoretically derived linear scaling ℓ for the Laplacian kernel, as well as the square-root + +16 +R.J. WEBBER AND M. MORZFELD +scaling of ℓ for the Gaussian kernel. We see that the theory is useful for finite ensemble size +n and finite dimension d, and it is robust across five different covariance models. +5. Properties of the Schur estimator and QC estimator. In this section, we prove math- +ematical results involving the Schur product estimator and the QC estimator. +Proposition 5.1. For any positive definite L, the Schur product estimator +(5.1) +ˆΣSchur = ˆΣsamp ◦ L. +cannot be the MAP estimator for a smooth Bayesian prior distribution. +Proof. We use a proof by contradiction. If there exists a smooth prior density function +whose logarithm is ℓ(Σ), we can write the log posterior density as +(5.2) +ℓ(Σ) − n +2 log |Σ| − n +2 tr(Σ−1 ˆΣsamp). +The gradient of (5.2) is given by +∇ℓ(Σ) − n +2 Σ−1 + n +2 Σ−1 ˆΣsampΣ−1. +We next assume that the log posterior density (5.2) achieves its maximum value at the matrix +Σ = ˆΣsamp◦L. The Schur product theorem tells us Σ is positive definite whenever the sample +covariance ˆΣsamp is positive definite, whereby Σ is a local maximum and the gradient equals +zero at this point: +(5.3) +0 = ∇ℓ( ˆΣsamp ◦ L) − n +2 ( ˆΣsamp ◦ L)−1 + n +2 ( ˆΣsamp ◦ L)−1 ˆΣsamp( ˆΣsamp ◦ L)−1. +We will use (5.3) to obtain a contradiction. To begin, we differentiate the (i, j) element of +(5.3) with respect to ˆΣsamp +kl +to yield +0 = Lkl∂2 +ΣijΣkl( ˆΣsamp ◦ L) + n +2 (Lkl + 1)( ˆΣsamp ◦ L)−1 +ik ( ˆΣsamp ◦ L)−1 +jl +− n +2 Lkl( ˆΣsamp ◦ L)−1 +ik e∗ +j( ˆΣsamp ◦ L)−1 ˆΣsamp( ˆΣsamp ◦ L)−1el +− n +2 Lkl( ˆΣsamp ◦ L)−1 +jl e∗ +i ( ˆΣsamp ◦ L)−1 ˆΣsamp( ˆΣsamp ◦ L)−1ek. +By multiplying through by Lij, we obtain an expression of the form +(5.4) +LijLkl∂2 +ΣijΣkl( ˆΣsamp ◦ L) = −n +2 Lij( ˆΣsamp ◦ L)−1 +ik ( ˆΣsamp ◦ L)−1 +jl + fsym((i, j), (k, l)), +where fsym((i, j), (k, l)) is a symmetric function of (i, j) and (k, l). Switching the roles of (i, j) +and (k, l) and using the symmetry of the second derivatives, we have +(5.5) +LijLkl∂2 +ΣijΣkl( ˆΣsamp ◦ L) = −n +2 Lkl( ˆΣsamp ◦ L)−1 +jl ( ˆΣsamp ◦ L)−1 +jl + fsym((i, j), (k, l)). +Subtracting (5.4) from (5.5) leads to the expression +0 = (Lij − Lkl)( ˆΣsamp ◦ L)−1 +ik ( ˆΣsamp ◦ L)−1 +jl , + +LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION +17 +valid for all positive definite matrices ˆΣsamp and all indices 1 ≤ i, j, k, l ≤ p. We note that +( ˆΣsamp ◦ L)−1 +ii +must be positive (it is the diagonal entry of a positive definite matrix); hence, +(5.6) +0 = (Lij − Lil)( ˆΣsamp ◦ L)−1 +jl , +Differentiating (5.6) with respect to ˆΣsamp +jl +, we find +0 = −(Lij − Lil)(Σ ◦ L)−1 +jj (Σ ◦ L)−1 +ll Ljl, +which implies that the j and l columns of L are identical when Ljl ̸= 0. Since L is positive +definite, we cannot have two identical columns and we conclude that L must be a diagonal +matrix. Last, L cannot be diagonal. To show this, we write ˆΣsamp = D + ϵ∆ where D is +diagonal and ∆ is a nontrivial off-diagonal matrix. Then, the gradient equality (5.3) yields +0 = ∇ℓ(D ◦ L) − n +2 (D ◦ L)−1 + n +2 (D ◦ L)−1(D + ϵ∆)(D ◦ L)−1. +Taking the derivative with respect to ϵ, we find that ∆ = 0, which cannot hold because ∆ +is nontrivial by assumption. We have arrived at a contradiction and conclude that the Schur +product estimator (5.1) cannot be the MAP estimator for a smooth Bayesian prior. +Proposition 5.2. If the QC density +(5.7) +p(Σ) ∝ exp +� +−1 +4tr +� +Σ−1� +Θ ◦ Σ−1��� +is a well-defined density (i.e., it integrates to one), then it solves the entropy maximization +problem +max +p +� +H[p] − 1 +4 +d +� +i,j=1 +Θij +� +p(Σ)|Σ−1 +ij |2dΣ +� +. +Proof. We introduce a Lagrange multiplier η to enforce the constraint that +� +p(Σ) dΣ = 1 +and then choose p to maximize +L[p] = H[p] − 1 +4 +d +� +i,j=1 +Θij +� +p(Σ)|Σ−1 +ij |2dΣ + η +�� +p(Σ) dΣ −1 +� +. +The first variation of L[p] is +δL[p] = − log p(Σ) − 1 − 1 +4tr +� +Σ−1� +Θ ◦ Σ−1�� ++ η. +Setting the first variation equal to zero yields +p(Σ) = exp +� +−1 +4tr +� +Σ−1� +Θ ◦ Σ−1�� +− 1 + η +� +, + +18 +R.J. WEBBER AND M. MORZFELD +which is a scalar multiple of the QC density (5.7). By selecting an appropriate Lagrange +multiplier η, we ensure that p is a probability density. Additionally, p �→ L[p] is a concave +functional on probability densities, because the entropy H[p] is concave on probability densities +and the other terms are linear in p. Since (5.7) is a stationary point of a concave Lagrangian, +we conclude that (5.7) maximizes L[p] and solves the entropy maximization problem. +Proposition 5.3. Assume diag( ˆΣsamp) > 0, and consider a penalization of the form Θ = +sΘref, where Θref +ij > 0 if and only if i ̸= j. Then for large enough s > 0, the log likelihood +ℓ(Σ) = n +2 log +��Σ−1�� − n +2 tr +� ˆΣsampΣ−1� +− 1 +4tr +� +Σ−1� +Θ ◦ Σ−1�� +has a unique positive definite global maximizer. +Proof. We decompose the sample covariance into diagonal and off-diagonal components +ˆΣsamp = ˆΣdiag + ˆΣoff +and similarly decompose the log likelihood as +ℓ(Σ) = +� +n +2 log +��Σ−1�� − n +2 tr +� ˆΣdiagΣ−1� +� ++ +� +−n +2 tr +� ˆΣoffΣ−1� +− 1 +4tr +� +Σ−1� +Θ ◦ Σ−1�� +� +. +Next, we optimize the two components of the log likelihood separately. Since the determinant +of a positive definite matrix is bounded from above by the product of its diagonal entries, we +calculate +n +2 log +��Σ−1�� − n +2 tr +� ˆΣdiagΣ−1� +≤ n +2 +d +� +i=1 +� +log Σ−1 +ii − ˆΣsamp +ii +Σ−1 +ii +� +≤ n +2 +d +� +i=1 +� +− log ˆΣsamp +ii +− 1 +� +. +The left-hand side does not come within n +2 units of the upper bound unless +1 +2 ≤ ˆΣsamp +ii +Σ−1 +ii ≤ 2 +for each 1 ≤ i ≤ d. Next observe that +−n +2 tr +� ˆΣoffΣ−1� +− 1 +4tr +� +Σ−1� +Θ ◦ Σ−1�� += 1 +4 +� +i̸=j +�n2| ˆΣsamp +ij +|2 +Θij +− Θij +����Σ−1 +ij + +n ˆΣsamp +ij +Θij +���� +2� +≤ n2 +4 +� +i̸=j +| ˆΣsamp +ij +|2 +Θij +. +In this case, the left-hand side does not come within n +2 units of the upper bound, unless +����Σ−1 +ij + +n ˆΣsamp +ij +Θij +���� +2 +≤ 2n +Θij + +LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION +19 +for each i ̸= j. Tying together the argument, a large log likelihood +ℓ(Σ) ≥ n +2 +d +� +i=1 +� +− log ˆΣsamp +ii +− 1 +� ++ n2 +4 +� +i̸=j +| ˆΣsamp +ij +|2 +Θij +− n +2 +implies the following conditions +� +� +� +� +� +� +� +1 +2 ≤ ˆΣsamp +ii +Σ−1 +ii ≤ 2, +1 ≤ i ≤ d +����Σ−1 +ij + +n ˆΣsamp +ij +Θij +���� +2 +≤ +2n +Θij , +i ̸= j. +For a large enough penalization strength s, these conditions confine Σ to a compact set of +positive definite matrices. Additionally, this set contains an isolated maximizer of ℓ, because +one point inside the set, Σ = ˆΣdiag, achieves a higher log likelihood +ℓ( ˆΣdiag) = n +2 +d +� +i=1 +� +− log ˆΣsamp +ii +− 1 +� +than all the points outside the set. Last, note that +Σ−1 �→ n +2 log +��Σ−1�� − n +2 tr +� ˆΣsampΣ−1� +− 1 +4tr +� +Σ−1� +Θ ◦ Σ−1�� +is concave, whereby any isolated maximum is the unique global maximum. +6. Conclusions. We have studied the problem of estimating a high-dimensional covari- +ance matrix from a small number of samples. This problem is difficult but also ubiquitous in +the setting of numerical weather prediction (NWP) when merging high-dimensional models +with real-world observations. We have developed a new mathematical theory to help jus- +tify the covariance estimators which are used in NWP practice, but which have little to no +mathematical justification. +NWP practitioners boost the accuracy of covariance estimates by enforcing the assumption +that correlations decay with distance. We have argued that this practical approach follows the +Bayesian paradigm of estimating an unknown (covariance matrix) from data (the samples we +have) and prior information (spatial decay of correlations). We have then investigated prior +distributions that lead to practically useful covariance estimators. Put differently, we have +interpreted common estimation procedures within a Bayesian framework and identified how +particular choices of prior distributions influence posterior estimates of covariance matrices. +The situation is clear in the case of hybrid estimators. These estimators can be understood +as Bayesian estimators with an inverse Wishart prior distribution. The Bayesian theory shows +that hybrid estimators converge at an asymptotically optimal rate and also reveals how to +adjust the hybrid estimator when the ensemble size changes. +The Bayesian interpretation of Schur product estimators is more complicated. We have +shown that Schur product estimators are not Bayesian. Nonetheless, we have proposed a +new “quadratically constrained” distribution that leads to a Schur product estimator in the + +20 +R.J. WEBBER AND M. MORZFELD +limit of increasing localization strength. Building on this interpretation, the Bayesian theory +suggests how best to adjust the length scale in a Schur product estimator as the ensemble +size increases. This scaling is new and relevant for practical NWP because it can reduce the +amount of tuning required for efficient covariance estimation. +Perhaps more importantly, our analysis suggests that covariance estimation via Schur +products may not be the most economical approach for covariance estimation. For one, it is +not Bayesian and therefore it has not been rigorously justified. Additionally, our theory, in line +with previous work [21], suggests that we can estimate covariance matrices more efficiently by +working with the inverse of the covariance matrix, i.e., by penalizing conditional correlations, +rather than correlations. +Acknowledgments. We thank Dr. Daniel Hodyss of the Naval Research Laboratory for +discussions of hybrid estimators. +REFERENCES +[1] J. Anderson and L. 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Vaart, Asymptotic Statistics, Cambridge Series in Statistical and Probabilistic Mathematics, +Cambridge University Press, 1998, https://doi.org/10.1017/CBO9780511802256. + diff --git a/v9E3T4oBgHgl3EQf-wsu/content/tmp_files/load_file.txt b/v9E3T4oBgHgl3EQf-wsu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7135ba348a3985868d30e3806b3347b8a3b18793 --- /dev/null +++ b/v9E3T4oBgHgl3EQf-wsu/content/tmp_files/load_file.txt @@ -0,0 +1,828 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf,len=827 +page_content='Localized covariance estimation: A Bayesian justification∗ Robert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Webber† and Matthias Morzfeld‡ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' A major problem in numerical weather prediction (NWP) is the estimation of high-dimensional co- variance matrices from a small number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Maximum likelihood estimators cannot provide reliable estimates when the overall dimension is much larger than the number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For- tunately, NWP practitioners have found ingenious ways to boost the accuracy of their covariance estimators by leveraging the assumption that the correlations decay with spatial distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In this work, Bayesian statistics is used to provide a new justification and analysis of the practical NWP covariance estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The Bayesian framework involves manipulating distributions over symmetric positive definite matrices, and it leads to two main findings: (i) the commonly used “hybrid estima- tor” for the covariance matrix has a naturally Bayesian interpretation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' (ii) the very commonly used “Schur product estimator” is not Bayesian, but it can be studied and understood within the Bayesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' As practical implications, the Bayesian framework shows how to reduce the amount of tuning required for covariance estimation, and it suggests that efficient covariance estimation should be rooted in understanding and penalizing conditional correlations, rather than correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Covariance estimation, Bayesian statistics, numerical weather prediction AMS subject classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 62H10, 65C20, 86-10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In this work, we provide new insights into the estimation of high- dimensional covariance matrices from a small number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Our work is motivated by numerical weather prediction (NWP), a setting in which covariance estimation arises nat- urally and at a vast scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The goal in NWP is to generate a set of weather forecasts based on global weather models and real-world observations of the Earth’s atmosphere [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' All the standard NWP techniques require estimating the covariance matrix for the near-term weather forecasts [19,28], but the dimensionality of the covariance estimation problem in NWP is im- mense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' A typical global weather model has billions of unknowns, which are updated by tens of millions of observations within less than six hours of computing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The large cost of running the weather model necessitates that the ensemble size (number of model integrations) is small compared to the number of unknown weather variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' A typical ensemble size is ≤ 100 and therefore six orders of magnitude smaller than the number of unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The sample covariance is not an accurate covariance estimator unless the ensemble size is larger than the number of unknowns [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Therefore, covariance estimation at the extreme scale of NWP can only be accomplished by using additional information and tricks, commonly referred to as “covariance localization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The basic idea is that covariances should decay with spatial distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' On six hour time scales, the weather in La Jolla, California, is uncorrelated ∗Submitted to the editors DATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Funding: RJW is supported by the Office of Naval Research through BRC award N00014-18-1-2363 and the National Science Foundation through FRG award 1952777, under the aegis of Joel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Tropp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' MM is supported by the US Office of Naval Research (ONR) grant N00014-21-1-2309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' †Computing & Mathematical Sciences, California Institute of Technology, Pasadena, CA (rwebber@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' ‡Institute of Geophysics and Planetary Physics, Sripps Institution of Oceanography, University of California, San Diego, San Diego, CA (matti@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='04828v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='ME] 12 Jan 2023 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' WEBBER AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' MORZFELD with the weather in Chicago, Illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Localization, in its simplest form, means damping estimates of long-range of covariances because large magnitudes are caused by sampling error, not by the existence of long-range covariances [16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Localization started off as an ad hoc procedure that perhaps grew out of desperation to make NWP work – early NWP attempts using the ensemble Kalman filter [9], for example, led to useless forecasts because of the large errors in the forecast covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' By now, however, localization has been widely accepted as a necessary ingredient within the NWP community [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Not surprisingly, localization has been studied extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In the NWP community, the- oretical work has focused on adaptive localization methods [1] and theories for optimal lo- calization [10, 20, 23], but implementing these techniques can be data-intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In practice, localization is often implemented using a Schur product estimator [16, 17, 25], a hybrid co- variance estimator [5,22,30] (also called a “shrinkage” estimator [24,27]), or a combination of the two estimators [22], but there is little to no mathematical justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Meanwhile, the statistical community has introduced localized estimators with rigorous guarantees [3,4,6,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' However, these estimators are only guaranteed to work in the asymptotic limit as the ensem- ble size and the state dimension jointly grow to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' It remains unclear which localized estimators work best with finite ensemble size and finite state dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' To our surprise, localization is not typically understood from the Bayesian perspective, even though localization is a naturally Bayesian procedure: We estimate an unknown (the covariance matrix) based on limited data (the ensemble/forecast states) and enforce prior information about the problem structure (the spatial decay of covariances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' This paper is about describing the Bayesian perspective and explaining why this interpretation of covariance localization may be useful in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Put simply, we ask and answer the following question: “Are there any Bayesian prior distributions that lead to existing localization methods?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' This question is equivalent to asking, “Which existing localization methods are rooted in a Bayesian framework?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' To answer this question, we consider two different Bayesian prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' First, we consider the inverse Wishart distribution [29, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2], a classical distribution over covariance matrices that leads to the hybrid covariance estimator [5,16,17,22,25,30] as a maximum a pos- teriori estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Second, we consider a new “quadratically constrained” (QC) distribution, which forces the off-diagonal entries of the precision matrix (inverse covariance matrix) to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We introduce the QC prior to study localization via Schur products [5, 22, 24, 27, 30], which is a common method for covariance estimation but is surprisingly not Bayesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We show that the QC covariance estimator converges to the Schur product estimator as the lo- calization strength parameter tends toward infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In summary, our work provides a new Bayesian justification for two commonly used localization estimators in NWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The Bayesian framework is useful for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' First, the framework is designed for finite ensemble size and, thus, more practically applicable than statistical techniques that are largely asymptotic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=', [3,4,6,12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Second, the Bayesian framework suggests how to adjust the parameters in the hybrid esti- mator and Schur product estimator as the ensemble size changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Typically in an operational setting, this adjustment is done implicitly, since the localization is re-tuned as the ensem- ble grows larger or smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Our Bayesian theory helps with reducing the amount of tuning required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION 3 Last, the Bayesian theory may help to construct new localization estimators for the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Every year, Earth models are becoming increasingly complex, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=', coupled atmosphere, ocean and sea ice models, or seasonal to sub-seasonal forecast models, and data assimilation is also being extended to geomagnetic models [11,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For all these models, traditional localization based on a single length scale parameter may no longer be appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The theoretical foundations laid here are not limited to a single length scale parameter and are more generally applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' As a central feature, our theory emphasizes building the localization scheme via the precision matrix that describes the conditional correlations between variables, not via the covariance matrix directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Section 2 reviews covariance estimation from the NWP perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Section 3 analyzes covariance estimation from the Bayesian perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Section 4 numerically tests the predictions of the Bayesian theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Section 5 proves mathematical theorems to support our Bayesian analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Section 6 offers a summary and some conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Throughout this paper, we use bold lower case letters to refer to vectors and bold capital letters to refer to matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The ij entry of the matrix A is written Aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The determinant of a matrix A is written |A| and the Schur (element-wise) product of compatible matrices A and B is written A ◦ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Last, the Frobenius norm of a matrix is written ∥A∥F = ��d i,j=1 |Aij|2�1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' A rapid review of NWP covariance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We start by briefly describing how covariance estimation is accomplished in NWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' To keep things simple, we assume that inde- pendent samples x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' , xn ∈ Rd are drawn from a mean-zero Gaussian distribution, xi ∼ N(0, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We assume that ensemble size n is much smaller than the dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Our goal is to estimate the d × d positive semidefinite covariance matrix Σ from n ≪ d samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' A classical estimator for Σ is the maximum likelihood (ML) estimator ˆΣ = arg max Σ n � i=1 p(xi|Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Under the mean zero assumption, the ML estimator is just the “sample covariance” or “em- pirical covariance” ˆΣsamp = 1 n n � i=1 xixT i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The ML estimator is unbiased (mean Σ), and as the number of data points n approaches infinity, the ML estimator converges to the true covariance Σ at the optimal rate [32, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 8] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1) √n( ˆΣsamp − Σ) D→ N(0, I(Σ)−1), where I(Σ)−1 denotes the inverse Fisher information tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' This means that the ML estima- tor achieves the optimal O(1/√n) error scaling, and the limiting distribution of √n( ˆΣsamp−Σ) is as tightly concentrated as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' However, since we work in a framework where n ≪ p, the sample covariance is known to be inaccurate [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' WEBBER AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' MORZFELD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Schur product estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Covariance localization is an approach for increasing the accuracy of the sample covariance when the ensemble size is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The basic idea is to damp long-range correlations based on the assumption that correlations decay with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Localization can be implemented via a Schur product with a symmetric positive definite localization matrix L: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2) ˆΣSchur = ˆΣsamp ◦ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' A simple example of a localization matrix is based on the Gaussian kernel and has elements Lij = exp � −(dij/ℓ)2� , where dij is the distance between grid points i and j, and where ℓ > 0 is a length scale parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Alternately, one can replace the Gaussian kernel function with a Laplacian kernel function (exponential decay), or with a Gaspari-Cohn kernel function [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The latter is zero at large distances and therefore promotes sparsity in the covariance matrix estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Compared to the sample covariance, the Schur product estimator creates an element-wise bias of E[ ˆΣSchur ij ] − Σij = LijE[ ˆΣsamp ij ] − Σij = (Lij − 1)Σij, and changes the element-wise variance by a factor of Var[ ˆΣSchur ij ] Var[ ˆΣsamp ij ] = L2 ijVar[ ˆΣsamp ij ] Var[ ˆΣsamp ij ] = L2 ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Since Li,j is small at large spatial separations, the effect of localization is clear: it introduces a small bias while drastically reducing the variance of the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Hybrid estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' A second important and widely used covariance estimator is the “hybrid” estimator ˆΣhyb, which is defined as a convex combination between the sample covariance ˆΣsamp and a prior covariance estimate Σprior: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3) ˆΣhyb = αΣprior + (1 − α) ˆΣsamp, for some α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Compared to the unbiased estimator ˆΣsamp, the hybrid estimator creates a bias of size E[ ˆΣhyb] − Σ = αΣprior + (1 − α)E[ ˆΣsamp] − Σ = α(Σprior − Σ) and changes the variance by a factor of Var[ ˆΣhyb] Var[ ˆΣsamp] = (1 − α)2Var[ ˆΣsamp] Var[ ˆΣsamp] = (1 − α)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Thus, the hybrid estimator typically adds a small bias while slightly reducing the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION 5 In NWP, the hybrid estimator is often presented in an equivalent form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='4) ˆΣhyb, NWP = w1Σclim + w2 ˆΣsamp, where Σclim is a climatological covariance matrix, derived from a long model run that re- veals covariance structure inherent to the physical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The NWP version of the hybrid estimator in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='4) and the version we presented in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3) are equivalent if we set α = 1 − w2, Σprior = w1 1 − w2 ˆΣclim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1, we will show that NWP researchers are using a principled Bayesian approach when applying the hybrid estimator, but the Bayesian ideas are somewhat hidden within the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' If we use the right symbols and notation, we can frame the practical NWP estimators within a rigorous Bayesian perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Tuning of covariance estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The accuracy of the Schur product and hybrid estimators depends on the various parameters that go into the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For the hybrid estimator in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3), one needs to specify the prior covariance matrix Σprior and the interpolation factor α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For the Schur product estimator in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2), one needs to specify the parameters that define the localization matrix L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' If we use Gaussian or Laplacian kernels to define the localization matrix, this means that one needs to determine an appropriate length scale ℓ for localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The parameters that define the covariance estimator are usually determined via parameter tuning, or, using more modern language, a “training” phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The idea is to simply try a few parameters and then determine which parameter combination gives the most useful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For example, one can run an ensemble data assimilation algorithm on a set of training observations and compute the forecast error that results from each choice of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' One then selects the parameters that lead to the smallest forecast errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' This tuning is expensive, computationally and otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In practice, localization and hybrid estimators are often combined [22], which means that a relatively large number of parameters needs to be tuned, which is even more costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Even worse, this entire tuning process must be repeated whenever the underlying model is modified, or if the ensemble size is increased because more computational power is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We will see in section 3 that the Bayesian perspective on covariance localization gives insights that can reduce the efforts that go into tuning covariance estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The Bayesian perspective on covariance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The main goal of Bayesian statistics is to combine prior information and data to estimate parameters in a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Bayesian statistics has three main components: the prior distribution, the likelihood, and the poste- rior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The Bayesian prior distribution encodes all information before any data are collected and the likelihood function infuses information from data into the posterior estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Here, we apply Bayesian statistics to the problem of estimating a positive definite covari- ance matrix Σ ∈ Rd×d from a set of n samples xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We assemble the n samples into a p × n data matrix X = � x1 · · xn � , and we express the posterior density function 6 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' WEBBER AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' MORZFELD as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1) p(Σ|X) � �� � posterior ∝ p(Σ) � �� � prior p(X|Σ) � �� � likelihood .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The symbol ∝ indicates that the left- and right-hand sides are proportional over all choices of Σ, but the proportionality constant is typically not needed for computing the covariance estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1), the prior density function p(Σ) is chosen to account for any structural knowledge of Σ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=', the decay of correlations with spatial distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The likelihood function p(X|Σ) accounts for information from the data, which in our case are the n samples assembled in the data matrix X, and the likelihood function takes the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2) p(X|Σ) = ��� 1 2πΣ−1��� n/2 exp � −1 2 n � i=1 xT i Σ−1xi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Last, the posterior density function p(Σ|X) gives a distribution of possible covariance ma- trices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Using the posterior density, we can calculate the “maximum a posterior” (MAP) estimator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3) ˆΣMAP = arg max Σ p(Σ|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The MAP estimator can be regarded as the single most likely value for the covariance under the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Other estimators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=', mean, median) are equally valid, but are harder to compute or analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' With the uniform prior distribution p(Σ) = Const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=', the MAP estimator is the same as the ML estimator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='4) ˆΣMAP = arg max Σ p(X|Σ) = 1 n n � i=1 xixT i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' However, more generally, the choice of prior distribution has a non-trivial effect on the Bayesian posterior — the whole purpose of imposing a prior is to fill in the gaps that the data leave open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The rest of this paper is about choices of non-uniform priors p(Σ) that promote structure in the covariance estimate ˆΣMAP, providing justification for existing but largely empirical covariance estimators in NWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The inverse Wishart prior and hybrid estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The inverse Wishart distribution [29, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2] is a classical distribution defined over symmetric positive definite matrices Σ ∈ Rd×d by the density (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='5) p(Σ) ∝ ��Σ−1��m/2 exp � −m 2 tr � ΣpriorΣ−1�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' There are two parameters in the inverse Wishart distribution: Σprior is the mode (most likely value) of the distribution, and m is the “sample size” parameter that controls the width of the distribution around the mode: a large m leads to tightly concentrated distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION 7 Given an inverse Wishart prior and a data matrix X = � x1 · · xn � , the Bayesian posterior is also an inverse Wishart distribution, because the inverse Wishart distribution is the “conjugate prior” [8] to the mean-zero multivariate Gaussian likelihood: p(Σ|X) ∝ ��Σ−1��(m+n)/2 exp � −m 2 tr � ΣpriorΣ−1� − 1 2 n � i=1 xT i Σ−1xi � ∝ ��Σ−1��(m+n)/2 exp � −m + n 2 tr � ˆΣIWΣ−1�� , where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='6) ˆΣIW = m m + nΣprior + n m + n ˆΣsamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In the posterior distribution, the two inverse Wishart parameters are updated in response to the data: the sample size parameter increases from m to m + n, and the mode changes from Σprior to ˆΣIW (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' It is now clear that the inverse Wishart prior leads to a covariance estimator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='6) that is identical to the hybrid estimator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3) with the parameter choice (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='7) α = m m + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In other words, the hybrid estimator is the same estimator that would result from selecting an inverse Wishart prior and systematically applying a Bayesian analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' This perspective provides a Bayesian justification for the hybrid estimator, assuming that the parameters m and Σprior represent reasonable prior knowledge about the covariance structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' A major benefit of Bayesian statistics is that it leads to a covariance estimator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='6) valid for any sample size n, whereas standard NWP covariance estimators require tuning parameters whenever the sample size changes (subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' When n is large, the Bayesian formula tells us to adjust our estimator according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='8) ˆΣIW = ˆΣsamp + O(n−1), and this scaling with n ensures that ˆΣIW converges to the true covariance at the optimal asymptotic rate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In NWP applications, we anticipate consistent accuracy in covariance estimation when the localized covariance estimate is adjusted according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We will revisit this idea in the numerical examples in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The QC prior and Schur product estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We now introduce a new “quadrat- ically contrained” (QC) distribution to study localization via Schur products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Surprisingly, localization via Schur products cannot result directly from a Bayesian prior (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' However, the QC prior allows us to study Schur product localization in a rigorous and mean- ingful way, in the asymptotic limit of increasing penalization strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The QC distribution is defined over symmetric positive definite matrices by the density function p(Σ) ∝ exp � −1 4tr � Σ−1� Θ ◦ Σ−1��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 8 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' WEBBER AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' MORZFELD The only parameter in the QC distribution is a symmetric nonnegative-valued matrix Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The QC prior can be “improper” [7], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=', the density can integrate to infinity for some Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' However, the corresponding Bayesian posterior distribution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='9) p(Σ|X) ∝ ��Σ−1��n/2 exp � −n 2 tr � ˆΣsampΣ−1� − 1 4tr � Σ−1� Θ ◦ Σ−1��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' is well-defined for every Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We further show in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3 that this posterior distribution has a unique positive definite global maximizer given a large localization strength, which justifies the use of the MAP as a covariance estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Motivation for the QC prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The QC distribution can be derived as the maximum entropy or most “random” [18] distribution that constrains the square entries of the precision matrix Σ−1 to be small (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' More specifically, with entropy defined as the amount of “randomness” in a density p via H[p] := − � p(Σ) log p(Σ) dΣ, the QC density solves the maximization problem max p � H[p] − 1 4 d � i,j=1 Θij � p(Σ)|Σ−1 ij |2dΣ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Here, Θ is the parameter that penalizes off-diagonal elements in Σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' At first, it is perhaps strange that we define the QC prior to target off-diagonal elements in the precision matrix Σ−1, while we aim to explain the Schur product estimator that constrains elements in the covariance matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' However, there is a systematic Bayesian explanation for why targeting the precision matrix is the right approach, based on the conditional correlation structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The conditional correlation between two variables xi and xj measures the degree of asso- ciation with the effects of all other components of x removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In many NWP applications, we expect that conditional correlations, even more so than correlations, should be confined to small neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For example, we expect the weather in La Jolla, California is condi- tionally uncorrelated with the weather in Chicago, Illinois, after accounting for the weather in all the in-between locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The fast decay of conditional correlations has been observed in many geophysical applications and has been described as the “screening effect” [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In a Gaussian model, the conditional correlations are described explicitly by corr � xi, xj | (xk)k/∈{i,j} � = − Σ−1 ij (Σ−1 ii Σ−1 jj )1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The magnitude of the conditional correlations is thus proportional to the magnitude of the Σ−1 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The QC prior can be interpreted as enforcing prior knowledge of the screening effect, by targeting the off-diagonal entries of Σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION 9 As an example of the screening effect, we consider a Gaussian process with covariances defined by the Laplacian kernel k(x, y) = exp � −|x − y| ℓ � , on a 1D spatial domain (not periodic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' When the data is generated from a uniform grid with mesh size ∆, the corresponding covariance matrix is Σ = � � � � � � � 1 e−∆ · · e−(d−2)∆ e−(d−1)∆ e−∆ 1 · · e−(d−3)∆ e−(d−2)∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' e−(d−2)∆ e−(d−3)∆ · · 1 e−∆ e−(d−1)∆ e−(d−2)∆ · · e−∆ 1 � � � � � � � , and the precision matrix is Σ−1 = 2 e∆ − e−∆ � � � � � � � � � e∆ −1 −1 e∆ + e−∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' e∆ + e−∆ −1 −1 e∆ � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The precision matrix Σ−1 is tridiagonal and, hence, has a faster off-diagonal decay than the covariance matrix (which has exponential decay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' This example thus supports the strategy of constraining off-diagonal entries in Σ−1, rather than in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' QC covariance estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Next, we study the MAP estimator corresponding to the QC prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We do so by maximizing the logarithm of the posterior distribution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='9) ℓ(Σ) = n 2 log ��Σ−1�� − n 2 tr � ˆΣsampΣ−1� − 1 4tr � Σ−1� Θ ◦ Σ−1�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' To find the unique global maximizer of ℓ we set its gradient ∇ℓ = n 2 Σ−1� −Σ + ˆΣsamp + 1 nΣ−1 ◦ Θ � Σ−1 equal to zero, and we obtain an implicit equation for the QC estimator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='10) ΣQC = ˆΣsamp + 1 n(ΣQC)−1 ◦ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In high dimensions, solving (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='10) is a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Nonetheless, we can extract useful asymptotic information from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='10) and make the connection to Schur product estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' First, we note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='10) implies the QC estimator is the same as the sample covariance ˆΣQC ij = ˆΣsamp ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 10 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' WEBBER AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' MORZFELD for any (i, j) entries such that Θij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In other words, the QC estimator trusts the sample covariance completely if we do not penalize the conditional correlation between xi and xj via Θi,j > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In NWP, it is unusual to penalize variances, so we assume for the rest of this section that Θij > 0 if and only if i ̸= j, which implies that ˆΣQC ii = ˆΣsamp ii , 1 ≤ i ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We now consider the asymptotic behavior of the QC estimator when we set Θ = sΘref and raise the penalization strength parameter s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In this limit, we may write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='11) ˆΣQC = D + s−1∆, where D is a diagonal matrix with elements Dii = ˆΣQC ii = ˆΣsamp ii and s−1∆ is the matrix containing all off-diagonal elements of ˆΣQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' As s → ∞, the inverse of the QC estimator is given by the Taylor series expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='12) ( ˆΣQC)−1 = D−1 − s−1D−1∆D−1 + O(s−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='12) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='10), and solving for s−1∆, we find ˆΣQC ij = ˆΣsamp ij � 1 + Θij n ˆΣsamp ii ˆΣsamp jj �−1 + O(s−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Therefore, the QC covariance estimator is asymptotically a Schur product estimator that damps the off-diagonal entries of the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The Bayesian framework thus begins to provide a theoretical foundation for the Schur product estimator used in NWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The QC estimator converges to a Schur product estimator with localization matrix (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='13) Lij = � 1 + Θij n ˆΣsamp ii ˆΣsamp jj �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' This approximation becomes increasingly accurate in the limit of a large penalization strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Adjusting the length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' When we use a localization matrix L with a single length scale parameter, the Bayesian perspective suggests how best to adjust the length scale with the ensemble size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' As n → ∞, the Schur product formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='13) satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='14) Lij = 1 + O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For example, if Lij = exp(−dij/ℓ), we need to adjust the length scale as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='15) exp(−dij/ℓ) = 1 + O(n−1) =⇒ 1 ℓ = O(n−1), to ensure that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='14) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Thus, we need to increase the length scale at a rate ℓ ∼ n or faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' If Lij = exp(−d2 ij/ℓ2), we need to adjust the length scale as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='16) exp � −(dij/ℓ)2� = 1 + O(n−1) =⇒ 1 ℓ2 = O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION 11 This means we need to increase the length scale at a rate ℓ ∼ n1/2 or faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The scaling of the length scale with the ensemble size is practically important in NWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Typically when the ensemble size changes, the localization length scale(s) are re-tuned, which is costly, both computationally and otherwise (subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The Bayesian perspective naturally provides a scaling of the localization length scale with ensemble size, thus reducing the amount of tuning necessary when the data assimilation system undergoes an upgrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Numerical illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In this section, we numerically test the scalings for the hybrid estimator and Schur product estimator that are predicted by our Bayesian theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We perform numerical tests with a variety of covariance matrices and a variety of ensemble sizes to check that the scalings are useful in practice and to reiterate that the theory holds at finite ensemble size, not only asymptotically (for n, p → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We estimate covariance matrices for five different Gaussian models, each with mean zero and a (spatial) dimension of d = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Following [20], the five covariance matrices are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Single-scale, Laplacian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The first covariance matrix has elements Σℓ ij = exp(−dij/ℓ), where ℓ is the length scale, which we set to ℓ = 5, and dij is the distance between grid points i and j, reflecting the periodic domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Single-scale, Gaussian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The second covariance matrix is similar to the first, but the decay of covariance is faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Namely, the matrix has elements Σℓ ij = exp(−(dij/ℓ)2) where ℓ and dij are the length scale and the distance between grid points on the periodic domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Here too, we set ℓ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Multiscale covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We define a multiscale covariance matrix as the sum of two single-scale covariances Σms = 1 2 � Σℓ1 + Σℓ2� , and we set ℓ1 = 2 and ℓ2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We use single-scale covariance matrices defined by a Gaussian kernel, but similar results can be obtained with a Laplacian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Nonstationary covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The next covariance matrix is “nonstationary” [26] and its elements are Σns i,j = (4ℓiℓj)1/4 (ℓi + ℓj)1/2 exp � −2|i − j|2 ℓi + ℓj � , where √ℓi can be thought of as a local length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Here, we consider the case where ℓi increases linearly from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1 to 22 over the domain (not periodic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Pressure-wind covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The final covariance matrix models two spatially extended 12 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' WEBBER AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' MORZFELD Single-scale Multiscale Nonstationary Pressure-wind 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='8 1 (a) (b) (c) (d) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Four of the five covariance matrices used in numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' fields, pressure and wind, that co-vary with each other according to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1) w = du dx, where u is pressure and w is wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The pressure has a single-scale covariance Σℓ (ℓ = 5, Gaussian kernel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1) then implies that the covariance matrix of both variables (pressure first and then wind) is Σpw = � � Σℓ DΣℓ ΣℓDT DΣℓDT � � where D ∈ R200×200 is a periodic, centered, second-order discretization of the first derivative operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Note that pressure and wind both have dimension 200 so that the overall dimension for this problem is 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Figure 1 shows four of the five covariance matrices used in our numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The figure does not include a plot of the single-scale covariance with a Laplacian kernel because it looks similar to the single-scale covariance matrix with a Gaussian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Benefits of localized covariance estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' To illustrate the benefit of localization, we consider estimating a single-scale covariance matrix (Gaussian kernel) of dimension 200 × LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION 13 Covariance matrix Sample estimate Hybrid estimate Localization estimate 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='8 1 (a) (b) (c) (d) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The true covariance matrix that we want to estimate from n = 30 samples is shown in the top left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The sample covariance, shown in the top right, contains large errors because the ensemble size is small compared to the dimension n = 30 ≪ d = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The hybrid (bottom left) and Schur product (bottom right) localized covariance estimates are more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 200 from n = 30 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Figure 2 shows the true covariance matrix, along with the sample covariance and the hybrid and Schur product estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The parameter tuning for the hybrid and Schur product estimators is described in subsections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The figure makes it intuitively clear that hybrid and Schur product estimates are more accurate than the sample covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The reason is that localized estimates take into account additional information about the covariance structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In section 3, we explained how this additional information can be understood as a Bayesian prior distribution that is strongly influencing the covariance estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Experiments with hybrid estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We now turn our attention to the hybrid estimator ˆΣhyb = αΣprior + (1 − α) ˆΣsamp, and evaluate whether or not it is appropriate to scale the interpolation factor α as α ∼ n−1, which is suggested by the Bayesian theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We use the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For each covariance matrix on our list, we apply interpo- 14 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' WEBBER AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' MORZFELD 0 50 100 n 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='5 1 Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 0 50 100 n 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='5 1 0 50 100 n 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='5 1 0 50 100 n 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='5 1 0 50 100 n 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='5 1 Shrinkage (a) (b) (c) (d) (e) Laplacian Gaussian Multiscale Nonstationary Pressure-wind Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Optimal interpolation factor α as a function of ensemble size n, with a comparison to the theoretical α ∼ n−1 scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' lation factors α ranging from 0 to 1, in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Then, we compute the error Error(α) := ∥ ˆΣhyb, α − Σ∥F ∥Σ∥F , where ˆΣhyb, α is the hybrid estimate of the covariance matrix Σ using the interpolation factor α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We repeat this procedure 103 times and average the error over independent data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The “optimal” interpolation factor is the one that leads to the smallest averaged error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Repeating this procedure for various sample sizes allows us to describe how the optimal interpolation factor α depends on the ensemble size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We define the prior covariance matrix Σprior using a single-scale, Gaussian kernel model, but we vary the length scale for each set of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For the single-scale experiments we choose ℓ = 1 (Laplacian kernel) and ℓ = 16 (Gaussian kernel);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' for the multiscale covariance we choose ℓ = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' for the nonstationary experiments, we choose ℓ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Finally, for the pressure- wind experiments we define a prior covariance by Σprior = �Σℓ Σℓ Σℓ Σℓ � , where Σℓ is a single-scale Gaussian kernel model with ℓ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We also performed experiments with different length scales for the various prior covariance matrices and obtained qualitatively similar results, which is not surprising, since the scaling of α in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='7) is largely independent of the choice of the prior covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Results of our experiments are summarized in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The figure shows the optimal interpolation factor as a function of the ensemble size n, along with a n−1 least squares fit to these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In all five cases, the fit of a n−1 polynomial is quite good, confirming the predictions from the Bayesian theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Remarkably, this scaling is independent of the underlying covariance structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION 15 50 100 n 10 15 20 25 Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' length scale 50 100 n 10 15 20 50 100 n 30 40 50 60 50 100 n 30 40 50 60 50 100 n 10 15 20 Laplacian loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' matrix (a) (b) (c) (d) (e) Laplacian Gaussian Multiscale Nonstationary Pressure-wind Gaussian loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' matrix 50 100 n 8 10 12 14 16 18 Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' length scale 50 100 n 6 8 10 12 50 100 n 20 25 30 35 50 100 n 20 25 30 35 50 100 n 6 8 10 12 (f) (g) (h) (i) (k) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Optimal length scale as a function of ensemble size, with theoretical ℓ ∼ n (top) and ℓ ∼ n1/2 (bottom) fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Experiments with Schur product estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We now consider the Schur product estimator ˆΣSchur = ˆΣsamp ◦ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' where the localization matrix is defined by the Laplacian kernel Lℓ ij = exp � −dij/ℓ � or the Gaussian kernel Lℓ ij = exp � −(dij/ℓ)2� , involving a single length-scale parameter ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For the pressure-wind experiments, we define the overall localization matrix by L = �Σℓ Σℓ Σℓ Σℓ � , where Σℓ is a 200×200 single-scale covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We evaluate whether or not the scalings ℓ ∼ n and ℓ ∼ n1/2 are appropriate, as predicted by the Bayesian theory (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We follow the same protocol as in the experiments with the hybrid estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For each ensemble size n and length scale ℓ, we perform 103 independent experiments and compute the average of the error defined by Error(ℓ) = ∥ ˆΣSchur, ℓ ℓ − Σ∥F ∥Σ∥F , where ˆΣSchur, ℓ ℓ is the Schur product estimate using the length scale ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We obtain an optimal length scale by minimizing this error over different length scales ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Varying the ensemble size n then allows us to describe how the optimal length scale varies with ensemble size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Results of our experiments are summarized in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The numerical experiments con- firm the theoretically derived linear scaling ℓ for the Laplacian kernel, as well as the square-root 16 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' WEBBER AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' MORZFELD scaling of ℓ for the Gaussian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We see that the theory is useful for finite ensemble size n and finite dimension d, and it is robust across five different covariance models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Properties of the Schur estimator and QC estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In this section, we prove math- ematical results involving the Schur product estimator and the QC estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For any positive definite L, the Schur product estimator (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1) ˆΣSchur = ˆΣsamp ◦ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' cannot be the MAP estimator for a smooth Bayesian prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We use a proof by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' If there exists a smooth prior density function whose logarithm is ℓ(Σ), we can write the log posterior density as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2) ℓ(Σ) − n 2 log |Σ| − n 2 tr(Σ−1 ˆΣsamp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The gradient of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2) is given by ∇ℓ(Σ) − n 2 Σ−1 + n 2 Σ−1 ˆΣsampΣ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We next assume that the log posterior density (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2) achieves its maximum value at the matrix Σ = ˆΣsamp◦L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The Schur product theorem tells us Σ is positive definite whenever the sample covariance ˆΣsamp is positive definite, whereby Σ is a local maximum and the gradient equals zero at this point: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3) 0 = ∇ℓ( ˆΣsamp ◦ L) − n 2 ( ˆΣsamp ◦ L)−1 + n 2 ( ˆΣsamp ◦ L)−1 ˆΣsamp( ˆΣsamp ◦ L)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We will use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3) to obtain a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' To begin, we differentiate the (i, j) element of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3) with respect to ˆΣsamp kl to yield 0 = Lkl∂2 ΣijΣkl( ˆΣsamp ◦ L) + n 2 (Lkl + 1)( ˆΣsamp ◦ L)−1 ik ( ˆΣsamp ◦ L)−1 jl − n 2 Lkl( ˆΣsamp ◦ L)−1 ik e∗ j( ˆΣsamp ◦ L)−1 ˆΣsamp( ˆΣsamp ◦ L)−1el − n 2 Lkl( ˆΣsamp ◦ L)−1 jl e∗ i ( ˆΣsamp ◦ L)−1 ˆΣsamp( ˆΣsamp ◦ L)−1ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' By multiplying through by Lij, we obtain an expression of the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='4) LijLkl∂2 ΣijΣkl( ˆΣsamp ◦ L) = −n 2 Lij( ˆΣsamp ◦ L)−1 ik ( ˆΣsamp ◦ L)−1 jl + fsym((i, j), (k, l)), where fsym((i, j), (k, l)) is a symmetric function of (i, j) and (k, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Switching the roles of (i, j) and (k, l) and using the symmetry of the second derivatives, we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='5) LijLkl∂2 ΣijΣkl( ˆΣsamp ◦ L) = −n 2 Lkl( ˆΣsamp ◦ L)−1 jl ( ˆΣsamp ◦ L)−1 jl + fsym((i, j), (k, l)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Subtracting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='4) from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='5) leads to the expression 0 = (Lij − Lkl)( ˆΣsamp ◦ L)−1 ik ( ˆΣsamp ◦ L)−1 jl , LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION 17 valid for all positive definite matrices ˆΣsamp and all indices 1 ≤ i, j, k, l ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We note that ( ˆΣsamp ◦ L)−1 ii must be positive (it is the diagonal entry of a positive definite matrix);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' hence, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='6) 0 = (Lij − Lil)( ˆΣsamp ◦ L)−1 jl , Differentiating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='6) with respect to ˆΣsamp jl , we find 0 = −(Lij − Lil)(Σ ◦ L)−1 jj (Σ ◦ L)−1 ll Ljl, which implies that the j and l columns of L are identical when Ljl ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Since L is positive definite, we cannot have two identical columns and we conclude that L must be a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Last, L cannot be diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' To show this, we write ˆΣsamp = D + ϵ∆ where D is diagonal and ∆ is a nontrivial off-diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Then, the gradient equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3) yields 0 = ∇ℓ(D ◦ L) − n 2 (D ◦ L)−1 + n 2 (D ◦ L)−1(D + ϵ∆)(D ◦ L)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Taking the derivative with respect to ϵ, we find that ∆ = 0, which cannot hold because ∆ is nontrivial by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We have arrived at a contradiction and conclude that the Schur product estimator (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='1) cannot be the MAP estimator for a smooth Bayesian prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' If the QC density (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='7) p(Σ) ∝ exp � −1 4tr � Σ−1� Θ ◦ Σ−1��� is a well-defined density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=', it integrates to one), then it solves the entropy maximization problem max p � H[p] − 1 4 d � i,j=1 Θij � p(Σ)|Σ−1 ij |2dΣ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We introduce a Lagrange multiplier η to enforce the constraint that � p(Σ) dΣ = 1 and then choose p to maximize L[p] = H[p] − 1 4 d � i,j=1 Θij � p(Σ)|Σ−1 ij |2dΣ + η �� p(Σ) dΣ −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The first variation of L[p] is δL[p] = − log p(Σ) − 1 − 1 4tr � Σ−1� Θ ◦ Σ−1�� + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Setting the first variation equal to zero yields p(Σ) = exp � −1 4tr � Σ−1� Θ ◦ Σ−1�� − 1 + η � , 18 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' WEBBER AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' MORZFELD which is a scalar multiple of the QC density (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' By selecting an appropriate Lagrange multiplier η, we ensure that p is a probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Additionally, p �→ L[p] is a concave functional on probability densities, because the entropy H[p] is concave on probability densities and the other terms are linear in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Since (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='7) is a stationary point of a concave Lagrangian, we conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='7) maximizes L[p] and solves the entropy maximization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Assume diag( ˆΣsamp) > 0, and consider a penalization of the form Θ = sΘref, where Θref ij > 0 if and only if i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Then for large enough s > 0, the log likelihood ℓ(Σ) = n 2 log ��Σ−1�� − n 2 tr � ˆΣsampΣ−1� − 1 4tr � Σ−1� Θ ◦ Σ−1�� has a unique positive definite global maximizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We decompose the sample covariance into diagonal and off-diagonal components ˆΣsamp = ˆΣdiag + ˆΣoff and similarly decompose the log likelihood as ℓ(Σ) = � n 2 log ��Σ−1�� − n 2 tr � ˆΣdiagΣ−1� � + � −n 2 tr � ˆΣoffΣ−1� − 1 4tr � Σ−1� Θ ◦ Σ−1�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Next, we optimize the two components of the log likelihood separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Since the determinant of a positive definite matrix is bounded from above by the product of its diagonal entries, we calculate n 2 log ��Σ−1�� − n 2 tr � ˆΣdiagΣ−1� ≤ n 2 d � i=1 � log Σ−1 ii − ˆΣsamp ii Σ−1 ii � ≤ n 2 d � i=1 � − log ˆΣsamp ii − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The left-hand side does not come within n 2 units of the upper bound unless 1 2 ≤ ˆΣsamp ii Σ−1 ii ≤ 2 for each 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Next observe that −n 2 tr � ˆΣoffΣ−1� − 1 4tr � Σ−1� Θ ◦ Σ−1�� = 1 4 � i̸=j �n2| ˆΣsamp ij |2 Θij − Θij ����Σ−1 ij + n ˆΣsamp ij Θij ���� 2� ≤ n2 4 � i̸=j | ˆΣsamp ij |2 Θij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' In this case, the left-hand side does not come within n 2 units of the upper bound, unless ����Σ−1 ij + n ˆΣsamp ij Θij ���� 2 ≤ 2n Θij LOCALIZED COVARIANCE ESTIMATION: A BAYESIAN JUSTIFICATION 19 for each i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Tying together the argument, a large log likelihood ℓ(Σ) ≥ n 2 d � i=1 � − log ˆΣsamp ii − 1 � + n2 4 � i̸=j | ˆΣsamp ij |2 Θij − n 2 implies the following conditions � � � � � � � 1 2 ≤ ˆΣsamp ii Σ−1 ii ≤ 2, 1 ≤ i ≤ d ����Σ−1 ij + n ˆΣsamp ij Θij ���� 2 ≤ 2n Θij , i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For a large enough penalization strength s, these conditions confine Σ to a compact set of positive definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Additionally, this set contains an isolated maximizer of ℓ, because one point inside the set, Σ = ˆΣdiag, achieves a higher log likelihood ℓ( ˆΣdiag) = n 2 d � i=1 � − log ˆΣsamp ii − 1 � than all the points outside the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Last, note that Σ−1 �→ n 2 log ��Σ−1�� − n 2 tr � ˆΣsampΣ−1� − 1 4tr � Σ−1� Θ ◦ Σ−1�� is concave, whereby any isolated maximum is the unique global maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We have studied the problem of estimating a high-dimensional covari- ance matrix from a small number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' This problem is difficult but also ubiquitous in the setting of numerical weather prediction (NWP) when merging high-dimensional models with real-world observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We have developed a new mathematical theory to help jus- tify the covariance estimators which are used in NWP practice, but which have little to no mathematical justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' NWP practitioners boost the accuracy of covariance estimates by enforcing the assumption that correlations decay with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We have argued that this practical approach follows the Bayesian paradigm of estimating an unknown (covariance matrix) from data (the samples we have) and prior information (spatial decay of correlations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We have then investigated prior distributions that lead to practically useful covariance estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Put differently, we have interpreted common estimation procedures within a Bayesian framework and identified how particular choices of prior distributions influence posterior estimates of covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The situation is clear in the case of hybrid estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' These estimators can be understood as Bayesian estimators with an inverse Wishart prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The Bayesian theory shows that hybrid estimators converge at an asymptotically optimal rate and also reveals how to adjust the hybrid estimator when the ensemble size changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' The Bayesian interpretation of Schur product estimators is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' We have shown that Schur product estimators are not Bayesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Nonetheless, we have proposed a new “quadratically constrained” distribution that leads to a Schur product estimator in the 20 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' WEBBER AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' MORZFELD limit of increasing localization strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Building on this interpretation, the Bayesian theory suggests how best to adjust the length scale in a Schur product estimator as the ensemble size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' This scaling is new and relevant for practical NWP because it can reduce the amount of tuning required for efficient covariance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Perhaps more importantly, our analysis suggests that covariance estimation via Schur products may not be the most economical approach for covariance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' For one, it is not Bayesian and therefore it has not been rigorously justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Additionally, our theory, in line with previous work [21], suggests that we can estimate covariance matrices more efficiently by working with the inverse of the covariance matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=', by penalizing conditional correlations, rather than correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQf-wsu/content/2301.04828v1.pdf'} +page_content=' 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a/w9FJT4oBgHgl3EQfgiwX/content/tmp_files/2301.11561v1.pdf.txt b/w9FJT4oBgHgl3EQfgiwX/content/tmp_files/2301.11561v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..591c130d5e5a4b115b833a1e203cade836913537 --- /dev/null +++ b/w9FJT4oBgHgl3EQfgiwX/content/tmp_files/2301.11561v1.pdf.txt @@ -0,0 +1,1152 @@ +Control Scheme of Polarization Circulation Speed Meter Using a Dual-Retardation +Waveplate +Yohei Nishino,∗ Tomotada Akutsu, Yoichi Aso, and Takayuki Tomaru +National Astronomical Observatory of Japan (NAOJ), Mitaka City, Tokyo 181-8588, Japan +(Dated: January 30, 2023) +In interferometric gravitational wave detectors, quantum radiation pressure noise, which is a back +action of the measurement, will limit their sensitivities at low frequencies. Speed meters are one of +the solutions to reduce the back action noise and improve the sensitivities, and furthermore, they can +surpass the standard quantum limit over a wide range of frequencies. The Polarization Circulation +Speed Meter is the latest incarnation of the speed meter concept in the sense that it requires a +slight modification in the conventional interferometer designs; however, its control scheme has not +been developed. The main difficulty is the length and alignment control of the cavity formed by the +polarization circulation mirror and the input test masses, whose round-trip phase shift should be +kept to π. In this article, we propose a new control scheme using a dual-retardation waveplate, called +Dual-Retardance Control (DRC). In addition, we compare the shot noise level of the DRC to another +simpler scheme by dithering. Finally, we design the experimental setup for the demonstration of +the DRC and show the expected results through the transfer function measurement. +I. +INTRODUCTION +The sensitivity of the gravitational wave (GW) detec- +tors will be fundamentally limited by quantum noise. +Especially at low frequencies, due to the low-noise con- +trol of the suspensions, improvement of thermal noises, +and installation of high-power lasers, it will be lim- +ited by quantum radiation pressure noise. +This low- +frequency-limiting noise provides the standard quantum +limit (SQL), which is one of the consequences of Heisen- +berg’s uncertainty relation [1]. The SQL is a fundamental +limit that we cannot overcome by conventional methods, +and many techniques to beat it, so-called quantum non- +demolition (QND) measurement, have been studied [1, 2]. +Speed meters are one of the QND measurements. The +concept was first proposed by Braginsky and Khalili [3], +and many practical implementations have been investi- +gated [4–13]. The amplitude fluctuations of vacuum fields +entering from the anti-symmetric (AS) port of the inter- +ferometer are coupled with the pump laser and kick the +mirrors randomly, which turns out to be the quantum ra- +diation pressure noise [2]. In speed meters, the vacuum +field interacts with the mirror twice with opposite signs. +Taking into account the sloshing time τ, the back-action +force applied on the mirror is [14]: +ˆFb.a.(Ω) ≃ −iΩτ 2ˆIc(Ω) +c +(1) +at low frequencies Ω ≪ 1/τ. +ˆIc is the fluctuation part +of the circulating laser power via the coupling between +vacuum fluctuation and the pump laser and c is the speed +of light. The signal is proportional to the velocity at low +∗ yohei.nishino@grad.nao.ac.jp; Also at Department of Astronomy, +University of Tokyo, Bunkyo, Tokyo 113-0033, Japan +frequencies +φ(t) ∝ ˆx(t + τ) − ˆx(t) +(2) +∼ τ ¯v. +(3) +Note that the velocity measurement reduces the amount +of signals, but in terms of the signal-to-noise ratio, it +shows better properties than position measurements at +low frequencies. +The advantage of speed meters is a broadband-sensitivity +improvement at low frequencies. +Also by combination +with a balanced homodyne detection (BHD), it can go +beyond the free mass SQL. It is worth noting that it does +not need frequency-dependent homodyne angles, which +means we do not need additional filter cavities [2]. Every +noise-reduction process happens inside the interferome- +ter, so it is more robust to losses [15]. +The Polarization Circulation Speed Meter (PCSM) is a +new speed meter design proposed by Danilishin et al [11] +(see Fig. 1). The PCSM only needs a small modifica- +tion in the AS port; there is no need to touch the cen- +tral interferometer. In the current situation that all the +current large-scale GW detectors are based on the Dual- +Recycled Fabry-P´erot Michelson Interferometer (DRF- +PMI) and the fact that it will not be largely changed, +this design is the most possible candidate for the practical +implementation of speed meters, but the control scheme +has not been investigated. +There are two main issues to be solved to achieve PCSM. +The first issue is that DC components of differential sig- +nals will be reduced to zero in an ideal speed meter, +making it hard to control the interferometer. +This is +a generic problem in speed meters since they reduce the +back action noise in return for deliberately decreasing +signals. To obtain enough DC signals one needs to add +some loss that deteriorates its performance from the ideal +case (detailed analysis has been done in ref. [16] in the +case of the Sagnac-type speed meter). Speed control, in +the sense of control at low frequencies, has already been +demonstrated in the proof-of-principal experiment of the +arXiv:2301.11561v1 [gr-qc] 27 Jan 2023 + +2 +Sagnac speed meter in Glasgow [17]. +The second issue inheres in PCSM, for one needs to keep +the round-trip phase shift from the input test masses +(ITMs) and the polarization circulation mirror (PCM) +to π to flip a sign of the second interaction. In this pa- +per, we focus on this issue and propose a new scheme +to control the phase shift using a dual-retardation wave- +plate and an auxiliary laser. These components enable +us to obtain a Pound-Drever-Hall (PDH) signal [18] of +the cavity length formed by the PCM and ITMs (Polar- +ization Circulation Cavity: PCC), which allows us to use +techniques that are commonly used in the GW detectors +such as wave-front sensing and gives us high stability of +the PCC length/alignment control. +Also, this scheme +will get along well with the BHD, which could be a fu- +ture update in the GW detectors. +The outline of this paper is as follows: in Section II we +show the details of the PCSM and the difficulties of its +control. In Section III, we propose a new control scheme, +DRC. and in Section IV we characterize the DRC and +compare the shot noise level of the control signals be- +tween the DRC and another candidate, the dithering +control. In Section V, we show the layout for the exper- +imental demonstration of the DRC, and finally, we give +discussions in Section VI and conclusions in Section VII. +II. +BACKGROUND AND ISSUES +In this section, we review the mechanism of the +PCSM, whose detailed study is shown in Danilishin et +al., 2018 [11], and show the inherent difficulties in the +PCC control. +A. +PCSM +The conceptual design of the PCSM is shown in Fig. 1. +The main interferometer is the same as the conventional +position meter, but the AS port has two polarization +components, the quarter-wave plate (QWP) and the po- +larization beam splitter (PBS). We call a set of compo- +nents, the QWP, PBS, and PCM the polarization cir- +culator (PC), and call the cavity formed by the PCM +and the Y-arm input test mass (ITMY) PCC. The lin- +ear p-polarized (p-pol) vacuum fluctuation that sneaks +into the interferometer is converted into the circular left- +polarization (l-pol) by the QWP (denoted by ˆal), couples +with the pump laser, kicks the mirror randomly, then re- +turns to the AS port (denoted by ˆbl) and is converted into +s-polarization (s-pol). The PBS will reflect all the s-pol +light and it is reflected by the PCM, then goes back to the +interferometer again as r-pol (denoted by ˆar). The right- +polarization (r-pol) kicks the mirror again, comes back +to the AS port (denoted by ˆbr), and finally goes through +the PBS. The round-trip phase shift between the ITMs +and PCM is kept to π, so the radiation pressure forces +given by ˆar and ˆal have opposite signs and cancel each +other. +FIG. 1. +Configuration of the PCSM [11]. The QWP +converts the polarization state of the vacuum so that it expe- +riences the interferometer twice. The PC is a set of the QWP, +PBS, and PCM, and the PCC is a cavity formed by the PCM +and the ITMY with the QWP and PBS inside. +(E)ITMs +stand for (end) input test masses. +B. +Difficulties in PCC control +In general, especially in the context of GW detectors, +the distance between two mirrors can be stabilized by the +PDH method [18]. This method is powerful because one +can stabilize the length of a cavity made by two high- +reflectivity mirrors on a nanometer scale. +All second- +generation GW detectors make full use of this technique +to control many degrees of freedom, including the sig- +nal recycling cavity (SRC). In the control of SRC in +the resonant sideband extraction configuration, a radio- +frequency (RF) sideband generated by an electro-optic +modulator (EOM) is used to sense the length fluctuation +of the SRC. One might think control of PCC should be +done by the same method with an additional RF side- +band. However, PCC cannot be locked in the same man- +ner, because the IR beam can circulate inside the PCC +twice at most due to the QWP and PBS. It means the +finesse of the PCC is ∼ 0. This is a serious problem since +one cannot effectively amplify signal sidebands. In short, +one cannot use the PDH method. +Hence, the PCSM needs a new control scheme for the +PCC. One simple solution is modulating the PCM me- +chanically to generate sidebands and demodulating the +output from the AS port. It is what we call ’dithering’ + +ETMY +r-pol +I-pol +p-pol +lod-s +ITMY +45°linear-pol +BS +Laser +Polarization +ai +ari!b, +b. +ITMX +ETMX +Circulation +Cavity +PCM +QWP +PBS +Polarization +Circulator +H3 +FIG. 2. +Conceptual illustration of the dithering control. (a) A local oscillator is connected to a PZT behind the PCM +and applies modulation. It generates local oscillator sidebands around the carrier. (b) Electrical fields in the same direction in +the phaser diagram interfere with each other. After demodulation, one can get an error signal of the PCC length. +(see Fig. 2). Detuning the arm cavities and leak some +amount of the DC light (DC offset, see ref. [19]), the DC +value of the output is zero if the round-trip phase shift in +the PCC is kept to π, but we will see non-zero DC signals +if it is shifted. Taking a beat between the sidebands and +the DC offset, one can obtain an error signal. +This method is simple but has several problems. In the +first place, mechanical modulation onto PCM will add +noises to the signal sideband that carries GW signals, +compared to electrical modulation. Secondly, one cannot +expect a high signal-to-noise ratio (SNR) in the error sig- +nal. The amount of light reaching the AS port is limited, +so DC offsets will be needed to increase the SNR. How- +ever, in future GW detectors, one might not need DC +offsets thanks to balanced homodyne detection (BHD), +which is also critical for speed meters. To make full use +of this advantage, a scheme without DC offsets is prefer- +able. Lastly, it is not sensible to alignment fluctuations. +Wave-front sensing (WFS) performed in all large-scale +interferometers is an extension of the PDH method, so +without forming a cavity it is difficult to guarantee that +the reflected s-pol is completely going back to the arm. +For these reasons alternative schemes are necessary. +III. +DUAL-RETARDANCE CONTROL +A. +Idea +The problems described above can be solved by a dual- +retardance waveplate that works as a HWP for a green +(GR) laser. The main obstacle is that the QWP changes +the polarization so that the PBS can transmit half of the +IR light. If the QWP does not change the polarization +state in one-way transmission or keep the state so that +the PBS does not discard any light, one can form a cavity +with the PCM and ITMs. +Primarily, the retardance of waveplates is described as: +φret = 2π (ns − nf)d +λ0 +, +(4) +where ns and nf are refractive indices along the slow and +fast axes respectively, d is the thickness of the waveplate +and λ0 is the wavelength of the light. +In this simple +assumption, when it works as a QWP at the wavelength +of λ0, it should work as a HWP at the wavelength of +λ0/2. +A HWP will not change the polarization state +by round-trip transmission (see Fig. 3), so if we inject a +half-wavelength beam from PCM with s-polarization, it +can resonate inside the PCC. Practically, the refractive +index has a wavelength dependence, so it is critical to +manufacturing a dual-retardation waveplate with small +retardation errors. Henceforth, we call this scheme the +dual-retardance control (DRC). +The DRC will solve the issues enumerated in the previous +section. DRC makes it possible to use the full advantage +of the PDH method, for one can assume a high SNR +without mechanical modulation like dithering and can +perform WFS. In the case of the current large-scale GW +detectors, a half wavelength light of 1064 nm is 532 nm +generated by second harmonic generation (SHG), which +has been employed for lock acquisition [20]. +B. +Lock acquisition +To acquire a stable lock of the PCC length, one should +follow some steps, which are called ’lock acquisition’. As +a preparation, the GR frequency (ωGR) should be phase- +locked with respect to the main IR frequency (ω0), which +means: +ωGR = 2(ω0 + ωoff) +(5) +where ωoff is a tunable offset to the GR frequency. Also, +the IR and GR beam paths have to overlap, so one will +prepare an additional cavity outside the PCC to make + +PZT +I +(a) +(b) +A +Phase +PCM +Sidebands +Carrier +I Po +Interfere +QWP +ri,ti +2 +Amp +Output +PBS +BS +ITMY +ri,ti +ITMX +Output +I4 +FIG. 3. +Conceptual illustration of the DRC. We pre- +pare a waveplate that works as a QWP for the carrier and +HWP for GR. It will change the polarization from s to p or p +to s, but it can be kept to only s between the PCM and the +HWP. +them share the same beam path. The arm cavity can also +be used for the path-sharing process. The transmissivity +of the BS for GR is set to ∼ 1 for simplicity. +Even though the paths seem to completely overlap, the +optical path length of the PCC for the IR (lIR +PCC) will not +exactly be the same as that for the GR (lGR +PCC) due to the +dispersion of refractive indices of materials: +lGR +PCC = lIR +PCC + δlPCC, +(6) +where δlPCC the difference of the optical path lengths. +Adding a frequency offset ωoff, the round-trip phase shift +in the PCC for the GR is: +φGR = 2ωGRlGR +PCC/c +(7) += 2 [2(ω0 + ωoff) + δω] (lIR +PCC + δlPCC)/c +(8) += 4ω0lIR +PCC +c ++ 4ω0δlPCC +c ++ 4ωofflGR +PCC +c ++ 2δωlGR +PCC +c +(9) +The first term is a phase shift in case of no dispersion. +The second term is a shift due to the dispersion, and +the third term is a phase compensation by the frequency +offset. +The fourth term is a phase noise in the Phase +Locked Loop (PLL), which results in the average PCC +fluctuation (see ϵ in Eq. (7) in ref. [11]). +The conceptual figure of the lock acquisition is shown in +Fig. 4. First, using the dithering method, the optimal +position of the PCM, which makes the round-trip phase +shift of the IR φIR, is determined to π (see the denotation +(i) in Fig. 4). This corresponds to the first term in Eq. (9) +satisfies the condition: +2φIR = 4ω0lIR +PCC +c +≡ 0 (mod 2π). +(10) +One needs to detune the arm cavity adding DC offset to +obtain enough DC signals if necessary. The IR error sig- +nal is fed-back to the mechanical actuator on the PCM +(a PZT, for example). +FIG. 4. +Toy picture of the lock acquisition. +a) The +DC output of the PBS transmission. b) The IR error signal +by dithering. c) The solid red line is the IR error signal by +dithering and the solid green line is the GR PDH signal. After +adding offsets, one can hand over the error signals to the GR +PDH which is steeper than the IR error signal. +Second, the GR resonates inside the PCC by adding the +offset frequency (see the denotation (ii) in Fig. 4), corre- +sponding to the round trip phase shift for the GR in the +PCC satisfies below: +φGR ≡ 0 (mod 2π). +(11) +The GR PDH signal is fed-back to the frequency actuator +on the GR (an acousto-optic modulator, for example). +Lastly, given the absolute frequency of the main IR (ω0) +and the optical path difference (δlPCC) is stable enough, +the round-trip phase fluctuations by the PCM motion are +proportional to the length fluctuation of the PCC for the +IR (φIR) and GR (φGR). Therefore, lastly, one can hand +over the error signals to the GR PDH which is steeper +than the IR error signal (see the denotation (iii) in Fig. 4). +In this final stage, the GR PDH is fed-back to the PCM. +Note that the last term in Eq. (9): +δφPCC = 2δωlGR +PCC +c +(12) +will contribute as a noise of the PCC length. After the +handover, the dithering and DC offset can be lifted. +IV. +CHARACTERIZATION +A. +Error signal +In this section, we analyze the electric fields of a cavity +with an HWP and PBS inside to derive the error signal. + +Phase +locked +Main IR +S-pol +PCM +HWP +for GR +PBS +BS +TGR +1) +ITMYa) IR output +i)Find and keep the best +positionofthePcM bydithering +wo +--- +[Hz] +b)GRintracavitypower +ii) Add offset +2wo +2wo+2woff +c) Error signals +ili);Hand over +GR PDH +IR dither5 +FIG. 5. +GR field amplitudes inside and outside the +PCC. The properties of all components are given for the +GR (super-scripted by ”GR”). The HWP is represented in +the Jones matrix, ˆJGR. The reflectivity of the PBS is also +represented in the reflectivity matrix, ˆρGR. We assume the +BS is transparent for the GR for simplicity +We define bases of p- and s-pol as: +ep = +� +1 +0 +� +, es = +� +0 +1 +� +. +(13) +Symbols used in the analysis are shown in Fig. 5. The +reflectivity matrix of the PBS is: +ˆρGR = +�� +RGR +p +0 +0 +� +RGR +s +� +(14) +where RGR +s +, RGR +p +is the power reflectivity of s-pol and p- +pol of the PBS. rGR +0 +, tGR +0 +are the amplitude reflectivity +and transmissivity of the PCM, and Φ is the round-trip +phase shift in the PCC. The Jones matrix for the 45◦ +rotated HWP can be written as: +ˆJGR = 1 +2 +� +1 + e−2iδφ 1 − e−2iδφ +1 − e−2iδφ 1 + e−2iδφ +� +, +(15) +where δφ is the retardation error. +The boundary conditions are written as: +E1 = tGR +0 +E0 + rGR +0 +E3, +(16) +E2 = eiΦ/2 ˆJGRˆρGRE1, +(17) +E3 = eiΦ/2ˆρGR ˆJGRE2, +(18) +Er = −rGR +0 +E0 + tGR +0 +E3. +(19) +Solving those equations, the reflectivity from the s-pol +input to s-pol reflection is: +rs→s(Φ +′) = E0,s +Er,s +(20) += −rGR +0 ++ (tGR +0 +)2(RGR +s +cos δφeiΦ +′ +− rGR +0 +RGR +p +RGR +s +e2iΦ +′ +) +det M +(21) +where +Φ +′ = Φ − δφ, +(22) +and +det M = 1 − rGR +0 +(RGR +s ++ RGR +p +) cos δφeiΦ +′ ++ (rGR +0 +)2RGR +s +RGR +p +e2iΦ +′ +. +(23) +Here we assumed the reflectivity of the ITMY and the +transmissivity of the BS are ∼ 1 for simplicity. The losses +in the BS and PBS are also imposed on the loss of the +PCM (denoted LGR in Fig. 6). We set RGR +p +to 0, which +means p-pol generated by the retardation error will be +discarded from the PBS, and we impose the imperfection +of the s−pol reflectivity: +LGR +s += 1 − RGR +s +(24) +on the PCM loss. We show the imaginary part of the +reflectivity in Fig. 6 with various round trip losses, which +deteriorate the slope of the error signals. +FIG. 6. +DRC error signals of the PCC. Red lines show +error signals with various round-trip losses with retardation +error of λ0/300. The black line is an error signal without any +retardation errors and losses. +B. +Estimation of shot noise level +We compare shot noise levels of two methods, the +dithering control and DRC. The detailed analysis is +shown in Appendix A. Using Eq. (A4), (A7), (A8), (A12) +and (A13), and choosing realistic parameters (see Ta- +ble I), the ratio of each shot noise level becomes: +SDither +L +SDRC +L +∼ 4 × 104. +(25) +This reflects an advantage of forming a cavity, that is, +the phase amplification by a factor of the finesse of the +cavity and the amount of the local oscillator power that +can be used for control. + +Eo +.GR +GR +E3 +BS +E2 +OGR +GR +DGR1.00 +No errors +Imaginary part of reflectivity +0.75 +5A = Ao/300, CGR = 0.1% +5A = Ao/300, CsR = 0.5% +0.50 +5A = Aa/300, CGR = 1% +A = Ao/300, CGR = 3% +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +0.10 +0.05 +0.00 +0.05 +0.10 +[pe] x6 +FIG. 7. +Setup of the experimental demonstration of the DRC. A basic configuration is FPMI with 15 cm arm cavities. +The GR is phase-locked with the main IR and injected from the AR side of the PCM. +V. +EXPERIMENTAL DEMONSTRATION OF +DRC +The experimental setup to demonstrate the DRC is +shown in Fig. 7 and parameters for IR and GR are shown +in Table I and II, respectively. We aim to see a ∝ f struc- +ture in a transfer function and check the DRC works. The +GR is generated by SHG and phase-locked with the main +IR laser. +The basic design is FPMI with 15 cm rigid +arm cavities with the flat ITMs and the curved ETMs +(R = 1.5m). The radius of curvature of the PCM is 1 m. +The basic control scheme is the pre-modulation method +performed in all the current GW detectors. The error +signal obtained by the GR PDH will be fed back to the +PCM. The GR frequency is also tunable by the frequency +offset in the PLL. +A small fraction of the main IR will be picked off after +the EOM and injected from the AR side of the ITMY. +This light gets phase-modulated through an EOM and +the generated sidebands play a role as pseudo-GW sig- +nals. +The expected transfer function from the phase modula- +tion to the DARM output is shown in Fig 8. Given the +carrier is resonant in the arm cavity, the amplitude re- +flectivity of the arm cavity can be written as: +r(Ω) ≃ γ1 − γ2 + iΩ +γ1 + γ2 − iΩ, +(26) +where +γ1 ≡ cTITM +4larm +(cavity pole), +(27) +γ2 ≡ cLarm +4larm +. +(28) +TITM is the power transmissivity of the input mirror and +Larm is the round-trip loss of the arm cavity and larm is +the length of the arm cavity. Assuming the round-trip +power loss in the PCC as LPCC, the output is propor- +tional to: +(Output) ∝ 1 − (1 − LPCC)r(Ω) +2 +≃ γ2 + LPCCγ1/2 − iΩ +γ1 − iΩ +. +(29) +This means losses will generate a zero at: +γcut = γ2 + LPCCγ1 +2 +(30) += +c +4larm +� +Larm + πLPCC +F +� +, +(31) +where F is the finesse of the arm cavity: +F ≡ +2π +TITM +. +(32) +The total PCC losses can be written as: +LPCC = 2(LBS + LQWP + LPBS + TSPBS + RPPBS) ++ LPCM + Lalign + Lmis + (δφPCC)2 +2 +. +(33) + +Phase-modulation +iniection +Invacuum +Ppick +Curved ETM +CARM,MICH control +(R=1.5m) +PLL +Flat ITM +Main laser +EOM +BS +L= 15 cm +Faraday HWP +Phase +Isolator +locked +Curved PCM +QWP/HWP +(R = 1.0 m) +PBS +Auxiliary laser +PCCL control +DARM control7 +Note that losses in the PCC train are doubled due to +the round-trip effect, except for the PCM. The mode- +mismatching due to the PCM misalignment and the +Schnupp asymmetry is also counted as a loss. The final +term δφPCC is the length fluctuation of the PCC, which +contributes as a form of cosine: +cos(δφPCC) ≃ 1 − (δφPCC)2 +2 +. +(34) +Definitions of each term are shown in Table III. +In Fig. 8, we show both lossless and loss-included cases. +The cutoffs at low frequencies are generated by the losses. +The ∝ 1/f structure above the cavity pole is due to the +first-order low-pass nature of the arm cavity. Note that +even in the lossless case (gray line), we still see a cutoff +but it is caused by the transmission of the ETM that is +necessary to inject the artificial phase-modulated light. +FIG. 8. +Simulated transfer function by the DARM +noise injection. Red lines show the transfer function with +losses and gray lines show that without losses. There is a flat +structure at low frequencies even in the lossless case. This +is due to the transmissivity of the ETMY to inject phase- +modulated light from behind. +VI. +DISCUSSION +One of the potential issues is the long-term stability +of the dispersion of the QWP. It might change due to +the heat effect of the laser or environmental temperature +fluctuation. Also, beam jitters might also be a source of +the noise. It should be tested how long the stability of +the PCC control is and how often we need to dither to +check if the condition of the interferometer is in the speed +meter or not. +From the perspective of practical implementation, the +DRC might conflict with the lock acquisition scheme of +the ongoing detectors. KAGRA, for example, injects the +auxiliary GR laser from the center part of the interfer- +ometer. To avoid the GR leaking and resonating inside +the arm cavity, the DRC sets the ITM transmissivity for +the GR as small as possible. Hence it is necessary to find +a compromise between them. +VII. +CONCLUSION +In this article, we propose a feasible control scheme for +the PCSM using a dual-retardation waveplate and auxil- +iary laser. We name it the DRC, which makes it possible +to control the PCC length and alignment in the same way +as other degrees of freedom of the GW detectors using the +PDH methods. In the DRC, we can get error signals with +a higher SNR than the dithering control. Also, the DRC +is compatible with the BHD because we do not need the +DC offset anymore after the full PCC lock is acquired. +After the experimental demonstration of the DRC with +rigid cavities, we will proceed to the fully-suspended sys- +tems to realize the PCSM in the future GW detectors +such as the Einstein Telescope [21]. +ACKNOWLEDGMENTS +We thank Stefan Danilishin, Marc Eisenmann, Ken- +taro Komori, and Kentaro Somiya for fruitful discussions. +Appendix A: Shot noise estimation +In the case of the PDH method, the output before +demodulation can be written as [22]: +P = PDC + Dδl sin ωmt + (2ωm), +(A1) +where δl is the length fluctuation of the cavity we control +and ωm is the sideband frequency. D corresponds to the +TABLE I. +Parameters for IR used for the design of +the experiment. +Parameters +value +Note +λ0[nm] +1064 +Nd:YAG +P0 [mW] +50 +IR laser intensity +Ppick [µW] +125 +Pick-off laser intensityb +TITM +0.004 +ITM transmissivitya +TETM +30 ppm +ETM transmissivity +TPCM +< 0.01% +PCM transmissivity +RITM [m] +∞ +Radius of curvature +RETM [m] +1.5 +- +RPCM [m] +1 +- +larm [m] +0.15 +Arm cavity length +lmichx [m] +0.075 +Michelson length of the X-arm +lmichy [m] +0.125 +Michelson length of the Y-arm +lSch [m] +0.050 +Schnupp asymmetry +lPCC [m] +0.307 +Mean PCC length +fm [MHz] +47.5 +RF frequency +A [nm] +0.1 +Modulation amplitude +∆LDC [pm] +10 +DC offset +F +∼ 1500 +Finesse +fc +3.2 × 105 [Hz] +Cavity pole +fcut +1.7 × 104 [Hz] +Cutoff frequency +a Fused Silica substrate +b for phase-noise injection + +IPI [W/rad] +10-7 +10-8 +10-1 +101 +103 +105 +107 + [degree] +100 +lossless +lossy +0 +Phase +-100 +10-1 +101 +103 +105 +107 +Frequency [Hz]8 +TABLE II. Parameters for GR. +Parameters +value +Note +λGR [nm] +532 +SHG by 1064 nm +PGR [mW] +20 +GR laser intensity +TPCM +0.01 +PCM transmissivity +TITM [m] +< 10 ppm +ITM transmissivity +lPCC [m] +0.332 +Length from the PCM to ITMY +βm +0.2 +Modulation index +δφGR +ret +2πλGR/300 +QWP retardation error for GR +LGR +3 % +Total losses in the PCC +FGR +150 +Finesse +slope of the error signal, which is proportional to the +carrier and sideband power Pc, Ps and the imaginary part +of the reflectivity Im[r(ω)]: +D ∝ +� +PcPsIm[r(ω)]. +(A2) +PDC is the DC power, which turns out to be the source +of the shot noise in the single-sided spectrum: +Sshot = +� +2eηPDC [A/ +√ +Hz], +(A3) +where e is the elementary charge and η is the quantum +efficiency of the photo detector [A/W]. Hence the shot- +noise-equivalent length noise is: +SL = Sshot +D . +(A4) +In the case of the dithering control, the carrier power Pc +depends on the amount of ∆LDC: +Pc = +�2ω0t2 +1r2∆LDC +c(1 − r1r2) +�2 +P0, +(A5) +where r1 and t1 are the ITM reflectivity and transmis- +sivity, r2 is the ETM reflectivity for the IR, ∆LDC is the +amount of DC offset and P0 is the power at the beam +splitter. The slope amplitude is: +DDitherδlPCC ∼ 2J1(β)PcIm[1 − e−iδφPCC] +(A6) += 16π2A +λ2 +0 +PcδlPCC, +(A7) +where A is the amplitude of the PCM modulation, λ0 is +the wavelength of the main laser, Jn is the n-th order +Bessel functions and β is the modulation index. For the +transformation from Eq. (A6) to Eq. (A7) we have used +J1(β) = β +2 = 2πA +λ0 +, +δφPCC = 4πδlPCC +λ0 +. +P Dither +DC +can be written as: +P Dither +DC += |1 − F(ψ0)|2Pc, +(A8) +where F is the arm cavity reflectivity: +F(ψ) = −r1 + +t2 +1r1e−iψ +1 − r1r2e−iψ +(A9) +and ψ0 is the round-trip phase shift of the arm cavity for +the IR. In the case of the DRC, the imaginary part of the +reflectivity rs→s behaves around Φ +′ as: +Im[rs→s(Φ +′)] +��� +Φ′=0 ≃ dIm[rs→s(Φ′)] +dΦ′ +���� +Φ′=0 +× δΦ +′ +(A10) += dIm[rs→s(Φ′)] +dΦ′ +���� +Φ′=0 +× 4ω0δlPCC +c +. +(A11) +The slope amplitude can be written as: +DDRCδlPCC = 4 +� +PcPs Im[rs→s(Φ +′)] +��� +Φ′=0 += 8βmω0PGR +c +dIm[rs→s(Φ′)] +dΦ′ +���� +Φ′=0 +δlPCC, +(A12) +where βm is the modulation index of the EOM, PGR is +the GR laser power. The DC power can be written as: +P DRC +DC += |rs→s(0)|2PGR. +(A13) +[1] V. Braginsky and F. Khalili, Quantum measurement +(Cambridge University Press, New York, 1992). +[2] H. J. Kimble, Y. Levin, A. B. Matsko, K. S. 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Ward, Laser phase +and frequency stabilization using an optical resonator, +Applied Physics B: Lasers and Optics 31, 97 (1983). +[19] P. Fritschel, +(2003), DC Readout for Advanced LIGO, +LSC meeting in Hannover, LIGO Document G030460-v1. +[20] T. Akutsu et al., An arm length stabilization system for +kagra and future gravitational-wave detectors, Classical +and Quantum Gravity 37, 035004 (2020). +[21] S. Hild, S. Chelkowski, A. Freise, J. Franc, N. Morgado, +R. Flaminio, and R. DeSalvo, A xylophone configuration +for a third-generation gravitational wave detector, Clas- +sical and Quantum Gravity 27, 015003 (2009). +[22] E. D. Black, An introduction to Pound-Drever-Hall laser +frequency stabilization, American Journal of Physics 69, +79 (2001). + diff --git a/w9FJT4oBgHgl3EQfgiwX/content/tmp_files/load_file.txt b/w9FJT4oBgHgl3EQfgiwX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f51bf226ce4f0ee5dbad842e7c42149fa43788f --- /dev/null +++ b/w9FJT4oBgHgl3EQfgiwX/content/tmp_files/load_file.txt @@ -0,0 +1,490 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf,len=489 +page_content='Control Scheme of Polarization Circulation Speed Meter Using a Dual-Retardation Waveplate Yohei Nishino,∗ Tomotada Akutsu, Yoichi Aso, and Takayuki Tomaru National Astronomical Observatory of Japan (NAOJ), Mitaka City, Tokyo 181-8588, Japan (Dated: January 30, 2023) In interferometric gravitational wave detectors, quantum radiation pressure noise, which is a back action of the measurement, will limit their sensitivities at low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Speed meters are one of the solutions to reduce the back action noise and improve the sensitivities, and furthermore, they can surpass the standard quantum limit over a wide range of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The Polarization Circulation Speed Meter is the latest incarnation of the speed meter concept in the sense that it requires a slight modification in the conventional interferometer designs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' however, its control scheme has not been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The main difficulty is the length and alignment control of the cavity formed by the polarization circulation mirror and the input test masses, whose round-trip phase shift should be kept to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In this article, we propose a new control scheme using a dual-retardation waveplate, called Dual-Retardance Control (DRC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In addition, we compare the shot noise level of the DRC to another simpler scheme by dithering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Finally, we design the experimental setup for the demonstration of the DRC and show the expected results through the transfer function measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' INTRODUCTION The sensitivity of the gravitational wave (GW) detec- tors will be fundamentally limited by quantum noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Especially at low frequencies, due to the low-noise con- trol of the suspensions, improvement of thermal noises, and installation of high-power lasers, it will be lim- ited by quantum radiation pressure noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' This low- frequency-limiting noise provides the standard quantum limit (SQL), which is one of the consequences of Heisen- berg’s uncertainty relation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The SQL is a fundamental limit that we cannot overcome by conventional methods, and many techniques to beat it, so-called quantum non- demolition (QND) measurement, have been studied [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Speed meters are one of the QND measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The concept was first proposed by Braginsky and Khalili [3], and many practical implementations have been investi- gated [4–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The amplitude fluctuations of vacuum fields entering from the anti-symmetric (AS) port of the inter- ferometer are coupled with the pump laser and kick the mirrors randomly, which turns out to be the quantum ra- diation pressure noise [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In speed meters, the vacuum field interacts with the mirror twice with opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Taking into account the sloshing time τ, the back-action force applied on the mirror is [14]: ˆFb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (Ω) ≃ −iΩτ 2ˆIc(Ω) c (1) at low frequencies Ω ≪ 1/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' ˆIc is the fluctuation part of the circulating laser power via the coupling between vacuum fluctuation and the pump laser and c is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The signal is proportional to the velocity at low ∗ yohei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='nishino@grad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='nao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='jp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Also at Department of Astronomy, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan frequencies φ(t) ∝ ˆx(t + τ) − ˆx(t) (2) ∼ τ ¯v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (3) Note that the velocity measurement reduces the amount of signals, but in terms of the signal-to-noise ratio, it shows better properties than position measurements at low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The advantage of speed meters is a broadband-sensitivity improvement at low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Also by combination with a balanced homodyne detection (BHD), it can go beyond the free mass SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' It is worth noting that it does not need frequency-dependent homodyne angles, which means we do not need additional filter cavities [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Every noise-reduction process happens inside the interferome- ter, so it is more robust to losses [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The Polarization Circulation Speed Meter (PCSM) is a new speed meter design proposed by Danilishin et al [11] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The PCSM only needs a small modifica- tion in the AS port;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' there is no need to touch the cen- tral interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In the current situation that all the current large-scale GW detectors are based on the Dual- Recycled Fabry-P´erot Michelson Interferometer (DRF- PMI) and the fact that it will not be largely changed, this design is the most possible candidate for the practical implementation of speed meters, but the control scheme has not been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' There are two main issues to be solved to achieve PCSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The first issue is that DC components of differential sig- nals will be reduced to zero in an ideal speed meter, making it hard to control the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' This is a generic problem in speed meters since they reduce the back action noise in return for deliberately decreasing signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' To obtain enough DC signals one needs to add some loss that deteriorates its performance from the ideal case (detailed analysis has been done in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' [16] in the case of the Sagnac-type speed meter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Speed control, in the sense of control at low frequencies, has already been demonstrated in the proof-of-principal experiment of the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='11561v1 [gr-qc] 27 Jan 2023 2 Sagnac speed meter in Glasgow [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The second issue inheres in PCSM, for one needs to keep the round-trip phase shift from the input test masses (ITMs) and the polarization circulation mirror (PCM) to π to flip a sign of the second interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In this pa- per, we focus on this issue and propose a new scheme to control the phase shift using a dual-retardation wave- plate and an auxiliary laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' These components enable us to obtain a Pound-Drever-Hall (PDH) signal [18] of the cavity length formed by the PCM and ITMs (Polar- ization Circulation Cavity: PCC), which allows us to use techniques that are commonly used in the GW detectors such as wave-front sensing and gives us high stability of the PCC length/alignment control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Also, this scheme will get along well with the BHD, which could be a fu- ture update in the GW detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The outline of this paper is as follows: in Section II we show the details of the PCSM and the difficulties of its control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In Section III, we propose a new control scheme, DRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' and in Section IV we characterize the DRC and compare the shot noise level of the control signals be- tween the DRC and another candidate, the dithering control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In Section V, we show the layout for the exper- imental demonstration of the DRC, and finally, we give discussions in Section VI and conclusions in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' BACKGROUND AND ISSUES In this section, we review the mechanism of the PCSM, whose detailed study is shown in Danilishin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=', 2018 [11], and show the inherent difficulties in the PCC control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' PCSM The conceptual design of the PCSM is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The main interferometer is the same as the conventional position meter, but the AS port has two polarization components, the quarter-wave plate (QWP) and the po- larization beam splitter (PBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' We call a set of compo- nents, the QWP, PBS, and PCM the polarization cir- culator (PC), and call the cavity formed by the PCM and the Y-arm input test mass (ITMY) PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The lin- ear p-polarized (p-pol) vacuum fluctuation that sneaks into the interferometer is converted into the circular left- polarization (l-pol) by the QWP (denoted by ˆal), couples with the pump laser, kicks the mirror randomly, then re- turns to the AS port (denoted by ˆbl) and is converted into s-polarization (s-pol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The PBS will reflect all the s-pol light and it is reflected by the PCM, then goes back to the interferometer again as r-pol (denoted by ˆar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The right- polarization (r-pol) kicks the mirror again, comes back to the AS port (denoted by ˆbr), and finally goes through the PBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The round-trip phase shift between the ITMs and PCM is kept to π, so the radiation pressure forces given by ˆar and ˆal have opposite signs and cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Configuration of the PCSM [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The QWP converts the polarization state of the vacuum so that it expe- riences the interferometer twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The PC is a set of the QWP, PBS, and PCM, and the PCC is a cavity formed by the PCM and the ITMY with the QWP and PBS inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (E)ITMs stand for (end) input test masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Difficulties in PCC control In general, especially in the context of GW detectors, the distance between two mirrors can be stabilized by the PDH method [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' This method is powerful because one can stabilize the length of a cavity made by two high- reflectivity mirrors on a nanometer scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' All second- generation GW detectors make full use of this technique to control many degrees of freedom, including the sig- nal recycling cavity (SRC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In the control of SRC in the resonant sideband extraction configuration, a radio- frequency (RF) sideband generated by an electro-optic modulator (EOM) is used to sense the length fluctuation of the SRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' One might think control of PCC should be done by the same method with an additional RF side- band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' However, PCC cannot be locked in the same man- ner, because the IR beam can circulate inside the PCC twice at most due to the QWP and PBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' It means the finesse of the PCC is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' This is a serious problem since one cannot effectively amplify signal sidebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In short, one cannot use the PDH method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Hence, the PCSM needs a new control scheme for the PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' One simple solution is modulating the PCM me- chanically to generate sidebands and demodulating the output from the AS port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' It is what we call ’dithering’ ETMY r-pol I-pol p-pol lod-s ITMY 45°linear-pol BS Laser Polarization ai ari!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='b, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' ITMX ETMX Circulation Cavity PCM QWP PBS Polarization Circulator H3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Conceptual illustration of the dithering control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (a) A local oscillator is connected to a PZT behind the PCM and applies modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' It generates local oscillator sidebands around the carrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (b) Electrical fields in the same direction in the phaser diagram interfere with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' After demodulation, one can get an error signal of the PCC length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Detuning the arm cavities and leak some amount of the DC light (DC offset, see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' [19]), the DC value of the output is zero if the round-trip phase shift in the PCC is kept to π, but we will see non-zero DC signals if it is shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Taking a beat between the sidebands and the DC offset, one can obtain an error signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' This method is simple but has several problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In the first place, mechanical modulation onto PCM will add noises to the signal sideband that carries GW signals, compared to electrical modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Secondly, one cannot expect a high signal-to-noise ratio (SNR) in the error sig- nal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The amount of light reaching the AS port is limited, so DC offsets will be needed to increase the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' How- ever, in future GW detectors, one might not need DC offsets thanks to balanced homodyne detection (BHD), which is also critical for speed meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' To make full use of this advantage, a scheme without DC offsets is prefer- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Lastly, it is not sensible to alignment fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Wave-front sensing (WFS) performed in all large-scale interferometers is an extension of the PDH method, so without forming a cavity it is difficult to guarantee that the reflected s-pol is completely going back to the arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' For these reasons alternative schemes are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' DUAL-RETARDANCE CONTROL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Idea The problems described above can be solved by a dual- retardance waveplate that works as a HWP for a green (GR) laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The main obstacle is that the QWP changes the polarization so that the PBS can transmit half of the IR light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' If the QWP does not change the polarization state in one-way transmission or keep the state so that the PBS does not discard any light, one can form a cavity with the PCM and ITMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Primarily, the retardance of waveplates is described as: φret = 2π (ns − nf)d λ0 , (4) where ns and nf are refractive indices along the slow and fast axes respectively, d is the thickness of the waveplate and λ0 is the wavelength of the light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In this simple assumption, when it works as a QWP at the wavelength of λ0, it should work as a HWP at the wavelength of λ0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' A HWP will not change the polarization state by round-trip transmission (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 3), so if we inject a half-wavelength beam from PCM with s-polarization, it can resonate inside the PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Practically, the refractive index has a wavelength dependence, so it is critical to manufacturing a dual-retardation waveplate with small retardation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Henceforth, we call this scheme the dual-retardance control (DRC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The DRC will solve the issues enumerated in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' DRC makes it possible to use the full advantage of the PDH method, for one can assume a high SNR without mechanical modulation like dithering and can perform WFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In the case of the current large-scale GW detectors, a half wavelength light of 1064 nm is 532 nm generated by second harmonic generation (SHG), which has been employed for lock acquisition [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Lock acquisition To acquire a stable lock of the PCC length, one should follow some steps, which are called ’lock acquisition’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' As a preparation, the GR frequency (ωGR) should be phase- locked with respect to the main IR frequency (ω0), which means: ωGR = 2(ω0 + ωoff) (5) where ωoff is a tunable offset to the GR frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Also, the IR and GR beam paths have to overlap, so one will prepare an additional cavity outside the PCC to make PZT I (a) (b) A Phase PCM Sidebands Carrier I Po Interfere QWP ri,ti 2 Amp Output PBS BS ITMY ri,ti ITMX Output I4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Conceptual illustration of the DRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' We pre- pare a waveplate that works as a QWP for the carrier and HWP for GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' It will change the polarization from s to p or p to s, but it can be kept to only s between the PCM and the HWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' them share the same beam path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The arm cavity can also be used for the path-sharing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The transmissivity of the BS for GR is set to ∼ 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Even though the paths seem to completely overlap, the optical path length of the PCC for the IR (lIR PCC) will not exactly be the same as that for the GR (lGR PCC) due to the dispersion of refractive indices of materials: lGR PCC = lIR PCC + δlPCC, (6) where δlPCC the difference of the optical path lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Adding a frequency offset ωoff, the round-trip phase shift in the PCC for the GR is: φGR = 2ωGRlGR PCC/c (7) = 2 [2(ω0 + ωoff) + δω] (lIR PCC + δlPCC)/c (8) = 4ω0lIR PCC c + 4ω0δlPCC c + 4ωofflGR PCC c + 2δωlGR PCC c (9) The first term is a phase shift in case of no dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The second term is a shift due to the dispersion, and the third term is a phase compensation by the frequency offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The fourth term is a phase noise in the Phase Locked Loop (PLL), which results in the average PCC fluctuation (see ϵ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (7) in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The conceptual figure of the lock acquisition is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' First, using the dithering method, the optimal position of the PCM, which makes the round-trip phase shift of the IR φIR, is determined to π (see the denotation (i) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' This corresponds to the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (9) satisfies the condition: 2φIR = 4ω0lIR PCC c ≡ 0 (mod 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (10) One needs to detune the arm cavity adding DC offset to obtain enough DC signals if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The IR error sig- nal is fed-back to the mechanical actuator on the PCM (a PZT, for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Toy picture of the lock acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' a) The DC output of the PBS transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' b) The IR error signal by dithering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' c) The solid red line is the IR error signal by dithering and the solid green line is the GR PDH signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' After adding offsets, one can hand over the error signals to the GR PDH which is steeper than the IR error signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Second, the GR resonates inside the PCC by adding the offset frequency (see the denotation (ii) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 4), corre- sponding to the round trip phase shift for the GR in the PCC satisfies below: φGR ≡ 0 (mod 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (11) The GR PDH signal is fed-back to the frequency actuator on the GR (an acousto-optic modulator, for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Lastly, given the absolute frequency of the main IR (ω0) and the optical path difference (δlPCC) is stable enough, the round-trip phase fluctuations by the PCM motion are proportional to the length fluctuation of the PCC for the IR (φIR) and GR (φGR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Therefore, lastly, one can hand over the error signals to the GR PDH which is steeper than the IR error signal (see the denotation (iii) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In this final stage, the GR PDH is fed-back to the PCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Note that the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (9): δφPCC = 2δωlGR PCC c (12) will contribute as a noise of the PCC length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' After the handover, the dithering and DC offset can be lifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' CHARACTERIZATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Error signal In this section, we analyze the electric fields of a cavity with an HWP and PBS inside to derive the error signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Phase locked Main IR S-pol PCM HWP for GR PBS BS TGR 1) ITMYa) IR output i)Find and keep the best positionofthePcM bydithering wo --- [Hz] b)GRintracavitypower ii) Add offset 2wo 2wo+2woff c) Error signals ili);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='Hand over GR PDH IR dither5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' GR field amplitudes inside and outside the PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The properties of all components are given for the GR (super-scripted by ”GR”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The HWP is represented in the Jones matrix, ˆJGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The reflectivity of the PBS is also represented in the reflectivity matrix, ˆρGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' We assume the BS is transparent for the GR for simplicity We define bases of p- and s-pol as: ep = � 1 0 � , es = � 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (13) Symbols used in the analysis are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The reflectivity matrix of the PBS is: ˆρGR = �� RGR p 0 0 � RGR s � (14) where RGR s , RGR p is the power reflectivity of s-pol and p- pol of the PBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' rGR 0 , tGR 0 are the amplitude reflectivity and transmissivity of the PCM, and Φ is the round-trip phase shift in the PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The Jones matrix for the 45◦ rotated HWP can be written as: ˆJGR = 1 2 � 1 + e−2iδφ 1 − e−2iδφ 1 − e−2iδφ 1 + e−2iδφ � , (15) where δφ is the retardation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The boundary conditions are written as: E1 = tGR 0 E0 + rGR 0 E3, (16) E2 = eiΦ/2 ˆJGRˆρGRE1, (17) E3 = eiΦ/2ˆρGR ˆJGRE2, (18) Er = −rGR 0 E0 + tGR 0 E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (19) Solving those equations, the reflectivity from the s-pol input to s-pol reflection is: rs→s(Φ ′) = E0,s Er,s (20) = −rGR 0 + (tGR 0 )2(RGR s cos δφeiΦ ′ − rGR 0 RGR p RGR s e2iΦ ′ ) det M (21) where Φ ′ = Φ − δφ, (22) and det M = 1 − rGR 0 (RGR s + RGR p ) cos δφeiΦ ′ + (rGR 0 )2RGR s RGR p e2iΦ ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (23) Here we assumed the reflectivity of the ITMY and the transmissivity of the BS are ∼ 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The losses in the BS and PBS are also imposed on the loss of the PCM (denoted LGR in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' We set RGR p to 0, which means p-pol generated by the retardation error will be discarded from the PBS, and we impose the imperfection of the s−pol reflectivity: LGR s = 1 − RGR s (24) on the PCM loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' We show the imaginary part of the reflectivity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 6 with various round trip losses, which deteriorate the slope of the error signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' DRC error signals of the PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Red lines show error signals with various round-trip losses with retardation error of λ0/300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The black line is an error signal without any retardation errors and losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Estimation of shot noise level We compare shot noise levels of two methods, the dithering control and DRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The detailed analysis is shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (A4), (A7), (A8), (A12) and (A13), and choosing realistic parameters (see Ta- ble I), the ratio of each shot noise level becomes: SDither L SDRC L ∼ 4 × 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (25) This reflects an advantage of forming a cavity, that is, the phase amplification by a factor of the finesse of the cavity and the amount of the local oscillator power that can be used for control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Eo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='GR +GR E3 BS E2 OGR GR DGR1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='00 No errors Imaginary part of reflectivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='75 5A = Ao/300, CGR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='1% 5A = Ao/300, CsR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='50 5A = Aa/300, CGR = 1% A = Ao/300, CGR = 3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='10 [pe] x6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Setup of the experimental demonstration of the DRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' A basic configuration is FPMI with 15 cm arm cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The GR is phase-locked with the main IR and injected from the AR side of the PCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' EXPERIMENTAL DEMONSTRATION OF DRC The experimental setup to demonstrate the DRC is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 7 and parameters for IR and GR are shown in Table I and II, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' We aim to see a ∝ f struc- ture in a transfer function and check the DRC works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The GR is generated by SHG and phase-locked with the main IR laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The basic design is FPMI with 15 cm rigid arm cavities with the flat ITMs and the curved ETMs (R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='5m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The radius of curvature of the PCM is 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The basic control scheme is the pre-modulation method performed in all the current GW detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The error signal obtained by the GR PDH will be fed back to the PCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The GR frequency is also tunable by the frequency offset in the PLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' A small fraction of the main IR will be picked off after the EOM and injected from the AR side of the ITMY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' This light gets phase-modulated through an EOM and the generated sidebands play a role as pseudo-GW sig- nals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The expected transfer function from the phase modula- tion to the DARM output is shown in Fig 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Given the carrier is resonant in the arm cavity, the amplitude re- flectivity of the arm cavity can be written as: r(Ω) ≃ γ1 − γ2 + iΩ γ1 + γ2 − iΩ, (26) where γ1 ≡ cTITM 4larm (cavity pole), (27) γ2 ≡ cLarm 4larm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (28) TITM is the power transmissivity of the input mirror and Larm is the round-trip loss of the arm cavity and larm is the length of the arm cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Assuming the round-trip power loss in the PCC as LPCC, the output is propor- tional to: (Output) ∝ 1 − (1 − LPCC)r(Ω) 2 ≃ γ2 + LPCCγ1/2 − iΩ γ1 − iΩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (29) This means losses will generate a zero at: γcut = γ2 + LPCCγ1 2 (30) = c 4larm � Larm + πLPCC F � , (31) where F is the finesse of the arm cavity: F ≡ 2π TITM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (32) The total PCC losses can be written as: LPCC = 2(LBS + LQWP + LPBS + TSPBS + RPPBS) + LPCM + Lalign + Lmis + (δφPCC)2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (33) Phase-modulation iniection Invacuum Ppick Curved ETM CARM,MICH control (R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='5m) PLL Flat ITM Main laser EOM BS L= 15 cm Faraday HWP Phase Isolator locked Curved PCM QWP/HWP (R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='0 m) PBS Auxiliary laser PCCL control DARM control7 Note that losses in the PCC train are doubled due to the round-trip effect, except for the PCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The mode- mismatching due to the PCM misalignment and the Schnupp asymmetry is also counted as a loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The final term δφPCC is the length fluctuation of the PCC, which contributes as a form of cosine: cos(δφPCC) ≃ 1 − (δφPCC)2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (34) Definitions of each term are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 8, we show both lossless and loss-included cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The cutoffs at low frequencies are generated by the losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The ∝ 1/f structure above the cavity pole is due to the first-order low-pass nature of the arm cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Note that even in the lossless case (gray line), we still see a cutoff but it is caused by the transmission of the ETM that is necessary to inject the artificial phase-modulated light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Simulated transfer function by the DARM noise injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Red lines show the transfer function with losses and gray lines show that without losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' There is a flat structure at low frequencies even in the lossless case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' This is due to the transmissivity of the ETMY to inject phase- modulated light from behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' DISCUSSION One of the potential issues is the long-term stability of the dispersion of the QWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' It might change due to the heat effect of the laser or environmental temperature fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Also, beam jitters might also be a source of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' It should be tested how long the stability of the PCC control is and how often we need to dither to check if the condition of the interferometer is in the speed meter or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' From the perspective of practical implementation, the DRC might conflict with the lock acquisition scheme of the ongoing detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' KAGRA, for example, injects the auxiliary GR laser from the center part of the interfer- ometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' To avoid the GR leaking and resonating inside the arm cavity, the DRC sets the ITM transmissivity for the GR as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Hence it is necessary to find a compromise between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' CONCLUSION In this article, we propose a feasible control scheme for the PCSM using a dual-retardation waveplate and auxil- iary laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' We name it the DRC, which makes it possible to control the PCC length and alignment in the same way as other degrees of freedom of the GW detectors using the PDH methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In the DRC, we can get error signals with a higher SNR than the dithering control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Also, the DRC is compatible with the BHD because we do not need the DC offset anymore after the full PCC lock is acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' After the experimental demonstration of the DRC with rigid cavities, we will proceed to the fully-suspended sys- tems to realize the PCSM in the future GW detectors such as the Einstein Telescope [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Stefan Danilishin, Marc Eisenmann, Ken- taro Komori, and Kentaro Somiya for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Appendix A: Shot noise estimation In the case of the PDH method, the output before demodulation can be written as [22]: P = PDC + Dδl sin ωmt + (2ωm), (A1) where δl is the length fluctuation of the cavity we control and ωm is the sideband frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' D corresponds to the TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Parameters for IR used for the design of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Parameters value Note λ0[nm] 1064 Nd:YAG P0 [mW] 50 IR laser intensity Ppick [µW] 125 Pick-off laser intensityb TITM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='004 ITM transmissivitya TETM 30 ppm ETM transmissivity TPCM < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='01% PCM transmissivity RITM [m] ∞ Radius of curvature RETM [m] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='5 RPCM [m] 1 larm [m] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='15 Arm cavity length lmichx [m] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='075 Michelson length of the X-arm lmichy [m] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='125 Michelson length of the Y-arm lSch [m] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='050 Schnupp asymmetry lPCC [m] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='307 Mean PCC length fm [MHz] 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='5 RF frequency A [nm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='1 Modulation amplitude ∆LDC [pm] 10 DC offset F ∼ 1500 Finesse fc 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='2 × 105 [Hz] Cavity pole fcut 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='7 × 104 [Hz] Cutoff frequency a Fused Silica substrate b for phase-noise injection IPI [W/rad] 10-7 10-8 10-1 101 103 105 107 [degree] 100 lossless lossy 0 Phase 100 10-1 101 103 105 107 Frequency [Hz]8 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Parameters for GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Parameters value Note λGR [nm] 532 SHG by 1064 nm PGR [mW] 20 GR laser intensity TPCM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='01 PCM transmissivity TITM [m] < 10 ppm ITM transmissivity lPCC [m] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='332 Length from the PCM to ITMY βm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='2 Modulation index δφGR ret 2πλGR/300 QWP retardation error for GR LGR 3 % Total losses in the PCC FGR 150 Finesse slope of the error signal, which is proportional to the carrier and sideband power Pc, Ps and the imaginary part of the reflectivity Im[r(ω)]: D ∝ � PcPsIm[r(ω)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (A2) PDC is the DC power, which turns out to be the source of the shot noise in the single-sided spectrum: Sshot = � 2eηPDC [A/ √ Hz], (A3) where e is the elementary charge and η is the quantum efficiency of the photo detector [A/W].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Hence the shot- noise-equivalent length noise is: SL = Sshot D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (A4) In the case of the dithering control, the carrier power Pc depends on the amount of ∆LDC: Pc = �2ω0t2 1r2∆LDC c(1 − r1r2) �2 P0, (A5) where r1 and t1 are the ITM reflectivity and transmis- sivity, r2 is the ETM reflectivity for the IR, ∆LDC is the amount of DC offset and P0 is the power at the beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The slope amplitude is: DDitherδlPCC ∼ 2J1(β)PcIm[1 − e−iδφPCC] (A6) = 16π2A λ2 0 PcδlPCC, (A7) where A is the amplitude of the PCM modulation, λ0 is the wavelength of the main laser, Jn is the n-th order Bessel functions and β is the modulation index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' For the transformation from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (A6) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (A7) we have used J1(β) = β 2 = 2πA λ0 , δφPCC = 4πδlPCC λ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' P Dither DC can be written as: P Dither DC = |1 − F(ψ0)|2Pc, (A8) where F is the arm cavity reflectivity: F(ψ) = −r1 + t2 1r1e−iψ 1 − r1r2e−iψ (A9) and ψ0 is the round-trip phase shift of the arm cavity for the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' In the case of the DRC, the imaginary part of the reflectivity rs→s behaves around Φ ′ as: Im[rs→s(Φ ′)] ��� Φ′=0 ≃ dIm[rs→s(Φ′)] dΦ′ ���� Φ′=0 × δΦ ′ (A10) = dIm[rs→s(Φ′)] dΦ′ ���� Φ′=0 × 4ω0δlPCC c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (A11) The slope amplitude can be written as: DDRCδlPCC = 4 � PcPs Im[rs→s(Φ ′)] ��� Φ′=0 = 8βmω0PGR c dIm[rs→s(Φ′)] dΦ′ ���� Φ′=0 δlPCC, (A12) where βm is the modulation index of the EOM, PGR is the GR laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' The DC power can be written as: P DRC DC = |rs→s(0)|2PGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' (A13) [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Braginsky and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Khalili, Quantum measurement (Cambridge University Press, New York, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Kimble, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Levin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Matsko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Thorne, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Vyatchanin, Conversion of conventional gravitational-wave interferometers into quantum nonde- molition interferometers by modifying their input and/or output optics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' D 65, 022002 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' [3] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Braginsky and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Khalili, Gravitational wave antenna with qnd speed meter, Physics Letters A 147, 251 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Braginsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Gorodetsky, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Khalili, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Thorne, Dual-resonator speed meter for a free test mass, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' D 61, 044002 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' [5] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Chen, Sagnac interferometer as a speed-meter- type, quantum-nondemolition gravitational-wave detec- tor, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' D 67, 122004 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' 9 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Losses and errors used for the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Parameters value Note Larm 130 ppm Arm cavity round-trip loss including ETM transmissivity, 30 ppm LBS,QWP,PBS,PCM 50 ppm PCC optics losses TSPBS 1 % PBS s-pol transmissivity RPPBS 1 % PBS p-pol reflectivity Lalign 1 % PCM misalignment Lmis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content='15 % Mode mismatching between X and Y arm δφPCC [rad] 10−7 PCC length fluctuation by the PLL noise [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Purdue and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FJT4oBgHgl3EQfgiwX/content/2301.11561v1.pdf'} +page_content=' Chen, Practical speed meter designs for quantum nondemolition gravitational-wave interfer- ometers, 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a/x9E3T4oBgHgl3EQflwp-/content/tmp_files/load_file.txt b/x9E3T4oBgHgl3EQflwp-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..929a51ae0e0ca0c96f9364c5ebe692b9238fab58 --- /dev/null +++ b/x9E3T4oBgHgl3EQflwp-/content/tmp_files/load_file.txt @@ -0,0 +1,1123 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf,len=1122 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='04610v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='FA] 11 Jan 2023 QUASI GELFAND TRIPLES NATHANAEL SKREPEK Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We generalize the notion of Gelfand triples (also called Banach- Gelfand triples or rigged Hilbert spaces) by dropping the necessity of a con- tinuous embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This means in our setting we lack of a chain inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We replace the continuous embedding by a closed embedding of a dense sub- space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This new notion will be called quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' These triples appear naturally, when we regard the boundary spaces of spatially multidimensional differential operators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' the Maxwell operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We will show that there is a smallest space where we can continuously embed the entire triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, we will show density results for intersections of members of the quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Finally, we show that every quasi Gelfand triple can be decomposed into two “ordinary” Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Introduction Normally, when we talk about Gelfand triples we have a Hilbert space X0 and a reflexive Banach space X+ that can be continuously and densely embedded into X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The third space X− is given by the completion of X0 with respect to ∥g∥X− := sup f∈X+\\{0} |⟨g, f⟩X0| ∥f∥X+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The duality between X+ and X− is given by ⟨g, f⟩X−,X+ = lim k→∞⟨gk, f⟩X0, where (gk)k∈N is a sequence in X0 that converges to g in X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The space X− is then isometrically isomorphic to X ′ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The theory of Gelfand triples was introduced by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Gelfand and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Kostyuchenko [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The concept has been refined over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In the introduction of [1] they give a short historical overview of Gelfand triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We want to weaken the assumptions such that the norm of X+ is not necessarily related to the norm of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, we cannot expect a continuous embedding of X+ into X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In [4] this weakened concept was introduced for the case, where X+ is also a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Here we will introduce these generalized triples also for reflexive Ba- nach spaces X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, in [4] the notion of quasi Gelfand triples was developed with the focus on a solution theory for port-Hamiltonian systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In particular to handle the boundary spaces of differential operators, like e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' the Maxwell operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In this work we will take a wider and deeper look at quasi Gelfand triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Even a little bit earlier also [1] introduced the notion of such quasi Gelfand triples for Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' They call it triples of closely embedded Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The motivation in [1] were weighted Sobolev and L2 spaces, were the positive weight is neither bounded from above nor from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Date: January 12, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 46A20, 46C07, 46E99, 47A70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Gelfand triples, quasi Gelfand triples, rigged Hilbert spaces, Banach- Gelfand triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' E-mail: nathanael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='skrepek@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='tu-freiberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 1 2 NATHANAEL SKREPEK In this work we will lift the setting of [4] to Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' So the beginning will be relatively similar to the introduction of quasi Gelfand triples in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This lifting has also be done in the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' thesis [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, we go beyond the refinements of [3] and show that there exists a smallest space where we can structure preservingly embed the entire quasi Gelfand triple in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Furthermore, we show a bijective relation between quasi Gelfand triples and Gram operators in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This connection to Gram operators has also been discovered in [1] or it was actually the starting point of their journey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' They call the Gram operator the Hamiltonian of the triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, we take the next step and utilize this connection to the Gram operator to construct a decomposition of the quasi Gelfand triple into two “ordinary” Gelfand triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Preliminary Since we will often switch between Hilbert space inner products and dual pairings, it is more convenient to always regard the anti-dual space instead of the dual space, which we will do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The anti-dual space is the space of all continuous antilinear mappings from the vectors space to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, we will use a generalized concept for (unbounded) linear operators, namely linear relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The following notion of linear relations, dual pairs and adjoints with respect to dual pairs are carefully covered in [3, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 1, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' A linear relation T between the vector spaces X and Y is a linear subspace of X ×Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Clearly, every linear operator is also a linear relation (we do not distinguish between a function and its graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For linear operators we have [ x y ] ∈ T is equivalent to T x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We will use the following notation ker T := {x ∈ X | [ x 0 ] ∈ T }, ran T := {y ∈ Y | ∃x : [ x y ] ∈ T }, mul T := � y ∈ Y �� � 0 y � ∈ T � , dom T := {x ∈ X | ∃y : [ x y ] ∈ T }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Thus, T is single-valued (an operator), if mul T = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The closure T of a linear relation T is the closure in X × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that every linear relation is closable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Also every operator has a closure as a linear relation, but its closure can be multi-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Therefore, showing mul T = {0} is necessary, even if mul T = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let X and Y be Banach spaces and let ⟨·, ·⟩Y,X : Y × X → C be continuous and sesquilinear (linear in the first argument and antilinear in the second argument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We define Ψ: � Y → X′, y �→ ⟨y, ·⟩Y,X, and Φ: � X → Y ′, x �→ ⟨·, x⟩Y,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If Ψ is isometric and bijective, then we say that (X, Y ) is a (anti-)dual pair and ⟨·, ·⟩Y,X is its (anti-)dual pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We define ⟨x, y⟩X,Y := ⟨y, x⟩Y,X, which is again a sesquilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If also Φ is isometric and bijective, then we say that (X, Y ) is a complete (anti- )dual pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Clearly, (X, X′) is a dual pair with the canonical dual pairing ⟨x′, x⟩X′,X = x′(x) and it is complete, if X is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For a Hilbert space (H, H) is a complete dual pair with the inner product as dual pairing ⟨x, y⟩H,H = ⟨x, y⟩H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, if we think of the Sobolev space H1(R) there are two “natural” possible dual pairings: the standard Hilbert space (complete) dual pair (H1(R), H1(R)) and the duality induced by the L2 inner product (H1(R), H−1(R)) given by ⟨x, y⟩H1(R),H−1(R) = limn→∞⟨x, yn⟩L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, in order to avoid saying both H1(R) and H−1(R) is the QUASI GELFAND TRIPLES 3 dual space of H1(R), which can lead to confusion, we prefer the term (complete) dual pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (X1, Y1), (X2, Y2) be dual pairs and A a linear relation between X1 and X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then we define the adjoint linear relation by A∗Y2×Y1 := �� y2 y1 � ∈ Y2 × Y1 ���� ⟨y2, x2⟩Y2,X2 = ⟨y1, x1⟩Y1,X1 for all � x1 x2 � ∈ A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We will just write A∗, if the dual pairs are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For a Banach space X, we will regard the dual pair (X, X′) for the adjoint, if no other dual pair is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Similar, for a Hilbert space H we will regard the dual pair (H, H), if no other dual pair is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that this definition matches the usual Hilbert space adjoint, if A is a densely defined operator between two Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If A is an operator (mul A = {0}) from X1 to X2, then we can characterize the domain of A∗ by y2 ∈ dom A∗ ⇔ dom A ∋ x1 �→ ⟨y2, Ax1⟩Y2,X2 is continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, we have the following relations ker A∗ = (ran A)⊥ and mul A∗ = (dom A)⊥, where M ⊥ denotes the annihilator space of M (which is the orthogonal complement in the Hilbert space case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Motivation Let Ω ⊆ R3 be a bounded open set with bounded Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For f, g ∈ C∞(R3) we have the following integration by parts formula: ⟨div f, g⟩L2(Ω) + ⟨f, gradg⟩L2(Ω) = ⟨ν · γ0f, γ0g⟩L2(∂Ω), where ν is the normal vector on ∂Ω and γ0 is the boundary trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' It is also well known that we can extend this formula for f ∈ H(div, Ω) and g ∈ H1(Ω): ⟨div f, g⟩L2(Ω) + ⟨f, grad g⟩L2(Ω) = ⟨γνf, γ0g⟩H−1/2(∂Ω),H1/2(∂Ω), where γν is the continuous extension of ν · γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In this extension we stumble over the Gelfand triple (H 1/2(∂Ω), L2(∂Ω), H−1/2(∂Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, in general such an in- tegration by parts formula will not automatically lead to such an extension where we can replace the L2 inner product on the boundary by dual pairing that comes from a Gelfand triple with L2(∂Ω) as pivot space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For example for f, g ∈ C∞(R3) we have ⟨rot f, g⟩L2(Ω) + ⟨f, rot g⟩L2(Ω) = ⟨ν × γ0f, (ν × γ0g) × ν⟩L2(∂Ω), (1) but contrary to the previous case neither ν ×γ0 nor (ν ×γ0)×ν can be continuously extended to H(rot, Ω) such that its codomain is still L2(∂Ω) (or can be continuously embedded into L2(∂Ω)), see [4, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, in order to better understand the relation between the extension of (1) to H(rot, Ω) and the L2(∂Ω) inner product we need a more general tool than Gelfand triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In order to try to find a suitable boundary space such that we can extend ν × γ0 on H(rot, Ω), we endow ran(ν × γ0) with the range norm that comes from H(rot, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This gives a norm on a dense subspace of L2 τ(∂Ω) = {φ ∈ L2(∂Ω) | ν · f = 0} that is unrelated to ∥·∥L2(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This setting will be our starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This particular problem was treated in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Here we want to discover the world of quasi Gelfand triples without any particular applications in mind (or maybe with Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 in mind).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 4 NATHANAEL SKREPEK 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We will have the following setting: Let X0 be a Hilbert space with the inner product ⟨·, ·⟩X0 and ⟨·, ·⟩X+ be another inner product on X0 (not necessarily related to ⟨·, ·⟩X0), which is defined on a dense (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0) subspace ˜D+ of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We denote the completion of ˜D+ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+ (∥f∥X+ := � ⟨f, f⟩X+) by X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This completion is, by construction a Hilbert space with the extension of ⟨·, ·⟩X+, for which we use the same symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Now we have ˜D+ is dense in X0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0 and dense in X+ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Figure 1 illustrates this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that X+, as a Hilbert space, is automatically reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For the further construction the crucial property of X+ is its reflexivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, we will weaken the previous setting such that X+ is only a reflexive Banach space: X0 Hilbert space endowed with ⟨·, ·⟩X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ˜D+ dense subspace of X0 (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+ another norm defined on ˜D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' X+ completion of ˜D+ with respect to ∥·∥X+ is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' X+ X0 ˜D+ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Setting of X0, ˜D+ and X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let X0 = ℓ2(Z \\ {0}) with the standard inner product ⟨x, y⟩X0 = �∞ n=1 xnyn + x−ny−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We define the inner product ⟨x, y⟩X+ := ∞ � n=1 n2xnyn + 1 n2 x−ny−n and the set ˜D+ := {f ∈ X0 | ∥f∥X+ < +∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Clearly, this inner product is well- defined on ˜D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let ei denote the sequence which is 1 on the i-th position and 0 elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since {ei | i ∈ Z \\ {0}} is a orthonormal basis of X0 and contained in ˜D+, ˜D+ is dense in X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The sequence � �n i=1 e−i � n∈N is a Cauchy sequence with respect to ∥·∥X+, but not with respect to ∥·∥X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We define ∥g∥X− := sup f∈ ˜ D+\\{0} |⟨g, f⟩X0| ∥f∥X+ for g ∈ X0 and D− := � g ∈ X0 ��� ∥g∥X− < +∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We denote the completion of D− w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X− by X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We will also denote the extension of ∥·∥X− to X− by ∥·∥X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By definition of D− we can identify every g ∈ D− with an element of X ′ + by the continuous extension of ψg : � D+ → C, f �→ ⟨g, f⟩X0, QUASI GELFAND TRIPLES 5 on X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We denote this extension again by ψg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By definition of D− we have ∥ψg∥X ′ + = ∥g∥X− for g ∈ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, we can extend the isometry Ψ: � D− → X ′ +, g �→ ψg, by continuity on X−, this is extension is again denoted by Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' So X− can be seen as the closure of D− in X ′ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We can define a dual pairing between X+ and X− by ⟨g, f⟩X−,X+ := ⟨Ψg, f⟩X ′ +,X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, this does not necessarily make (X+, X−) a dual pair in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='1, because we do not know whether Ψ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' D− is complete with respect to ∥g∥X−∩X0 := � ∥g∥2 X0 + ∥g∥2 X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (gn)n∈N be a Cauchy sequence in D− with respect to ∥·∥X−∩X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then (gn)n∈N is a convergent sequence in X0 (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0) and a Cauchy sequence in D− (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We denote the limit in X0 by g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By definition of ∥·∥X− we obtain for f ∈ ˜D+ |⟨g0, f⟩X0| = lim n→∞|⟨gn, f⟩X0| ≤ lim n→∞∥gn∥X−∥f∥X+ ≤ C∥f∥X+ and consequently g0 ∈ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let ǫ > 0 be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since (gn)n∈N is a Cauchy sequence with respect to ∥·∥X−, there is an n0 ∈ N such that for all f ∈ ˜D+ with ∥f∥X+ = 1 |⟨gn − gm, f⟩X0| ≤ ǫ 2, if n, m ≥ n0 holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Furthermore, for every f ∈ ˜D+ there exists an mf ≥ n0 such that |⟨g0 − gmf , f⟩X0| ≤ ǫ∥f∥X+ 2 , because gm → g0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This yields |⟨g0 − gn, f⟩X0| ∥f∥X+ ≤ |⟨g0 − gmf , f⟩X0| ∥f∥X+ + |⟨gmf − gn, f⟩X0| ∥f∥X+ ≤ ǫ, if n ≥ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since the right-hand-side is independent of f, we obtain ∥g0 − gn∥X− = sup f∈ ˜ D+\\{0} |⟨g0 − gn, f⟩X0| ∥f∥X+ ≤ ǫ, if n ≥ n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, g0 is also the limit of (gn)n∈N with respect to ∥·∥X− and consequently the limit of (gn)n∈N with respect to ∥·∥X−∩X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Strictly speaking ˜D+ and D− are subsets of X0, but sometimes we rather want to regard them as subsets of X+ and X−, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, introduce the following embedding mappings ˜ι+ : � ˜D+ ⊆ X+ → X0, f �→ f, and ι− : � D− ⊆ X− → X0, g �→ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This allows us to distinguish between f ∈ ˜D+ as element of X+ and ˜ι+(f) as element of X0, if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Clearly, the same for g ∈ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The embedding ˜ι+ is a densely defined operator with ran˜ι+ is dense in X0 and ker˜ι+ = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Furthermore, the embedding ι− is closed and ker ι− = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By assumption on ˜D+ and definition of X+ the embedding ˜ι+ is densely de- fined and has a dense range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Clearly, ker˜ι+ = {0} and ker ι− = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4 ι− is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ 6 NATHANAEL SKREPEK Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let ˜ι∗ + = ˜ι ∗X0×X′ + + denote the adjoint relation (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' the dualities (X0, X0) and (X+, X ′ +)) of ˜ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then ˜ι∗ + is an operator (single-valued, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' mul˜ι∗ + = {0}) and ker ˜ι∗ + = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Its domain coincides with D− and ˜ι∗ +ι− : D− ⊆ X− → X ′ + is isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If ker˜ι+ = {0}, then ran˜ι∗ + is dense in X ′ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The density of the domain of ˜ι+ yields mul ˜ι∗ + = (dom˜ι+)⊥ = {0}, and ran˜ι+ X0 = X0 yields ker˜ι∗ + = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The following equivalences show dom˜ι∗ + = D−: g ∈ dom˜ι∗ + ⇔ ˜D+ ∋ f �→ ⟨g,˜ι+f⟩X0 is continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+ ⇔ sup f∈ ˜ D+\\{0} |⟨g, f⟩X0| ∥f∥X+ < +∞ ⇔ g ∈ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For g ∈ D− ⊆ X− we have ∥g∥X− = sup f∈ ˜ D+\\{0} |⟨ι−g, f⟩X0| ∥f∥X+ = sup f∈ ˜ D+\\{0} |⟨˜ι∗ +ι−g, f⟩X ′ +,X+| ∥f∥X+ = ∥˜ι∗ +ι−g∥X ′ +, which proves that ˜ι∗ +ι− is isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that the reflexivity of X+ implies ˜ι+ = ˜ι∗∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If ker ˜ι+ = {0}, then the following equation implies the density of ran˜ι∗ + in X ′ + {0} = ker ˜ι+ = ker ˜ι∗∗ + = (ran˜ι∗ +)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' As mentioned in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3 every g ∈ D− can be regarded as an element of X ′ + by ψg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let g ∈ D−, f ∈ X+ and (fn)n∈N in ˜D+ converging to f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since D− = dom ˜ι∗ +, we have ⟨ψg, f⟩X ′ +,X+ = lim n→∞⟨g, fn⟩X0 = lim n→∞⟨ι−g,˜ι+fn⟩X0 = ⟨˜ι∗ +ι−g, f⟩X ′ +,X+ and consequently ψg = ˜ι∗ +ι−g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, ΨD− = ˜ι∗ +ι−D− = ran˜ι∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (i) There is a Hausdorff topological vector space (Z, T ) and two continuous embeddings φX+ : X+ → Z and φX0 : X0 → Z such that the diagram ˜D+ X+ Z ˜D+ X0 ˜ι+ id φX+ ˜ι−1 + id φX0 commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (ii) If ˜D+ ∋ fn → 0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+ and limn→∞ fn exists w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0, then this limit is also 0 and if ˜D+ ∋ fn → 0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0 and limn→∞ fn exists w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+, then this limit is also 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (iii) ˜ι+ : ˜D+ ⊆ X+ → X0, f �→ f is closable (as an operator) and its closure is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (iv) D− is dense in X0 and dense in X ′ +, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ΨD− is dense in X ′ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We will follow the strategy (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (i) ⇒ (ii): Let (fn)n∈N be a sequence in ˜D+ such that fn → ˆf w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' X+ and fn → f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since T is coarser than both of the topologies induced by these norms, QUASI GELFAND TRIPLES 7 we also have ˆf fn f T T in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since T is Hausdorff, we conclude f = ˆf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, if either ˆf or f is 0, then also the other is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (ii) ⇒ (iii): If (fn, fn)n∈N is a sequence in ˜ι+ that converges to (0, f) ∈ X+ × X0, then f = 0 by (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, mul ˜ι+ = {0} and consequently ˜ι+ is closable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' On the other hand, if (fn, fn)n∈N is a sequence in ˜ι+ that converges to (f, 0), then f = 0 by (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Consequently, ker ˜ι+ = {0} and ˜ι+ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (iii) ⇒ (iv): We have (dom ˜ι∗ +)⊥ = mul ˜ι∗∗ + = mul ˜ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since ˜ι+ is closable, we have mul˜ι+ = {0}, which implies that dom˜ι∗ + is dense in X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='6 dom˜ι∗ + coincides with D−, which gives the density of D− in X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The second assertion of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='6 yields that ran˜ι∗ + is dense in X ′ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='7 we have ran˜ι∗ + = ΨD− and therefore the density of ΨD− in X ′ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (iv) ⇒ (i): Let Y := D− be equipped with ∥g∥Y := ∥g∥X−∩X0 = � ∥g∥2 X− + ∥g∥2 X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We define Z := Y ′ as the (anti-)dual space of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then we have |⟨f, g⟩X0| ≤ ∥f∥X0∥g∥X0 ≤ ∥f∥X0∥g∥Y for f ∈ X0, g ∈ Y and |⟨f,˜ι∗ +g⟩X+,X ′ +| ≤ ∥f∥X+ ∥˜ι∗ +g∥X ′ + � �� � =∥g∥X− ≤ ∥f∥X+∥g∥Y for f ∈ X+, g ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, φX0 : f �→ ⟨f, ·⟩X0 and φX+ : f �→ ⟨f,˜ι∗ +·⟩X+,X ′ + are continuous mappings from X0 and X+, respectively, into Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The injectivity of these mappings follows from the density of D− in X0 and D− in X ′ + (˜ι∗ +D− dense in X ′ +), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For f ∈ ˜D+ we have φX+f = ⟨f,˜ι∗ +·⟩X+,X ′ + = ⟨˜ι+f, ·⟩X0 = φX0 ◦ ˜ι+f and consequently the diagram in (i) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ If one and therefore all assertions in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 are satisfied, then X+ ∩ X0 is defined as the intersection in Z and complete with the norm ∥·∥X+∩X0 := � ∥·∥2 X+ + ∥·∥2 X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, we define D+ as the closure of ˜D+ in X+ ∩ X0 (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+∩X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that although X+ ∩ X0 may depend on Z, D+ is independent of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We will denote the extension of ˜ι+ to D+ by ι+, which can be expressed by ι+ = ˜ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The adjoint ι∗ + coincides with ˜ι∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Also D− does not change, if we replace ˜D+ by D+ in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2 and all previous results in this section also hold for D+ and ι+ instead of ˜D+ and ˜ι+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If ˜ι+ is already closed, then D+ = ˜D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let one assertion in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let Z = Y ′, where Y = D− endowed with ∥g∥Y := ∥g∥X−∩X0 = � ∥g∥2 X− + ∥g∥2 X0 (from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 (iv) ⇒ (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then we have the following characterization for D+: D+ = dom ι∗ −, D+ = X+ ∩ X0 in Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 8 NATHANAEL SKREPEK Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that for g ∈ D− we have g = (ι∗ +)−1ι∗ +g and that ι∗ +ι− is isometric from D− = dom ι− ⊆ X− onto ran ι∗ + = dom(ι∗ +)−1 ⊆ X ′ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The following equivalences show the first assertion: f ∈ dom ι∗ − ⇔ D− ∋ g �→ ⟨f, ι−g⟩X0 is continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X− ⇔ D− ∋ g �→ ⟨f, (ι∗ +)−1ι∗ +ι−g⟩X0 is continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X− ⇔ dom(ι∗ +)−1 ∋ h �→ ⟨f, (ι∗ +)−1h⟩X0 is continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X ′ + ⇔ f ∈ dom � (ι∗ +)−1�∗ = dom ι−1 + = ran ι+ = D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For the second characterization we define P+ := X+ ∩ X0 and we define P− analogously to D− in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2: ∥g∥P− := sup f∈P+\\{0} |⟨g, f⟩X0| ∥f∥X+ and P− := {g ∈ X0 | ∥g∥P− < +∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Clearly, ∥g∥X− ≤ ∥g∥P− for g ∈ P− and consequently P− ⊆ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Furthermore, we can define ιP+ : P+ ⊆ X+ → X0, f �→ f analogously to ˜ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that ιP+ is closed due the completeness of (X+ ∩ X0, ∥·∥X+∩X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then we have dom ι∗ P+ = P− and ˜ι+ ⊆ ιP+ and therefore ι∗ P+ ⊆ ˜ι∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For g ∈ D− and f ∈ P+ we have, by definition of P+ = X+ ∩ X0 in Z, |⟨g, f⟩X0| = |⟨˜ι∗ +g, f⟩X ′ +,X+| ≤ ∥˜ι∗ +g∥X ′ +∥f∥X+ = ∥g∥X−∥f∥X+, which yields ∥g∥P− ≤ ∥g∥X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, P− = D−, ι∗ P+ = ˜ι∗ + and ιP+ = ˜ι+, which is equivalent to P+ = X+ ∩ X0 = ˜D+ X+∩X0 = D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let one assertion in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then the con- tinuous extension of ι∗ +ι− denoted by ι∗ +ι− equals Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, Ψ is surjective and (X+, X−) is a complete dual pair with ⟨g, f⟩X−,X+ := ⟨Ψg, f⟩X ′ +,X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We have already shown, that ι∗ +ι−g = Ψg for g ∈ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since D− is dense in X−, we also have ι∗ +ι−g = Ψg for g ∈ X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If one assertion in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 is true, then all of them are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, ΨD− is dense in X ′ + and because Ψ is isometric ran Ψ is closed and therefore ran Ψ = X ′ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since Ψ is an isomorphism between X− and X ′ +, it immediately follows that (X+, X−) is a complete dual pair with the dual pairing ⟨·, ·⟩X−,X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For f ∈ D+ and g ∈ D− we have ⟨g, f⟩X−,X+ = ⟨Ψg, f⟩X ′ +,X+ = ⟨ι∗ +ι−g, f⟩X ′ +,X+ = ⟨ι−g, ι+f⟩X0 = ⟨g, f⟩X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since these two sets are dense in X+ and X− respectively, we have for f ∈ X+ and g ∈ X− ⟨g, f⟩X−,X+ = lim (n,m)→(∞,∞)⟨gn, fm⟩X0, where (fm)m∈N is a sequence in D+ that converges to f in X+ and (gn)n∈N is a sequence in D− that converges to g in X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Definition and Results The previous section leads to the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (X+, X−) be a complete dual pair and X0 be a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Furthermore, let ι+ : dom ι+ ⊆ X+ → X0 and ι− : dom ι− ⊆ X− → X0 be QUASI GELFAND TRIPLES 9 X0 ran ι+ ran ι− ran ι+ ∩ ran ι− X+ dom ι+ ι+ X− dom ι− ι− Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Illustration of a quasi Gelfand triple densely defined, closed, and injective linear mappings with dense range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We call (X+, X0, X−) a pre-quasi Gelfand triple, if ⟨g, f⟩X−,X+ = ⟨ι−g, ι+f⟩X0 (2) for all f ∈ dom ι+ and g ∈ dom ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The space X0 will be referred as pivot space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If we additionally have dom ι∗ + = ran ι−, then we call (X+, X0, X−) a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Figure 2 illustrates the setting of a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Contrary to the previous section we will regard the adjoint of ι+ and ι− with respect to the complete dual pairs (X+, X−) and (X0, X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Therefore, ι∗ + is a densely defined operator from X0 to X− and ι∗ − is a densely defined operator from X0 to X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We could not do this before, because we did not know from the beginning that (X+, X−) is a complete dual pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let X+ = Lp(R), X− = Lq(R) and X0 = L2(R), where p ∈ (1, +∞) and 1 p + 1 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then (X+, X−) is a complete dual pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that Lp(R) ∩ L2(R) is already well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We can define ι+ : � Lp(R) ∩ L2(R) ⊆ Lp(R) → L2(R), f �→ f, and ι− : � Lq(R) ∩ L2(R) ⊆ Lq(R) → L2(R), g �→ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' These mapping are densely defined, injective and closed with dense range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By definition of the dual pairing of (Lp(R), Lq(R)) we have ⟨g, f⟩Lq(R),Lp(R) = � R gf dλ = ⟨g, f⟩X0 = ⟨ι−g, ι+f⟩X0 for g ∈ Lq(R) ∩ L2(R) and f ∈ Lp(R) ∩ L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By the H¨older inequality we also have dom ι∗ + = ran ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, (Lp(R), L2(R), Lq(R)) is a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that the mapping ι+ gives us an identification of dom ι+ and ran ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, we can introduce the norm of X+ on ran ι+ by ∥f∥X+ = ∥ι−1 + f∥X+ for f ∈ ran ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then the completion of ran ι+ with respect to ∥·∥X+ is isometrically isomorphic to X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Accordingly, we can do the same for X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This justifies the following definition and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 10 NATHANAEL SKREPEK X+ X− X0 D+ D− D+∩D− Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Illustration of a quasi Gelfand triple, where D+ = ran ι+ and D− = ran ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For a quasi Gelfand triple (X+, X0, X−) we define X+ ∩ X0 := ran ι+ and X− ∩ X0 := ran ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If either ι+ or ι− is continuous, then a quasi Gelfand triple is an “ordinary” Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Clearly, every “ordinary” Gelfand triple is also a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The additional condition dom ι∗ + = ran ι− that makes a pre-quasi Gelfand triple a quasi Gelfand triple is not crucial as it can always be forced, which we will see later in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 we ask ourselves, if this condition is automatically fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, the next lemma shows that we can also ask for the converse condition dom ι∗ − = ran ι+ instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that from (2) we can immediately see that dom ι∗ + ⊇ ran ι− and dom ι∗ − ⊇ ran ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, for f ∈ dom ι+ and g ∈ dom ι− we have ⟨g, f⟩X−,X+ = ⟨ι−g, ι+f⟩X0 = � ⟨ι∗ +ι−g, f⟩X−,X+, ⟨g, ι∗ −ι+f⟩X−,X+, (3) which implies ι∗ +ι−g = g and ι∗ −ι+f = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (X+, X0, X−) be a pre-quasi Gelfand triple with the embeddings ι+ and ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then dom ι∗ + = ran ι− ⇔ dom ι∗ − = ran ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In particular, if (X+, X0, X−) is a quasi Gelfand triple, then also dom ι∗ − = ran ι+ holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The proof of this is basically the first part of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let dom ι∗ + = ran ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The following equivalences f ∈ dom ι∗ − ⇔ dom ι− ∋ g �→ ⟨f, ι−g⟩X0 is continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X− ⇔ dom ι− ∋ g �→ ⟨f, (ι∗ +)−1 ι∗ +ι−g � �� � =g ⟩X0 is continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X− ⇔ f ∈ dom � (ι∗ +)−1�∗ = dom ι−1 + = ran ι+ imply dom ι∗ − = ran ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The other implication follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ In contrast to “ordinary” Gelfand triple, the setting for quasi Gelfand triple is somehow “symmetric”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' the roles of X+ and X− are interchangeable, since neither of the embeddings ι+ and ι− has to be continuous, as indicated in the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' QUASI GELFAND TRIPLES 11 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (X+, X0, X−) be a pre-quasi Gelfand triple with the embeddings ι+ and ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then there exists an extension ˆι− of ι− that respects (2) and satisfies dom ι∗ + = ranˆι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In particular, (X+, X0, X−) with ι+ and ˆι− forms a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that ι∗ +ι−g = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, ι∗ + ⊇ ι−1 − and (ι∗ +)−1 ⊇ ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We define ˆι− as (ι∗ +)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then clearly ranˆι− = dom ι∗ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For f ∈ dom ι+ and g ∈ domˆι− we have ⟨ˆι−g, ι+f⟩X0 = ⟨ι∗ +˜ι−g, f⟩X−,X+ = ⟨g, f⟩X−,X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Alternatively, we could have extended ι+ by setting ˆι+ := (ι∗ −)−1 in the previous lemma to obtain a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If (X+, X0, X−) is a quasi Gelfand triple and (X+, � X−) is another dual pair for X+, then also (X+, X0, � X−) is a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (X+, X0, X−) be a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then ι∗ + = ι−1 − and ι∗ − = ι−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By (3) we have ι∗ +ι−g = g for all g ∈ dom ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since ran ι− = dom ι∗ + (by assumption), we conclude that ι∗ + = ι−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Analogously, the second equality can be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let X+ be a reflexive Banach space and X0 be a Hilbert space and ι+ : dom ι+ ⊆ X+ → X0 be a densely defined, closed, and injective linear mapping with dense range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then there exists a Banach space X− and a mapping ι− such that (X+, X0, X−) is a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In particular, X− is given by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2, where D+ = ran ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We will identify dom ι+ with ran ι+ and denote it by D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then item (iii) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, the corresponding D− (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2) is dense in X0 and its completion X− (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' to ∥·∥X−) establishes the complete dual pair (X+, X−), by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The mapping ι− : � D− ⊆ X− → X0, g �→ g, is densely defined and injective by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By the already shown ran ι− = D− is dense in X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Finally, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='5 ι− is closed and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='6 dom ι∗ + = D− = ran ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 the setting in the beginning of Section 3 establishes a quasi Gelfand triple, if one assertion of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' From now on we will assume that (X+, X0, X−) is a quasi Gelfand triple and we will identify dom ι+ with ran ι+ and denote it by D+ as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Anal- ogously, we identify dom ι− with ran ι− and denote it with D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' These identifications are really meaningful as we can endow D+ (as a subset of X0) with ∥f∥X+ := ∥ι−1 + f∥X+ for f ∈ D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then the completion of D+ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' to this norm is clearly isomorphic to X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The same goes for D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The set D− = ran ι− (previous identification) coincides with the set D− defined in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2 for ˜D+ := D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The space D+ ∩ D− is complete with respect to ∥·∥X+∩X− := � ∥·∥2 X+ + ∥·∥2 X− and ∥f∥X0 ≤ ∥f∥X+∩X− ∀f ∈ D+ ∩ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 12 NATHANAEL SKREPEK Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For f ∈ D+ ∩ D− we have ∥f∥2 X0 = |⟨f, f⟩X0| = |⟨f, f⟩X−,X+| ≤ ∥f∥X−∥f∥X+ ≤ ∥f∥2 X+∩X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, every Cauchy sequence in D+ ∩ D− with respect to ∥·∥X+∩X− is also a Cauchy sequence with respect to ∥·∥X0, ∥·∥X+ and ∥·∥X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (fn)n∈N be a Cauchy sequence in D+∩D− with respect to ∥·∥X+∩X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By the closedness of ι+ the limit with respect to ∥·∥X0 and the limit with respect to ∥·∥X+ coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The same argument for ι− yields that the limit with respect to ∥·∥X0 and the limit with respect ∥·∥X− also coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Therefore, all these limits have to coincide and (fn)n∈N converges to that limit in D+ ∩ D− w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+∩X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The operator �ι+ ι− � : \uf8f1 \uf8f2 \uf8f3 D+ × D− ⊆ X+ × X− → X0, �f g � �→ f + g, is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let ��� fn gn � , zn �� n∈N be a sequence in � ι+ ι− � that converges to �� f g � , z � in X+ × X− × X0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' lim n→∞ fn = f (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+), lim n→∞ gn = g (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X−), and lim n→∞ fn + gn = lim n→∞ zn = z (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then we have ∥z∥2 X0 = lim n→∞∥fn + gn∥2 X0 = lim n→∞ � ∥fn∥2 X0 + ∥gn∥2 X0 + 2 Re⟨fn, gn⟩X0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since 2 Re⟨fn, gn⟩X0 converges to 2 Re⟨f, g⟩X+,X−, we conclude that ∥fn∥X0 and ∥gn∥X0 are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, there exists a subsequence of (fn)n∈N that converges weakly (in X0) to an ˜f ∈ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2 we can pass on to a further subsequence (fn(k))k∈N such that � 1 j �j k=1 fn(k) � j∈N converges to ˜f strongly (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The sequence � 1 j �j k=1 fn(k) � j∈N has still the limit f in X+ (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+) and because ι+ is closed we conclude that f = ˜f ∈ D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By linearity of the limit we also have 1 j �j k=1 gn(k) → z − f in X0 for the same subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since 1 j �j k=1 gn(k) is a Cauchy sequence in both X− and X0, the closedness of ι− gives that g = z − f ∈ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, z = �ι+ ι− � � f g � and the operator �ι+ ι− � is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' D+ ∩ D− is dense in X0 with respect to ∥·∥X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By dom ι∗ ± = ran ι∓ = D∓ (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4) and mul �ι+ ι− � = {0} we have X0 = � mul �ι+ ι− � �⊥ = dom �ι+ ι− �∗ = dom ι∗ + ∩ dom ι∗ − = D− ∩ D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Quasi Gelfand Triples with Hilbert Spaces In this section we will regard a quasi Gelfand triple (X+, X0, X−), where X+ and X− (and of course X0) are Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Maybe also some these results can be proven for general quasi Gelfand triple, but we would need a replacement for Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For a quasi Gelfand triple (X+, X0, X−) consisting of Hilbert spaces, there exists a unitary mapping Ψ from X− to X+ satisfying ⟨g, f⟩X−,X+ = ⟨Ψg, f⟩X+ and ⟨f, g⟩X+,X− = ⟨Ψ−1f, g⟩X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' QUASI GELFAND TRIPLES 13 We will refer to this mapping Ψ as the duality map of the quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that we previously regarded the adjoint of ι+ with respect to the dual pairs (X0, X0) and (X+, X−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The main reason for this choice was, that if X+ is not a Hilbert space, then the dual pair (X+, X+) is not available, but also sometimes the adjoint with respect to the dual pair (X+, X−) is more natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, now that X+ is a Hilbert space, the dual pairs (X+, X+) and (X−, X−) are available and seem reasonable when it comes to calculating adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, if we have an additional dual pair (Y, Z) and a linear operator A from X+ to Y , then we have two choices for the adjoint: A∗Z×X+ : dom A∗ ⊆ Z → X+ and A∗Z×X− : dom A∗ ⊆ Z → X−, as defined in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In order to have a short notation we will denote the adjoints that are taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' the dual pairs (X+, X+) and (X−, X−) by A∗h (h for Hilbert space duality) and the adjoints w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (X+, X−) still by A∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' A∗h : dom A∗h ⊆ Z → X+ and A∗ : dom A∗ ⊆ Z → X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Clearly, the same goes for mappings, where X+ is the codomain and analogously for X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that for X0 we regard only the dual pair (X0, X0), therefore we always take adjoints with respect to this dual pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In particular for ι+ we have ι∗h + : dom ι∗h + ⊆ X0 → X+ and ι∗ + : dom ι∗ + ⊆ X0 → X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3 we have the following relations between the adjoints: ι∗h + = Ψι∗ + and ι∗h − = Ψ−1ι∗ −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The set D+ ∩ D− is dense in X+ and X− with respect to their corresponding norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' More precisely dom ι∗ +ι+ = ι−1 + (D+ ∩ D−) is dense in X+ and dom ι∗ −ι− = ι−1 − (D+ ∩ D−) is dense in X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Furthermore, ι−1 + (D+ ∩ D−) is a core of ι+ and ι−1 − (D+ ∩ D−) is a core of ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Applying Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4 to ι+ yields ι∗h + ι+ is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3 we have ι∗h + = Ψι∗ +, where Ψ is the duality map introduced in the begin- ning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, dom ι∗h + ι+ = dom ι∗ +ι+ is dense in X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4 dom ι∗ + = D−, consequently dom ι∗ +ι+ = ι−1 + (dom ι∗ + ∩ ran ι+) = ι−1 + (D− ∩ D+) = D+ ∩ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (4) Finally, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='6 and (4) gives that ι−1 + (D+ ∩ D−) is a core of ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' An analogous argument for ι− yields D+ ∩ D− is dense in X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' D+ + D− = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Applying Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4 to ι+ gives that (IX0 + ι+ι∗h + ) is onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, for every x ∈ X0 there exists a gx ∈ dom ι+ι∗h + ⊆ D− such that x = gx ���� ∈D− + ι+ι∗h + gx � �� � ∈D+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since gx ∈ dom ι+ι∗h + , we have ι∗h + gx ∈ D+ and consequently x ∈ D+ + D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Next we will show that we can embed an entire quasi Gelfand triple structure preservingly into a larger space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We will even give the smallest possible space that contains the entire quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, before we start we give a proper definition of what we mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 14 NATHANAEL SKREPEK Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let H be a Hausdorff topological vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We say the quasi Gelfand triple (X+, X0, X−) can be structure preservingly embedded into H, if there exist linear, injective and continuous mappings φX+ : X+ → H, φX0 : X0 → H and φX− : X− → H such that φX+ �� dom ι+ = φX0ι+ and φX− �� dom ι− = φX0ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (5) Basically the previous definition means that the following diagram commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' H X+ X0 X− dom ι+ ran ι+ ran ι− dom ι− φX+ φX0 φX− id ι+ ι−1 + id id ι−1 − ι− id Since we identify dom ι+ and ran ι+ with each other and denote it as D+ and the same for ι−, we can reduce the previous diagram to the following diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' H X+ X0 X− D+ D− φX+ φX0 φX− id id id id From this point of view the compatibility condition (5) can be seen as φX+f = φX0f ∀f ∈ D+ and φX−g = φX0g ∀g ∈ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note if (X+, X0, X−) is an “ordinary” Gelfand triple (where ι+ is continuous), then it is usually denoted by X+ ⊆ X0 ⊆ X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' To be precise these inclusions are actually identifications via the mappings ι+ and ι−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The continuity and closedness of ι+ implies dom ι+ = X+ and that ι∗ + is also continuous and everywhere defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since ι∗ + = ι−1 − (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='7), we have the following setting: X+ X0 X−, ι+ ι−1 − which suggests that X− contains the entire Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Defining φX+ = ι−1 − ι+, φX0 = ι−1 − and φX− = idX− justifies that X− contains the Gelfand triple in a structure preserving manner as defined in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For quasi Gelfand triple the construction of a space that covers the entire quasi Gelfand triple needs a bit more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='10, D+∩D− with ∥·∥X+∩X− is complete and therefore a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since X+ and X− are Hilbert spaces (in this section) we can define the inner product ⟨g, f⟩X+∩X− := ⟨g, f⟩X+ + ⟨g, f⟩X− on D+ ∩ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This inner product induces the previous norm ∥·∥X+∩X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Conse- quently D+ ∩ D− is a Hilbert space with ⟨·, ·⟩X+∩X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For shorter notation we denote D+ ∩ D− by Z+, the corresponding inner product and norm by ⟨·, ·⟩Z+ and ∥·∥Z+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' QUASI GELFAND TRIPLES 15 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let Z+ = D+ ∩ D− be the space defined in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then the triple (Z+, X0, Z′ +) forms an “ordinary” Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In particular Z′ + is isometrically isomorphic to Z−, the completion of X0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥h∥Z− := sup z∈Z+\\{0} |⟨h, z⟩X0| ∥z∥Z+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='12 we know that Z+ is dense in X0 and by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='10 that the mapping ιZ+ : Z+ → X0, z �→ z is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, “ordinary” Gelfand triple theory or Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 gives the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We can structure preservingly embed the quasi Gelfand triple (X+, X0, X−) into the space Z′ + by the embeddings ψX+f = ⟨f, ι−1 − ·⟩X+,X−, ψX0h = ⟨h, ·⟩X0 and ψX−g = ⟨g, ι−1 + ·⟩X−,X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that by our identifications of D+ and D− we have ι−1 + z = z and ι−1 − z = z for z ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, making this change of spaces visible can sometimes help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Nevertheless, most of the time this is only additional dead weight, this is why we will often just write φX+(f)(z) = ⟨f, z⟩X+,X−, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='. Clearly, since Z′ + and Z− are isometrically isomorphic we can also structure preservingly embed (X+, X0, X−) into Z−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For notational harmony we prefer to use Z− instead of Z′ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, for our purpose there is no need to strictly distin- guish between them, this is why we will use these symbols as synonyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Figure 4 illustrates the meaning of the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' First we have to check that these mappings are well-defined: Let z ∈ Z+, f ∈ X+, h ∈ X0 and g ∈ X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then |ψX+(f)(z)| = |⟨f, z⟩X+,X−| ≤ ∥f∥X+∥z∥X− ≤ ∥f∥X+∥z∥Z+, |ψX0(h)(z)| = |⟨h, z⟩X0| ≤ ∥h∥X0∥z∥X0 ≤ ∥h∥X0∥z∥Z+, |ψX−(g)(z)| = |⟨g, z⟩X−,X+| ≤ ∥g∥X−∥z∥X+ ≤ ∥g∥X−∥z∥Z+, which implies ψX+(f), ψX0(h) and ψX−(g) are in Z′ +, and ψX+, ψX0 and ψX− are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The linearity of ψX+, ψX0 and ψX− follows from the sequilinearity of a dual pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If ψX+(f) = 0, then f ⊥ ι−1 − Z+ = ι−1 − (D+ ∩ D−) = dom ι∗ −ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since dom ι∗ −ι− is dense in X−, we conclude f = 0, which proves φX+ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Analogously, we can show that ψX− is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If ψX0(h) = 0, then h ⊥ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since Z+ is dense in X0, h has to be 0, which gives the injectivity of ψX0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The compatibility condition (5) follows from ψX0 ◦ ι+(f)(z) = ⟨ι+f, z⟩X0 = ⟨f, ι∗ +z⟩X+,X− = ⟨f, ι−1 − z⟩X+,X− = ψX+(f)(z), ψX0 ◦ ι−(g)(z) = ⟨ι−g, z⟩X0 = ⟨g, ι∗ −z⟩X−,X+ = ⟨g, ι−1 + z⟩X−,X+ = ψX−(g)(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Now since we can always structure preservingly embed a quasi Gelfand triple into Z− (Z′ +) we can regard this quasi Gelfand triple as subsets of Z−, see Figure 4a, and do not have to deal with all this embeddings (most of the time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, we will not get completely rid of these embeddings, as they are sometimes helpful, but we can always regard them as identity mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Z− = X+ + X− and ∥h∥Z− = inf f+g=h f∈X+,g∈X− � ∥f∥2 X+ + ∥g∥2 X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that Z+ is a Hilbert space with ⟨z1, z2⟩Z+ = ⟨z1, z2⟩X+ + ⟨z1, z2⟩X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, there is a duality map Φ from Z− to Z+ and we can write ⟨h, z⟩Z−,Z+ = ⟨Φh, z⟩Z+ = ⟨Φh, z⟩X+ + ⟨Φh, z⟩X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 16 NATHANAEL SKREPEK X+ X− X0 D+ D− D+∩D− = Z+ Z− (a) Venn diagram Z− X+ X0 X− D+ D− Z+ (b) Commutative diagram Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' quasi Gelfand triple embedded in Z− Furthermore, with the duality map Ψ from X− to X+ we have ⟨h, z⟩Z−,Z+ = ⟨Ψ−1Φh, z⟩X−,X+ + ⟨ΨΦh, z⟩X+,X− and h = Ψ−1Φh + ΨΦh in Z−, where Ψ−1Φh ∈ X− and ΨΦh ∈ X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let h ∈ Z−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then for every f ∈ X+, g ∈ X− that satisfy h = f + g in Z− we have |⟨h, z⟩Z−,Z+| = |⟨f, z⟩X+,X− + ⟨g, z⟩X−,X+| ≤ |⟨f, z⟩X+,X−| + |⟨g, z⟩X−,X+| ≤ ∥f∥X+∥z∥X− + ∥g∥X−∥z∥X+ ≤ � ∥f∥2 X+ + ∥g∥2 X− � ∥z∥2 X− + ∥z∥2 X+ = � ∥f∥2 X+ + ∥g∥2 X−∥z∥Z+, which implies ∥h∥Z− ≤ infh=f+g � ∥f∥2 X+ + ∥g∥2 X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' On the other hand ∥h∥2 Z− = ∥Φh∥2 Z+ = ∥Φh∥2 X+ + ∥Φh∥2 X− = ∥Ψ−1Φh∥2 X− + ∥ΨΦh∥2 X+ finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ The next result reinforces Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The intersection X+ ∩ X0 in Z− equals D+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ran ψX+ ∩ ran ψX0 = ran(ψX0 ◦ ι+), and the intersection X− ∩ X0 in Z− equals D−, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ran ψX− ∩ ran ψX0 = ran(ψX0 ◦ ι−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let h ∈ X+ ∩ X0 ⊆ Z−, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' it exists an f ∈ X+ and a k ∈ X0 such that ⟨h, z⟩Z−,Z+ = ⟨f, ι−1 − z⟩X+,X− = ⟨k, z⟩X+ for all z ∈ Z+ = D+ ∩ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We define x = ι−1 − z, which leads to ⟨f, x⟩X+,X− = ⟨k, ι−x⟩X+ for all x ∈ ι−1 − (D+ ∩ D−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since ι−1 − (D+ ∩ D−) is a core of ι− (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='1), this equation is also true for all x ∈ dom ι−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, this implies f = ι∗ −k and k ∈ dom ι∗ − = D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By ι∗ − = ι−1 + we obtain ι+f = k ∈ D+ and ⟨h, z⟩Z−,Z+ = ⟨f, ι−1 − z⟩X+,X− = ⟨k, z⟩X0 = ⟨ι+f, z⟩X0, QUASI GELFAND TRIPLES 17 which gives h = f = k = ι+f in Z− and h ∈ D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The same steps can also be done for X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The intersection X+ ∩ X− in Z− is D+ ∩ D−(= Z+), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ran ψX+ ∩ ran ψX− = ran(ψX0 ◦ ι+) ∩ ran(ψX0 ◦ ι−) = ψX0(Z+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This means that area of X+ ∩ X− in Figure 4 outside of X0 is actually empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let h ∈ X+ ∩ X− ⊆ Z−, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' it exists an f ∈ X+ and a g ∈ X− such that ⟨h, z⟩Z−,Z+ = ⟨f, ι−1 − z⟩X+,X− = ⟨g, ι−1 + z⟩X−,X+ for all z ∈ D+ ∩ D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We define x := ι−1 + z, which leads to z = ι+x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since z ∈ dom ι−1 − , we have x ∈ dom ι−1 − ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Recall that ι−1 − = ι∗ + and ι−1 + Z+ = ι−1 + (D+ ∩ D−) = dom ι∗ +ι+ (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='7 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, ⟨f, ι∗ +ι+x⟩X+,X− = ⟨g, x⟩X−,X+ for all x ∈ dom ι∗ +ι+, which implies (ι∗ +ι+)∗f = g and f ∈ dom(ι∗ +ι+)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='6 (ι∗ +ι+)∗ = ι∗ +ι+ and therefore f ∈ dom ι∗ +ι+ and in particular, ι+f ∈ ι+(dom ι∗ +ι+) = D+∩D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that again by ι−1 − = ι∗ + we have ι−1 − ι+f = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Thus, g ∈ dom ι− and ι+f = ι−g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This gives ⟨h, z⟩Z−,Z+ = ⟨ι+f, z⟩X0 = ⟨ι−g, z⟩X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Therefore, h = f = g = ι+f = ι−g in Z−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For f ∈ X+ and g ∈ X− we have ∥f + g∥Z− = inf z∈Z+ � ∥f + z∥2 X+ + ∥g − z∥2 X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='6 we have ∥f + g∥Z− = inf ˜f+˜g=f+g ˜f∈X+,˜g∈X− � ∥ ˜f∥2 X+ + ∥˜g∥2 X− Note that f + g = ˜f + ˜g implies z := f − ˜f � �� � ∈X+ = − (g − ˜g) � �� � ∈X− ∈ X+ ∩ X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We can write ˜f = f − z and ˜g = f + z and by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 we have z ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Consequently, ∥f + g∥ = inf z∈Z+ � ∥f − z∥2 X+ + ∥g + z∥2 X− = inf z∈Z+ � ∥f + z∥2 X+ + ∥g − z∥2 X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ The space Z− is the smallest space where we can embed the quasi Gelfand triple structure preservingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The following theorem makes this statement precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let H be a Hausdorff topological vector space such that we can structure preservingly embed the quasi Gelfand triple (X+, X0, X−) into H and let φX+, φX0 and φX− denote the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then also Z− can be continuously embedded into H by a mapping φZ−, such that φZ− ◦ ψX+ = φX+, φZ− ◦ ψX0 = φX0 and φZ− ◦ ψX− = φX−, 18 NATHANAEL SKREPEK i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' the following diagram commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' H Z− X+ X0 X− D+ D− Z+ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Recall that we can assume that X+, X0, X− ⊆ Z− and ψX+f = f, ψX0h = h, and ψX−g = g, by simply replacing the quasi Gelfand triple (X+, X0, X−) by (ψX+(X+), ψX0(X0), ψX−(X−)), see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For conveniece we define ˆ X+ = φX+(X+), ˆ X0 = φX0(X0) and ˆ X− = φX−(X−) with ∥f∥ ˆ X+ = ∥φ−1 X+f∥X+, ∥h∥ ˆ X0 = ∥φ−1 X0 h∥X0 and ∥g∥ ˆ X− = ∥φ−1 X−g∥X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We will show as a first step that we can endow ˆ X+ + ˆ X− in H with ∥h∥ ˆ X++ ˆ X− = inff+g=h � ∥f∥2 ˆ X+ + ∥g∥2 ˆ X− such that the corresponding topology of ∥·∥ ˆ X++ ˆ X− is finer than the topology TH of H (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' whenever (hn)n∈N converges w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥ ˆ X++ ˆ X−, it also converges w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' TH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that we can alternatively write the norm as ∥f + g∥ ˆ X++ ˆ X− = inf �� ∥ ˜f∥2 ˆ X+ + ∥˜g∥2 ˆ X− ��� ˜f + ˜g = f + g � = inf �� ∥f + z∥2 ˆ X+ + ∥g − z∥2 ˆ X− ��� z ∈ X+ ∩ X− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, the mapping Λ: � X+ × X− → H, � f g � �→ φX+f + φX−g, is continuous as composition of the continuous embeddings into H and the contin- uous addition in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, ker Λ is closed in X+ × X− and the quotient space X+ × X−/ker Λ is a Hilbert space and is isometrically isomorphic to ˆ X+ + ˆ X− with ∥·∥ ˆ X++ ˆ X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The quotient mapping Λ/ker Λ : X+ × X−/ker Λ → H is injective and continuous, which implies that topology of ∥·∥ ˆ X++ ˆ X− is finer than the trace topology of TH on ˆ X+ + ˆ X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We can regard ˆ Z+ := φX0(Z+) ⊆ ˆ X0 ⊆ H and endow this space with ∥z∥ ˆ Z+ := � ∥z∥2 ˆ X+ + ∥z∥2 ˆ X− = ∥φ−1 X0 z∥Z+ for z ∈ ˆZ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Furthermore, we can define a new norm on X0 by ∥h∥ ˆ Z− := supz∈ ˆ Z+\\{0} |⟨h,z⟩| ˆ X0 ∥z∥ ˆ Z+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that every h ∈ X0 can be written as h = f + g, where f ∈ D+ and g ∈ D−, see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, also every h ∈ ˆ X0 can be written as h = f + g, where f ∈ φX0(D+) = φX+(D+) ⊆ ˆ X+ ∩ ˆ X0 and g ∈ φX0(D−) = φX−(D−) ⊆ ˆ X0 ∩ ˆ X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' QUASI GELFAND TRIPLES 19 We know by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='9 for every f + g ∈ ˆ X0 (f ∈ φX0(D+), g ∈ φX0(D−)) that ∥f + g∥ ˆ Z− = ∥φ−1 X0 (f + g)∥Z− = inf z∈Z+ � ∥φ−1 X0 (f) + z∥2 X+ + ∥φ−1 X0 (g) − z∥2 X− = inf z∈ ˆ Z+ � ∥φ−1 X0 (f) + φ−1 X0 (z) � �� � =φ−1 X+ (f+z) ∥2 X+ + ∥φ−1 X0 (g) − φ−1 X0 (z) � �� � =φ−1 X−(g−z) ∥2 X− = inf z∈ ˆ Z+ � ∥f + z∥2 ˆ X+ + ∥g − z∥2 ˆ X− ≥ inf z∈X+∩X− � ∥f + z∥2 ˆ X+ + ∥g − z∥2 ˆ X− = ∥f + g∥ ˆ X++ ˆ X−, because ˆZ+ ⊆ ˆ X+ ∩ ˆ X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, the completion of ˆ X0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥ ˆ Z− can also be continuously embedded into ˆ X+ + ˆ X−, because ˆ X+ + ˆ X− is complete, and therefore also into H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In particular the mapping (ψX0 does not do anything by assumption) φX0 ◦ ψ−1 X0 : ψX0(X0) ⊆ Z− → H is continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' the ∥·∥Z− topology on ψX0(X0) and TH on H and injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By the density of X0 in Z− we can continuously extend this mapping, denoted by φZ− := φX0 ◦ ψ−1 X0 : Z− → H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By construction we already have φZ− ◦ ψX0 = φX0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that for z ∈ Z+ we have z = ψX+z = ψX0z = ψX−z and φX+z = φX0z = φX−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Now for f ∈ X+ there exists a sequence (zn)n∈N in Z+ that converges to f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, the continuity of φZ−, ψX+ and φX+ gives φZ− ◦ ψX+f = lim n→∞ φZ− ◦ ψX+zn = lim n→∞ φZ− ◦ ψX0zn = lim n→∞ φX0zn = lim n→∞ φX+zn = φX+f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Analogously, we can show φZ− ◦ ψX− = φX−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let H be a topological Hausdorff vector space such that we can structure preservingly embed the quasi Gelfand triple (X+, X0, X−) into H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then X+ ∩ X− in H equals D+ ∩ D−, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' φX+(X+) ∩ φX−(X−) = φX0(D+ ∩ D−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='10 we can also embed Z− into H such that X+ X− ⊆ Z− ⊆ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, X+ ∩X− in H is the same as X+ ∩X− in Z−, which equals, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8, D+ ∩ D− = Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Gram operators Every quasi Gelfand triple (X+, X0, X−) is fully determined (up to isomorphic identifications) by X0, ran ι+ and ∥·∥X+ on ran ι+ (or ran ι− with ∥·∥X−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, in the Hilbert space case (X+ is a Hilbert space) we can even encode the entire information of a quasi Gelfand triple in a single (so called Gram) operator G on X0, that is self-adjoint, positive and injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This means that ⟨Gf, g⟩X0 defines a new inner product on X0, which gives rise to ⟨f, g⟩X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In particular, we will see that D+ = dom G 1/2 and ⟨G 1/2f, G 1/2g⟩ = ⟨f, g⟩X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 20 NATHANAEL SKREPEK Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (X+, X0, X−) be a quasi Gelfand triple of Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then we define the Gram operator G+ : dom G+ ⊆ X0 → X0 of the quasi Gelfand triple by G+ := (ι−1 + )∗hι−1 + = (ι+ι∗h + )−1, where here the adjoint is taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' the dual pairs (X0, X0) and (X+, X+), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (ι−1 + )∗h = (ι−1 + )∗X+×X0 and ι∗h + = ι ∗X0×X+ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4 G+ is self-adjoint and positive (not necessarily strictly positive (coercive)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, by the functional calculus for unbounded self-adjoint opera- tors on Hilbert spaces there exists a root G 1/2 + of G+, which is also self-adjoint and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Clearly, we can do the same for ι− and define G− := (ι−1 − )∗hι−1 − , where again here the adjoint is taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' the dual pairs (X0, X0) and (X−, X−), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (ι−1 − )∗h = (ι−1 − )∗X−×X0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In fact we will see that G− = G−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (X+, X0, X−) be a quasi Gelfand triple of Hilbert spaces and G+ its Gram operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then ran ι+ = dom G 1/2 + and ⟨f, g⟩X+ = ⟨G 1/2 + f, G 1/2 + g⟩X0 for all f, g ∈ dom G 1/2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In particular, ∥f∥X+ = ∥G 1/2 + f∥X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that dom G+ = dom(ι−1 + )∗hι−1 + is a core of ι−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This implies that for every f ∈ ranι+ there exists a sequence (fn)n∈N in dom G+ such that fn → f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0 and ι−1 + fn → ι−1 + f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In order words dom G+ is dense in D+ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+∩X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For f, g ∈ dom G+ ⊆ dom G 1/2 + we have ⟨ι−1 + f, ι−1 + g⟩X+ = � (ι−1 + )∗hι−1 + f, g � X+ = ⟨G+f, g⟩X0 = ⟨G 1/2 + f, G 1/2 + g⟩X0 (6) and in particular we have ∥ι−1 + f∥X+ = ∥G 1/2 + f∥X0 for all f ∈ dom G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For every f ∈ ran ι+ there exists a sequence (fn)n∈N in dom G+ that converges to f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+∩X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, we have ∥G 1/2 + fn∥X0 = ∥ι−1 + fn∥X+ → ∥ι−1 + f∥X+ and in particular (G 1/2 + fn)n∈N is a bounded sequence in X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Therefore, there exists a weakly convergent subsequence and by taking a convex combination Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2 we end up with a sequence ( ˜fn)n∈N that still converges to f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+∩X0 and additionally (G 1/2 + ˜fn)n∈N converges to some ˜f ∈ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By the closedness of G 1/2 + the limit ˜f has to coincide with f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This implies ran ι+ ⊆ dom G+ and we can extend (6) by continuity to ⟨ι−1 + f, ι−1 + g⟩X+ = � G 1/2 + f, G 1/2 + g � X0 for all f, g ∈ ran ι+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that dom G+ = dom(G 1/2 + )∗G 1/2 + is a core of G 1/2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Now we will the repeat the previous step with switched roles of G 1/2 + and ι−1 + : For every f ∈ dom G 1/2 + there exists a sequence (fn)n∈N in dom G+ such that fn → f and G 1/2 + fn → G 1/2 + f both w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This gives ∥ι−1 + fn∥X+ = ∥G 1/2 + fn∥X0 → ∥G 1/2 + f∥X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Now (ι−1 + fn)n∈N is a bounded sequence in X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Therefore there exists a weakly convergent subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover a convex combination of this subsequence con- verges even w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t ∥·∥X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In total we have a sequence ( ˜fn)n∈N such that ˜fn → f, G 1/2 + ˜fn → G 1/2 + f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0 and ι−1 + ˜fn → ˜f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+ for an ˜f ∈ X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By QUASI GELFAND TRIPLES 21 the closedness of ι+ we conclude ˜f = ι−1 + f and in turn dom G 1/2 + ⊆ ran ι+, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (X+, X0, X−) be a quasi Gelfand triple of Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then G− = G−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let Ψ: X− → X+ denote the duality mapping between X− and X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Recall G− = (ι−1 − )∗hι−1 − = (ι−ι∗h − )−1, ι∗h + = Ψι∗ + = Ψι−1 − and ι∗h − = Ψ−1ι∗ − = Ψ−1ι−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, we have G−1 − = ι−ι∗h − = ι−Ψ−1ι−1 + = (Ψι−1 − )−1ι−1 + = (ι∗h + )−1ι−1 + = (ι−1 + )∗hι−1 + = G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (X+, X0, X−) be a quasi Gelfand triple of Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then ran ι− = dom G 1/2 − = dom G−1/2 + = ran G 1/2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' So far we have shown that there is a self-adjoint positive and injective operator with dense range for every quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Now the next theorem will show that also the reverse is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' That is, every self-adjoint positive and injective oper- ator G with dense range establishes a quasi Gelfand triple whose Gram operator is G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let X0 be a Hilbert space and G a self-adjoint positive and injective operator on X0 with dense range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then there exists a quasi Gelfand triple whose Gram operator is G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In particular, if we denote the corresponding quasi Gelfand triple by (X+, X0, X−) we have ran ι+ = dom G 1/2 and ran ι− = ran G 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, G coincides with the Gram operator G+ of (X+, X0 X−), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' G = G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that dense range and injectivity are for a self-adjoint operator equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, the density of the range (or the injectivity of the operator) is not really a necessity as we can always split X0 = ker G ⊕ ran G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, we just replace X0 with ran G and G with G �� ran G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We define ⟨f, g⟩X+ := ⟨G 1/2f, G 1/2g⟩X0 and the corresponding norm ∥f∥X+ = ∥G 1/2f∥X0 for f, g ∈ dom G 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since G 1/2 is positive ⟨·, ·⟩X+ is really an inner product and ∥·∥X+ a norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, dom G 1/2 with ⟨·, ·⟩X+ is a pre-Hilbert space and its completion X+ is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We define ι+ : � dom G 1/2 ⊆ X+ → X0, f �→ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let �� fn fn �� n∈N be a sequence in ι+ that converges to � g f � ∈ X+ × X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then �� fn G 1/2fn �� n∈N is a Cauchy sequence in X0 × X0, and therefore convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The closedness of G 1/2 implies f ∈ dom G 1/2 = D+ and � fn G 1/2fn � → � f G 1/2f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This leads to ∥fn − f∥X+ = ∥G 1/2(fn − f)∥X0 → 0 and consequently f = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Now we can apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 and see that there is a space X− such that (X+, X0, X−) forms a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Now we have for f, g ∈ dom G 1/2 = ran ι+ = dom G 1/2 + ⟨G 1/2f, G 1/2g⟩X0 = ⟨f, g⟩X+ = ⟨G 1/2 + f, G 1/2 + g⟩X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 22 NATHANAEL SKREPEK Note that dom G ⊆ dom G 1/2 and therefore for f ∈ dom G we have ⟨Gf, g⟩X0 = ⟨G 1/2 + f, G 1/2 + g⟩X0, which implies G 1/2 + f ∈ dom G 1/2 + and G 1/2 + G 1/2 + f = Gf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence G ⊆ G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The same argument with G and G+ switched gives G+ ⊆ G and thus G = G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3 we have G− = G−1 + = G−1 and therefore, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2 for G−, ran ι− = dom G 1/2 − = ran G−1/2 − = ran G 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ There is a bijection between the set of quasi Gelfand triples with pivot space X0 and all self-adjoint positive and injective operators with dense range on X0, see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (X+, X0, X−) G X+, ι+ D+, ⟨·, ·⟩X+ (ι+ι∗ +)−1 dom G 1/2, ⟨G 1/2·,G 1/2·⟩X0 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 completion Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Illustration of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='5 Since all infinite dimensional separable Hilbert spaces are isomorphic, it is clear that there exists a dual pairing ⟨·, ·⟩X+,X0 such that also (X+, X0) is a complete dual pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, we can explicitly write this mapping by ⟨f, g⟩X+,X0 = � G 1/2 + ι+f, g � X0 = � f, G 1/2 + ι+ −1 g � X+, where G 1/2 + ι+ is the continuous extension of the isometric mapping G 1/2 + ι+ : dom ι+ ⊆ X+ → X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Decomposition into two “ordinary” Gelfand triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In this section we will see that every quasi Gelfand triple of Hilbert spaces can be decomposed into two “ordinary” Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This means for a quasi Gelfand triple (X+, X0, X−) there exist “ordinary” Gelfand triples X 1 + ⊆ X 1 0 ⊆ X 1 − and X 2 + ⊆ X 2 0 ⊆ X 2 − such that X+ = X 1 + ⊕ X 2 −, X0 = X 1 0 ⊕ X 2 0 and X− = X 1 − ⊕ X 2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (X+, X0, X−) be a quasi Gelfand triple of Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then there exist two “ordinary” Gelfand triple X 1 + ⊆ X 1 0 ⊆ X 1 − and X 2 + ⊆ X 2 0 ⊆ X 2 − such that X+ = X 1 + ⊕ X 2 −, X0 = X 1 0 ⊕ X 2 0 and X− = X 1 − ⊕ X 2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This means that every quasi Gelfand triple (of Hilbert spaces) is the result of two “ordinary” Gelfand triple that are cross-wise composed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We will step the proof in several steps: QUASI GELFAND TRIPLES 23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Step: Decomposition of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let G+ be the Gram operator of the quasi Gelfand triple and G 1/2 + its positive square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then there exists a spectral measure E for G 1/2 + such that G 1/2 + = � R+ λ dE(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We can decompose X0 into X0 = ran E((1, ∞)) � �� � =:X 1 0 ⊕ ran E((0, 1]) � �� � =:X 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By spectral theory X 2 0 = ran E((0, 1]) ⊆ dom G 1/2 + = ran ι+ = D+, as (0, 1] is a bounded set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We can write every f ∈ D+ as f = E((1, ∞))f + E((0, 1])f and since E((0, 1])f ∈ D+, we conclude that also E((1, ∞))f ∈ D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For an arbi- trary f ∈ X 1 0 ⊆ X0 there exists a sequence (fn)n∈N in D+ such that fn → f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since also (E((1, ∞))fn)n∈N converges to E((1, ∞))f = f by continuity, and E((1, ∞))f ∈ D+, we conclude that X 1 0 ∩ D+ is dense in X 1 0 (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' On the other hand, X 2 0 ⊆ D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Step: Decomposition of X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For f ∈ D+ we have ∥E((0, 1])f∥2 X+ = ∥G 1/2 + E((0, 1])f∥2 X0 = � (0,1] |λ|2 dEf,f ≤ � (0,∞) |λ|2 dEf,f = ∥f∥2 X+ ∥E((1, ∞))f∥2 X+ = ∥G 1/2 + E((1, ∞))f∥2 X0 = � (1,∞) |λ|2 dEf,f ≤ � (0,∞) |λ|2 dEf,f = ∥f∥2 X+ Hence, the spectral projections E((0, 1]) and E((1, ∞)) are also continuous on D+ with respect to ∥·∥X+ and we can extend these projections continuously on X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that for f ∈ D+ we have G 1/2E(∆)f = E(∆)G 1/2 for all ∆ in the Borel sets of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, we have for f, g ∈ D+ ⟨E((0, 1])f, E((1, ∞))g⟩X+ = ⟨G 1/2 + E((0, 1])f, G 1/2 + E((1, ∞))g⟩X0 = ⟨G+E((1, ∞))E((0, 1]) � �� � =0 f, g⟩X0 = 0, which implies that the extensions of E((0, 1]) �� D+ and E((1, ∞)) �� D+ are orthogonal projections on X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, for f ∈ X+ there exists a sequence (fn)n∈N in D+ that converges to f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By the continuity of projections we conclude that (E((0, 1])fn)n∈N and (E((1, ∞))fn)n∈N converges and therefore f = lim n→∞ fn = lim n→∞ E((0, 1])fn + E((1, ∞))fn = lim n→∞ E((0, 1])fn + lim n→∞ E((1, ∞))fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This leads to: the extensions of these projections are also complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We denote these extensions by E((0, 1])+ and E((1, ∞))+ and we have X+ = ran E((1, ∞))+ � �� � =:X 1 + ⊕ ran E((0, 1])+ � �� � =:X 2 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Step: Relationship between the decompositions of X0 and X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that E((1, ∞))+D+ = E((1, ∞))D+ = X 1 0 ∩ D+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Furthermore, for f ∈ X 1 0 ∩ D+ we have ∥f∥2 X+ = ∥E((1, ∞))f∥2 X+ = ∥G 1/2 + E((1, ∞))f∥2 X0 = � (1,∞) |λ|2 dEf,f ≥ inf λ∈(1,∞)|λ|2∥f∥2 X0 ≥ ∥f∥2 X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (7) 24 NATHANAEL SKREPEK Now for f ∈ X 1 + there exists a sequence (fn)n∈N in D+ that converges to f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+ and therefore also ( ˜fn)n∈N = (E((1, ∞))+fn)n∈N converges to f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By (7) we have ∥ ˜fn − ˜fm∥X0 ≤ ∥ ˜fn − ˜fm∥X+ → 0, which implies that ( ˜fn)n∈N is a Cauchy sequence in X 1 0 (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By the closedness of ι+ the limit of this sequence (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X0) has to coincide with f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, X 1 + = X 1 0 ∩D+ and the restricted embedding ι+ �� X 1 + : X 1 + → X 1 0 is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' On the other hand, since X 2 0 ⊆ D+ we automatically have X 2 0 ⊆ X 2 −, by con- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Furthermore, for f ∈ X 2 0 we have ∥f∥2 X+ = ∥E((0, 1])f∥2 X+ = ∥G 1/2 + E((0, 1])f∥2 X0 = � (0,1] |λ|2 dEf,f ≤ sup λ∈(0,1] |λ|2∥f∥2 X0 ≤ ∥f∥2 X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (8) This implies that the inverse embedding ι−1 + restricted to X 2 0 is continuous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ι−1 + �� X 2 0 : X 2 0 → X 2 − is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, we have X 2 0 ⊆ X 2 − and X 1 + ⊆ X 1 0 densely with continuous embeddings 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Step: Decomposition of X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that for g ∈ D− we have ∥g∥X− = ∥G 1/2 − g∥X0 = ∥G−1/2 + g∥X0 and additionally by the rules for the spectral calculus we have G 1/2 − = G−1/2 + = � (0,∞) 1 λ dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, the exact same construction as in the second step (replace X+ by X−, D+ by D−, G+ by G− and |λ| by | 1 λ|) gives the decomposition X− = ran E((1, ∞))− � �� � =:X 1 − ⊕ ran E((0, 1])− � �� � =:X 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Step: Relation ship between the decompositions of X0 and X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Again repeating the arguments of the third step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In particular, for g ∈ D− we have ∥E((0, 1])g∥2 X− = ∥G 1/2 − E((0, 1])g∥2 X0 = � (0,1] ���� 1 λ ���� 2 dEg,g ≥ inf λ∈(0,1] ���� 1 λ ���� 2 ∥g∥2 X0 = ∥g∥2 X0 and ∥E((1, ∞))g∥2 X− = ∥G 1/2 − E((1, ∞))g∥2 X0 = � (1,∞) ���� 1 λ ���� 2 dEg,g ≤ inf λ∈(1,∞) ���� 1 λ ���� 2 ∥g∥2 X0 = ∥g∥2 X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This implies ι− �� X 2 + : X 2 + → X 2 0 and ι−1 − �� X 1 0 : X 1 0 → X 1 − are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In particular, we have X 2 + ⊆ X 2 0 and X 1 0 ⊆ X 1 − densely with continuous embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Step: Dualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Hahn-Banach we can identify (X 2 −)′ with X ′ + �� X 2 −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, for f ∈ X+ and g ∈ X− there exist sequences (fn)n∈N in D+ and (gn)n∈N in D− QUASI GELFAND TRIPLES 25 such that fn → f w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X+ and gn → g w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ∥·∥X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, ⟨E((0, 1])+f, E((1, ∞))−g⟩X+,X− = lim n→∞⟨E((0, 1])+fn, E((1, ∞))−gn⟩X+,X− = lim n→∞⟨E((0, 1])fn, E((1, ∞))gn⟩X0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Clearly, we also have ⟨E((1, ∞))+f, E((0, 1])−g⟩X+,X− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For φ ∈ (X 2 −)′ there exists a g ∈ X− such that φ(f) = ⟨g, f⟩X−,X+ = ⟨E((0, 1])−g, f⟩X−,X+ + ⟨E((1, ∞))−g, f⟩X−,X+ � �� � =0 ∀f ∈ X 2 −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, ∥φ∥(X 2 −)′ = sup f∈X 2 −\\{0} |φ(f)| ∥f∥X+ = sup f∈X 2 −\\{0} |⟨E((0, 1])−g, f⟩X−,X+| ∥f∥X+ = sup f∈X+\\{0} |⟨E((0, 1])−g, f⟩X−,X+| ∥f∥X+ = ∥E((0, 1])−g∥X− On the other hand, if ⟨E((0, 1])−g, f⟩X−,X+ = 0 for all f ∈ X 2 −, then we automati- cally have ⟨E((0, 1])−g, f⟩X−,X+ = 0 for all f ∈ X+ and therefore E((0, 1])−g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In conclusion (X 2 −, X 2 +) is a complete dual pair and (X 2 −, X 2 0 , X 2 +) is a quasi Gelfand triple with the embeddings ι+ �� X 2 − and ι− �� X 2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, since ι− �� X 2 + is continuous, it is even an “ordinary” Gelfand triple (X 2 + ⊆ X 2 0 ⊆ X 2 −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We can show completely analogously that also (X 1 +, X 1 0 , X 1 −) is an “ordinary” Gelfand triple (X 1 + ⊆ X 1 0 ⊆ X 1 −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Note that this decomposition is not unique as we could have split the space X0 by any two subspaces ran E(∆) and ran E(∆∁), where ∆ ⊆ R+ is a bounded non-empty Borel set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Finally, we end with two conjectures Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='7 (weak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Every pre-quasi Gelfand triple of Hilbert spaces is a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='8 (strong).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Every pre-quasi Gelfand triple is a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' At least the weak conjecture seems to be true, but all attempts failed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' In fact Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='10 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='6 are the result of failed attempts to prove the weak conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The strong conjecture seems much more difficult, as a lot of Hilbert space theory is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Conclusion We have introduces a generalization of Gelfand triple that does not need con- tinuous embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This was done by replacing the continuity of the embeddings by closedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We showed that D+ ∩ D−, the set that is in the intersection of the quasi Gelfand triple, is dense in the pivot space X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' If we regard quasi Gelfand triples of Hilbert spaces, then we can show that D+ ∩ D− is also dense in X+ and X− w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' their norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Furthermore, we have shown that there exists a smallest space were we can embed the entire quasi Gelfand triple structure preservingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Finally, we have shown that every quasi Gelfand triple is associated to a Gram operator and the other way round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This led us to a decomposition of the quasi Gelfand triple into two “ordinary” Gelfand triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We ended with the weak and strong version of the conjecture that every pre-quasi Gelfand triple is in fact already a quasi Gelfand triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 26 NATHANAEL SKREPEK One application that we did not cover, that is still worth mentioning: Quasi Gelfand triples can be used to properly define boundary spaces and characterizing suitable boundary conditions for partial differential equations, see [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Auxiliary Results Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (xn)n∈N be a sequence in a normed vector space X that con- verges w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' the weak topology to an x0 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then (xn)n∈N is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' supn∈N∥xn∥X < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let ι denote the canonical embedding from X into X′′ that maps x to ⟨x, ·⟩X,X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then, by assumption, for every fixed φ ∈ X′ (ιxn)(φ) → (ιx0)(φ), in particular supn∈N|(ιxn)(φ)| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The principle of uniform boundedness yields supn∈N∥ιxn∥X′′ < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since ∥ιx∥X′′ = ∥x∥X for every x ∈ X, this proves the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (xn)n∈N be a weak convergent sequence in a Hilbert space H with limit x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then there exists a subsequence (xn(k))k∈N such that ���� 1 N N � k=1 xn(k) − x ���� → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We assume that x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For the general result we just need to replace xn by xn − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We define the subsequence inductively: n(1) = 1 and for k > 1 we choose n(k) such that |⟨xn(k), xn(j)⟩| ≤ 1 k for all j < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This is possible, because (xn)n∈N converges weakly to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='1 supn∈N∥xn∥ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This yields ���� 1 N N � k=1 xn(k) ���� 2 = 1 N 2 N � k=1 N � j=1 ⟨xn(k), xn(j)⟩ = 1 N 2 N � k=1 ∥xn(k)∥2 + 1 N 2 N � j=1 N � k=j+1 2 Re⟨xn(k), xn(j)⟩ ≤ 1 N C2 + 2 N 2 N � j=1 N � k=j+1 1 k ≤ C2 N + 1 N ln(N) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ The next lemma is also true for general linear relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' However, since densely defined linear operators are enough for our purpose we restrict ourselves to these operators, also to use commonly known techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let (X1, Y1), (X1, Z1), (X2, Y2) and (X2, Z2) be dual pairs and Ψ1 : Y1 → Z1 and Ψ2 : Y2 → Z2 be the isomorphisms between Y1 and Z1, and Y2 and Z2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then for a densely defined linear operator A from X1 to X2 we have A∗Z2×Z1 = Ψ1A∗Y2×Y1 Ψ−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let z2 ∈ Z2 be such that Ψ−1 2 z ∈ dom A∗Y2×Y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then ⟨Ax1, z2⟩X2,Z2 = ⟨Ax1, Ψ−1 2 z2⟩X2,Y2 = ⟨Ax1, Ψ−1 2 z2⟩X2,Y2 = ⟨x1, A∗Y2×Y1 Ψ−1 2 z2⟩X1,Y1 = ⟨x1, Ψ1A∗Y2×Y1 Ψ−1 2 z2⟩X1,Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This implies Ψ1A∗Y2×Y1 Ψ−1 2 ⊆ A∗Z2×Z1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The same steps with Z2 and Z1 replaced with Y2 and Y1 yield the reversed inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ QUASI GELFAND TRIPLES 27 Z1 Z2 X1 X2 Y1 Y2 Ψ1 A∗Z2×Z1 Ψ2 A Ψ−1 1 A∗Y2×Y1 Ψ−1 2 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' A∗Z2×Z1 = Ψ1A∗Y2×Y1 Ψ−1 2 The following theorem can be found in [5, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' 200], we just changed that the operator maps into a different space, which does not change the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4 (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' von Neumann).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let T be a closed linear operator from the Hilbert space X to the Hilbert space Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then T ∗T and T T ∗ are self-adjoint, and (IX +T ∗T ) and (IY + T T ∗) are boundedly invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that here the adjoint T ∗ is calculated with respect to the “natural” dual pairs (X, X) and (Y, Y ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' T ∗ = T ∗Y ×X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Since T ∗ = � 0 IY −IX 0 � T ⊥, we have T ⊕ � 0 −IX IY 0 � T ∗ = X × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, for [ h 0 ] ∈ X × Y there are unique x ∈ dom T and y ∈ dom T ∗ such that �h 0 � = � x T x � + �−T ∗y y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (9) Consequently, h = x − T ∗y and y = −T x, which implies x ∈ dom T ∗T and h = x + T ∗T x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Because of the uniqueness of the decomposition in (9), x ∈ dom T ∗T is uniquely determined by h ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Therefore, (IX + T ∗T )−1 is a well-defined and everywhere defined operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For h1, h2 ∈ X, we define x1 := (IX + T ∗T )−1h1 and x2 := (IX + T ∗T )−1h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then x1, x2 ∈ dom T ∗T and, by the closedness of T , T ∗∗ = T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, ⟨h1, (IX + T ∗T )−1h2⟩ = ⟨(IX + T ∗T )x1, x2⟩ = ⟨x1, x2⟩ + ⟨T ∗T x1, x2⟩ = ⟨x1, x2⟩ + ⟨T x1, T x2⟩ = ⟨x1, x2⟩ + ⟨x1, T ∗T x2⟩ = ⟨x1, (IX + T ∗T )x2⟩ = ⟨(IX + T ∗T )−1h1, h2⟩, which yields that (IX + T ∗T )−1 is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Therefore (IX + T ∗T ) and T ∗T are also self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, (IX + T ∗T )−1 is bounded as a closed and everywhere defined operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By T T ∗ = (T ∗)∗(T ∗) the other statements follow by the already shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let T be the operator from the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then dom T ∗T is a core of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Note that dom T ∗T is a core of T is equivalent that to dom T ∗T is dense in dom T with respect to the graph norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Hence, it is sufficient to show that the orthogonal complement of dom T ∗T is {0} w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' the graph inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Suppose dom T ∗T is not a core, then there exists an x ∈ dom T \\ {0} such that 0 = ⟨x, y⟩T = ⟨x, y⟩X + ⟨T x, T y⟩Y = ⟨x, y + T ∗T y⟩X for all y ∈ dom T ∗T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4 (I + T ∗T )y is surjective, which implies x = 0 and contradicts the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ❑ 28 NATHANAEL SKREPEK In the next proposition we will look at the situation where we deal with Hilbert spaces, but work with another dual pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' We will denote the adjoint with respect to the canonical Hilbert space dual pair by ∗h and the adjoint with respect to the other dual pair by ∗d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let X, H be Hilbert spaces, (X, Y ) be a complete dual pair and T : dom T ⊆ X → H be a densely defined and closed linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then T ∗dT : dom T ∗dT ⊆ X → Y is self-adjoint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' (T ∗dT )∗d = T ∗dT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Moreover, dom T ∗dT is a core of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' For x, y ∈ dom T ∗dT we have ⟨T ∗dT x, y⟩Y,X = ⟨T x, T y⟩H = ⟨x, T ∗dT y⟩X,Y , which leads to T ∗dT ⊆ (T ∗dT )∗d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' By Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='4 we already know that T ∗hT is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Let Ψ: X → Y the duality mapping, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ⟨Ψx, y⟩Y,X = ⟨x, y⟩X for x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Then T ∗d = ΨT ∗h (by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='3) and therefore T ∗dT = ΨT ∗hT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Now for x ∈ dom(T ∗dT )∗d and y ∈ dom T ∗dT = dom T ∗hT we have ⟨Ψ−1(T ∗dT )∗dx, y⟩X = ⟨(T ∗dT )∗dx, y⟩Y,X = ⟨x, T ∗dT y⟩X,Y = ⟨x, Ψ−1T ∗d � �� � =T ∗h T y⟩X = ⟨x, T ∗hT y⟩X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' This implies Ψ−1(T ∗dT )∗d ⊆ (T ∗hT )∗h = T ∗hT and applying Ψ on both sides gives (T ∗dT )∗d ⊆ ΨT ∗hT = T ∗dT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' The last assertion follows from dom T ∗dT = dom T ∗hT and dom T ∗hT is a core of T , ❑ References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Cojuhari and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Gheondea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Triplets of closely embedded Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Integral Equations Operator Theory, 81(1):1–33, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' doi:10.' metadata={'source': 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eigenfunctions of differential and other operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Nauk SSSR (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' ), 103:349–352, 1955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Skrepek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' Linear port-Hamiltonian Systems on Multidimensional Spatial Domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' PhD thesis, University of Wuppertal, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='25926/g7h8-bd50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' [4] N.' metadata={'source': 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1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content=' TU Bergakademie Freiberg, Institute of Applied Analysis, Akademiestraße 6, D- 09596 Freiberg, Germany Email address: nathanael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='skrepek@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='tu-freiberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} +page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E3T4oBgHgl3EQflwp-/content/2301.04610v1.pdf'} diff --git a/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf b/x9FJT4oBgHgl3EQfhSy6/content/2301.11565v1.pdf new file mode 100644 index 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has helped uncover real-world attacks, find crypto- +graphic weaknesses, and understand both operator and mis- +creant behavior. Studies that employ scanning have largely +assumed that services are hosted on their IANA-assigned +ports, overlooking the study of services on unusual ports. In +this work, we investigate where Internet services are deployed +in practice and evaluate the security posture of services on +unexpected ports. We show protocol deployment is more dif- +fuse than previously believed and that protocols run on many +additional ports beyond their primary IANA-assigned port. +For example, only 3% of HTTP and 6% of TLS services run +on ports 80 and 443, respectively. Services on non-standard +ports are more likely to be insecure, which results in studies +dramatically underestimating the security posture of Inter- +net hosts. Building on our observations, we introduce LZR +(“Laser”), a system that identifies 99% of identifiable unex- +pected services in five handshakes and dramatically reduces +the time needed to perform application-layer scans on ports +with few responsive expected services (e.g., 5500% speedup +on 27017/MongoDB). We conclude with recommendations +for future studies. +1 +Introduction +Internet-wide scanning—the process of connecting to ev- +ery public IPv4 address on a targeted port—is a standard +research technique for understanding real-world service con- +figuration and deployment. Leveraging tools like ZMap [26] +and Masscan [29], more than 300 papers have used Internet- +wide scanning to discover weaknesses in TLS, SSH, and the +Web PKI [6,9,11,13,15,17,24,36–38], to uncover real-world +attacks [22,50,60], and to better understand botnets [10,46], +ICS/IoT deployment [19,51,67], censorship [42,52,53], and +operator behavior [23,25,47]. +∗Work done while visiting Stanford University. +Past scanning studies have largely assumed that services +are hosted on their IANA-assigned ports (e.g., HTTPS on +TCP/443) and have overlooked scanning additional ports for +unexpected services. Yet, many of these same studies have +also observed that a non-negligible fraction of the hosts that re- +spond to a SYN scan never complete the expected application- +layer handshake [21,24,26,36,51,67]. It is unclear whether +operators hide services on unexpected ports, whether scanners +fail to account for protocol inconsistencies or server-side im- +plementation errors, or whether firewalls detect scanning and +block further interaction. In this work, we investigate where +Internet services are deployed in practice, and we evaluate the +security posture of services hosted in unexpected places. +We start by investigating services that do not appear to +speak the expected IANA-assigned protocol. We confirm that +up to 96% of services (by port) do not complete the expected +application-layer (L7) handshake on 37 popular ports (Sec- +tion 2). We introduce a heuristic that infers server-side TCP +state, which we use to show that 28% of initially-responsive +services do not allow any L7 data exchange. Rather, 12% im- +mediately tear down the connection, 5% prevent an L7 hand- +shake by specifying a zero TCP window, 0.6% are blocked +from receiving our ACK, and 11% “shun” our IP between the +discovery and application-layer scan phases. We trace these +behaviors to middleboxes and firewalls, and we evaluate their +efficacy at enabling scan evasion. +While network defenses account for most L7 unresponsive +services, a significant number of services are TCP compliant, +but fail the expected L7 handshake (e.g., 14% on TCP/80 +and 96% on TCP/102). We show that this is due to services +running on unexpected ports, protocol handshakes that re- +quire pre-established secrets, and network-based protections +that acknowledge data on every port but speak no detectable +protocol (Sections 3–4). Notably, protocol deployment is ex- +ceptionally diffuse. For example, only 3.0% of HTTP and +6.4% of TLS services run on ports 80 and 443, respectively. +Achieving 90% coverage of TLS-based services requires scan- +ning 40K ports. Worryingly, services deployed on unexpected +arXiv:2301.04841v1 [cs.CR] 12 Jan 2023 + +ports have worse security postures, which we trace back to +IoT devices that host insecure services on non-standard ports. +To enable researchers to more comprehensively find In- +ternet services, we introduce LZR (“Laser”), a system that +efficiently filters hosts that do not speak any L7 protocol +and identifies unexpected services (Section 5). LZR can fin- +gerprint 88% of identifiable services with a single packet +and 99% of identifiable unexpected services with five hand- +shakes. LZR also speeds up scans by quickly filtering the +bulk of seemingly-responsive hosts that SYN-ACK but cannot +complete an application layer handshake. For example, on +port 27017, LZR filters out 80% of hosts that SYN-ACK, de- +creasing the time to complete scans of MongoDB by 55 times, +while still identifying 99.6% of MongoDB services and iden- +tifying an additional 23K hosts running unexpected protocols +(a 31% coverage increase for the port). +Our work concludes with recommendations for future stud- +ies. We hope that by shedding light on the ecosystem of unex- +pected services, and by releasing LZR as an open-source tool, +we enable security researchers to more accurately understand +Internet services. +2 +Identifying Real TCP Services +Fast research scans of the Internet are typically conducted in +two phases today [21, 26, 36, 38]. In the first stage, a scan- +ner like ZMap [26] statelessly sends SYN packets to public +IPv4 addresses. Then, in a second process, a stateful scan- +ner like ZGrab [21] performs complex follow-up handshakes +using the kernel TCP/IP stack. The two-phased nature of +Internet scanning is largely attributable to ZMap’s architec- +ture, which uses a stateless network stack to efficiently probe +services, but is unable to complete handshakes that require +maintaining local state. The biases and unintended conse- +quences from scanning in two phases have not been inves- +tigated, and worryingly, prior studies have repeatedly noted +that more than half of the IPv4 hosts that respond to a SYN +scan never complete a follow-up application-layer handshake +(e.g., [24,26,36,51,67]). +In this section, we investigate this discrepancy. We show +that TCP liveness does not accurately indicate the presence of +an application-layer service due to several common security +protections, including middleboxes and user-space firewalls. +Guided by TCP’s design [54], we uncover five defensive be- +haviors that degrade the signal provided by L4 responsiveness. +We quantify the deployment of these defenses, and we eval- +uate their efficacy at protecting against DDoS attacks and +evading Internet scans. We then go on to develop a better L4 +heuristic to approximate application-layer liveness, which we +use to better understand service deployment in Section 3. +2.1 +Layer 4 versus Layer 7 Liveness +We start our investigation by confirming whether TCP- +responsive hosts (i.e., hosts that reply with a SYN-ACK packet) +complete the IANA assigned [39] application-layer hand- +shake. Mimicking prior Internet scans (e.g., [6,9,16,36,72]), +we perform a two-phase scan in which we send a SYN packet +to a random 1% sample of public IPv4 addresses using +ZMap [26] and immediately attempt a follow-up application +handshake using ZGrab [21]. We scan all IANA-assigned +ports with available ZGrab scanners (i.e., 37 ports in +Appendix A) on November 12–14, 2019. We follow the +best practices set forth by Durumeric et al. [26] to minimize +scan impact, and we exclude networks that have previously +contacted us. We receive no complaints, but note that we +have used our network in the past for other experiments and +exclude operators who previously requested removal. +Consistent with prior studies [24,26,36,51,67], we find that +a considerable fraction of TCP-responsive hosts never com- +plete the expected L7 handshake (Figure 1). The raw number +of L7-unresponsive hosts varies from 21K unresponsive hosts +on 502/Modbus to 201K hosts on 443/HTTPS (µ = 54,542, +σ2 = 31,002). We see this heavy-tail distribution throughout +our investigation and we present our results for both popular +and unpopular ports. We split ports into the two categories +using Grubbs’s test for outliers [30] with a 99.9% confidence +interval based on the total number SYN-ACKs and the presence +of an expected service. Our popular set contains ports 80, 443, +7547, 22, 21, and 25; the unpopular set contains the remaining +31 ports. Popular protocols are most likely to complete the +expected L7 handshake:1 86% and 80% of TCP-responsive +hosts on ports 80 and 443 complete an HTTP(S) handshake +while only 9% and 4% of hosts on ports 502 and 102 speak +Modbus and Siemens S7 (two SCADA protocols). +In the following section we start our investigation of L7- +unresponsivess by analyzing the changing state of services +between the two phases of scanning. +2.2 +Connection Shunning +About 1.6% of services on popular ports and 5% of services +on unpopular ports do not respond with a SYN-ACK during +our follow-up ZGrab TCP handshake. This could be due to +DHCP churn, transient network failure, or the destination +host blocking the scanner between handshakes (“connection +shunning”). To determine whether hosts “shun” scanners, we +connect to TCP-responsive hosts found by ZMap from two IP +addresses: the original IP address used by ZMap to identify +the host and a fresh IP that has not previously contacted the +host. We scan a random ephemeral port, 48302, because we +see the largest fraction of disappearing hosts on unpopular +ports. We find that 70% of IPs that do not respond a second +1Spearman’s Correlation p-value of port rank (based on number of SYN- +ACK) relative to L7 and SYN-ACK percent difference is 5×10−11. + +80/HTTP +443/TLS +7547/HTTP +22/SSH +21/FTP +25/SMTP +8080/HTTP +4567/HTTP +53/DNS +110/POP3 +3306/MYSQL +143/IMAP +3389/RDP +587/SMTP +993/IMAPS +995/POP3S +465/SMTP +23/TELNET +8443/TLS +1723/PPTP +5432/POSTGRES +1883/MQTT +5672/AMQP +8883/MQTT +1521/Oracle +6379/redis +5900/VNC +20000/DNP3 +1433/MSSQL +445/SMB +631/IPP +6443/Kubernetes +623/IPMI +27017/Mongodb +502/Modbus +102/Siemens +11211/memcached +Port/Service +0 +2 +4 +6 +8 +IPs (100,000s) +SYN-ACK only +L7 Handshake +Figure 1: L4 vs. L7 Responsiveness—A significant frac- +tion of hosts that respond with a SYN-ACK packet never com- +plete the expected application-layer handshake. The differ- +ence varies dramatically across ports by both percent differ- +ence (14–96%) and raw count (21,050–200,902). +time on the used IP do respond to the fresh IP, indicating that +most hosts that go missing between scan stages are typically +not lost due to churn or network failure. +In the case that the fresh IP receives a SYN-ACK, we ob- +serve two types of responses from the previously-used IP: no +response (93%) and RST packet (7%). This blocking occurs at +the IP granularity: once a scanner has been blocked by a host, +the host will not respond with a SYN-ACK on any port. We +further confirm that connection shunning is not a defensive re- +action—triggered by failing to complete an application layer +handshake—by running a 1% IPv4 scan of all popular ports +using ZGrab for the initial host discovery. The same fraction +of connections are shunned as when ZMap is used. +We find that connection shunning is deployed at both the +host and network granularity by computing the largest blocks +of consecutive TCP-Responsive IPs that show shunning be- +havior on a random ephemeral port: 40% of networks that +shun scanners are /32s (i.e., individual hosts) and 10% of +IPs block in groups larger than a /24 (Figure 4). The largest +network to deploy connection shunning is a /20 owned by +Alestra Net (ASN 11172), a Mexican ISP. +Both network hardware (e.g., Cisco IOS-based routers [34]) +and host software (e.g., Snort [59]) document connection +shunning and dynamic blocking as features where connec- +tions are blocked after an IP is classified as malicious. Connec- +tion shunning prevents clients from using a single source-IP +to scan the network and forces scanners to use multiple source +IPs to reach the end-host, thereby dramatically increasing the +cost for an attacker. We compare the number of legitimate ser- +vices found when using both single and multiple source-IPs +during scanning and find no evidence that any hosts that shun +connections host legitimate services. We thereby conclude +that they can be safely ignored in security studies if they can +be efficiently filtered. +2.3 +Do TCP-Responsive Hosts Speak TCP? +The vast majority of services (average of 96% across ports) +that do not complete an application-layer handshake respond +with a SYN-ACK during the second (ZGrab) handshake. In the +remainder of the section, we explore whether these hosts reach +a state where they can exchange application-layer data or sim- +ply stop responding after sending a SYN-ACK. In Figure 2, we +provide a modified TCP state diagram based on RFC 793 [54] +that captures what a scanner can infer about a server’s TCP +state, which we use to guide our investigation. For a TCP +connection to enter the ESTABLISHED state, the server sends +only a single packet (SYN-ACK). Once the client has sent an +ACK, it can normally send data—the amount specified by the +server window size in the SYN-ACK packet. +We note that TCP has an edge case in which the server can +respond with a zero-sized window in its SYN-ACK [54]. In this +situation, the client is expected to send follow-up ACK packets +to probe when the server is ready to accept data. We add a new +ACCEPTS DATA state in Figure 2 to capture whether a server +is ready for data. Once the server has reached the ACCEPTS +DATA state, it is expected to keep the TCP connection open +long enough to receive data and to acknowledge receipt. We +define ACKNOWLEDGES DATA as the server allowing the client +to send data and acknowledging client data. +LISTEN +SYN RECEIVED +ESTABLISHED +Receive : SYN +Send: SYN-ACK +Receive: ACK +Receive: Data or Close or Timeout +Send: Close or Timeout +Send: SYN-ACK +Receive : Close or Timeout +ACKNOWLEDGES DATA +Receive: Data +Send: ACK +Window Size > 0? +Yes +No +Receive: Timeout +Send: Close or Timeout +ACCEPTS DATA +Figure 2: Client Perspective of Server TCP State—We +investigate L7 service liveness based on a modified version +of the TCP state machine in RFC 793 [54]. We introduce two +new states: “accepts data” and “acknowledges data” because +an established connection cannot necessarily exchange data. +To test how far into a TCP session servers reach, we de- +velop a new scanner based on ZGrab [5] that establishes a TCP +connection, sends two newlines, and deduces the server TCP +state (Algorithm 1). We scan random 1% samples of IPv4 +addresses on a random 2,000 ports as well as the 37 IANA +assigned ports that host protocols with ZGrab scanners (Ap- +pendix A). An average 16% of services on popular ports and +40% of services on unpopular ports fail to acknowledge data +(Figure 3a). We detail why in the remainder of this section. + +(a) Portion of TCP-responsive hosts that fail to acknowledge data +0 +50 +100 +# (1000s) +Connection Shunning +Dropping Connections Mid-Handshake +Zero Window +Dynamic Blocking (Handshake) +Reset Connection +Leftover Non-ACK Hosts +80 +443 +7547 +22 +30005 +5060 +21 +25 +2000 +8080 +50805 +4567 +53 +49154 +49152 +8081 +8089 +110 +3306 +8085 +8000 +143 +51005 +3389 +587 +58000 +993 +995 +465 +23 +8443 +1723 +179 +5432 +1883 +5672 +8883 +1521 +53194 +62220 +6379 +5900 +20000 +161 +65535 +1433 +445 +631 +6443 +623 +47808 +27017 +502 +102 +11211 +Port Number +0 +1 +Fraction +Non-Acking IPs +(b) Reasons SYN-ACK-only hosts fail to acknowledge data +Figure 3: Unexpected TCP Behavior of IPv4 Hosts—An average 16% of services on popular ports and 40% of services on +unpopular ports that respond to a TCP SYN scan with a SYN-ACK packet do not fully speak TCP. Here, we show the portion of +hosts by port that do not acknowledge client data and the breakdown of reasons why. +Algorithm 1: Deducing Server TCP State +Send SYN +if receive RST or FIN or Timeout then +return NO_ACK_HOST +end +// checking for zero window sizes +Print syn-ack.window_size +// sending protocol-agnostic data +Send "\n\n" +// Time for 8 re-transmissions (RFC 1122 rec.) +while timeout < 100 seconds do +if received ACK then +return ACK_HOST +end +if received RST or FIN then +return NO_ACK_HOST +end +end +return NO_ACK_HOST // host has timed out +2.4 +Zero Window DDoS Protections +Of the services that never acknowledge data, 13% of services +on popular ports and 26% on unpopular ports actively prevent +clients from sending data by specifying a zero-sized TCP +window and never increasing it. Across all scanned ports, +at least 99.94% of hosts with a zero window never increase +it; 90% do not respond to secondary probes and 10% reset +the connection. The behavior appears to be network- or host- +based rather than service-based: 99% of hosts that respond +Figure 4: Network Granularity of TCP Blocking—Some +protections appear to be host-based while others are more +prevalent on large networks. Zero Window DDoS protections +are most likely to appear at a large network granularity, while +connection shunning is more likely a host-level behavior. +with a zero-window on one port will send a zero-sized window +on all ports. Offhand, this behavior appears self-defeating. +Hosts that respond and never increase window size might +as well never respond. However, we find the feature in a +Juniper networks patent [66] and used in Juniper’s Secure +Service Gateway Proxy [41] to prevent DDoS attacks through +network-based SYN cookies. The protection responds to all +SYN packets with a zero-window SYN-ACK. Once the client +completes the three-way handshake by sending an ACK, the +firewall sends a SYN packet to the backend server to establish +the connection. By maintaining a zero-sized TCP window +with the client, the middlebox prevents the client from sending +data it cannot yet forward to the backend server. + +Zero-window SYN-ACKs are deployed across entire sub- +networks: 90% of IPs that SYN-ACK with a zero window +do so in a network larger than a /24 (Figure 4). The largest +network, the State of Florida Department of Management Ser- +vices (ASN 8103), is responsible for 16% of all zero-windows +Internet-wide and accounts for around 3% of all SYN-ACKs +on a random port. The TTL for SYN-ACK is consistently one +hop closer than the later RST, further confirming a network +appliance is responsible. +2.5 +Dropping Connections Mid-Handshake +Beyond specifying a zero window, an average 2% of the +hosts per port that never acknowledge data do not appear +to complete a three-way handshake, despite the client sending +an ACK (Figure 3b). We infer that the server never reaches +the ESTABLISHED state based on a continual stream of SYN- +ACK packets (average 7.8 SYN-ACK re-transmissions). Hosts +do not simply have broken TCP stacks; in the case of MCI +Communication Services, for example, IPs that re-transmit +SYN-ACKs on port 4567 have compliant behavior on other +ports (e.g., RDP on TCP/3389). Real services respond with a +TTL over twice as large as the TTL value which re-transmits +the SYN-ACK, suggesting that a middlebox selectively drops +packets. Dropping connections mid-handshake is a defensive +behavior exhibited primarily by ISPs protecting consumer +premise equipment: CenturyLink (AS 209), Frontier Com- +munications (AS 5650), and MCI Communications Services +(AS 701) all drop inbound traffic to port 4567/TRAM post- +SYN (accounting for 96% of dropped connections). Korea +Telecom (AS 4766) and Axtel (AS 6503)—accounting for +73%—interrupt connections on 7547/CWMP. The behavior +is rare on common ports (e.g., only 5% of TCP-responsive +hosts that do not acknowledge data drop connections mid- +handshake on port 80). +2.6 +Reset Connections +An average 73% of services on popular ports and 34% of +services on unpopular ports that do not acknowledge data +reach the ESTABLISHED state but will immediately reset the +connection after the client completes the three-way handshake +(Figure 3b). Per RFC 793 [54], if a server does not want to +communicate with a client (e.g., due to mismatches in “secu- +rity clearances”), the server should close the TCP connection +after the client acknowledges the SYN-ACK. This is also how +user-space firewalls like DenyHosts [63] appear to scanners. +While we cannot detect what software closes a connection, +we note that networks that RST on port 22 are 10 times more +likely to do so in block-sizes of /32 than port 80, implying that +blocking happens more often on hosts running SSH compared +to HTTP, consistent with Wan et al.’s findings [69]. Network- +level behavior looks to be caused by DDoS protections similar +to the networks that send zero-window SYN-ACKs. To pro- +tect against SYN-flooding, middleboxes send a SYN-ACK on +behalf of the server and later establish a connection with the +server after the client has finished the three-way handshake. +If the server refuses the connection, the middlebox terminates +the client connection. This functionality is available in Cisco +IOS-based routers as a part of their threat detection logic [58]. +The behavior is visible in prominent networks, with more +than 40% of such IPs located in Korea Telecom, Vodaphone +Australia, OVH, and Akamai. Hosts are 20% more likely to +close a connection on popular ports because Google load bal- +ancers in AS 19527 come with a standard firewall policy that +accept traffic on these ports by default—in order to be able +to perform service health checks—and rely on the backend +virtual machine to reset connections if the port is closed [1,2]. +2.7 +Dynamic Blocking after Handshake +Not all hosts that fail to acknowledge data send RSTs or contin- +ually re-transmit SYN-ACKs. Many simply never acknowledge +any data. An average of 10% services on popular ports and +18% of services on unpopular ports do not acknowledge client +data (Figure 3b). These hosts frequently do not respond to +later follow-up handshakes either. This “shunning” behav- +ior is similar—but not identical—to the behavior we found +in Section 2.2 and has previously been documented in the +Great Firewall of China [18] where it is used to stop future +connections, triggered only when data is sent. +To differentiate between hosts that shun the scanner after +a handshake from those that simply never acknowledge data, +we simultaneously attempt an L7 handshake with initially- +responsive hosts that did not acknowledge data from two IP +addresses, one that matches the initial connection and one +that differs. Of the initially unresponsive IPs, 98% respond +to the fresh IP, indicating the behavior is not likely due to +transient network failure, but rather explicit blocking of in- +coming connections. In total, post-handshake dynamic block- +ing accounts for 6% and 12% of the remaining hosts that do +not acknowledge data for common port and uncommon port +hosts respectively. Note that this behavior only occurs after +a three-way handshake, thereby differing from connection +shunning (Section 2.2). The largest network to dynamically +block after a handshake is Coming ABCDE HK (AS 133201), +which accounts for 48% of all IPs that block after a handshake. +We also discover a similar TTL phenomenon as described in +Section 2.4 implying a middlebox-based protection. +We deduce that the rest of the hosts that fail to acknowledge +data are not performing dynamic blocking because though +they will not respond to anything after the actual handshake, +they do consistently respond to all scans (no matter the source +IP). Vodaphone (AS 133612) and Webclassit (AS 34358) have +this behavior across all scanned ports and make up 66% of +all IPs with such a behavior. We find similar evidence of +mismatching TTL values, which indicate a middlebox. + +2.8 +Efficacy of Middlebox Protections +Identifiable middlebox protections are common. About 16% +of the services on popular and 40% of the services on unpop- +ular ports that respond to a SYN packet—but do not speak +any identifiable L7 protocol—are artifacts of DDoS and scan- +ning protections; 40% of routed ASes contain at least one +such protection. Reset connections after a handshake—a be- +havior found in software like DenyHosts [63]—is by far the +most common behavior by both IP and AS, and is present in +34% of ASes. Middleboxes employing connection shunning +or dynamic blocking are each used by 6% of networks, and +Juniper’s patented zero-window DDoS protection appears +in 2% of networks. These protections prevent clients from +directly connecting to servers—at least initially—and all +middleboxes succeed at doing so, even if the protection is +identifiable. However, with the use of more than one source +IP address, an adversary can bypass connection shunning and +dynamic blocking and still solicit SYN-ACKs from the end- +host, albeit rate-limited by the number of scanner addresses. +Beyond actively preventing DDoS attacks and some scan- +ning, each protection inadvertently slows down the discovery +of new services through Internet scanning and can slow down +the spread of malware. Dynamic blocking (completing the +handshake without acknowledging data) is the most effective +at doing so. The technique slows scans by up to 55 times as +in the case of host discovery on 27017/MongoDB (Section 5), +by forcing the scanner to timeout upon not receiving an ACK +for each scanned host. Though zero window SYN-ACKs also +cause a scanner to eventually timeout, zero-sized windows +are easy to filter. Immediately closing the connection after the +handshake causes only a negligible slowdown, bounded only +by the time it takes to complete a handshake (about 100 ms). +Connection shunning is the least effective at slowing down +stateless scanners but slows down stateful scanners at the +same rate as dynamic blocking. +2.9 +Summary +Our results establish that SYN-ACKs are a poor indicator for +the presence of a service. In the worst case, SYN-ACKs overes- +timate the hosts that acknowledge data by 533% on port 11211 +(memcached). We also discover that an average 16% of ser- +vices on popular ports and 40% of services on unpopular ports +fail to acknowledge data, which is a likely indicator for the +presence of a middlebox protection. We investigate why hosts +that appear to fully speak TCP do not always complete L7 +handshakes in the next section. +3 +Application-Layer Service Deployment +In the last section, we investigated L4-responsive services that +do not appear to speak any L7 service and are artifacts of DoS +and scanning protections. After excluding the 28% of pseudo- +80/HTTP +443/TLS +7547/HTTP +22/SSH +21/FTP +25/SMTP +8080/HTTP +4567/HTTP +53/DNS +110/POP3 +3306/MYSQL +143/IMAP +3389/RDP +587/SMTP +993/IMAPS +995/POP3S +465/SMTP +23/TELNET +8443/TLS +1723/PPTP +5432/POSTGRES +1883/MQTT +5672/AMQP +8883/MQTT +1521/Oracle +6379/redis +5900/VNC +20000/DNP3 +1433/MSSQL +445/SMB +631/IPP +6443/Kubernetes +623/IPMI +27017/Mongodb +502/Modbus +102/Siemens +11211/memcached +Port/Service +0 +2 +4 +6 +8 +IPs (100,000s) +SYN-ACK only +ACK Data +L7 Handshake +Figure 5: SYN-ACK vs. Ack. Data vs. L7 Handshake— +There are up to three orders of magnitude fewer IPs that +acknowledge data than respond with a SYN-ACK packet. +services, we discover 27% of services on popular ports and +63% services on unpopular ports that acknowledge data do +not run the expected application-layer protocol (Figure 5). In +this section, we analyze services that complete unexpected +application-layer handshakes or acknowledge data but do not +speak any identifiable application-layer protocol. We show +that while IANA-assigned services are prominent on popular +ports, unexpected but identifiable services dominate other +ports. Moreover, assigned ports only host a tiny fraction of the +services that run popular protocols. For example, only 6.4% +of TLS services run on TCP/443. Services on unexpected +ports are commonly hosted by IoT devices and have weaker +security postures, which suggests the need for the security +community to study the services on unassigned ports. +3.1 +Finding Unexpected Services +To determine the extent to which unexpected services co- +reside on ports with assigned services, we scan 1% random +samples of the IPv4 address space on the set of ports from +Section 2.3 (37 ports with an expected service and 18 ports +without an unexpected service or implemented scanner). For +each responsive service, we first attempt to complete an L7 +handshake using the expected protocol, if one exists. Upon +failure, we attempt follow-up handshakes using the 30 proto- +col scanners—the total number of unique protocol scanners— +implemented in ZGrab (Appendix A) with default parameters. +Ethical considerations. +Prior studies have primarily per- +formed Internet scans that target only expected protocols; to +minimize the potential impact of our experiment, we scan only +1% of the IPv4 address space. We received zero abuse com- +plaints, requests to be blocked from future scans, or questions +from operators from this set of experiments. +Data acknowledging firewalls. +The number of data- ac- +knowledging services per IP follows a bi-modal distribution: +98% of IPs serve fewer than four unidentifiable services and +2% of IPs host unidentifiable services on over 60K ports. +About 75% of all unidentifiable services on unpopular ports + +are hosted by IPs with unidentifiable services on nearly every +port (“Unknown Service - across ports” in Figure 6). Hosts +have unidentifiable services on most but not all ports because +some networks drop all traffic to security-sensitive ports. For +example, out of the top 50 networks that send back the most +SYN-ACK responses across all ports, 28% drop all traffic to +port 445 (SMB) and 10% drop port 23 (Telnet). Hosts with +unidentifiable services on nearly every port are concentrated +in a small number of networks; five ASes belonging to the +Canadian government (74, 25689, 818, 2680, and 806) ac- +count for 77% of all IPs that host unidentifiable services on +nearly every port. +We trace this behavior to the F5 Big-IP Firewall based on a +RST fingerprint [3] that contains the words “BIG-IP System.” +An F5 DevCentral blog post [4] speculates that IPs respond on +every port due to the accidental use of a wildcard when config- +uring the firewall or an overload of the firewall’s SYN-cookie +cache. We identify and exclude these hosts, to avoid biasing +our analysis, by checking whether hosts acknowledge data +on five random ephemeral ports, which effectively filters out +99.9% of such hosts. Nonetheless, an average of 10% of popu- +lar and 25% of unpopular services remain unidentifiable (i.e., +do not respond to any of the 30 handshakes) after filtering. +3.2 +Characterizing Unexpected Services +After filtering out hosts with unknown services on nearly all +ports, we investigate unexpected services on assigned ports +and services on ports without any assigned service. We sum- +marize our results in Figure 6 and describe them here. +Unexpected services. +Services on popular ports typically +run the expected protocol: 93% of hosts that acknowledge +data on port 80 respond to an HTTP GET request and 89% +on port 443 complete an HTTPS handshake (Figure 6). Only +1.6% of the services on port 80 and 4.25% of services on +port 443 respond to one of the other 30 unqiue handshakes. +The majority (75%) of unexpected services on port 80 are +TLS-based and nearly all on port 443 are HTTP-based (Fig- +ure 7). This implies that operator recommendations to run +services on ports 80 or 443 to bypass firewall restrictions [49] +are not widespread. As ports decrease in popularity, the frac- +tion of IPs that speak the expected service approaches zero. +For example, on port 623, only 1% of services that acknowl- +edge data speak IPMI and 18.9% speak other identifiable +protocols. Consequently, the number of additionally identi- +fiable services diminishes after the first few protocols and +appears to converge at 96% (Figure 8). Each port contains its +own long-tail of unexpected services, but for many ports, this +number plateaus quickly—just not at 100%. +The number of identifiable services on ports without an as- +signed service varies between 2–97% based on port. Among +random ephemeral ports, our 30 handshakes identify the pro- +tocol for an average 21% of services that acknowledge data +and an average of 10 unique protocols per port. Across all +scanned ports, nearly 65% of unexpected, but identifiable, +services speak HTTP and 30% speak TLS. IoT devices are +a prominent culprit behind unexpected services; unexpected +TLS services are 5 times more likely and unexpected SSH +2 times more likely to belong to an IoT device than 443/TLS +and 22/SSH services, respectively. We also find evidence of +operators attempting to hide services. For example, 70% of +hosts serving TLS on the random ephemeral ports 49227, +47808, and 49152 are issued certificates by BBIN Interna- +tional Limited, a Philippine offshore online gambling plat- +form [56]. We further detail the types of services hosted on +unassigned ports in Sections 3.3. +Long tail of ports by protocol. +Our results suggest that +protocols run on many additional ports beyond their primary +IANA-assigned port. To quantify how many ports researchers +need to scan to achieve coverage of a protocol, we conduct +a new scan targeting 0.1% of the IPv4 address space on +10 popular protocols on all 65,535 ports and compute the +fraction of hosts running a given service across multiple +ports (Figure 9). We find that port 80 contains only 3.0% of +hosts running HTTP; another 1.2% of HTTP hosts run on +port 7547 and 0.7% on port 30005. To cover approximately +90% of HTTP, one must scan 25,000 ports. Only 5.5% of +Telnet resides on TCP/23, with the assigned alternative +port TCP/2323 being only the 10th most popular; other +unexpected ports dominate the top-10 ports with the most +Telnet services (Table 1). Previous work tracking botnet +behavior [10,44] has primarily studied assigned Telnet ports +(i.e., 23, 2323); our findings imply that the attack surface and +number of potentially vulnerable devices is potentially over +15 times worse than previously shown. +Some protocols are still relatively clustered around their as- +signed ports. For example, 83.1% of all AMQP is on port 5672 +and an additional 3.1% is on port 5673. HTTP and TLS are +the only two protocols which appear on every port in our 0.1% +IPv4 scan. The set of most popular ports also varies per pro- +tocol and is often not correlated with the popularity of ports +that send data (i.e., across all protocols), as most services are +drowned out by the overwhelming popularity of HTTP and +TLS. For example, 7 of the top 10 ports most likely to host Tel- +net are ranked above 12,000 in overall popularity. As a result, +when choosing which popular ports to study for a specific +protocol, we recommend researchers conduct a lightweight +sub-sampled scan across all ports. +3.3 +Security of Unexpected Services +Services on unexpected ports are more likely to be insecure +than services on assigned ports. We use the results from our +experiment in Section 3.1 (scanning 30 protocols on 55 ports) +to show four examples of how unexpected services affect the +results of previous and future security studies. + +Figure 6: Distribution of Types of Services—A smaller fraction of services run the assigned protocol on less popular ports. +For example, only 4% of services on TCP/102 speak the assigned S7 protocol. The fraction of services that can be identified on +unassigned ports (on the right hand side) varies widely. +TLS +REDIS +SMTP +HTTP +TLS +TLS +HTTP +HTTP +TLS +HTTP +TLS +VNC +FTP +SSH +MQTT +TELNET +SSH +90% +80% +60% 70% +50% +40% +30% +20% +10% +Fraction of IPs (known unassigned service) +Port +80 +7547 +22 +21 +all +443 +Figure 7: Distribution of Unexpected Services—HTTP and +TLS are the most popular unexpected services, with 65% of +unexpected services speaking HTTP and 30% speaking TLS. +IoT devices. +IoT devices are frequent targets due to their +consistently weak security designs [28, 48, 70]. While pas- +sive measurement has shown that a significant number of +IoT devices inhabit non-standard ports [45], active mea- +surement of IoT devices has largely studied only standard +ports [14,20,27,55,62,71]. By manually identifying server +certificates belonging to an IoT manufacturer, we find IoT +interfaces on unexpected ports are widespread; 50% of TLS +server certificates on unexpected ports belong to IoT devices +and unexpected TLS is 5 times more likely to belong to an +IoT device than on port 443. For example, 35% of 8000/TLS +are icctv devices (i.e., surveillance cameras) in Korea Tele- +com and 38% of 80/TLS are Huawei network nodes spread +across 1% of all international networks. About 5% of TLS on +port 8443 belongs to Android TVs in Korean networks and at +least 20% belongs to routers. Unassigned ports also contain +more TCP/UPnP devices. For example, there are 12 times +more TCP/UPnP devices on port 49152 (primarily in Latin +America and Asian Telecoms) and 2 times as many on ports +58000 and 30005 than on port 80. +Vulnerable TLS. +TLS services on unassigned ports are +1.17 times more likely to have a certificate with a known +Figure 8: Protocol Coverage Convergence—The marginal +gain of scanning additional protocols is negligible beyond +the top 10 protocols. Still, for most ephemeral ports (e.g., +port 49227) the majority of services remain unknown. +private key than on assigned ports. When scanning unassigned +ports, we find over twice as many certificates have a known +private key than reported in prior work [32,36]. For example, +40.2% of TLS hosts on port 8081 are DOCSIS 3.1 Wireless +Gateways in Telecom Argentina (AS 10481 and 10318) using +the same OpenSSL Test Certificate with a known private key +and 39% of TLS hosts on port 58000 are Qno wireless devices +with the same self-signed certificate with a known private key. +Across 23% of scanned ports, public keys are more likely— +up to 1.7 times more—to be shared than those on port 443 +(e.g., 80/TLS is 1.5 times more likely). Nonetheless, previous +work studying cryptographic keys on the Internet [26, 32, +36] has limited analysis to 443/HTTPS, 22/SSH, 995/POP3S, +993/IMAPS, and 25/SMTPS. +Login pages. +Over half of unexpected ports scanned host a +higher fraction of public-facing login pages (i.e., HTML con- +taining a login, username, or password field) than 80/HTTP +and 443/HTTPS. Though the total number of HTTP login +pages is greatest on port 80, a page on 8080/HTTP is 2.4 times +more likely to be a login page, thus offering an additional +25% of such pages compared to port 80. Furthermore, all the +aforementioned IoT devices (e.g., icctv, routers) hosting TLS +also serve a login HTTPS page on their respective ports. + +Measure Names +80 +Bacnet + Memcached +I Modbus +443 +Snmp +Oracle +7547 +Siemens +Port +Prometheus +Mongodb +Dns +Postgres +Elasticsearch +Dnp3 +Smb +Amqp +Imap +Ftp +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 + Kubernetes +FractionofKnownUnassigned Services +Ipp +Vnc +Amqp,Bacnet, Dnp3, Dns, Elasticsearch, Ftp,Http, Imap,Ipmi,Ipp, Kubernetes, Memcached, Modbus, Mongodb, Mqtt, +Mysql +Port. Color shows details about Amqp, Bacnet, Dnp3, Dns, Elasticsearch, Ftp, Http, Imap, Ipmi, Ipp, Kubernetes, Memcached +Pptp +Telnet +Telnet,Tls and Vnc. +Rdp +Smtp +Mqtt +Redis +Ssh +Pop3 +TIs +Http +IpmiFigure 9: Protocol Coverage Across Ports—Only 3.0% of +HTTP services are served on port 80. Researchers must scan +25K ports to achieve 90% coverage of HTTP services. On the +other hand, 83.1% of AMQP services are on port 5672. +Port +Hosts +Top AS +% of Hosts +in Top AS +23 +2,606 +Telecom Argentina (10318) +8.7% +5523 +521 +Claro S.A (28573) +87% +9002 +396 +Fastweb Italia (12874) +4% +6002 +232 +Fastweb Italia (12874) +6% +8000 +158 +Powercomm KR (17858) +89% +Table 1: Top 5 Ports Hosting Telnet—While Telnet is most +often seen on its assigned port (TCP/23), the majority of +Telnet services are served on unassigned ports. Unexpected +Telnet devices are sometimes spread across a large number +of ASes (e.g., port 9002) and are therefore likely not due to a +single operator decision. +SSH hygiene. +Unexpected ports hosting SSH are 15% more +likely to allow non-public key authentication methods (e.g., +password, host-based, challenge-response) than 22/SSH and +2.4 times less likely to be using only public key authentica- +tion (11% vs. 26%). 60% of scanned ports are on average +2 times more likely (9% vs. 18%) to be running a software +implementation of SSH that is likely to be on an IoT device +(e.g., Dropbear, Cisco, Huawei). +3.4 +Summary and Implications +Most services that acknowledge data on popular IANA- +assigned ports run the expected L7 protocol, but this drops +to nearly zero for less popular protocols with assigned ports. +The majority of services that speak popular protocols (e.g., +TLS, Telnet, HTTP) are spread across all 65K ports rather +than on their assigned port(s). For example, only 3% of HTTP +services listen on port 80. Many of the services listening on +random ports belong to IoT devices and/or have a weak se- +curity posture, and it behooves the security community to +consider these services when quantifying risk. +4 +Efficiently Identifying Services +L7 scanning is more challenging when there is no assigned +protocol for a port or when the expected L7 handshake fails. +Though Section 3.3 demonstrates the importance of scan- +ning for unexpected services, the naive method we used tests +30 unique L7 handshakes and is too intrusive and slow for +large-scale experiments. In this section, we explore how to +most efficiently detect unexpected L7 services. Encourag- +ingly, only five handshake messages are needed to uncover +99% of unexpected services running identifiable protocols. +4.1 +Protocol Discovery +We investigate two directions for accelerating protocol discov- +ery: (1) methods that trigger protocol-identifying responses +on a large number of protocols and (2) attempting handshakes +in an order that optimizes for efficient service discovery. +Wait and fingerprint. +The most efficient first step for de- +tecting the protocol on a port is to simply wait to send any +handshake message and to see what the server sends first. A +total of 8 of the 30 protocols implemented in ZGrab—POP3, +IMAP, MySQL, FTP, VNC, SSH, Telnet, and SMTP—are +“server-first” protocols: after a TCP handshake concludes, the +server will send a banner to the client, which allows the client +to parse and identify the actual service. For example, 99.99% +of hosts which complete an SSH handshake have the keyword +sshin the SSH banner, 90% of SMTP banners contains smtp, +72% of Telnet contains login or user, and 100% of VNC re- +sponses contain RFB. We are able to identify banner signatures +for all implemented binary and ASCII-based protocols. +We also find that many protocols respond to incorrect hand- +shake messages, including HTTP and TLS. Through 1% scans +of the IPv4 space, we find that 16 of 30 protocols respond to +an HTTP GET request or two newline characters for at least +50% of public services that speak the protocol (Figure 10). +In general, most services that respond to the wrong hand- +shake respond to both a GET request and TLS Client Hello, +but MongoDB, and Redis do not send data in response to +a TLS handshake. Though sending two newline characters +is protocol-compliant for many ASCII protocols, doing so +discovers fewer services than TLS and HTTP. We discover +a similar phenomenon when sending 50 newline characters, +thereby implying that the contents of the newline message— +rather than the length—causes the lack of responses. +A total of 75% of binary (i.e., non-ASCII) services, in- +cluding MQTT, Postgres, PPTP, Oracle DB, Microsoft SQL, +Siemens S7, DNS, and SMB, send no data back unless we +scan with their specific protocol. We note that our selection +of tested protocols are biased towards ASCII protocols, and +that it is likely that many binary protocols do not respond +to these handshake messages. However, as discussed in Sec- +tion 3.2, the long tail of binary protocols on the Internet are + +Scan +IANA-Assigned Ports +Ephemeral Ports +Order +Protocol +∆ Coverage +Protocol +∆ Coverage +1 +wait +51.3% +wait +66.3% +2 +TLS +29.0% +HTTP +17.1% +3 +HTTP +13.6% +TLS +15.9% +4 +DNS +3.4% +Oracle DB +0.23% +5 +PPTP +1.8% +PPTP +0.14% +Table 2: Optimal Handshake Order—For IANA-assigned +ports, waiting and then sending a TLS Client Hello discovers +80.3% of unexpected services. Five handshakes can identify +over 99% of identifiable unexpected services. +less spread out across a large number of ports compared to +common protocols like HTTP. +Figure 10: Scanning L7 With Different Handshakes— +Sending an HTTP handshake (i.e., a GET Request) prompts +the most number of services to send back data. The data can +then be used to fingerprint the actual service running. +Optimal handshake order. +We compute the optimal order +of L7 handshakes that maximize the chances of identifying the +service running on a port using a greedy approach across two +sets of ports: (1) all IANA-assigned ports and (2) five random +ephemeral ports (62220, 53194, 49227, 47808, and 65535). Of +the 30 protocols with ZGrab scanners that we can identify, we +find that five handshake messages elicit responses from over +99% of identifiable unexpected services on both sets of ports. +We show the top-five L7 handshakes that discover the most +unexpected services for the two sets of ports, excluding the +expected services in Table 2. Across both IANA-assigned and +ephemeral ports, merely opening a connection to the client +(i.e., waiting) can immediately fingerprint more than half +of unexpected services. For IANA-assigned ports, waiting +and then sending a TLS Client Hello discovers 80.3% of +unexpected services. For ephemeral ports, waiting and HTTP +discover 83.4% of services. It is not surprising that DNS and +PPTP provide the 4th and 5th most additional coverage for +IANA-assigned ports, as these are relatively popular protocols +that do not answer to other handshakes (e.g., HTTP GET). +4.2 +Impact of L7 Filtering +One reason that we may not be able to identify all services is +that even if our protocol guess is correct, our selected hand- +shake parameters might be rejected. For example, in SNMP, +servers may reject requests that do not specify the correct +community string in the first packet by first acknowledging +the data, but then sending a TCP RST. To estimate whether L7 +filtering decisions cause a service to not send any data back to +the client, thereby hindering fingerprinting efforts, we run two +sets of scans, each with different handshake options, for each +of the following ports and protocols: 8081/HTTP, 443/TLS, +and 1723/PPTP. +For HTTP, in one scan we send a GET request and +in another we specify the OPTIONS request. For TLS, +in one scan we advertise the insecure cipher suite +TLS_RSA_EXPORT_WITH_RC4_40_MD5 and in the other +we advertise modern Chrome cipher suites. For PPTP, in +one scan the first message is crafted to contain the speci- +fied “Magic Cookie” value (a specific constant used to syn- +chronize the TCP datastream) according to RFC 2637 [31], +0x1A2B3C4D, and in another we specify the Magic Cookie to +be 0x11111111. RFC 2637 states that “Loss of synchroniza- +tion must result in immediate closing of the control connec- +tion’s TCP session;” we thus expect that fewer IPs will send +data to the client if the magic cookie is incorrect and use this +as a “control” experiment. +Port (Service) +Handshake Option +IPs that send data +Only GET Request +27% +8081 (HTTP) +Only OPTIONS Request +7.3% +Both +65.7% +Only Good Cookie +67.1% +1723 (PPTP) +Only Bad Cookie +0.001% +Both +32.8% +Only Secure Cipher +2.65% +443 (TLS) +Only Insecure Cipher +0.05% +Both +97.3% +Table 3: Impact of Handshake Options—Handshake pa- +rameters influence the services that send back identifiable +data. For example, an HTTP OPTIONS request on port 8081 +results in 7.3% more IPs to respond with data than an HTTP +GET request. 65.7% of IPs will respond to both types of +requests on port 8081. + +An HTTP OPTIONS request discovers an additional 7.3% +IPs that speak HTTP compared to a GET request on port 8081. +Responsive IPs will acknowledge data and close the connec- +tion after receiving a GET request, hindering a scanner’s abil- +ity to fingerprint the service as HTTP. However, by sending +an OPTIONS request, 72% of IPs will respond with a 501 +status (method not implemented) and 17% will respond with +a 405 status (method not allowed), thereby confirming they +do speak HTTP. IPs that exclusively respond to an OPTIONS +request are not constrained to a particular network and are +present across 5.3% of ASes. The discrepancy is less pro- +nounced on port 80 where only 0.02% of IPs will respond to +an OPTIONS request but not GET and only 1.1% of IPs will +respond to GET but not an OPTIONS request. +For TLS, per RFC 8446 [57], a handshake failure should +generate an error message and notify the application before +closing the connection. However, 2.65% of IPs will simply +close the connection without any application-layer error when +an incompatible cipher is given. As expected for PPTP, speci- +fying an incorrect magic cookie results in 67.1% of IPs failing +to respond (Table 3). Hosts practicing their own Layer 7 filter- +ing depending upon certain handshake options—and thereby +not sending any data to the client—presents an unavoidable +challenge for any L7 scanner to guess the perfect parameters +to speak the appropriate Layer 7 with every single host. In +Figure 6, we estimate all unknown services to be due to not +having the expected handshake options. +4.3 +Consequences of Handshake Order +Similar to how handshake options might prevent a server from +responding, trying repeated incorrect handshakes prior to the +correct one might also prevent the identification of services. +We evaluate whether hosts filter or refuse connections after +receiving incorrect L7 messages by (1) sending successive +HTTP GET and TLS Client Hello messages to all IANA- +assigned ports for 1% of the IPv4 space and (2) comparing +the number of hosts that successfully complete a follow-up +handshake when being sent the expected L7 data to the num- +ber of hosts that successfully complete a follow-up handshake +when being sent unexpected L7 data. +Depending on the protocol, we find that sending unexpected +L7 data causes up to 30% of follow-up handshakes to fail +compared to the hosts found when directly scanning for the +protocol (Figure 11). For example, sending non-Telnet data to +Telnet servers causes 17% to fail a follow-up handshake; 65% +send a TCP RST and 35% do not SYN-ACK to a follow up TCP +handshake. Sending an HTTP GET request to TLS servers +causes 29% of follow-up TLS handshakes to fail. We find this +behavior to be similar to a Cisco IOS feature, Login Block, +which allows administrators to temporarily block connections +to L7 services after unsuccessful login attempts [33]. Sur- +prisingly, this phenomenon only affects hosts after they send +protocol-identifying data—likely because this is when they +first store server-side application-layer state about the connec- +tion. As such, this blocking does not prevent any servers from +being fingerprinted. It only prevents a follow-up handshake af- +ter identifying data has been sent back to the scanner. Failure +is generally temporary: 75% of hosts will successfully com- +plete the L7 handshake within 5 seconds and 99% of hosts +will take less than 2 minutes. Nonetheless, waiting between +fingerprinting and completing the follow-up handshake can +reduce this filtering effect. +Figure 11: Impact of Sending Incorrect Handshakes— +Sending unexpected data to hosts causes some services to +fail the follow-up expected handshake even when fingerprint- +ing was successful. For example, only 71% of TLS hosts +successfully complete a handshake when initially being sent +an HTTP handshake message. We provide the fraction of total +hosts successfully fingerprinted in the third column. +4.4 +Summary and Implications +One fundamental limitation of L7 scanning is that services +may require specific handshake options to respond. Nonethe- +less, our results indicate that the vast majority of identifiable +Internet services can be easily identified during scans. Many +hosts respond to the “wrong” L7 handshake and send data +that help fingerprint the service: 16 of 30 protocols can be +detected with a single HTTP GET request and 99% of unex- +pected services can be identified with five handshakes. We +use these optimizations to build a scanner (LZR) dedicated to +accurate and efficient unexpected service discovery. +5 +LZR: A System for Identifying Services +In this section, we introduce LZR, a scanner that accurately +and efficiently identifies Internet services based on the lessons +learned from Sections 2–4. LZR can be used with ZMap to +quickly identify protocols running on a port, or as a shim +between ZMap and an application-layer scanner like ZGrab, +to instruct the scanner what follow-up handshake to perform. +LZR’s novelty and performance gain is primarily due to its + +“fail-fast” approach to scanning and “fingerprint everything” +approach to identifying protocols. It builds on two main ideas: +Ignore non-acknowledging hosts. +About 40% of services +that send a SYN-ACK never acknowledge data. None of these +services can complete an L7 handshake and can be safely +ignored during Internet scans. Quickly identifying and ig- +noring these services can significantly reduce costs because +non-acknowledging services force stateful scanners to open +an OS socket and wait for the full timeout period to elapse, +which typically takes much longer than completing a normal +handshake. Non-acknowledging hosts can be filtered out by +sending a single packet—an ACK with data—similar to how +ZMap statelessly SYN scans. +Listen more. +Up to 96% of services per port run unexpected +protocols. In 8 of the 30 protocols we scanned, the server +sends data first, and 10 protocols send fingerprint-able data +when sent an incorrect L7 handshake. By always waiting and +then fingerprinting invalid server responses, we can identify +up to 16 of the 30 protocols by sending a single packet. A +scanner only needs to perform minimal computation to fin- +gerprint a service: the first packet from a server identifies the +running protocol, which does not require a full TCP/IP stack. +5.1 +Scan Algorithm +We outline LZR’s logic in Figure 12. LZR accepts a stream +of SYN-ACK packets from ZMap or tuples of (IP, port) to scan. +In the case that LZR has full connection details from ZMap, +LZR will start by filtering hosts that send SYN-ACKs with +a zero window. Otherwise, it will initiate a new connection. +For non-zero windows, LZR will continue the connection by +sending an ACK packet containing the expected protocol’s +first-packet handshake data. If LZR receives any type of data +in response from the host, it will fingerprint the data and close +the connection. If a host neither acknowledges the data nor +closes the connection, LZR re-transmits the data with the +PUSH flag (further discussed in Section 5.3). If a host does +not acknowledge the data (e.g., never responds or RSTs the +connection without an acknowledgement), LZR fingerprints +the host as likely not hosting a real service and does not pro- +ceed with further connection attempts. Otherwise, if a host +acknowledges the data but does not send any data in response +(i.e., server is unresponsive or closes the connection immedi- +ately afterwards), LZR proceeds to close the connection, start +a new connection, and send the next handshake. The process +continues until LZR identifies the running protocol or runs +out of additional handshakes to try. LZR can also optionally +filter IPs that respond on nearly every port (Section 3.1) by si- +multaneously sending SYN packets to a user-specified number +of random ephemeral ports and checking for a SYN-ACK. +5.2 +Architecture +LZR is written in 3.5K lines of Go and implements all unique +protocols (i.e., handshakes) in Appendix A. Similar to ZMap, +LZR uses libpcap [68] to send and receive raw Ethernet pack- +ets rather than rely on the OS TCP/IP stack. This allows LZR +to efficiently fingerprint services because a single socket can +be used for the duration of a scan and it allows LZR to adopt +and continue connections initiated by a stateless scanner like +ZMap. Because LZR only needs to send and receive a single +packet to fingerprint services, a full TCP stack is not needed. +LZR takes as input a command-line argument list of proto- +cols to test and a stream of SYN-ACKs from ZMap or IP/ports +to scan. Internally, a small pool of Go routines send followup +ACK packets containing handshake messages and fingerprint +their responses. Adding new protocols/handshakes to LZR is +easy; each handshake implements a Handshake interface that +specifies (1) the data to attach to the ACK packet and (2) what +to search for in a response packet to fingerprint the protocol. +Once LZR receives data to fingerprint, LZR first checks if the +data matches the fingerprint (specified using the Handshake +interface) of the protocol being attempted. If not, LZR checks +all the remaining fingerprints for a match. We note that be- +cause ZMap sends probes using a raw Ethernet socket, LZR +users need to install an iptables rule to prevent the Linux ker- +nel from sending RST packets in response to the SYN-ACKs +it receives. Otherwise, LZR cannot adopt and continue these +connections. We have released LZR under the Apache 2.0 +license at https://github.com/stanford-esrg/lzr. +5.3 +Evaluation +We evaluate both the accuracy and performance of LZR by +comparing protocol-specific ZGrab handshakes with four +LZR configurations. The first two are the expected use cases: +1. ZMap/LZR: We use LZR with ZMap to identify the +service running on a port that ZMap finds. +2. ZMap/LZR + ZGrab: We use LZR as a shim between +ZMap and ZGrab to instruct ZGrab what full L7 hand- +shake to complete for hosts that ZMap finds. +During experiments with these configurations at 1gbE, we +find that LZR is able to filter hosts much faster than ZMap is +able to find hosts—especially on ephemeral ports with low +hitrates. ZMap artificially limits how fast LZR and ZGrab +operate. As such, we introduce two additional metrics that +approximate LZR’s performance under the premise of ZMap +finding hosts infinitely quickly. This allows us to compute +how quickly LZR can find hosts as scan speeds increase and +how much time ZGrab can save in an environment where there +are many hosts to scan because the researcher is investigating +multiple ports simultaneously. +3. Offline ZMap/LZR + ZGrab: We perform scans in two +phases. In the first, we use ZMap and LZR to identify + +No +Window +0? +ZMap +S/A +Receive +ACK? +Yes +Try all +Fingerprinting +Modules +No +Receive +Data? +Yes +No +Yes +Max +retransmits +reached? +Receive +RST? +No +Receive +FIN? +No +More +handshakes +given at +runtime? +Yes +Yes +Yes +No +Send Ack +w/ Handshake[i] +i++ +Send RST +End +Send SYN +Receive +S/A? +No +End +Yes +Yes +End +No +Send Ack w/ PSH +w/ Handshake[i] +Filter +Unknown +Service +Across +Ports? +From +Random +Ephemeral +Port? +No +Yes +Send SYN on eph_limit +# of random +ephemeral ports +No +End +Max +retransmits +reached? +Yes +No +num_received +>= eph_limit ? +No +Yes +Yes +i == 1 +Yes +No +Figure 12: LZR Algorithm—LZR efficiently identifies real Internet services by sending application-layer data with the ACK of +a TCP handshake to filter out non-acknowledging hosts and fingerprint the responding protocol. +Internet hosts that speak a known protocol and exclude +this phase from our benchmarking. Then, in a second +phase, we allow ZGrab to process services at full speed. +4. Offline ZMap + LZR: We perform scans in two phases. +In the first, we find candidate services with ZMap, and +exclude this phase from our benchmarking. In the second +phase, we benchmark how quickly LZR can fingerprint +services operating at full speed. +We report L4 and L7 behavior breakdown, CPU time, and +bandwidth savings of LZR from 100% scans of the IPv4 +address space completed during June 2020 in Table 4. We cal- +culate runtime performance using CPU cycles per second for +ZGrab and LZR as both tools are CPU bound: ZGrab’s com- +pletion of a full handshake (e.g., encryption/decryption for +TLS) and LZR’s fingerprinting (e.g., pattern matching) create +the biggest performance bottlenecks for each. When bench- +marking LZR, we receive complaints from seven different +organizations, but there is no indication that the complaints +are the result of a particular LZR optimization; we follow- +up with all responsive network operators and learn that the +complaints are simply due to the 100% coverage of the scans. +How many additional services does LZR find? +One of +LZR’s key features is that it can identify additional services, +while filtering out unresponsive ones by analyzing the re- +sponse to the data included in the ACK packet. Using the +keyword-fingerprinting strategy, LZR identifies an average +of 12 additional unique protocols across ports in our exper- +iment by using only the expected 1–2 handshakes; for ex- +ample, 1.3 million IPs hosting an additional 16 protocols +on port 443 and 238,000 IPs hosting an additional 18 pro- +tocols on port 80 are found with just the single expected +handshake. Furthermore, LZR finds over 2 times more unex- +pected than expected services when sending a single AMQP +handshake to 5672/AMQP. The breakdown of the unexpected +services is, unsurprisingly, nearly identical to the distribu- +tion in Figure 6 (i.e., HTTP and TLS dominate). Across all +ports in Appendix A, LZR identifies 88% of all identifiable +services with just a single HTTP handshake message. The +exact signatures LZR uses for fingerprinting services can +be found at https://github.com/stanford-esrg/lzr/ +tree/master/handshakes. +Does LZR filter out appropriate hosts? +LZR does +not find a statistically significantly different set of hosts +than scanning with just ZMap and ZGrab (Table 4). The +Kolmogorov–Smirnov (KS) test [40] finds p > 0.05, rejecting +the hypothesis that the approaches find a different number +of services for all tested ports. We also verify that sending +data with an ACK during the handshake does not produce +a statistically significant difference in the total number of +hosts that acknowledge data or the total number of IPs that +send back data across three trials of 1% IPv4 samples for +80/HTTP, 443/TLS and 27017/MongoDB. However, we do +find that an additional average of 0.18% of hosts respond +when setting the PUSH flag during the retransmission. Though +the addition of the PUSH flag causes the follow-up packet to +not qualify as an exact TCP retransmission per RFC 793 [54], +we confirm that there is no increase in the number of closed +connections when re-transmitting with a PUSH flag compared +to an identical retransmission. We do not set the PUSH flag +immediately during the handshake as that causes about 0.6% +of IPs to close the connection. +How much faster is L7 scanning with LZR? +ZMap/LZR +performance is always faster than ZGrab due to LZR’s ability +to identify service presence without completing an L7 hand- +shake, which often requires a large number of CPU cycles for + +Port +80 +443 +21 +23 +5672 +5900 +27017 +62220 +80 +443 +47808 +Protocol(s) +HTTP +TLS +FTP +TEL +AMQP +VNC +Mongo +HTTP +HTTP +TLS +HTTP +(Consecutively Scanned) +TLS +HTTP +TLS +Number of Hosts Found +SYN-ACK +62.6M +51.8M +14M +6.4M +3.5M +3.5M +2.4M +2.6M +63M +51.6M +2.8M +Zero Window +1.3M +2.1M +1.7M +1M +899K +1.2M +695K +737K +1.2M +1.8M +742K +RST +1.7M +2.3M +1.1M +673K +502K +730K +166K +349K +1.3M +1.9M +31K +ACKs Data +55M +45M +9.5M +4.6M +1.4M +1.4M +505K +628K +56.3M +45M +1.1M +L7 Handshake +Expected (LZR) +54.66M +43.7M +9.2M +2.71M +123K +277K +73.3K +38K +56M +44.3M +22.6K +Expected (ZGrab) +54.63M +43.7M +9.3M +2.73M +123K +277K +73.6K +36K +56M +44.4M +22.7K +Unexpected (LZR) +238K +1.3M +113K +230K +260K +56K +23K +23K +207K +758K +26.5K +Unique Unexpected +18 +16 +10 +10 +11 +8 +14 +12 +18 +16 +14 +Speed Up (Time) +ZMap/LZR +3.3× +4.7× +2.8× +3.9× +1.9× +2× +1.6× +2.7× +3.3× +6.3× +2× +ZMap/LZR + ZGrab +1.2× +1.1× +1.2× +2.5× +1.8× +1.9× +1.4× +2.6× +1.1× +0.95× +2× +Offline ZMap/LZR + ZGrab +1.1× +1.1× +2.1× +1.6× +3.3× +4× +7× +5.4× +1.1× +1.1× +2.5× +Offline ZMap + LZR +4.1× +4.1× +5× +10.7× +11.4× +13.3× +55× +25.3× +5.6× +3.4× +29× +Bandwidth Savings +ZMap/LZR +60% +75% +67% +78% +70% +79% +66% +68% +79% +84% +87% +ZMap/LZR + ZGrab +-28% +-16% +3% +3% +41% +46% +46% +54% +-16% +-9% +75% +Offline ZMap/LZR + ZGrab +12% +10% +36% +67% +72% +68% +81% +79% +5% +7% +98% +Offline ZMap + LZR +49% +60% +56% +69% +75% +78% +87% +85% +58% +68% +99% +Table 4: LZR Performance—Filtering for IPs that acknowledge data increases service fingerprinting speed by up to 55 times +while finding up to 30% more unexpected services. All relative performance numbers are compared to ZGrab and measured at a +1 Gb/s scanning rate. +expensive operations (e.g., cryptographic functions in TLS). +At minimum, LZR is 1.9 times faster than ZGrab when scan- +ning 5672/AMQP and, at maximum, 6.3 times faster when +scanning 443/TLS+HTTP—equivalent to a 40 CPU hour +speed-up of a 100% scan of IPv4 when using ZGrab’s default +number of senders (1,000) and scanning at ZMap’s calculated +sending rate that minimizes ZGrab’s packet loss (50K pps). +The performance of LZR as ZGrab’s shim (i.e., ZMap/LZR + +ZGrab) varies based on a port’s service makeup. When a port +contains a large raw number of hosts that do not consistently +establish a TCP connection (e.g., zero window), there is sub- +stantial performance improvement: ZMap/LZR + ZGrab is +2.6 times faster than ZGrab when scanning 62220/HTTP. On +the contrary, since the relative number of hosts that do not +consistently establish a TCP connection on port 443 is small, +there is little improvement (1.1 times). +When a significant fraction of candidate services do not +acknowledge data, there is significant improvement when us- +ing LZR to filter hosts offline (i.e., when ZGrab can run at +full speed). On a 100% IPv4 scan of 27017/MongoDB, only +21% of hosts that SYN-ACK acknowledge data and an addi- +tional 30% of hosts send a zero window, which allows LZR +to increase ZGrab performance by 7 times and a LZR scan +by 55 times. Unpopular ports are expected to have the same +performance improvement as 62220/HTTP (e.g., a 25 times +speed-up) because IPs on the majority of ports are more likely +to not acknowledge data when sending a SYN-ACK. +How much bandwidth does LZR save? +Using LZR alone +to fingerprint services always saves bandwidth (up to 87% +on 47808/HTTP+TLS) when the reasonably-expected data +is sent during the initial handshake, as (1) LZR does not +attempt to re-transmit ACKs to zero-window hosts to check +for an increase in window size, and (2) LZR does not need +to complete full L7 handshakes. However, when using LZR +alongside ZGrab when scanning a port where the majority +of TCP-responsive hosts serve the expected protocol, there +exists an overhead in the number of total packets sent—even +when there is a speed-up in time—due to LZR sending at +least one extra ACK to fingerprint before re-attempting the +actual handshake (e.g., LZR + ZGrab together send 28% more +packets than ZMap+ZGrab for 80/HTTP even though LZR + +ZGrab run 1.2 times faster than ZMap+ZGrab). +6 +Related Work +Fast Internet-wide scanning has been used in hundreds of +academic papers in the past seven years. While we cannot +enumerate every paper that has used the technique, we empha- +size that scanning is now common in the security, network- +ing, and Internet measurement communities. Data collected + +through Internet-wide scans has been used to understand cen- +sorship [42, 52, 53], botnet behavior [10, 46], patching be- +havior [23, 25, 47] as well as to uncover vulnerabilities in +IoT and SCADA devices [19, 51, 67], cryptographic proto- +cols like TLS [9,11,13,17,37], SSH [6,36], and SMTP [22], +and the Web PKI [25]. Multiple tools have emerged in the +space, most notably ZMap [26] and Masscan [29]. As of 2020, +more than 300 papers used ZMap and in 2014, Durumeric +et al. found that a significant fraction of all Internet scanning +uses ZMap [23]. Prior to the development of these tools in +2013, groups performed smaller-scale studies to measure a +multitude of Internet dynamics (e.g., [35]). +Despite the growing popularity of the technique, there has +been relatively little work specifically investigating the dy- +namics of Internet-wide scanning. Several works have noted +the large discrepancy between L4 and L7 responses [21,24, +26,36,51,67]. Clayton et al. [18] find evidence of dynamic +blocking within the Great Firewall of China—but do not for- +mally quantify how wide-spread the behavior is—and Wan +et al. [69] find evidence of dynamic blocking within SSH. +Alt et al. introduced degreaser [8] to locate “tarpits”—fake +services that attempt to trick network scanners; tarpits may +use some of the same techniques we see middleboxes use at +the start of a connection. In a similar vein to our work, in 2018, +Bano et al. [12] studied the notion of host liveness. As part +of their taxonomy, they considered the relationship between +live services on different points, showing that the responses +on popular ports are correlated with one another. In 2014, +Durumeric et al. investigated server blacklisting and how +operators respond to Internet-wide scanning; at the time they +found that blacklisting behavior was negligible [12]. Rüth et +al. considered the ICMP responses received in response to +ZMap IPv4 SYN scans [61]. +One contribution of our work is the introduction of LZR, +which reduces the time needed to scan less populous ports. +Prior work has similarly attempted to reduce the time required +to complete Internet-wide scans, though through starkly dif- +ferent approaches. Klick et al. [43] show that much of the +IP address space does not need to be continually scanned by +services like Censys [21]. Adrian et al. introduce a faster +version of ZMap that operates at 10gbE [7]. LZR solves +a different problem and can be used in coordination with +these other performance improvements. Similar to how we +use a single packet to identify services, several works have +focused on single-packet fingerprinting to identify operator +systems [64,65]. +7 +Recommendations and Conclusion +We began our analysis by investigating the troubling obser- +vation that a significant fraction of hosts on the Internet that +respond to a SYN scan never complete an application-layer +handshake [21,24,26,36,51,67]. We found that middleboxes +are responsible for the majority of responses with no real ser- +vices. We also showed that a significant fraction of services +are also located on unexpected ports. For example, 97% of +HTTP and 93% of TLS services are not located on ports 80 +and 443, respectively. Worryingly, unexpected services often +have weaker security postures than those on standard ports. +Building on these observations, we introduced LZR, a scan- +ner that dramatically reduces the time required to perform an +application-layer scan on ports with few expected services +(e.g., 5500% speedup on 27017/MongoDB) while simultane- +ously identifying many unexpected services running on the +port. LZR can identify 16 protocols and 88% of identifiable +services with one packet and 99% of identifiable unexpected +services with 5 handshakes. Nonetheless, there are two addi- +tional challenges to scanning unassigned ports: (1) scanning +100% of all 65,535 ports is not feasible, and (2) it is not clear +which subset of ports is worth scanning (e.g., contain a sig- +nificant fraction of the particular behavior being studied). We +therefore recommend that researchers conduct lightweight +sub-sampled (e.g., 0.1%) application-layer scans across all +ports to detect the prevalence of targeted protocols. We em- +phasize that merely using the top n most popular ports is not +sufficient to evaluate which ports are most likely to host par- +ticular services, as most protocols are drowned out by the +overwhelming popularity of HTTP and TLS. We hope that +researchers find LZR helpful in accurately and efficiently +identifying services in Internet-wide scans. +Acknowledgements +The authors thank Tatyana Izhikevich, Katherine Izhikevich, +Kimberly Ruth, Deepak Kumar, David Adrian, Deepti Ragha- +van, Jeff Cody, members of the Stanford University and UC +San Diego security and networking groups, and the anony- +mous reviewers for providing insightful discussion and com- +ments on various versions of this work. We further thank +Sadjad Fouladi and Katherine Izhikevich for using their artis- +tic talent to greatly improve the visual graphics in this work. +This work was supported in part by the National Science +Foundation under award CNS-1823192, Cisco Systems, Inc., +Google., Inc., the NSF Graduate Fellowship DGE-1656518 +and a Stanford Graduate Fellowship. +References +[1] External HTTP(S) load balancing overview. https://cloud.google. +com/load-balancing/docs/https/#firewall_rules. +[2] Is there any way to block ports of a loadbalancer on GKE? +https://stackoverflow.com/questions/54757395/is-there- +any-way-to-block-ports-of-a-loadbalancer-on-gke. +[3] TCP +RST +from +remote +system +error in +F5. +https:// +devcentral.f5.com/s/question/0D51T00006i7iWK/ +tcp-rst-from-remote-system-error-in-f5. +[4] Vulnerability scan lists all IP’s and port as open. +https:// +devcentral.f5.com/s/question/0D51T00006i7iKu/ +vulnerability-scan-lists-all-ips-and-port-as-open. + +[5] ZGrab 2.0. https://github.com/zmap/zgrab2. +[6] D. 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In NDSS, 2014. + +A +Protocols Scanned +Top 30 +Port +Expected Protocol +IANA-Assigned +Scanner +x +80 +HTTP +HTTP +HTTP +x +443 +HTTPS +HTTPS +TLS +x +7547 +CWMP (HTTP) +CWMP (HTTP) +HTTP +x +22 +SSH +SSH +SSH +x +30005 +- +- +- +x +5060 +SIP +SIP +- +x +21 +FTP +FTP +FTP +x +25 +SMTP +SMTP +SMTP +x +2000 +sccp +cisco-sccp +- +x +8080 +HTTP +HTTP +HTTP +x +50805 +- +- +- +x +4567 +HTTP +tram +HTTP +x +53 +DNS +DNS +DNS (TCP) +x +49154 +- +- +- +x +49152 +- +- +- +x +8081 +- +sunproxyadmin +- +x +8089 +- +- +- +x +110 +POP3 +POP3 +POP3 +x +3306 +MYSQL +MYSQL +MYSQL +x +8085 +- +- +- +x +8000 +- +irdmi +- +x +143 +IMAP +IMAP +IMAP +x +51005 +- +- +- +x +3389 +RDP +RDP +RDP +x +587 +SMTP +submission +SMTP +x +58000 +- +- +- +x +993 +IMAPS +IMAPS +IMAPS +x +995 +POP3S +POP3S +POP3S +Top 30 +Port +Expected Protocol +IANA-Assigned +Scanner +x +465 +SMTP +SMTP +SMTP +x +23 +Telnet +Telnet +Telnet +8443 +HTTPS +pcsync-https +TLS +1723 +PPTP +PPTP +PPTP +179 +BGP +BGP +- +5432 +Postgres +Postgres +Postgres +1883 +MQTT +MQTT +MQTT +5672 +AMQP +AMQP +AMQP +8883 +mqtt +secure-mqtt +mqtt +1521 +Oracle DB +Oracle DB +Oracle DB +53194 +- +- +- +62220 +- +- +- +49227 +- +- +- +6379 +redis +redis +redis +5900 +VNC +VNC +VNC +20000 +DNP3 +DNP3 +DNP3 +65535 +- +- +- +1433 +mssql +mssql +mssql +445 +SMB +SMB +SMB +631 +IPP +IPP +IPP +6443 +Kubernetes +sun-sr-https +Kubernetes +623 +IPMI +IPMI +IPMI +47808 +- +Bacnet +- +27017 +Mongodb +Mongodb +Mongodb +502 +Modbus +Modbus +Modbus +102 +Siemens S7 +iso-tsap +Siemens S7 +11211 +memcached +memcached +memcached +Figure 13: Port Selection—Three categories of ports are scanned: (1) The top 30 ports determined by a SYN-ACK scan conducted across all 65K ports of 1% of IPv4. +(2) Ports for which a ZGrab-scanner exists (i.e., to be able to complete the full L7 handshake). (3) A random selection of 5 ephemeral ports. We label the expected +service being hosted on the port, as well as the IANA-assigned service. Note that each of these categories contain overlapping ports. + diff --git a/zNE4T4oBgHgl3EQfAQs4/content/tmp_files/load_file.txt b/zNE4T4oBgHgl3EQfAQs4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..65625d14096c3c4e57334d4d7930e3e71097db8c --- /dev/null +++ b/zNE4T4oBgHgl3EQfAQs4/content/tmp_files/load_file.txt @@ -0,0 +1,1636 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf,len=1635 +page_content='LZR: Identifying Unexpected Internet Services Liz Izhikevich Stanford University Renata Teixeira Inria, Paris∗ Zakir Durumeric Stanford University Abstract Internet-wide scanning is a commonly used research tech- nique that has helped uncover real-world attacks, find crypto- graphic weaknesses, and understand both operator and mis- creant behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Studies that employ scanning have largely assumed that services are hosted on their IANA-assigned ports, overlooking the study of services on unusual ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In this work, we investigate where Internet services are deployed in practice and evaluate the security posture of services on unexpected ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We show protocol deployment is more dif- fuse than previously believed and that protocols run on many additional ports beyond their primary IANA-assigned port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, only 3% of HTTP and 6% of TLS services run on ports 80 and 443, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Services on non-standard ports are more likely to be insecure, which results in studies dramatically underestimating the security posture of Inter- net hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Building on our observations, we introduce LZR (“Laser”), a system that identifies 99% of identifiable unex- pected services in five handshakes and dramatically reduces the time needed to perform application-layer scans on ports with few responsive expected services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', 5500% speedup on 27017/MongoDB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We conclude with recommendations for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 1 Introduction Internet-wide scanning—the process of connecting to ev- ery public IPv4 address on a targeted port—is a standard research technique for understanding real-world service con- figuration and deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Leveraging tools like ZMap [26] and Masscan [29], more than 300 papers have used Internet- wide scanning to discover weaknesses in TLS, SSH, and the Web PKI [6,9,11,13,15,17,24,36–38], to uncover real-world attacks [22,50,60], and to better understand botnets [10,46], ICS/IoT deployment [19,51,67], censorship [42,52,53], and operator behavior [23,25,47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' ∗Work done while visiting Stanford University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Past scanning studies have largely assumed that services are hosted on their IANA-assigned ports (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', HTTPS on TCP/443) and have overlooked scanning additional ports for unexpected services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Yet, many of these same studies have also observed that a non-negligible fraction of the hosts that re- spond to a SYN scan never complete the expected application- layer handshake [21,24,26,36,51,67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' It is unclear whether operators hide services on unexpected ports, whether scanners fail to account for protocol inconsistencies or server-side im- plementation errors, or whether firewalls detect scanning and block further interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In this work, we investigate where Internet services are deployed in practice, and we evaluate the security posture of services hosted in unexpected places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We start by investigating services that do not appear to speak the expected IANA-assigned protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We confirm that up to 96% of services (by port) do not complete the expected application-layer (L7) handshake on 37 popular ports (Sec- tion 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We introduce a heuristic that infers server-side TCP state, which we use to show that 28% of initially-responsive services do not allow any L7 data exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Rather, 12% im- mediately tear down the connection, 5% prevent an L7 hand- shake by specifying a zero TCP window, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6% are blocked from receiving our ACK, and 11% “shun” our IP between the discovery and application-layer scan phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We trace these behaviors to middleboxes and firewalls, and we evaluate their efficacy at enabling scan evasion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' While network defenses account for most L7 unresponsive services, a significant number of services are TCP compliant, but fail the expected L7 handshake (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', 14% on TCP/80 and 96% on TCP/102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We show that this is due to services running on unexpected ports, protocol handshakes that re- quire pre-established secrets, and network-based protections that acknowledge data on every port but speak no detectable protocol (Sections 3–4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Notably, protocol deployment is ex- ceptionally diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='0% of HTTP and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4% of TLS services run on ports 80 and 443, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Achieving 90% coverage of TLS-based services requires scan- ning 40K ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Worryingly, services deployed on unexpected arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='04841v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='CR] 12 Jan 2023 ports have worse security postures, which we trace back to IoT devices that host insecure services on non-standard ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' To enable researchers to more comprehensively find In- ternet services, we introduce LZR (“Laser”), a system that efficiently filters hosts that do not speak any L7 protocol and identifies unexpected services (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' LZR can fin- gerprint 88% of identifiable services with a single packet and 99% of identifiable unexpected services with five hand- shakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' LZR also speeds up scans by quickly filtering the bulk of seemingly-responsive hosts that SYN-ACK but cannot complete an application layer handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, on port 27017, LZR filters out 80% of hosts that SYN-ACK, de- creasing the time to complete scans of MongoDB by 55 times, while still identifying 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6% of MongoDB services and iden- tifying an additional 23K hosts running unexpected protocols (a 31% coverage increase for the port).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Our work concludes with recommendations for future stud- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We hope that by shedding light on the ecosystem of unex- pected services, and by releasing LZR as an open-source tool, we enable security researchers to more accurately understand Internet services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 2 Identifying Real TCP Services Fast research scans of the Internet are typically conducted in two phases today [21, 26, 36, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In the first stage, a scan- ner like ZMap [26] statelessly sends SYN packets to public IPv4 addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Then, in a second process, a stateful scan- ner like ZGrab [21] performs complex follow-up handshakes using the kernel TCP/IP stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The two-phased nature of Internet scanning is largely attributable to ZMap’s architec- ture, which uses a stateless network stack to efficiently probe services, but is unable to complete handshakes that require maintaining local state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The biases and unintended conse- quences from scanning in two phases have not been inves- tigated, and worryingly, prior studies have repeatedly noted that more than half of the IPv4 hosts that respond to a SYN scan never complete a follow-up application-layer handshake (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', [24,26,36,51,67]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In this section, we investigate this discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We show that TCP liveness does not accurately indicate the presence of an application-layer service due to several common security protections, including middleboxes and user-space firewalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Guided by TCP’s design [54], we uncover five defensive be- haviors that degrade the signal provided by L4 responsiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We quantify the deployment of these defenses, and we eval- uate their efficacy at protecting against DDoS attacks and evading Internet scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We then go on to develop a better L4 heuristic to approximate application-layer liveness, which we use to better understand service deployment in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1 Layer 4 versus Layer 7 Liveness We start our investigation by confirming whether TCP- responsive hosts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', hosts that reply with a SYN-ACK packet) complete the IANA assigned [39] application-layer hand- shake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Mimicking prior Internet scans (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', [6,9,16,36,72]), we perform a two-phase scan in which we send a SYN packet to a random 1% sample of public IPv4 addresses using ZMap [26] and immediately attempt a follow-up application handshake using ZGrab [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We scan all IANA-assigned ports with available ZGrab scanners (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', 37 ports in Appendix A) on November 12–14, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We follow the best practices set forth by Durumeric et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' [26] to minimize scan impact, and we exclude networks that have previously contacted us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We receive no complaints, but note that we have used our network in the past for other experiments and exclude operators who previously requested removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Consistent with prior studies [24,26,36,51,67], we find that a considerable fraction of TCP-responsive hosts never com- plete the expected L7 handshake (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The raw number of L7-unresponsive hosts varies from 21K unresponsive hosts on 502/Modbus to 201K hosts on 443/HTTPS (µ = 54,542, σ2 = 31,002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We see this heavy-tail distribution throughout our investigation and we present our results for both popular and unpopular ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We split ports into the two categories using Grubbs’s test for outliers [30] with a 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='9% confidence interval based on the total number SYN-ACKs and the presence of an expected service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Our popular set contains ports 80, 443, 7547, 22, 21, and 25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' the unpopular set contains the remaining 31 ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Popular protocols are most likely to complete the expected L7 handshake:1 86% and 80% of TCP-responsive hosts on ports 80 and 443 complete an HTTP(S) handshake while only 9% and 4% of hosts on ports 502 and 102 speak Modbus and Siemens S7 (two SCADA protocols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In the following section we start our investigation of L7- unresponsivess by analyzing the changing state of services between the two phases of scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2 Connection Shunning About 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6% of services on popular ports and 5% of services on unpopular ports do not respond with a SYN-ACK during our follow-up ZGrab TCP handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' This could be due to DHCP churn, transient network failure, or the destination host blocking the scanner between handshakes (“connection shunning”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' To determine whether hosts “shun” scanners, we connect to TCP-responsive hosts found by ZMap from two IP addresses: the original IP address used by ZMap to identify the host and a fresh IP that has not previously contacted the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We scan a random ephemeral port, 48302, because we see the largest fraction of disappearing hosts on unpopular ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We find that 70% of IPs that do not respond a second 1Spearman’s Correlation p-value of port rank (based on number of SYN- ACK) relative to L7 and SYN-ACK percent difference is 5×10−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 80/HTTP 443/TLS 7547/HTTP 22/SSH 21/FTP 25/SMTP 8080/HTTP 4567/HTTP 53/DNS 110/POP3 3306/MYSQL 143/IMAP 3389/RDP 587/SMTP 993/IMAPS 995/POP3S 465/SMTP 23/TELNET 8443/TLS 1723/PPTP 5432/POSTGRES 1883/MQTT 5672/AMQP 8883/MQTT 1521/Oracle 6379/redis 5900/VNC 20000/DNP3 1433/MSSQL 445/SMB 631/IPP 6443/Kubernetes 623/IPMI 27017/Mongodb 502/Modbus 102/Siemens 11211/memcached Port/Service 0 2 4 6 8 IPs (100,000s) SYN-ACK only L7 Handshake Figure 1: L4 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' L7 Responsiveness—A significant frac- tion of hosts that respond with a SYN-ACK packet never com- plete the expected application-layer handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The differ- ence varies dramatically across ports by both percent differ- ence (14–96%) and raw count (21,050–200,902).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' time on the used IP do respond to the fresh IP, indicating that most hosts that go missing between scan stages are typically not lost due to churn or network failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In the case that the fresh IP receives a SYN-ACK, we ob- serve two types of responses from the previously-used IP: no response (93%) and RST packet (7%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' This blocking occurs at the IP granularity: once a scanner has been blocked by a host, the host will not respond with a SYN-ACK on any port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We further confirm that connection shunning is not a defensive re- action—triggered by failing to complete an application layer handshake—by running a 1% IPv4 scan of all popular ports using ZGrab for the initial host discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The same fraction of connections are shunned as when ZMap is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We find that connection shunning is deployed at both the host and network granularity by computing the largest blocks of consecutive TCP-Responsive IPs that show shunning be- havior on a random ephemeral port: 40% of networks that shun scanners are /32s (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', individual hosts) and 10% of IPs block in groups larger than a /24 (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The largest network to deploy connection shunning is a /20 owned by Alestra Net (ASN 11172), a Mexican ISP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Both network hardware (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', Cisco IOS-based routers [34]) and host software (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', Snort [59]) document connection shunning and dynamic blocking as features where connec- tions are blocked after an IP is classified as malicious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Connec- tion shunning prevents clients from using a single source-IP to scan the network and forces scanners to use multiple source IPs to reach the end-host, thereby dramatically increasing the cost for an attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We compare the number of legitimate ser- vices found when using both single and multiple source-IPs during scanning and find no evidence that any hosts that shun connections host legitimate services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We thereby conclude that they can be safely ignored in security studies if they can be efficiently filtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3 Do TCP-Responsive Hosts Speak TCP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The vast majority of services (average of 96% across ports) that do not complete an application-layer handshake respond with a SYN-ACK during the second (ZGrab) handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In the remainder of the section, we explore whether these hosts reach a state where they can exchange application-layer data or sim- ply stop responding after sending a SYN-ACK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In Figure 2, we provide a modified TCP state diagram based on RFC 793 [54] that captures what a scanner can infer about a server’s TCP state, which we use to guide our investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For a TCP connection to enter the ESTABLISHED state, the server sends only a single packet (SYN-ACK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Once the client has sent an ACK, it can normally send data—the amount specified by the server window size in the SYN-ACK packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We note that TCP has an edge case in which the server can respond with a zero-sized window in its SYN-ACK [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In this situation, the client is expected to send follow-up ACK packets to probe when the server is ready to accept data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We add a new ACCEPTS DATA state in Figure 2 to capture whether a server is ready for data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Once the server has reached the ACCEPTS DATA state, it is expected to keep the TCP connection open long enough to receive data and to acknowledge receipt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We define ACKNOWLEDGES DATA as the server allowing the client to send data and acknowledging client data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' LISTEN SYN RECEIVED ESTABLISHED Receive : SYN Send: SYN-ACK Receive: ACK Receive: Data or Close or Timeout Send: Close or Timeout Send: SYN-ACK Receive : Close or Timeout ACKNOWLEDGES DATA Receive: Data Send: ACK Window Size > 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Yes No Receive: Timeout Send: Close or Timeout ACCEPTS DATA Figure 2: Client Perspective of Server TCP State—We investigate L7 service liveness based on a modified version of the TCP state machine in RFC 793 [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We introduce two new states: “accepts data” and “acknowledges data” because an established connection cannot necessarily exchange data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' To test how far into a TCP session servers reach, we de- velop a new scanner based on ZGrab [5] that establishes a TCP connection, sends two newlines, and deduces the server TCP state (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We scan random 1% samples of IPv4 addresses on a random 2,000 ports as well as the 37 IANA assigned ports that host protocols with ZGrab scanners (Ap- pendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' An average 16% of services on popular ports and 40% of services on unpopular ports fail to acknowledge data (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We detail why in the remainder of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='(a) Portion of TCP-responsive hosts that fail to acknowledge data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='# (1000s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='Connection Shunning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='Dropping Connections Mid-Handshake ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='Zero Window ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='Dynamic Blocking (Handshake) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='Reset Connection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='Leftover Non-ACK Hosts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='443 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7547 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='502 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='11211 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='Port Number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='Fraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='Non-Acking IPs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='(b) Reasons SYN-ACK-only hosts fail to acknowledge data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='Figure 3: Unexpected TCP Behavior of IPv4 Hosts—An average 16% of services on popular ports and 40% of services on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='unpopular ports that respond to a TCP SYN scan with a SYN-ACK packet do not fully speak TCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Here, we show the portion of hosts by port that do not acknowledge client data and the breakdown of reasons why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Algorithm 1: Deducing Server TCP State Send SYN if receive RST or FIN or Timeout then return NO_ACK_HOST end // checking for zero window sizes Print syn-ack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='window_size // sending protocol-agnostic data Send "\\n\\n" // Time for 8 re-transmissions (RFC 1122 rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=') while timeout < 100 seconds do if received ACK then return ACK_HOST end if received RST or FIN then return NO_ACK_HOST end end return NO_ACK_HOST // host has timed out 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4 Zero Window DDoS Protections Of the services that never acknowledge data, 13% of services on popular ports and 26% on unpopular ports actively prevent clients from sending data by specifying a zero-sized TCP window and never increasing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Across all scanned ports, at least 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='94% of hosts with a zero window never increase it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 90% do not respond to secondary probes and 10% reset the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The behavior appears to be network- or host- based rather than service-based: 99% of hosts that respond Figure 4: Network Granularity of TCP Blocking—Some protections appear to be host-based while others are more prevalent on large networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Zero Window DDoS protections are most likely to appear at a large network granularity, while connection shunning is more likely a host-level behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' with a zero-window on one port will send a zero-sized window on all ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Offhand, this behavior appears self-defeating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Hosts that respond and never increase window size might as well never respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' However, we find the feature in a Juniper networks patent [66] and used in Juniper’s Secure Service Gateway Proxy [41] to prevent DDoS attacks through network-based SYN cookies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The protection responds to all SYN packets with a zero-window SYN-ACK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Once the client completes the three-way handshake by sending an ACK, the firewall sends a SYN packet to the backend server to establish the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' By maintaining a zero-sized TCP window with the client, the middlebox prevents the client from sending data it cannot yet forward to the backend server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Zero-window SYN-ACKs are deployed across entire sub- networks: 90% of IPs that SYN-ACK with a zero window do so in a network larger than a /24 (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The largest network, the State of Florida Department of Management Ser- vices (ASN 8103), is responsible for 16% of all zero-windows Internet-wide and accounts for around 3% of all SYN-ACKs on a random port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The TTL for SYN-ACK is consistently one hop closer than the later RST, further confirming a network appliance is responsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='5 Dropping Connections Mid-Handshake Beyond specifying a zero window, an average 2% of the hosts per port that never acknowledge data do not appear to complete a three-way handshake, despite the client sending an ACK (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We infer that the server never reaches the ESTABLISHED state based on a continual stream of SYN- ACK packets (average 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='8 SYN-ACK re-transmissions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Hosts do not simply have broken TCP stacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' in the case of MCI Communication Services, for example, IPs that re-transmit SYN-ACKs on port 4567 have compliant behavior on other ports (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', RDP on TCP/3389).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Real services respond with a TTL over twice as large as the TTL value which re-transmits the SYN-ACK, suggesting that a middlebox selectively drops packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Dropping connections mid-handshake is a defensive behavior exhibited primarily by ISPs protecting consumer premise equipment: CenturyLink (AS 209), Frontier Com- munications (AS 5650), and MCI Communications Services (AS 701) all drop inbound traffic to port 4567/TRAM post- SYN (accounting for 96% of dropped connections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Korea Telecom (AS 4766) and Axtel (AS 6503)—accounting for 73%—interrupt connections on 7547/CWMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The behavior is rare on common ports (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', only 5% of TCP-responsive hosts that do not acknowledge data drop connections mid- handshake on port 80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6 Reset Connections An average 73% of services on popular ports and 34% of services on unpopular ports that do not acknowledge data reach the ESTABLISHED state but will immediately reset the connection after the client completes the three-way handshake (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Per RFC 793 [54], if a server does not want to communicate with a client (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', due to mismatches in “secu- rity clearances”), the server should close the TCP connection after the client acknowledges the SYN-ACK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' This is also how user-space firewalls like DenyHosts [63] appear to scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' While we cannot detect what software closes a connection, we note that networks that RST on port 22 are 10 times more likely to do so in block-sizes of /32 than port 80, implying that blocking happens more often on hosts running SSH compared to HTTP, consistent with Wan et al.’s findings [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Network- level behavior looks to be caused by DDoS protections similar to the networks that send zero-window SYN-ACKs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' To pro- tect against SYN-flooding, middleboxes send a SYN-ACK on behalf of the server and later establish a connection with the server after the client has finished the three-way handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' If the server refuses the connection, the middlebox terminates the client connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' This functionality is available in Cisco IOS-based routers as a part of their threat detection logic [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The behavior is visible in prominent networks, with more than 40% of such IPs located in Korea Telecom, Vodaphone Australia, OVH, and Akamai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Hosts are 20% more likely to close a connection on popular ports because Google load bal- ancers in AS 19527 come with a standard firewall policy that accept traffic on these ports by default—in order to be able to perform service health checks—and rely on the backend virtual machine to reset connections if the port is closed [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7 Dynamic Blocking after Handshake Not all hosts that fail to acknowledge data send RSTs or contin- ually re-transmit SYN-ACKs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Many simply never acknowledge any data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' An average of 10% services on popular ports and 18% of services on unpopular ports do not acknowledge client data (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' These hosts frequently do not respond to later follow-up handshakes either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' This “shunning” behav- ior is similar—but not identical—to the behavior we found in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2 and has previously been documented in the Great Firewall of China [18] where it is used to stop future connections, triggered only when data is sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' To differentiate between hosts that shun the scanner after a handshake from those that simply never acknowledge data, we simultaneously attempt an L7 handshake with initially- responsive hosts that did not acknowledge data from two IP addresses, one that matches the initial connection and one that differs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Of the initially unresponsive IPs, 98% respond to the fresh IP, indicating the behavior is not likely due to transient network failure, but rather explicit blocking of in- coming connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In total, post-handshake dynamic block- ing accounts for 6% and 12% of the remaining hosts that do not acknowledge data for common port and uncommon port hosts respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Note that this behavior only occurs after a three-way handshake, thereby differing from connection shunning (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The largest network to dynamically block after a handshake is Coming ABCDE HK (AS 133201), which accounts for 48% of all IPs that block after a handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We also discover a similar TTL phenomenon as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4 implying a middlebox-based protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We deduce that the rest of the hosts that fail to acknowledge data are not performing dynamic blocking because though they will not respond to anything after the actual handshake, they do consistently respond to all scans (no matter the source IP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Vodaphone (AS 133612) and Webclassit (AS 34358) have this behavior across all scanned ports and make up 66% of all IPs with such a behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We find similar evidence of mismatching TTL values, which indicate a middlebox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='8 Efficacy of Middlebox Protections Identifiable middlebox protections are common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' About 16% of the services on popular and 40% of the services on unpop- ular ports that respond to a SYN packet—but do not speak any identifiable L7 protocol—are artifacts of DDoS and scan- ning protections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 40% of routed ASes contain at least one such protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Reset connections after a handshake—a be- havior found in software like DenyHosts [63]—is by far the most common behavior by both IP and AS, and is present in 34% of ASes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Middleboxes employing connection shunning or dynamic blocking are each used by 6% of networks, and Juniper’s patented zero-window DDoS protection appears in 2% of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' These protections prevent clients from directly connecting to servers—at least initially—and all middleboxes succeed at doing so, even if the protection is identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' However, with the use of more than one source IP address, an adversary can bypass connection shunning and dynamic blocking and still solicit SYN-ACKs from the end- host, albeit rate-limited by the number of scanner addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Beyond actively preventing DDoS attacks and some scan- ning, each protection inadvertently slows down the discovery of new services through Internet scanning and can slow down the spread of malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Dynamic blocking (completing the handshake without acknowledging data) is the most effective at doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The technique slows scans by up to 55 times as in the case of host discovery on 27017/MongoDB (Section 5), by forcing the scanner to timeout upon not receiving an ACK for each scanned host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Though zero window SYN-ACKs also cause a scanner to eventually timeout, zero-sized windows are easy to filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Immediately closing the connection after the handshake causes only a negligible slowdown, bounded only by the time it takes to complete a handshake (about 100 ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Connection shunning is the least effective at slowing down stateless scanners but slows down stateful scanners at the same rate as dynamic blocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='9 Summary Our results establish that SYN-ACKs are a poor indicator for the presence of a service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In the worst case, SYN-ACKs overes- timate the hosts that acknowledge data by 533% on port 11211 (memcached).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We also discover that an average 16% of ser- vices on popular ports and 40% of services on unpopular ports fail to acknowledge data, which is a likely indicator for the presence of a middlebox protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We investigate why hosts that appear to fully speak TCP do not always complete L7 handshakes in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 3 Application-Layer Service Deployment In the last section, we investigated L4-responsive services that do not appear to speak any L7 service and are artifacts of DoS and scanning protections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' After excluding the 28% of pseudo- 80/HTTP 443/TLS 7547/HTTP 22/SSH 21/FTP 25/SMTP 8080/HTTP 4567/HTTP 53/DNS 110/POP3 3306/MYSQL 143/IMAP 3389/RDP 587/SMTP 993/IMAPS 995/POP3S 465/SMTP 23/TELNET 8443/TLS 1723/PPTP 5432/POSTGRES 1883/MQTT 5672/AMQP 8883/MQTT 1521/Oracle 6379/redis 5900/VNC 20000/DNP3 1433/MSSQL 445/SMB 631/IPP 6443/Kubernetes 623/IPMI 27017/Mongodb 502/Modbus 102/Siemens 11211/memcached Port/Service 0 2 4 6 8 IPs (100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='000s) SYN-ACK only ACK Data L7 Handshake Figure 5: SYN-ACK vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Ack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Data vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' L7 Handshake— There are up to three orders of magnitude fewer IPs that acknowledge data than respond with a SYN-ACK packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' services, we discover 27% of services on popular ports and 63% services on unpopular ports that acknowledge data do not run the expected application-layer protocol (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In this section, we analyze services that complete unexpected application-layer handshakes or acknowledge data but do not speak any identifiable application-layer protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We show that while IANA-assigned services are prominent on popular ports, unexpected but identifiable services dominate other ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Moreover, assigned ports only host a tiny fraction of the services that run popular protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, only 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4% of TLS services run on TCP/443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Services on unexpected ports are commonly hosted by IoT devices and have weaker security postures, which suggests the need for the security community to study the services on unassigned ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1 Finding Unexpected Services To determine the extent to which unexpected services co- reside on ports with assigned services, we scan 1% random samples of the IPv4 address space on the set of ports from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3 (37 ports with an expected service and 18 ports without an unexpected service or implemented scanner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For each responsive service, we first attempt to complete an L7 handshake using the expected protocol, if one exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Upon failure, we attempt follow-up handshakes using the 30 proto- col scanners—the total number of unique protocol scanners— implemented in ZGrab (Appendix A) with default parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Ethical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Prior studies have primarily per- formed Internet scans that target only expected protocols;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' to minimize the potential impact of our experiment, we scan only 1% of the IPv4 address space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We received zero abuse com- plaints, requests to be blocked from future scans, or questions from operators from this set of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Data acknowledging firewalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The number of data- ac- knowledging services per IP follows a bi-modal distribution: 98% of IPs serve fewer than four unidentifiable services and 2% of IPs host unidentifiable services on over 60K ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' About 75% of all unidentifiable services on unpopular ports are hosted by IPs with unidentifiable services on nearly every port (“Unknown Service - across ports” in Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Hosts have unidentifiable services on most but not all ports because some networks drop all traffic to security-sensitive ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, out of the top 50 networks that send back the most SYN-ACK responses across all ports, 28% drop all traffic to port 445 (SMB) and 10% drop port 23 (Telnet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Hosts with unidentifiable services on nearly every port are concentrated in a small number of networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' five ASes belonging to the Canadian government (74, 25689, 818, 2680, and 806) ac- count for 77% of all IPs that host unidentifiable services on nearly every port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We trace this behavior to the F5 Big-IP Firewall based on a RST fingerprint [3] that contains the words “BIG-IP System.” An F5 DevCentral blog post [4] speculates that IPs respond on every port due to the accidental use of a wildcard when config- uring the firewall or an overload of the firewall’s SYN-cookie cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We identify and exclude these hosts, to avoid biasing our analysis, by checking whether hosts acknowledge data on five random ephemeral ports, which effectively filters out 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='9% of such hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Nonetheless, an average of 10% of popu- lar and 25% of unpopular services remain unidentifiable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', do not respond to any of the 30 handshakes) after filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2 Characterizing Unexpected Services After filtering out hosts with unknown services on nearly all ports, we investigate unexpected services on assigned ports and services on ports without any assigned service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We sum- marize our results in Figure 6 and describe them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Unexpected services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Services on popular ports typically run the expected protocol: 93% of hosts that acknowledge data on port 80 respond to an HTTP GET request and 89% on port 443 complete an HTTPS handshake (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6% of the services on port 80 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='25% of services on port 443 respond to one of the other 30 unqiue handshakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The majority (75%) of unexpected services on port 80 are TLS-based and nearly all on port 443 are HTTP-based (Fig- ure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' This implies that operator recommendations to run services on ports 80 or 443 to bypass firewall restrictions [49] are not widespread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' As ports decrease in popularity, the frac- tion of IPs that speak the expected service approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, on port 623, only 1% of services that acknowl- edge data speak IPMI and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='9% speak other identifiable protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Consequently, the number of additionally identi- fiable services diminishes after the first few protocols and appears to converge at 96% (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Each port contains its own long-tail of unexpected services, but for many ports, this number plateaus quickly—just not at 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The number of identifiable services on ports without an as- signed service varies between 2–97% based on port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Among random ephemeral ports, our 30 handshakes identify the pro- tocol for an average 21% of services that acknowledge data and an average of 10 unique protocols per port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Across all scanned ports, nearly 65% of unexpected, but identifiable, services speak HTTP and 30% speak TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' IoT devices are a prominent culprit behind unexpected services;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' unexpected TLS services are 5 times more likely and unexpected SSH 2 times more likely to belong to an IoT device than 443/TLS and 22/SSH services, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We also find evidence of operators attempting to hide services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, 70% of hosts serving TLS on the random ephemeral ports 49227, 47808, and 49152 are issued certificates by BBIN Interna- tional Limited, a Philippine offshore online gambling plat- form [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We further detail the types of services hosted on unassigned ports in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Long tail of ports by protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Our results suggest that protocols run on many additional ports beyond their primary IANA-assigned port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' To quantify how many ports researchers need to scan to achieve coverage of a protocol, we conduct a new scan targeting 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1% of the IPv4 address space on 10 popular protocols on all 65,535 ports and compute the fraction of hosts running a given service across multiple ports (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We find that port 80 contains only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='0% of hosts running HTTP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' another 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2% of HTTP hosts run on port 7547 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7% on port 30005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' To cover approximately 90% of HTTP, one must scan 25,000 ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Only 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='5% of Telnet resides on TCP/23, with the assigned alternative port TCP/2323 being only the 10th most popular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' other unexpected ports dominate the top-10 ports with the most Telnet services (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Previous work tracking botnet behavior [10,44] has primarily studied assigned Telnet ports (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', 23, 2323);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' our findings imply that the attack surface and number of potentially vulnerable devices is potentially over 15 times worse than previously shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Some protocols are still relatively clustered around their as- signed ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1% of all AMQP is on port 5672 and an additional 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1% is on port 5673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' HTTP and TLS are the only two protocols which appear on every port in our 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1% IPv4 scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The set of most popular ports also varies per pro- tocol and is often not correlated with the popularity of ports that send data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', across all protocols), as most services are drowned out by the overwhelming popularity of HTTP and TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, 7 of the top 10 ports most likely to host Tel- net are ranked above 12,000 in overall popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' As a result, when choosing which popular ports to study for a specific protocol, we recommend researchers conduct a lightweight sub-sampled scan across all ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3 Security of Unexpected Services Services on unexpected ports are more likely to be insecure than services on assigned ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We use the results from our experiment in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1 (scanning 30 protocols on 55 ports) to show four examples of how unexpected services affect the results of previous and future security studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Figure 6: Distribution of Types of Services—A smaller fraction of services run the assigned protocol on less popular ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, only 4% of services on TCP/102 speak the assigned S7 protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The fraction of services that can be identified on unassigned ports (on the right hand side) varies widely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' TLS REDIS SMTP HTTP TLS TLS HTTP HTTP TLS HTTP TLS VNC FTP SSH MQTT TELNET SSH 90% 80% 60% 70% 50% 40% 30% 20% 10% Fraction of IPs (known unassigned service) Port 80 7547 22 21 all 443 Figure 7: Distribution of Unexpected Services—HTTP and TLS are the most popular unexpected services, with 65% of unexpected services speaking HTTP and 30% speaking TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' IoT devices are frequent targets due to their consistently weak security designs [28, 48, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' While pas- sive measurement has shown that a significant number of IoT devices inhabit non-standard ports [45], active mea- surement of IoT devices has largely studied only standard ports [14,20,27,55,62,71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' By manually identifying server certificates belonging to an IoT manufacturer, we find IoT interfaces on unexpected ports are widespread;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 50% of TLS server certificates on unexpected ports belong to IoT devices and unexpected TLS is 5 times more likely to belong to an IoT device than on port 443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, 35% of 8000/TLS are icctv devices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', surveillance cameras) in Korea Tele- com and 38% of 80/TLS are Huawei network nodes spread across 1% of all international networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' About 5% of TLS on port 8443 belongs to Android TVs in Korean networks and at least 20% belongs to routers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Unassigned ports also contain more TCP/UPnP devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, there are 12 times more TCP/UPnP devices on port 49152 (primarily in Latin America and Asian Telecoms) and 2 times as many on ports 58000 and 30005 than on port 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Vulnerable TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' TLS services on unassigned ports are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='17 times more likely to have a certificate with a known Figure 8: Protocol Coverage Convergence—The marginal gain of scanning additional protocols is negligible beyond the top 10 protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Still, for most ephemeral ports (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', port 49227) the majority of services remain unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' private key than on assigned ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' When scanning unassigned ports, we find over twice as many certificates have a known private key than reported in prior work [32,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2% of TLS hosts on port 8081 are DOCSIS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1 Wireless Gateways in Telecom Argentina (AS 10481 and 10318) using the same OpenSSL Test Certificate with a known private key and 39% of TLS hosts on port 58000 are Qno wireless devices with the same self-signed certificate with a known private key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Across 23% of scanned ports, public keys are more likely— up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7 times more—to be shared than those on port 443 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', 80/TLS is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='5 times more likely).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Nonetheless, previous work studying cryptographic keys on the Internet [26, 32, 36] has limited analysis to 443/HTTPS, 22/SSH, 995/POP3S, 993/IMAPS, and 25/SMTPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Login pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Over half of unexpected ports scanned host a higher fraction of public-facing login pages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', HTML con- taining a login, username, or password field) than 80/HTTP and 443/HTTPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Though the total number of HTTP login pages is greatest on port 80, a page on 8080/HTTP is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4 times more likely to be a login page, thus offering an additional 25% of such pages compared to port 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Furthermore, all the aforementioned IoT devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', icctv, routers) hosting TLS also serve a login HTTPS page on their respective ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Measure Names 80 Bacnet Memcached I Modbus 443 Snmp Oracle 7547 Siemens Port Prometheus Mongodb Dns Postgres Elasticsearch Dnp3 Smb Amqp Imap Ftp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='0 Kubernetes FractionofKnownUnassigned Services Ipp Vnc Amqp,Bacnet, Dnp3, Dns, Elasticsearch, Ftp,Http, Imap,Ipmi,Ipp, Kubernetes, Memcached, Modbus, Mongodb, Mqtt, Mysql Port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Color shows details about Amqp, Bacnet, Dnp3, Dns, Elasticsearch, Ftp, Http, Imap, Ipmi, Ipp, Kubernetes, Memcached Pptp Telnet Telnet,Tls and Vnc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Rdp Smtp Mqtt Redis Ssh Pop3 TIs Http IpmiFigure 9: Protocol Coverage Across Ports—Only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='0% of HTTP services are served on port 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Researchers must scan 25K ports to achieve 90% coverage of HTTP services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' On the other hand, 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1% of AMQP services are on port 5672.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Port Hosts Top AS % of Hosts in Top AS 23 2,606 Telecom Argentina (10318) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7% 5523 521 Claro S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='A (28573) 87% 9002 396 Fastweb Italia (12874) 4% 6002 232 Fastweb Italia (12874) 6% 8000 158 Powercomm KR (17858) 89% Table 1: Top 5 Ports Hosting Telnet—While Telnet is most often seen on its assigned port (TCP/23), the majority of Telnet services are served on unassigned ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Unexpected Telnet devices are sometimes spread across a large number of ASes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', port 9002) and are therefore likely not due to a single operator decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' SSH hygiene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Unexpected ports hosting SSH are 15% more likely to allow non-public key authentication methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', password, host-based, challenge-response) than 22/SSH and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4 times less likely to be using only public key authentica- tion (11% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 26%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 60% of scanned ports are on average 2 times more likely (9% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 18%) to be running a software implementation of SSH that is likely to be on an IoT device (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', Dropbear, Cisco, Huawei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4 Summary and Implications Most services that acknowledge data on popular IANA- assigned ports run the expected L7 protocol, but this drops to nearly zero for less popular protocols with assigned ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The majority of services that speak popular protocols (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', TLS, Telnet, HTTP) are spread across all 65K ports rather than on their assigned port(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, only 3% of HTTP services listen on port 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Many of the services listening on random ports belong to IoT devices and/or have a weak se- curity posture, and it behooves the security community to consider these services when quantifying risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 4 Efficiently Identifying Services L7 scanning is more challenging when there is no assigned protocol for a port or when the expected L7 handshake fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Though Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3 demonstrates the importance of scan- ning for unexpected services, the naive method we used tests 30 unique L7 handshakes and is too intrusive and slow for large-scale experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In this section, we explore how to most efficiently detect unexpected L7 services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Encourag- ingly, only five handshake messages are needed to uncover 99% of unexpected services running identifiable protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1 Protocol Discovery We investigate two directions for accelerating protocol discov- ery: (1) methods that trigger protocol-identifying responses on a large number of protocols and (2) attempting handshakes in an order that optimizes for efficient service discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Wait and fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The most efficient first step for de- tecting the protocol on a port is to simply wait to send any handshake message and to see what the server sends first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' A total of 8 of the 30 protocols implemented in ZGrab—POP3, IMAP, MySQL, FTP, VNC, SSH, Telnet, and SMTP—are “server-first” protocols: after a TCP handshake concludes, the server will send a banner to the client, which allows the client to parse and identify the actual service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='99% of hosts which complete an SSH handshake have the keyword sshin the SSH banner, 90% of SMTP banners contains smtp, 72% of Telnet contains login or user, and 100% of VNC re- sponses contain RFB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We are able to identify banner signatures for all implemented binary and ASCII-based protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We also find that many protocols respond to incorrect hand- shake messages, including HTTP and TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Through 1% scans of the IPv4 space, we find that 16 of 30 protocols respond to an HTTP GET request or two newline characters for at least 50% of public services that speak the protocol (Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In general, most services that respond to the wrong hand- shake respond to both a GET request and TLS Client Hello, but MongoDB, and Redis do not send data in response to a TLS handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Though sending two newline characters is protocol-compliant for many ASCII protocols, doing so discovers fewer services than TLS and HTTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We discover a similar phenomenon when sending 50 newline characters, thereby implying that the contents of the newline message— rather than the length—causes the lack of responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' A total of 75% of binary (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', non-ASCII) services, in- cluding MQTT, Postgres, PPTP, Oracle DB, Microsoft SQL, Siemens S7, DNS, and SMB, send no data back unless we scan with their specific protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We note that our selection of tested protocols are biased towards ASCII protocols, and that it is likely that many binary protocols do not respond to these handshake messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' However, as discussed in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2, the long tail of binary protocols on the Internet are Scan IANA-Assigned Ports Ephemeral Ports Order Protocol ∆ Coverage Protocol ∆ Coverage 1 wait 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3% wait 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3% 2 TLS 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='0% HTTP 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1% 3 HTTP 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6% TLS 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='9% 4 DNS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4% Oracle DB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='23% 5 PPTP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='8% PPTP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='14% Table 2: Optimal Handshake Order—For IANA-assigned ports, waiting and then sending a TLS Client Hello discovers 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3% of unexpected services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Five handshakes can identify over 99% of identifiable unexpected services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' less spread out across a large number of ports compared to common protocols like HTTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Figure 10: Scanning L7 With Different Handshakes— Sending an HTTP handshake (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', a GET Request) prompts the most number of services to send back data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The data can then be used to fingerprint the actual service running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Optimal handshake order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We compute the optimal order of L7 handshakes that maximize the chances of identifying the service running on a port using a greedy approach across two sets of ports: (1) all IANA-assigned ports and (2) five random ephemeral ports (62220, 53194, 49227, 47808, and 65535).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Of the 30 protocols with ZGrab scanners that we can identify, we find that five handshake messages elicit responses from over 99% of identifiable unexpected services on both sets of ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We show the top-five L7 handshakes that discover the most unexpected services for the two sets of ports, excluding the expected services in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Across both IANA-assigned and ephemeral ports, merely opening a connection to the client (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', waiting) can immediately fingerprint more than half of unexpected services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For IANA-assigned ports, waiting and then sending a TLS Client Hello discovers 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3% of unexpected services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For ephemeral ports, waiting and HTTP discover 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4% of services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' It is not surprising that DNS and PPTP provide the 4th and 5th most additional coverage for IANA-assigned ports, as these are relatively popular protocols that do not answer to other handshakes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', HTTP GET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2 Impact of L7 Filtering One reason that we may not be able to identify all services is that even if our protocol guess is correct, our selected hand- shake parameters might be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, in SNMP, servers may reject requests that do not specify the correct community string in the first packet by first acknowledging the data, but then sending a TCP RST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' To estimate whether L7 filtering decisions cause a service to not send any data back to the client, thereby hindering fingerprinting efforts, we run two sets of scans, each with different handshake options, for each of the following ports and protocols: 8081/HTTP, 443/TLS, and 1723/PPTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For HTTP, in one scan we send a GET request and in another we specify the OPTIONS request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For TLS, in one scan we advertise the insecure cipher suite TLS_RSA_EXPORT_WITH_RC4_40_MD5 and in the other we advertise modern Chrome cipher suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For PPTP, in one scan the first message is crafted to contain the speci- fied “Magic Cookie” value (a specific constant used to syn- chronize the TCP datastream) according to RFC 2637 [31], 0x1A2B3C4D, and in another we specify the Magic Cookie to be 0x11111111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' RFC 2637 states that “Loss of synchroniza- tion must result in immediate closing of the control connec- tion’s TCP session;”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' we thus expect that fewer IPs will send data to the client if the magic cookie is incorrect and use this as a “control” experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Port (Service) Handshake Option IPs that send data Only GET Request 27% 8081 (HTTP) Only OPTIONS Request 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3% Both 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7% Only Good Cookie 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1% 1723 (PPTP) Only Bad Cookie 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='001% Both 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='8% Only Secure Cipher 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='65% 443 (TLS) Only Insecure Cipher 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='05% Both 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3% Table 3: Impact of Handshake Options—Handshake pa- rameters influence the services that send back identifiable data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, an HTTP OPTIONS request on port 8081 results in 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3% more IPs to respond with data than an HTTP GET request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7% of IPs will respond to both types of requests on port 8081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' An HTTP OPTIONS request discovers an additional 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3% IPs that speak HTTP compared to a GET request on port 8081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Responsive IPs will acknowledge data and close the connec- tion after receiving a GET request, hindering a scanner’s abil- ity to fingerprint the service as HTTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' However, by sending an OPTIONS request, 72% of IPs will respond with a 501 status (method not implemented) and 17% will respond with a 405 status (method not allowed), thereby confirming they do speak HTTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' IPs that exclusively respond to an OPTIONS request are not constrained to a particular network and are present across 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3% of ASes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The discrepancy is less pro- nounced on port 80 where only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='02% of IPs will respond to an OPTIONS request but not GET and only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1% of IPs will respond to GET but not an OPTIONS request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For TLS, per RFC 8446 [57], a handshake failure should generate an error message and notify the application before closing the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' However, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='65% of IPs will simply close the connection without any application-layer error when an incompatible cipher is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' As expected for PPTP, speci- fying an incorrect magic cookie results in 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1% of IPs failing to respond (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Hosts practicing their own Layer 7 filter- ing depending upon certain handshake options—and thereby not sending any data to the client—presents an unavoidable challenge for any L7 scanner to guess the perfect parameters to speak the appropriate Layer 7 with every single host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In Figure 6, we estimate all unknown services to be due to not having the expected handshake options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3 Consequences of Handshake Order Similar to how handshake options might prevent a server from responding, trying repeated incorrect handshakes prior to the correct one might also prevent the identification of services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We evaluate whether hosts filter or refuse connections after receiving incorrect L7 messages by (1) sending successive HTTP GET and TLS Client Hello messages to all IANA- assigned ports for 1% of the IPv4 space and (2) comparing the number of hosts that successfully complete a follow-up handshake when being sent the expected L7 data to the num- ber of hosts that successfully complete a follow-up handshake when being sent unexpected L7 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Depending on the protocol, we find that sending unexpected L7 data causes up to 30% of follow-up handshakes to fail compared to the hosts found when directly scanning for the protocol (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, sending non-Telnet data to Telnet servers causes 17% to fail a follow-up handshake;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 65% send a TCP RST and 35% do not SYN-ACK to a follow up TCP handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Sending an HTTP GET request to TLS servers causes 29% of follow-up TLS handshakes to fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We find this behavior to be similar to a Cisco IOS feature, Login Block, which allows administrators to temporarily block connections to L7 services after unsuccessful login attempts [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Sur- prisingly, this phenomenon only affects hosts after they send protocol-identifying data—likely because this is when they first store server-side application-layer state about the connec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' As such, this blocking does not prevent any servers from being fingerprinted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' It only prevents a follow-up handshake af- ter identifying data has been sent back to the scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Failure is generally temporary: 75% of hosts will successfully com- plete the L7 handshake within 5 seconds and 99% of hosts will take less than 2 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Nonetheless, waiting between fingerprinting and completing the follow-up handshake can reduce this filtering effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Figure 11: Impact of Sending Incorrect Handshakes— Sending unexpected data to hosts causes some services to fail the follow-up expected handshake even when fingerprint- ing was successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, only 71% of TLS hosts successfully complete a handshake when initially being sent an HTTP handshake message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We provide the fraction of total hosts successfully fingerprinted in the third column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4 Summary and Implications One fundamental limitation of L7 scanning is that services may require specific handshake options to respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Nonethe- less, our results indicate that the vast majority of identifiable Internet services can be easily identified during scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Many hosts respond to the “wrong” L7 handshake and send data that help fingerprint the service: 16 of 30 protocols can be detected with a single HTTP GET request and 99% of unex- pected services can be identified with five handshakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We use these optimizations to build a scanner (LZR) dedicated to accurate and efficient unexpected service discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 5 LZR: A System for Identifying Services In this section, we introduce LZR, a scanner that accurately and efficiently identifies Internet services based on the lessons learned from Sections 2–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' LZR can be used with ZMap to quickly identify protocols running on a port, or as a shim between ZMap and an application-layer scanner like ZGrab, to instruct the scanner what follow-up handshake to perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' LZR’s novelty and performance gain is primarily due to its “fail-fast” approach to scanning and “fingerprint everything” approach to identifying protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' It builds on two main ideas: Ignore non-acknowledging hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' About 40% of services that send a SYN-ACK never acknowledge data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' None of these services can complete an L7 handshake and can be safely ignored during Internet scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Quickly identifying and ig- noring these services can significantly reduce costs because non-acknowledging services force stateful scanners to open an OS socket and wait for the full timeout period to elapse, which typically takes much longer than completing a normal handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Non-acknowledging hosts can be filtered out by sending a single packet—an ACK with data—similar to how ZMap statelessly SYN scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Listen more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Up to 96% of services per port run unexpected protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In 8 of the 30 protocols we scanned, the server sends data first, and 10 protocols send fingerprint-able data when sent an incorrect L7 handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' By always waiting and then fingerprinting invalid server responses, we can identify up to 16 of the 30 protocols by sending a single packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' A scanner only needs to perform minimal computation to fin- gerprint a service: the first packet from a server identifies the running protocol, which does not require a full TCP/IP stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1 Scan Algorithm We outline LZR’s logic in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' LZR accepts a stream of SYN-ACK packets from ZMap or tuples of (IP, port) to scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In the case that LZR has full connection details from ZMap, LZR will start by filtering hosts that send SYN-ACKs with a zero window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Otherwise, it will initiate a new connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For non-zero windows, LZR will continue the connection by sending an ACK packet containing the expected protocol’s first-packet handshake data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' If LZR receives any type of data in response from the host, it will fingerprint the data and close the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' If a host neither acknowledges the data nor closes the connection, LZR re-transmits the data with the PUSH flag (further discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' If a host does not acknowledge the data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', never responds or RSTs the connection without an acknowledgement), LZR fingerprints the host as likely not hosting a real service and does not pro- ceed with further connection attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Otherwise, if a host acknowledges the data but does not send any data in response (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', server is unresponsive or closes the connection immedi- ately afterwards), LZR proceeds to close the connection, start a new connection, and send the next handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The process continues until LZR identifies the running protocol or runs out of additional handshakes to try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' LZR can also optionally filter IPs that respond on nearly every port (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1) by si- multaneously sending SYN packets to a user-specified number of random ephemeral ports and checking for a SYN-ACK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2 Architecture LZR is written in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='5K lines of Go and implements all unique protocols (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', handshakes) in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Similar to ZMap, LZR uses libpcap [68] to send and receive raw Ethernet pack- ets rather than rely on the OS TCP/IP stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' This allows LZR to efficiently fingerprint services because a single socket can be used for the duration of a scan and it allows LZR to adopt and continue connections initiated by a stateless scanner like ZMap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Because LZR only needs to send and receive a single packet to fingerprint services, a full TCP stack is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' LZR takes as input a command-line argument list of proto- cols to test and a stream of SYN-ACKs from ZMap or IP/ports to scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Internally, a small pool of Go routines send followup ACK packets containing handshake messages and fingerprint their responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Adding new protocols/handshakes to LZR is easy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' each handshake implements a Handshake interface that specifies (1) the data to attach to the ACK packet and (2) what to search for in a response packet to fingerprint the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Once LZR receives data to fingerprint, LZR first checks if the data matches the fingerprint (specified using the Handshake interface) of the protocol being attempted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' If not, LZR checks all the remaining fingerprints for a match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We note that be- cause ZMap sends probes using a raw Ethernet socket, LZR users need to install an iptables rule to prevent the Linux ker- nel from sending RST packets in response to the SYN-ACKs it receives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Otherwise, LZR cannot adopt and continue these connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We have released LZR under the Apache 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='0 license at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='com/stanford-esrg/lzr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3 Evaluation We evaluate both the accuracy and performance of LZR by comparing protocol-specific ZGrab handshakes with four LZR configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The first two are the expected use cases: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' ZMap/LZR: We use LZR with ZMap to identify the service running on a port that ZMap finds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' ZMap/LZR + ZGrab: We use LZR as a shim between ZMap and ZGrab to instruct ZGrab what full L7 hand- shake to complete for hosts that ZMap finds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' During experiments with these configurations at 1gbE, we find that LZR is able to filter hosts much faster than ZMap is able to find hosts—especially on ephemeral ports with low hitrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' ZMap artificially limits how fast LZR and ZGrab operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' As such, we introduce two additional metrics that approximate LZR’s performance under the premise of ZMap finding hosts infinitely quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' This allows us to compute how quickly LZR can find hosts as scan speeds increase and how much time ZGrab can save in an environment where there are many hosts to scan because the researcher is investigating multiple ports simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Offline ZMap/LZR + ZGrab: We perform scans in two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In the first, we use ZMap and LZR to identify No Window 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' ZMap S/A Receive ACK?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Yes Try all Fingerprinting Modules No Receive Data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Yes No Yes Max retransmits reached?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Receive RST?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' No Receive FIN?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' No More handshakes given at runtime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Yes Yes Yes No Send Ack w/ Handshake[i] i++ Send RST End Send SYN Receive S/A?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' No End Yes Yes End No Send Ack w/ PSH w/ Handshake[i] Filter Unknown Service Across Ports?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' From Random Ephemeral Port?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' No Yes Send SYN on eph_limit # of random ephemeral ports No End Max retransmits reached?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Yes No num_received >= eph_limit ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' No Yes Yes i == 1 Yes No Figure 12: LZR Algorithm—LZR efficiently identifies real Internet services by sending application-layer data with the ACK of a TCP handshake to filter out non-acknowledging hosts and fingerprint the responding protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Internet hosts that speak a known protocol and exclude this phase from our benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Then, in a second phase, we allow ZGrab to process services at full speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Offline ZMap + LZR: We perform scans in two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In the first, we find candidate services with ZMap, and exclude this phase from our benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In the second phase, we benchmark how quickly LZR can fingerprint services operating at full speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We report L4 and L7 behavior breakdown, CPU time, and bandwidth savings of LZR from 100% scans of the IPv4 address space completed during June 2020 in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We cal- culate runtime performance using CPU cycles per second for ZGrab and LZR as both tools are CPU bound: ZGrab’s com- pletion of a full handshake (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', encryption/decryption for TLS) and LZR’s fingerprinting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', pattern matching) create the biggest performance bottlenecks for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' When bench- marking LZR, we receive complaints from seven different organizations, but there is no indication that the complaints are the result of a particular LZR optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' we follow- up with all responsive network operators and learn that the complaints are simply due to the 100% coverage of the scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' How many additional services does LZR find?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' One of LZR’s key features is that it can identify additional services, while filtering out unresponsive ones by analyzing the re- sponse to the data included in the ACK packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Using the keyword-fingerprinting strategy, LZR identifies an average of 12 additional unique protocols across ports in our exper- iment by using only the expected 1–2 handshakes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' for ex- ample, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3 million IPs hosting an additional 16 protocols on port 443 and 238,000 IPs hosting an additional 18 pro- tocols on port 80 are found with just the single expected handshake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Furthermore, LZR finds over 2 times more unex- pected than expected services when sending a single AMQP handshake to 5672/AMQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The breakdown of the unexpected services is, unsurprisingly, nearly identical to the distribu- tion in Figure 6 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', HTTP and TLS dominate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Across all ports in Appendix A, LZR identifies 88% of all identifiable services with just a single HTTP handshake message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The exact signatures LZR uses for fingerprinting services can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='com/stanford-esrg/lzr/ tree/master/handshakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Does LZR filter out appropriate hosts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' LZR does not find a statistically significantly different set of hosts than scanning with just ZMap and ZGrab (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The Kolmogorov–Smirnov (KS) test [40] finds p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='05, rejecting the hypothesis that the approaches find a different number of services for all tested ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We also verify that sending data with an ACK during the handshake does not produce a statistically significant difference in the total number of hosts that acknowledge data or the total number of IPs that send back data across three trials of 1% IPv4 samples for 80/HTTP, 443/TLS and 27017/MongoDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' However, we do find that an additional average of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='18% of hosts respond when setting the PUSH flag during the retransmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Though the addition of the PUSH flag causes the follow-up packet to not qualify as an exact TCP retransmission per RFC 793 [54], we confirm that there is no increase in the number of closed connections when re-transmitting with a PUSH flag compared to an identical retransmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We do not set the PUSH flag immediately during the handshake as that causes about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6% of IPs to close the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' How much faster is L7 scanning with LZR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' ZMap/LZR performance is always faster than ZGrab due to LZR’s ability to identify service presence without completing an L7 hand- shake, which often requires a large number of CPU cycles for Port 80 443 21 23 5672 5900 27017 62220 80 443 47808 Protocol(s) HTTP TLS FTP TEL AMQP VNC Mongo HTTP HTTP TLS HTTP (Consecutively Scanned) TLS HTTP TLS Number of Hosts Found SYN-ACK 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6M 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='8M 14M 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4M 3.' 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+page_content='4M 505K 628K 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3M 45M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1M L7 Handshake Expected (LZR) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='66M 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7M 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='71M 123K 277K 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3K 38K 56M 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3M 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6K Expected (ZGrab) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='63M 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7M 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='73M 123K 277K 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6K 36K 56M 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4M 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7K Unexpected (LZR) 238K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3M 113K 230K 260K 56K 23K 23K 207K 758K 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='5K Unique Unexpected 18 16 10 10 11 8 14 12 18 16 14 Speed Up (Time) ZMap/LZR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3× 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7× 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='8× 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='9× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='9× 2× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6× 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7× 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3× 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3× 2× ZMap/LZR + ZGrab 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2× 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='5× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='8× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='9× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4× 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1× 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='95× 2× Offline ZMap/LZR + ZGrab 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1× 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6× 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3× 4× 7× 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1× 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='5× Offline ZMap + LZR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1× 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1× 5× 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='7× 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4× 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3× 55× 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3× 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6× 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='4× 29× Bandwidth Savings ZMap/LZR 60% 75% 67% 78% 70% 79% 66% 68% 79% 84% 87% ZMap/LZR + ZGrab 28% 16% 3% 3% 41% 46% 46% 54% 16% 9% 75% Offline ZMap/LZR + ZGrab 12% 10% 36% 67% 72% 68% 81% 79% 5% 7% 98% Offline ZMap + LZR 49% 60% 56% 69% 75% 78% 87% 85% 58% 68% 99% Table 4: LZR Performance—Filtering for IPs that acknowledge data increases service fingerprinting speed by up to 55 times while finding up to 30% more unexpected services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' All relative performance numbers are compared to ZGrab and measured at a 1 Gb/s scanning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' expensive operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', cryptographic functions in TLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' At minimum, LZR is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='9 times faster than ZGrab when scan- ning 5672/AMQP and, at maximum, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='3 times faster when scanning 443/TLS+HTTP—equivalent to a 40 CPU hour speed-up of a 100% scan of IPv4 when using ZGrab’s default number of senders (1,000) and scanning at ZMap’s calculated sending rate that minimizes ZGrab’s packet loss (50K pps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' The performance of LZR as ZGrab’s shim (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', ZMap/LZR + ZGrab) varies based on a port’s service makeup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' When a port contains a large raw number of hosts that do not consistently establish a TCP connection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', zero window), there is sub- stantial performance improvement: ZMap/LZR + ZGrab is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='6 times faster than ZGrab when scanning 62220/HTTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' On the contrary, since the relative number of hosts that do not consistently establish a TCP connection on port 443 is small, there is little improvement (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1 times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' When a significant fraction of candidate services do not acknowledge data, there is significant improvement when us- ing LZR to filter hosts offline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', when ZGrab can run at full speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' On a 100% IPv4 scan of 27017/MongoDB, only 21% of hosts that SYN-ACK acknowledge data and an addi- tional 30% of hosts send a zero window, which allows LZR to increase ZGrab performance by 7 times and a LZR scan by 55 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Unpopular ports are expected to have the same performance improvement as 62220/HTTP (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', a 25 times speed-up) because IPs on the majority of ports are more likely to not acknowledge data when sending a SYN-ACK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' How much bandwidth does LZR save?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Using LZR alone to fingerprint services always saves bandwidth (up to 87% on 47808/HTTP+TLS) when the reasonably-expected data is sent during the initial handshake, as (1) LZR does not attempt to re-transmit ACKs to zero-window hosts to check for an increase in window size, and (2) LZR does not need to complete full L7 handshakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' However, when using LZR alongside ZGrab when scanning a port where the majority of TCP-responsive hosts serve the expected protocol, there exists an overhead in the number of total packets sent—even when there is a speed-up in time—due to LZR sending at least one extra ACK to fingerprint before re-attempting the actual handshake (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', LZR + ZGrab together send 28% more packets than ZMap+ZGrab for 80/HTTP even though LZR + ZGrab run 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='2 times faster than ZMap+ZGrab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 6 Related Work Fast Internet-wide scanning has been used in hundreds of academic papers in the past seven years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' While we cannot enumerate every paper that has used the technique, we empha- size that scanning is now common in the security, network- ing, and Internet measurement communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Data collected through Internet-wide scans has been used to understand cen- sorship [42, 52, 53], botnet behavior [10, 46], patching be- havior [23, 25, 47] as well as to uncover vulnerabilities in IoT and SCADA devices [19, 51, 67], cryptographic proto- cols like TLS [9,11,13,17,37], SSH [6,36], and SMTP [22], and the Web PKI [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Multiple tools have emerged in the space, most notably ZMap [26] and Masscan [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' As of 2020, more than 300 papers used ZMap and in 2014, Durumeric et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' found that a significant fraction of all Internet scanning uses ZMap [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Prior to the development of these tools in 2013, groups performed smaller-scale studies to measure a multitude of Internet dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Despite the growing popularity of the technique, there has been relatively little work specifically investigating the dy- namics of Internet-wide scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Several works have noted the large discrepancy between L4 and L7 responses [21,24, 26,36,51,67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Clayton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' [18] find evidence of dynamic blocking within the Great Firewall of China—but do not for- mally quantify how wide-spread the behavior is—and Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' [69] find evidence of dynamic blocking within SSH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Alt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' introduced degreaser [8] to locate “tarpits”—fake services that attempt to trick network scanners;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' tarpits may use some of the same techniques we see middleboxes use at the start of a connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In a similar vein to our work, in 2018, Bano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' [12] studied the notion of host liveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' As part of their taxonomy, they considered the relationship between live services on different points, showing that the responses on popular ports are correlated with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In 2014, Durumeric et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' investigated server blacklisting and how operators respond to Internet-wide scanning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' at the time they found that blacklisting behavior was negligible [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Rüth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' considered the ICMP responses received in response to ZMap IPv4 SYN scans [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' One contribution of our work is the introduction of LZR, which reduces the time needed to scan less populous ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Prior work has similarly attempted to reduce the time required to complete Internet-wide scans, though through starkly dif- ferent approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Klick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' [43] show that much of the IP address space does not need to be continually scanned by services like Censys [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Adrian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' introduce a faster version of ZMap that operates at 10gbE [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' LZR solves a different problem and can be used in coordination with these other performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Similar to how we use a single packet to identify services, several works have focused on single-packet fingerprinting to identify operator systems [64,65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' 7 Recommendations and Conclusion We began our analysis by investigating the troubling obser- vation that a significant fraction of hosts on the Internet that respond to a SYN scan never complete an application-layer handshake [21,24,26,36,51,67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We found that middleboxes are responsible for the majority of responses with no real ser- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We also showed that a significant fraction of services are also located on unexpected ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' For example, 97% of HTTP and 93% of TLS services are not located on ports 80 and 443, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Worryingly, unexpected services often have weaker security postures than those on standard ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Building on these observations, we introduced LZR, a scan- ner that dramatically reduces the time required to perform an application-layer scan on ports with few expected services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', 5500% speedup on 27017/MongoDB) while simultane- ously identifying many unexpected services running on the port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' LZR can identify 16 protocols and 88% of identifiable services with one packet and 99% of identifiable unexpected services with 5 handshakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Nonetheless, there are two addi- tional challenges to scanning unassigned ports: (1) scanning 100% of all 65,535 ports is not feasible, and (2) it is not clear which subset of ports is worth scanning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', contain a sig- nificant fraction of the particular behavior being studied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We therefore recommend that researchers conduct lightweight sub-sampled (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='1%) application-layer scans across all ports to detect the prevalence of targeted protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We em- phasize that merely using the top n most popular ports is not sufficient to evaluate which ports are most likely to host par- ticular services, as most protocols are drowned out by the overwhelming popularity of HTTP and TLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We hope that researchers find LZR helpful in accurately and efficiently identifying services in Internet-wide scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Acknowledgements The authors thank Tatyana Izhikevich, Katherine Izhikevich, Kimberly Ruth, Deepak Kumar, David Adrian, Deepti Ragha- van, Jeff Cody, members of the Stanford University and UC San Diego security and networking groups, and the anony- mous reviewers for providing insightful discussion and com- ments on various versions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We further thank Sadjad Fouladi and Katherine Izhikevich for using their artis- tic talent to greatly improve the visual graphics in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' This work was supported in part by the National Science Foundation under award CNS-1823192, Cisco Systems, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', the NSF Graduate Fellowship DGE-1656518 and a Stanford Graduate Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' References [1] External HTTP(S) load balancing overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' https://cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='google.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Heninger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Springall, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Thomé, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Valenta, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Imperfect forward secrecy: How Diffie-Hellman fails in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In 22nd ACM Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' on Computer and Communications Security, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Adrian, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Durumeric, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Singh, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Halderman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Zippier ZMap: Internet-wide scanning at 10 gbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' In USENIX Workshop on Offensive Technologies, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Alt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Beverly, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Dainotti.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='Siemens S7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='11211 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='memcached ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='memcached ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='memcached ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='Figure 13: Port Selection—Three categories of ports are scanned: (1) The top 30 ports determined by a SYN-ACK scan conducted across all 65K ports of 1% of IPv4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' (2) Ports for which a ZGrab-scanner exists (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=', to be able to complete the full L7 handshake).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' (3) A random selection of 5 ephemeral ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' We label the expected service being hosted on the port, as well as the IANA-assigned service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} +page_content=' Note that each of these categories contain overlapping ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE4T4oBgHgl3EQfAQs4/content/2301.04841v1.pdf'} diff --git a/ztE0T4oBgHgl3EQftwFw/content/tmp_files/2301.02596v1.pdf.txt b/ztE0T4oBgHgl3EQftwFw/content/tmp_files/2301.02596v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..32c639c8c334e5d874e576e293a58b8324e339f1 --- /dev/null +++ b/ztE0T4oBgHgl3EQftwFw/content/tmp_files/2301.02596v1.pdf.txt @@ -0,0 +1,5450 @@ +Benchmark solutions for radiative transfer with a moving mesh and +exact uncollided source treatments +William Bennett +Ryan G. McClarren +January 9, 2023 +Abstract +The set of benchmark solutions used in the thermal radiative transfer community suffer some coverage +gaps, in particular nonlinear, nonequilibrium problems. +Also, there are no nonequilibrium, optically +thick benchmarks. These shortcomings motivated the origination of a numerical method unfettered by +preconditions of linearity and easily able to converge on smooth optically thick problems, a moving +mesh Discontinuous Galerkin (DG) framework that utilizes an uncollided source treatment. +Having +already proven this method on time dependent scattering transport problems, we present here solutions +to non-equilibrium thermal radiative transfer problems for familiar linearized systems and more physical +nonlinear systems in both optically thin and thick regimes, including both the full transport and the +S2/P1 solution. Geometric convergence is observed for smooth sources at all times and some nonsmooth +sources at late times when there is local equilibrium and accurate solutions are achieved for step sources +when the solution is not smooth. +1 +Introduction +The Stefan-Boltzmann law [1], which describes the relationship between radiation emitted from a material +and its temperature as proportional to temperature to the fourth power, is to blame for the obdurate +nonlinearity in high-energy density radiative heat transfer models. +For this reason, the extant analytic +benchmarks with space and time dependence in this field are predicated on assumptions of linearity or +equilibrium. +There are solutions that assume the radiation energy and internal energy of the material +instantly equilibrate, inducing a Marshak Wave [2, 3, 4], and solutions for non-equilibrium problems that +linearize the system in T 4 by invoking a form for the the material heat capacity that is proportional to +temperature cubed, an innovation of Pomraning [5]. This technique of defining a heat capacity to linearize +the system has been used to produce an abundance of solutions, including transport treatments for the P1 +equations [6], full transport solutions with one speed [7, 8], and non-grey problems [9]. In the diffusion limit, +benchmarks solutions have been provided for one temperature [10], three temperature [11], and a non-grey +treatment [12]. +While these solutions are invaluable to code developers for verification, it is necessary to point out +that there are certain drawbacks to using linear problems to verify codes whose purpose it is to solve +nonlinear systems. While ideally the numerical code in question would solve the fully nonlinear equations and +implement a special equation of state when running these verification problems, there is nothing to prohibit +the curators of these codes from simply solving a linearized system when the benchmark is being run. The +result is the same in both cases. Also, solutions to linear systems can be scaled to match benchmarks, unlike +the unforgiving solutions to nonlinear systems, and the solution to the linearized equations equilibrates more +quickly than a nonlinear problem as the temperature increases as a result of the special equation of state. +Solving a linear problem does not completely verify the functionality of a radiative transfer code. +Although nonlinearity is an impediment for analytic methods, it is not necessarily a source of difficulty +for spectral methods. +This was the impetus for our development of a moving mesh, uncollided source +treatment Discontinuous Galerkin (DG) method for solving transport problems [13]. The time dependent +cell edges, which we call a moving mesh, and the uncollided source treatment were added on to the DG +1 +arXiv:2301.02596v1 [cs.CE] 6 Jan 2023 + +implementation because the transport equation with its finite wavespeeds admits discontinuities that inhibit +DG methods from attaining their higher order convergence potential. The moving mesh and uncollided +source can present a smoother problem for the method to solve: the moving mesh by matching edges to +moving wavefront discontinuities and the uncollided source by analytically resolving the most structured +part of the solution. +As documented in [13], we have already conducted extensive tests with this method on time-dependent +transport problems, which allowed for a detailed analysis of the efficacy of the moving mesh and uncollided +mesh for different source types. For example, for finite width, nonsmooth sources that induce a nonsmooth +solution that is smoothed over time, the method proved the most beneficial when compared to a standard +DG implementation, but displayed only algebraic error convergence, not the optimal geometric convergence +that DG methods are capable of. For smooth, Gaussian sources however, we we able to achieve spectral +convergence, but the importance of the moving mesh was diminished. +With an understanding of the effectiveness of this method on linear systems, we apply it to nonlinear +radiative transfer problems and obtain results with accuracy comparable to an analytic solution, which is the +stated intent of this work. Before attempting fully nonlinear problems however, we first apply our method +the existing linear radiative transfer problems. This will allow us to test our method on problems with known +solutions and uncover deficiencies in a more forgiving arena. For nonlinear problems, we can still gauge the +precision of our solution by inspecting magnitude of the expansion coefficients and the accuracy by checking +against existing numerical Sn solvers. +We selected the Su and Olson transport benchmark [8] as an ideal verification solution for our method. +Unfortunately, the results are not given to enough digits to fully demonstrate the effectiveness of our scheme +and recalculation of these results is non-trivial. Therefore, we rely on integration of a P1 version of this +benchmark [6] to create solutions which are not necessarily as physically accurate as the full transport +solution, but can be calculated out to more digits. +There are no existing radiative transfer transport verification solutions for optically thick problems outside +of the equilibrium diffusion limit. By optically thick, we mean that the source width or the support of an +initial condition is orders of magnitude larger than a mean free path. Conversely, optically thin problems +have source widths comparable to a mean free path. For a transport code to have sufficient coverage of +verification problems, converging to a diffusion benchmark of a thick problem while not resolving a mean +free path is a good test. If the code resolves a mean free path, however, it will converge to the diffusion +problem plus a transport correction. It is for the purpose of verifying this transport correction that we +include transport solutions and the S2/P1 solutions for optically thick problems. +The remaining sections of the paper are organized as follows. Section 2 contains an introduction to our +model equations, nondimensionalization, and derivation of the uncollided source. Our DG implementation +is laid out briefly in Section 3, but a more detailed derivation is left to [13]. Section 4 is devoted to the +calculation of S2 benchmarks and the corresponding uncollided solutions used in verifying our method. +Following this is a description of how the convergence of the error in the results section is calculated (Section +5. then our results (Sections 6 and 7). The results sections also contain specific details of the methods used +in each problem and discussion of the solution characteristics. +2 +Equations +We study non-equilbrium time dependent radiative heat transfer in an infinite, purely absorbing, stationary +medium with an internal radiation source. The radiation transport and material balance equations for this +system are, +�1 +v +∂ +∂τ + µ ∂ +∂z + σa +� +ψ(z, τ, µ) = σa +�1 +2avT(z, τ)4 +� ++ 1 +2S(z, τ), +(1) +∂ +∂τ e(z, τ) = σa +� +φ(z, τ) − avT(z, τ)4� +, +(2) +where the general form of the equation of state is, +e = +� T +0 +dT ′ Cv(T ′). +(3) +2 + +The variables in these equations are, ψ, the angular flux or intensity, φ = +� 1 +−1dµ′ψ(x, t, µ′) the scalar +flux, T, the temperature, and e, the material energy density. ψ and φ have units of energy per area per time +([GJ·cm−2ns−1]) and e has units of energy density ([GJ·cm−3]). S is a source term with units of energy +density per time. µ ∈ [−1, 1] is the cosine of the particle direction with respect to the z axis. v is the particle +velocity, which is the speed of light in a vacuum for our application, v = 29.998 cm ns−1. The radiation +constant is a = 4σSB/v = 0.0137225 GJ cm−3 keV−4, where σSB is the Stefan-Boltzman constant. The +absorption cross section, σa, is in units of inverse length. +We seek a non-dimensionalization for these equations that is compatible with the non-dimensionalization +given the in Su-Olson benchmark [9] and that may be used in optically thick problems without enlarging +the non-dimensionalized length to accommodate the larger opacity, +x = lσaz +t = lvσaτ. +(4) +l is a dimensionless scaling variable that is set to one for thin problems and a small number to offset the +greater σa in optically thick problems. +Each equation is transformed into the new variables and divided by avσaT 4 +H, where TH is the reference +temperature, called the hohlraum temperature in previous work, +� +l ∂ +∂t + µl ∂ +∂x + 1 +� +ψ(x, t, µ) = ca +�1 +2T(x, t)4 +� ++ 1 +2Q(x, t), +(5) +l ∂ +∂te(x, t) = ca +� +φ(x, t) − T(x, t)4� +. +(6) +Our non-dimensional dependent variables are now, +ψ(x, t) = ψ(x, t) +avT 4 +H +φ(x, t) = φ(x, t) +avT 4 +H +T(x, t) = T(x, t) +TH +e(x, t) = e(x, t) +aT 4 +H +, +(7) +the non-dimensional source is, +Q(x, t) = S(x, t) +σaavT 4 +H +(8) +and the absorption ratio is defined, +ca = 1. +(9) +In this work, we consider two functional forms for Cv. To solve the Su-Olson benchmark problem, we use +the familiar form, +Cv = αT 3. +(10) +which renders Eq. (6) linear in T 4. With the conventional choice of α = 4a, now +eSU = T +4, +(11) +where the subscript “SU” indicates that this is the equation of state for the linear Su-Olson problem. +While it is important for our investigation to solve these linear problems the novel aspect of this paper is +results for nonlinear problems. For these, we choose a more physical, constant specific heat, Cv = Cv0, with +units of of energy density per temperature. This choice renders e = Cv0T. To find the relationship between +the nondimensional variables with this equation of state, we define Cv0 = Cv0 +aT 3 +H . Now we can write +eN = Cv0T, +(12) +where the subscript “N” indicates that this is our equation of state for the nonlinear problems. +Other forms of Cv are certainly allowable. +For instance, we could implement a specific heat that is +dependent on a non-constant opacity. However, this will be left for future work. +3 + +2.1 +Uncollided solutions +In time dependent transport trials, we found that the deployment of an an uncollided source treatment, +where using the solution to the equation, +� +l ∂ +∂t + µl ∂ +∂x + 1 +� +ψu(x, t, µ) = 1 +2Q(x, t), +(13) +as a source term to solve for the collided flux is a significant boon for accuracy when the solution is not +smooth. The Green’s solution to Eq. (13) with l = 1 was provided by [14]. This solution is integrated for +different source configurations in [15], including a square and a Gaussian source. Using these solutions, we +can say that ψu is known and can be integrated analytically to find φu. For problems where l ̸= 1, a simple +scaling is required. For optically thick problems when l ≪ 1, the uncollided solution is not as useful since it +has decayed to zero by the pertinent evaluation times. +To solve for the remaining collided portion of the flux, we have the system, +� +l ∂ +∂t + µl ∂ +∂x + 1 +� +ψc(x, t, µ) = ca +�1 +2T(x, t)4 +� +(14) +l ∂ +∂te(x, t) = ca +� +φc(x, t) + φu(x, t) − T(x, t)4� +. +(15) +In linear transport applications, it is possible to decompose the flux infinitely, not just into uncolllided and +collided flux, but uncollided, first collided, second collided, etc. Even though the radiative transfer equations +are nonlinear, we are able to use this linear solution technique since the uncollided flux has no interaction +with the material. However, we cannot not further decompose the flux as we could in a linear transport +problem. +Answers obtained with an “uncollided source” treatment refer to solutions to Eqs. (14) and (15). A +“standard source” treatment refers to Eqs. (5) and (6). The most useful source treatment used in a specific +problem is determined by the behavior of the uncollided flux during the solution time window. Tests run +in [13] showed that integrating the uncollided source could require more computation time than a standard +source. This is because the uncollided source is a complex function of space and more difficult to integrate +with quadrature than the standard source. As a rule, problems at times where the uncollided flux has not +decayed enough to be a negligible portion of the flux are good candidates for an uncollided source treatment. +In these problems, [13] showed increase in accuracy and rate of convergence. At times where the uncollided +solution has decayed, the uncollided source treatment is not as helpful. +While the problems investigated in this paper are in purely absorbing media, the coupling between +the material energy density and the radiation energy density acts as a scatterer in that it can smooth +discontinuities over time. For this reason, we expect that the insights derived from solving purely scattering +transport problems with uncollided source treatments will extend to these purely absorbing radiative transfer +problems. +3 +Moving Mesh DG spatial discretization +Similar to the procedure in [13], we define a DG spatial discretization with a moving mesh to solve equations +of the form (5) and (6). We leave some of the details of the derivation to [13]. To solve for the integral +over µ to find the scalar flux, we discretize in angle via the method of discrete ordinates, where the solid +angle µ ∈ [−1, 1] is discretized by choosing the points with a Gauss-Lobatto quadrature rule [16] for our full +transport solution or, in the case of the S2 solution, a Gauss-Legendre rule. With the corresponding weights +from our chosen quadrature, we can define the scalar flux as a weighted sum, +φ ≈ +N +� +n′=1 +wn′ψ +n′ +, +(16) +where wn are the weights and ψn is the scalar flux evaluated at a given angle. This choice makes Eq. (5) +and (6), +4 + +� +l ∂ +∂t + µnl ∂ +∂x + 1 +� +ψ +n(x, t) = ca +�1 +2T(x, t)4 +� ++ 1 +2Q(x, t) for n = 1 . . . N, +(17) +∂ +∂te(x, t) = ca +� N +� +n′=1 +wn′ψn′ − T(x, t)4 +� +. +(18) +To discretize the spatial domain, we define K non-overlapping cells with time dependent edges xL(k, t) +and xR(k, t). To allow simplifications to the coming weak form of the equations, we define a mapping variable +x′(k, t) that maps x to [-1,1] inside a cell, +x′(k, t) ≡ xL(k, t) + xR(k, t) − 2x +xL(k, t) − xR(k, t) +, +k = 1 . . . K. +(19) +Now we define an orthonormalized Legendre polynomial basis function in x′ for each cell, +Bi,k(x′) = +√2i + 1 +� +xR(k, t) − xL(k, t) +Pi(x′). +(20) +Therefore, the weak solution of the angular flux in a cell for a given angle is +ψ +n(x, t) ≈ +M +� +j=0 +Bj,k(x′) un +k,j. +(21) +where u is an entry in our three dimensional solution matrix. Likewise, the solution for the energy density +in a given cell is, +e(x, t) ≈ +M +� +j=0 +Bj,k(x′) uN+1 +k,j , +(22) +The standard DG procedure for finding the weak form of the equations involves multiplying each equation +by a basis function, integrating over a cell, invoking integration by parts to shift the spatial derivative onto +the basis function, and taking advantage of orthogonality to simplify the system. Our moving mesh method +is similar to this, but with the added step of invoking the Reynolds Transport Theorem [17] since our +integration domain is time dependent. Leaving the general outline of this procedure to [13], we arrive at, +d +dtUn − GUn + +� +LUn +�(surf) − µnLUn + 1 +l Un = ca +2l H + 1 +2lQ for n = 1 . . . N, +(23) +d +dtUN+1 + RU surf − GU N+1 = ca +l +� N +� +n′=1 +w′ +nU n′ − H +� +, +(24) +where the time dependent solution vector is +U n,k = [un +k,0, un +k,1, ..., un +k,M]T , +where M + 1 is the number of basis functions. We also define +Li,j = +� xR +xL +dx Bj,k(x′) dBi,k(x′) +dx +, +(25) +Gi,j = +� xR +xL +dx Bj,k(x′) dBi(x′) +dt +, +(26) +Qi = +� xR +xL +dx Bi,k(x′) Q(x, t), +(27) +Hi = +� xR(k,t) +xL(k,t) +dx Bi(x′) T +4(x, t). +(28) +5 + +The numerical flux terms, which calculate the direction of flow of the solution with an upwinding scheme +based on the relative velocity of a particle with the cell edges +(LU)surf +i += +� +µn − dxR +dt +� +Bi,k(x′ = 1)ψ +n+ − +� +µl − dxL +dt +� +Bi,k(x′ = −1)ψ +n−. +(29) +(RU)surf +i += +� +−dxR +dt +� +Bi,k(x′ = 1)e+ − +� +−dxL +dt +� +Bi,k(x′ = −1)e−. +(30) +ψ +l+ and ψ +l− are found by evaluating Eq. (21) and e+ and e− are found by evaluating Eq. (22). +If we choose to employ an uncollided source treatment, Eqs. (23) and (24) change slightly in that the +source term Q disappears from the RHS of Eq. (23) and ca +l φ is added to the RHS of Eq. (24) where, +φu = +� xR +xL +dx Bj,k(x′) φu(x, t). +(31) +In this case, the numerical solution is for the collided flux, so it is necessary to add the uncollided flux at +the final step to obtain the full solution. +The general solution procedure is as follows. First, parameters such as the number of basis functions, the +number of spaces, and the Sn order are set. A source is specified and depending on the source treatment, the +uncollided solution or the standard source is integrated at each timestep with a standard Gaussian integrator +with points equal to 2M + 1. The edges of the mesh are governed by a function designed to optimize the +solution for the specific source. This function also returns velocities of the mesh edges in order to calculate +the numerical flux. The temperature balance terms are found by the equation of state and integrated in +the same way as the source term. The solver returns the coefficient arrays and the scalar flux and material +energy are reconstructed via the expansions defined in Eqs. (21) and (22) and, depending on the source +treatment, the uncollided flux is added onto the scalar flux. +To obtain solutions from our equations (23) and (24), we calculate the quadrature weights with the +python package quadpy [18] and integrate the ODES in with a built in integrator from scipy [19]. Our +python implementation can be found on Github1. +4 +Benchmarks and uncollided solutions for the S2 radiative trans- +fer equations +In order to show more decimal places of accuracy in our linear problems than are given in [8], it was expedient +to calculate our own analytic benchmarks. Since we already include results to each problem calculated with +S2, we choose to verify our solver by using a S2/P1 benchmark given by [6]. This benchmark gives the +analytic expression for the scalar flux and energy density solutions to Eq. (17) and (18) with N = 2, angles +and Gauss-Legendre weighting, ([µ1, µ2] = [ −1 +√ +3, +1 +√ +3], and [w1, w2] = [1, 1]) when the source is a delta function +in space and time. +The Green’s function given by [6] for φ = ψ1 + ψ2 for a delta function source at position s is, +G(x, s, t) = +v +2 +√ +3e−t +� +� +tI1 +�� +t2 − 3(x − s)2 +� +t2 − 3(x − s)2 +Θ +� +t − +√ +3|x − s| +� ++ I0 +�� +t2 − 3(x − s)2 +� +δ +� +t − +√ +3|x − s| +� +� +� , +(32) +where Θ is a step function, δ is a Dirac delta function and I0 and I1 are modified Bessel functions of the +first kind. The Green’s function for the material energy density is, +GU(x, s, t) = +√ +3 +2 e−t � +I0 +�� +t2 − 3(x − s)2 +� +Θ +� +t − +√ +3|x − s| +�� +. +(33) +1www.github.com/wbennett39/moving mesh radiative transfer +6 + +We choose to find solutions for a square source and a Gaussian source, to test our method on both smooth +and nonsmooth problems. Therefore, the solution to the integral, +φss,gs = +� ∞ +−∞ +ds +� ∞ +0 +dt′ Sss,gs(s, t′) G(x, s, t − t′) +(34) +gives the scalar flux. The energy density is likewise obtained by +ess,gs = +� ∞ +−∞ +ds +� ∞ +0 +dt′ Sss,gs(s, t′) Gu(x, s, t − t′). +(35) +Where the subscript in the solution and the source is either “ss” for square source or “gs” for Gaussian. The +source term is, +Sss(x, t) = Θ(x0 − x)Θ(t0 − t), +(36) +for the square source, or +Sgs(x, t) = exp +�x2 +x2 +0 +� +Θ(t0 − t), +(37) +for the Gaussian source. +Since finding a benchmark for an optically thick problem requires evaluating these integrals at extremely +late times where the integrand is not well behaved, we only calculate S2 benchmarks for our thin problems. +4.1 +S2 uncollided solutions +The uncollided solutions that we have utilized so far for the uncollided source treatment have been full +transport solutions from [15]. We cannot use these to solve the S2 transport equation, since the two uncollided +fluxes are not equal. The full transport solutions are based on the assumption that the Sn order of the +ODE’s sufficiently resolves the angular error and that the collided flux calculated with quadrature is a good +approximation of the analytic integral over µ, i.e. +N +� +n′=1 +wn′ψ +n′ +≈ +� 1 +−1 +dµ′ ψ(x, t, µ′), +(38) +so that it is acceptable to employ the uncollided scalar flux found by integrating analytically the solution for +the uncollided angular flux. In the S2 equations, the assumption of Eq. (38) does not hold and the uncollided +scalar flux must be found by numerical quadrature of the angular flux. +Therefore, the process for finding the uncollided scalar flux to use as a source in the S2 solutions to our +radiative transfer problems is to find the Green’s solution for the angular flux, integrate that solution with +quadrature, and then integrate again over the given source. The uncollided solution to Eq. 13 with a delta +function source (δ(x)δ(t)) is [14], +ψu(x, t) = e−t +2t δ +� +µ − x +t +� +. +(39) +To find the S2 uncollided scalar flux, the integral is done by Gauss-Legendre quadrature with N = 2 to give +the uncollided scalar flux, +φ +pl +u (x, t) = e−t +2t +� +δ +� +− 1 +√ +3 − x +t +� ++ δ +� 1 +√ +3 − x +t +�� +. +(40) +To finally find the uncollided scalar flux that corresponds to the benchmark solutions calculated with +Eqs. (34), we integrate +φ +ss,gs +u +(x, t) = +� ∞ +−∞ +ds +� ∞ +0 +dt′ φ +pl +u (x − s, t − t′) Sss,gs(s, t′), +(41) +where Sss,gs is given by Eq. (36) or Eq. (37). Solutions to Eq. (41) are given in Appendix A. +7 + +5 +Error estimation methods +In the problems presented, two methods are used to estimate the solution accuracy. For problems with a +benchmark solution, we use the root mean square error (RMSE) as our error metric. This is calculated by, +RMSE = +� +� +� +� +N +� +i +|yi − ˆyi|2 +N +, +(42) +where yi is either the calculated scalar flux or the calculated material energy density at a given node, ˆyi is +the corresponding benchmark solution, and N is the total number of nodes in the computational solution. +For problems that demonstrate geometric spectral convergence, as M → ∞, the error can be modeled as +ERROR = C exp(−c1M), +(43) +where M is the highest polynomial order of the basis and C and c1 are constants that could depend on the +number of cells used in the problem. This curve is a straight line on a logarithmic-linear scale. +For all of the problems in the following section, we plot the average of the absolute value of the coefficients +in the solution expansion to characterize the solution convergence. We define the average value of the jth +coefficient in the solution expansion, +|cj| = +�K +k=1 |ak| +K +, +(44) +where j corresponds to the order of the Legendre polynomial in the basis and K is the number of cells. +When characterizing the error of φ, since we are interested in the residual error of scalar flux, ak is the +weighted average using the weights from Eq. (16), +ak = +�N +l′=1 wl′u(l′,k,j) +�N +l′=1 wl′ +. +(45) +For the material energy density, ak is, +ak = u(N+1,k,j). +(46) +6 +Optically thin results +The results in this section are for problems where the source width is equal to a mean free path, meeting our +definition of an optically thin problem. These problems are characterized by solutions where the uncollided +solution is a significant portion of the flux and travelling wavefronts. Therefore, the problems in this section +all use an uncollided source and the square sources whhich have travelling discontinuities employ a moving +mesh. +6.1 +Su-Olson problem with a square source +We first replicate the the Su-Olson problem using the same square source originally presented in [8] with +σa = 1 cm−1, the source width, x0 = 0.5 and the source duration t0 = 10. The uncollided solution for this +source has already been presented in [15]. For the S2 treatment of this problem, the uncollided source is +given by Eq. (58). The temperature is calculated by Eq. (11). +Some modifications were made to the original mesh function invented to solve the square source transport +problem in [13]. In that mesh, the mesh edges inside the source never moved while the edges outside travelled +outwards with the wavespeed. This was done to resolve the static discontinuities at the source edge and the +travelling discontinuities at the wavefront. In the original Su-Olson results, the source turns off at t0 = 10 +and solutions are required long afterwards (t = 31.6228, 100). +With our previous square source mesh, +the edges would remain clustered around the source region long after the source has ceased to introduce +nonsmoothness. This is not the optimal distribution of computational zones. +8 + +Therefore, the mesh function used in this problem is as follows. If the mesh edges are defined as the +vector, +X(t) = +� +x0(t), x1(t), ..., xK(t) +� +, +(47) +and initialized to be, +if K +4 ≤ k ≤ 3K +4 , xk +o = 1 +2yj, +(48) +if k < K +4 , xk +o = sk(δx) + 2x0 + δx +2 +(49) +if k > 3K +4 , xk +o = sl(δx) − 2x0 − δx +2 +(50) +where yj are the Gauss-Lobatto evaluation points with N, the number of points, equal to K +2 + 1 numbered +from 0 and sm are the Gauss-Lobatto evaluation points for N = K +4 + 1. The indices j and m are equal +to k − K +4 and k − K +2 + 1 respectively. δx is a small initial width, and K is always an even number. This +initialization assigns one third of the edges to the source and the other two thirds to cover the rest of the +solution domain. Each subdomain is spanned by edges with Gauss-Lobatto spacing, which has the effect of +concentrating cells near the source edges and the outgoing wavefronts, where discontinuities are most likely. +As time progresses and the outside edges move outwards with the solution, their position is defined as, +if t ≤ t0, xk(t) = xk +o + xk +xko +× vt, +(51) +where v is the wavespeed, one for the transport problems and +1 +√ +3 for the S2 problems. The edge velocity is +defined, +if t ≤ t0, dxk +dt = xk +xko +× vt, +(52) +Defining the edge positions and velocities this way preserves the relative spacing of the initialized edges, +meaning that the edges are clustered at the source edges and the leading wavefronts. +At later times when the source is off, the solution to a square source in an optically thin problem will +behave more like the solution for a Gaussian source since the solution will become smoother without the +source emitting uncollided particles. Information flow is no longer dominated by the wavespeed. The solution +will be practically zero some distance from the origin that is much less than vt. For instance, in the Su-Olson +problem at t = 100 the solution is practically zero past x = ±30. +For these reasons, when the source turns off, a constant acceleration will divert the trajectory of each +edge so that at the final time, they are evenly spaced over a specified width. This width is an estimate to +how far the solution will have traveled by the evaluation time. We chose a constant acceleration instead +of a instantaneous velocity change because the latter induced numerical errors which resulted in failure to +converge to the benchmark solutions. The acceleration for each edge is found by, +ck = 2 +� +dxk +dt +���� +t0 +(t0 − tfinal) − xk +���� +t0 ++ xk +���� +tfinal +� +(t0 − tfinal)2 +(53) +where we find xk +���� +tfinal +by specifying that the final positions vector, X(tfinal), evenly span [−xf/2, xf/2] where +xf is our estimate for the width of the solution domain at the evaluation time. +With the acceleration defined, we calculate positions of the edges after the source has turned off with, +if t > t0, xk(t) = 1 +2ck (t − t0)2 + dxk +dt +���� +t0 +(t − t0) + xk +���� +t0 +, +(54) +and the velocities, +if t > t0, dxk +dt = ck (t − t0) + dxk +dt +���� +t0 +. +(55) +9 + +−10 +0 +10 +x +0 +10 +20 +30 +t +t0 +Figure 1: Edge position for each edge in the thin square source mesh run out to tfinal = 31.6228 with 8 +spaces, a wavespeed, v = +1 +√ +3, and a final domain width, xf = 30. +We refer to this method for governing the mesh edges and velocities as the “thin square source mesh”. +An example x vs t diagram of the mesh edges is given in Figure 1 for clarification. +After completing the necessary steps of defining a source, choosing a functional form for the temperature, +and defining a mesh, we may present our results for this problem. Tables 1 and 2 give our solutions for +the same points and evaluation times as in Table 1 and 2 of [8]. The convergence results for the coefficient +expansions of these results are plotted in Figures 2 and 3 . The solutions at a few selected times are plotted +in Figure 4 For each case, a moving mesh and the uncollided source was used except in the S2 solution at +times greater than t0 since the S2 uncollided solution becomes sharp and difficult to resolve via quadrature. +In these cases, the standard square source was integrated. +The convergence results (Figures 2 and 3) show that for this problem, S2 is considerably smoother at +early times. Twice as many spatial divisions were required at early times in the full transport solution 256 to +achieve similar levels of convergence to the S2 solutions. Both cases exhibited similar behavior over time. At +early times the significantly nonsmooth uncollided flux induced discontinuities in the material energy density +and required far more spatial divisions to resolve, stiffening the problem and limiting the number of basis +functions that could reasonably be used in the solution. After the source turned off, the solution smoothed +and equilibrated locally (complete equilibrium is impossible with an infinite material). The solution became +smoother and could easily be resolved with fewer spaces and more basis functions. We also note that at later +times after the source has turned off, the full transport and S2 solutions become more similar. +Since we claim to present benchmark quality results, we are obliged to discuss the accuracy of Tables 1 +and 2. While [8] claims to converge their solution to four digits, the observant reader will see that in some +cases only 3 digits match. Given that our solutions are converged beyond this point, we believe that our +reported digits are correct. +10 + +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +256 cells, t = 0.1 +256 cells, t = 0.31623 +256 cells, t = 1.0 +256 cells, t = 3.16228 +128 cells, t = 10.0 +32 cells, t = 31.6228 +32 cells, t = 100.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +256 cells, t = 0.1 +256 cells, t = 0.31623 +256 cells, t = 1.0 +256 cells, t = 3.16228 +128 cells, t = 10.0 +32 cells, t = 31.6228 +32 cells, t = 100.0 +(b) Material energy density, e +Figure 2: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for the +optically thin (σa = 1 cm−1) Su-Olson square source problem where x0 = 0.5, t0 = 10. The quadrature +order for all results is S256. All results were calculated with a moving mesh and uncollided source treatment. +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.1 +128 cells, t = 0.31623 +128 cells, t = 1.0 +128 cells, t = 3.16228 +128 cells, t = 10.0 +32 cells, t = 31.6228 +32 cells, t = 100.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.1 +128 cells, t = 0.31623 +128 cells, t = 1.0 +128 cells, t = 3.16228 +128 cells, t = 10.0 +32 cells, t = 31.6228 +32 cells, t = 100.0 +(b) Material energy density, e +Figure 3: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44))) for the +optically thin (σa = 1 cm−1) S2 Su-Olson square source problem where x0 = 0.5, t0 = 10. All results were +calculated with a moving mesh and uncollided source treatment except for the t = 31.6228 and t = 100 cases +where a standard source treatment was used. +11 + +−0.5 +0.0 +0.5 +0.00 +0.05 +0.10 +S2 +Transport +(a) t = 0.1 +−0.5 +0.0 +0.5 +0.0 +0.1 +0.2 +S2 +Transport +(b) t = 0.31623 +−1 +0 +1 +0.0 +0.2 +0.4 +0.6 +S2 +Transport +(c) t = 1 +−2 +0 +2 +0.0 +0.5 +1.0 +S2 +Transport +(d) t = 3.16228 +−10 +0 +10 +0 +1 +2 +S2 +Transport +(e) t = 10 +−20 +0 +20 +0.0 +0.1 +0.2 +0.3 +S2 +Transport +(f) t = 100 +Figure 4: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thin Su-Olson +square source problem with x0 = 0.5, t0 = 10. Solid lines are scalar flux, φ, and dashed are material energy +density, e. +6.2 +Constant Cv problem with a square source +This problem uses the same source as the problem of the last section but with a different functional form of +the heat capacity. Using Eq. (12) our system becomes nonlinear. We choose Cv0 = 0.03 GJ · cm−3 · keV−1. +This value was chosen to see an appreciable change in temperature during the selected time window. Now +that we no longer have the convenient condition that e = T +4, the local equilibrium condition is not φ = e as +in the Su-Olson problem but φ +1/4 = T. For this reason, the solution plots for this problem and all subsequent +constant Cv problems do not show scalar flux and material energy density but rather radiation temperature +and material temperature. +Though we can no longer rely on benchmark solutions for this problem,, we can be confident that our +solution is converged by plotting the magnitude of the coefficients and check for systematic errors with a Sn +solver. Also, since the mesh method employed here is the same method described in Section 6.1, we can be +confident that the mesh is not introducing error. +Solutions to this problem are plotted in Figure 7. +We note that the problem is not everywhere at +equilibrium by t = 100 as the Su-Olson problem is. Also, we note that the solution does not travel as far. +In the Su-Olson problem, the specific heat is very small when temperature is small and increases with the +cube of the temperature. This has the effect of attracting the solution to equilibrium. This effect is not +present in a constant Cv case and there is less incentive for the solution to fall into local equilibrium. It is +also noteworthy that at very early times (t < 1) the scalar flux has not interacted with the material as much +as in the Su-Olson problem and is mostly made up of the uncollided flux. This is apparent in Figures 5a +and 6a. +Since the solution has not fully equilibrated at later times, the solutions are less smooth compared to the +Su-Olson problem. The repercussions of this can be observed by comparing the convergence results at late +times for this problem in Figures 5 and 6 to the convergence results of the Su-Olson problem in Figure 2. +Also, the convergence results show that the material energy density is generally more nonsmooth than the +scalar flux. Nevertheless, we are satisfied with the convergence of these results and present them in Tables +3 and 4. +12 + +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +256 cells, t = 0.1 +256 cells, t = 0.31623 +256 cells, t = 1.0 +256 cells, t = 3.16228 +128 cells, t = 10.0 +32 cells, t = 31.6228 +64 cells, t = 100.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +256 cells, t = 0.1 +256 cells, t = 0.31623 +256 cells, t = 1.0 +256 cells, t = 3.16228 +128 cells, t = 10.0 +32 cells, t = 31.6228 +64 cells, t = 100.0 +(b) Material energy density, e +Figure 5: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for the +optically thin (σa = 1 cm−1) constant Cv square source problem where x0 = 0.5, t0 = 10. The quadrature +order for all results is S256. All results were calculated with a moving mesh and uncollided source treatment. +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.1 +128 cells, t = 0.31623 +128 cells, t = 1.0 +128 cells, t = 3.16228 +128 cells, t = 10.0 +32 cells, t = 31.6228 +32 cells, t = 100.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.1 +128 cells, t = 0.31623 +128 cells, t = 1.0 +128 cells, t = 3.16228 +128 cells, t = 10.0 +32 cells, t = 31.6228 +32 cells, t = 100.0 +(b) Material energy density, e +Figure 6: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for the +optically thin (σa = 1 cm−1) S2 constant Cv square source problem where x0 = 0.5, t0 = 10. All results +were calculated with a moving mesh and uncollided source treatment except for the t = 31.6228 and t = 100 +cases where a standard source treatment was used. +13 + +Table 1: Transport (top) and S2 (bottom) results for the scalar flux, φ for the thin square source Su-Olson +problem with x0 = 0.5, t0 = 10. RMSE values included in the last row of the S2 table are calculated from +the S2 benchmark from [6]. Convergence results for these answers are plotted in Figures 2 and 3. Bolded +digits are those that agree with the published Su-Olson solution in [8]. +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.095317 +0.275294 +0.643151 +1.20069 +2.235815 +0.690187 +0.357195 +0.1 +0.095317 +0.275294 +0.635943 +1.188724 +2.219553 +0.689743 +0.357137 +0.17783 +0.095317 +0.275294 +0.619626 +1.162044 +2.183558 +0.688773 +0.357011 +0.31623 +0.095317 +0.262715 +0.561896 +1.071861 +2.064534 +0.685719 +0.356612 +0.45 +0.08824 +0.203128 +0.447114 +0.909526 +1.860758 +0.681168 +0.356016 +0.5 +0.047658 +0.137647 +0.358083 +0.799027 +1.731816 +0.679072 +0.35574 +0.56234 +0.003762 +0.062776 +0.253722 +0.666804 +1.574955 +0.67616 +0.355355 +0.75 +- +0.002793 +0.114315 +0.446752 +1.273984 +0.665459 +0.353929 +1.0 +- +- +0.036471 +0.275396 +0.987815 +0.646922 +0.351409 +1.33352 +- +- +0.002894 +0.145309 +0.708221 +0.615381 +0.346972 +1.77828 +- +- +- +0.059674 +0.450163 +0.563509 +0.339223 +3.16228 +- +- +- +0.001155 +0.096453 +0.369659 +0.303466 +5.62341 +- +- +- +- +0.003632 +0.108305 +0.213818 +10.0 +- +- +- +- +- +0.003914 +0.072059 +17.78279 +- +- +- +- +- +- +0.002721 +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.095317 +0.275294 +0.661668 +1.0451 +1.918396 +0.659852 +0.352728 +0.1 +0.095317 +0.275294 +0.645629 +1.034885 +1.906562 +0.659502 +0.352673 +0.17783 +0.095317 +0.275294 +0.604866 +1.012559 +1.88071 +0.658738 +0.352555 +0.31623 +0.095317 +0.275294 +0.515344 +0.941841 +1.798924 +0.656328 +0.352181 +0.45 +0.089213 +0.179984 +0.405572 +0.835497 +1.67622 +0.652732 +0.351621 +0.5 +0.047658 +0.137647 +0.358083 +0.786079 +1.619313 +0.651071 +0.351362 +0.56234 +0.000000 +0.085419 +0.299433 +0.722866 +1.545756 +0.648764 +0.351001 +0.75 +- +0.0000000 +0.155119 +0.552606 +1.338003 +0.640252 +0.349662 +1.0 +- +- +0.027203 +0.369788 +1.092239 +0.625397 +0.347295 +1.33352 +- +- +0.000000 +0.19493 +0.816475 +0.599795 +0.343124 +1.77828 +- +- +- +0.056328 +0.532488 +0.556761 +0.335829 +3.16228 +- +- +- +- +0.09838 +0.384521 +0.30198 +5.62341 +- +- +- +- +0.00028 +0.116245 +0.215656 +10.0 +- +- +- +- +- +0.002009 +0.073828 +17.78279 +- +- +- +- +- +- +0.002307 +RMSE +2.656e-07 +1.747e-07 +1.642e-06 +3.589e-07 +2.647e-07 +9.157e-08 +6.128e-09 +The difference between the full transport solution and our S2 result is also of interest, as it provides +insight into the physical characteristics of the system. We note that the two solutions only begin to look +similar at later times as the solution equilibrates. This tells us that the solution becomes less angularly +dependent and better approximated by only two angles. +6.3 +Su-Olson problem with a Gaussian source +Returning to the linearized Su-Olson problem, we consider a Gaussian source. Here the source is defined by +Eq. 37 where the uncollided solution is taken from [15] for the full transport solution or Eq.(56) for the S2 +solution. We set x0 = 0.5 and the source duration is still t0 = 10. In [13], smooth Gaussian sources allowed +for geometric convergence of the solution at all times. We expect the same result in this application. +Since there are no discontinuities induced by nonsmoothness in the source, we are able to employ a far +simpler mesh function than what was used for the thin square source problems. We only guess the edge +14 + +Table 2: Transport (top) and S2 (bottom) results for the material energy density, e for the thin square +source Su-Olson problem with x0 = 0.5, t0 = 10. RMSE values included in the last row of the S2 table are +calculated from the S2 benchmark from [6]. Convergence results for these answers are plotted in Figures 2 +and 3. Bolded digits are those that agree with the published Su-Olson solution in [8]. +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.004682 +0.040935 +0.271307 +0.94687 +2.111923 +0.704991 +0.359136 +0.1 +0.004682 +0.040935 +0.268692 +0.937154 +2.095970 +0.704514 +0.359078 +0.17783 +0.004682 +0.040935 +0.26264 +0.915402 +2.060646 +0.703474 +0.358949 +0.31623 +0.004682 +0.04034 +0.239814 +0.840926 +1.943709 +0.700198 +0.358544 +0.45 +0.004552 +0.033142 +0.188264 +0.702883 +1.742967 +0.69532 +0.357938 +0.5 +0.002342 +0.020469 +0.141918 +0.604935 +1.615402 +0.693073 +0.357657 +0.56234 +0.00005 +0.00635 +0.08838 +0.48846 +1.460394 +0.689954 +0.357266 +0.75 +- +0.000063 +0.030141 +0.306558 +1.165912 +0.678498 +0.355816 +1.0 +- +- +0.00625 +0.175192 +0.889908 +0.658685 +0.353254 +1.33352 +- +- +0.000162 +0.08352 +0.625213 +0.625066 +0.348744 +1.77828 +- +- +- +0.029349 +0.386884 +0.570027 +0.34087 +3.16228 +- +- +- +0.000183 +0.076146 +0.367269 +0.304561 +5.62341 +- +- +- +- +0.002412 +0.103114 +0.213768 +10.0 +- +- +- +- +- +0.003426 +0.071226 +17.78279 +- +- +- +- +- +- +0.002609 +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.004682 +0.040935 +0.280241 +0.847357 +1.808991 +0.672725 +0.354597 +0.1 +0.004682 +0.040935 +0.273782 +0.837875 +1.797308 +0.672354 +0.354541 +0.17783 +0.004682 +0.040935 +0.261727 +0.817144 +1.771785 +0.671544 +0.354421 +0.31623 +0.004682 +0.040936 +0.225609 +0.751409 +1.691033 +0.668989 +0.354041 +0.45 +0.004642 +0.030469 +0.169249 +0.652361 +1.569863 +0.665177 +0.353472 +0.5 +0.002341 +0.020467 +0.141916 +0.606256 +1.513661 +0.663418 +0.353209 +0.56234 +0.000000 +0.008542 +0.10839 +0.547588 +1.441078 +0.660972 +0.352842 +0.75 +- +0.000000 +0.038396 +0.393514 +1.23688 +0.651954 +0.351482 +1.0 +- +- +0.001762 +0.236801 +0.997162 +0.636229 +0.349077 +1.33352 +- +- +0.000000 +0.101119 +0.731352 +0.609161 +0.344841 +1.77828 +- +- +- +0.019412 +0.462775 +0.563768 +0.337432 +3.16228 +- +- +- +0.000000 +0.074037 +0.383609 +0.303078 +5.62341 +- +- +- +- +0.000086 +0.110507 +0.215664 +10.0 +- +- +- +- +- +0.001626 +0.072987 +17.78279 +- +- +- +- +- +- +0.002196 +RMSE +1.233e-08 +2.372e-08 +9.65e-08 +1.366e-07 +4.052e-08 +9.937e-08 +5.936e-09 +of the problem domain and span the given space evenly with stationary edges. The moving mesh was not +used in this case because earlier tests in [13] revealed the mesh to be non-useful in smooth problems. The +uncollided solution however, was employed. We include Gaussian sources though they do not prove to be +challenging enough problems to require the full application of our method because we can achieve very +accurate solutions. +While S256 was used for the full transport solutions for the thin square sources, we only use S64 on the +Gaussian sources. This choice is informed by tests run in [13] that showed that far fewer quadrature points +are required to resolve the angular error. +With the temperature defined by Eq (11) we present solutions in Tables 5 and 6 with convergence results +shown in Figures 8 and 18. Solutions are shown in Figure 10. +As illustrated in the aforementioned convergence plots, the problem converges geometrically even with +a standard static mesh. We quickly note that for t = 31.6228 and t = 100 in the S2 results in Figure 18, +a moving mesh was employed. In this case, the mesh moved with a constant speed from the initial width +15 + +−0.5 +0.0 +0.5 +0.0 +0.2 +0.4 +S2 +Transport +(a) t = 0.1 +−0.5 +0.0 +0.5 +0.0 +0.2 +0.4 +0.6 +S2 +Transport +(b) t = 0.31623 +−1 +0 +1 +0.00 +0.25 +0.50 +0.75 +S2 +Transport +(c) t = 1 +−2 +0 +2 +0.0 +0.5 +S2 +Transport +(d) t = 3.16228 +−10 +0 +10 +0.0 +0.5 +1.0 +S2 +Transport +(e) t = 10 +−20 +0 +20 +0.0 +0.2 +0.4 +S2 +Transport +(f) t = 100 +Figure 7: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thin constant Cv +square source problem with x0 = 0.5, t0 = 10. Solid lines are radiation temperature φ +1/4, and dashed are +temperature, T. +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +64 cells, t = 0.1 +64 cells, t = 0.31623 +64 cells, t = 1.0 +64 cells, t = 3.16228 +64 cells, t = 10.0 +64 cells, t = 31.6228 +64 cells, t = 100.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +64 cells, t = 0.1 +64 cells, t = 0.31623 +64 cells, t = 1.0 +64 cells, t = 3.16228 +64 cells, t = 10.0 +64 cells, t = 31.6228 +64 cells, t = 100.0 +(b) Material energy density, e +Figure 8: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for the +optically thin (σa = 1 cm−1) Su-Olson Gaussian source problem where x0 = 0.5, t0 = 10. The quadrature +order for all results is S16. All results were calculated with a static mesh and uncollided source treatment. +16 + +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +64 cells, t = 0.1 +64 cells, t = 0.31623 +64 cells, t = 1.0 +64 cells, t = 3.16228 +64 cells, t = 10.0 +64 cells, t = 31.6228 +32 cells, t = 100.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +64 cells, t = 0.1 +64 cells, t = 0.31623 +64 cells, t = 1.0 +64 cells, t = 3.16228 +64 cells, t = 10.0 +64 cells, t = 31.6228 +32 cells, t = 100.0 +(b) Material energy density, e +Figure 9: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for the +optically thin (σa = 1 cm−1) S2 Su-Olson Gaussian source problem where x0 = 0.5, t0 = 10. All results were +calculated with a moving mesh and uncollided source treatment except for the t = 31.6228 and t = 100 cases +where a standard source treatment was used. The dashed lines represent solutions found with a moving +mesh. +−2 +0 +2 +0.00 +0.05 +S2 +Transport +(a) t = 0.1 +−2 +0 +2 +0.0 +0.1 +0.2 +S2 +Transport +(b) t = 0.31623 +−2.5 +0.0 +2.5 +0.0 +0.2 +0.4 +0.6 +S2 +Transport +(c) t = 1 +−2.5 +0.0 +2.5 +0.0 +0.5 +1.0 +S2 +Transport +(d) t = 3.16228 +−10 +0 +10 +0 +1 +2 +S2 +Transport +(e) t = 10 +−25 +0 +25 +0.0 +0.1 +0.2 +0.3 +S2 +Transport +(f) t = 100 +Figure 10: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thin Su-Olson +Gaussian source problem with x0 = 0.5, t0 = 10. Solid lines are scalar flux, φ, and dashed are material +energy density, e. +17 + +Table 3: Transport (top) and S2 (bottom) results for the scalar flux, φ, for the thin square source constant +Cv problem with x0 = 0.5, t0 = 10, and Cv0 = 0.03 GJ · cm−3 · keV−1. Convergence results for these answers +are plotted in Figures 5 and 6 +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.095162 +0.271108 +0.563683 +0.765084 +1.96832 +0.267247 +0.085108 +0.1 +0.095162 +0.271108 +0.557609 +0.756116 +1.950367 +0.266877 +0.085054 +0.17783 +0.095162 +0.271108 +0.543861 +0.736106 +1.910675 +0.266071 +0.084937 +0.31623 +0.095162 +0.258592 +0.495115 +0.668231 +1.779896 +0.263527 +0.084565 +0.45 +0.08809 +0.199962 +0.396442 +0.543721 +1.558248 +0.259729 +0.084008 +0.5 +0.047581 +0.135554 +0.316071 +0.453151 +1.420865 +0.257976 +0.08375 +0.56234 +0.00376 +0.061935 +0.222261 +0.349209 +1.252213 +0.255538 +0.083392 +0.75 +- +0.002788 +0.102348 +0.21078 +0.908755 +0.246543 +0.082061 +1.0 +- +- +0.034228 +0.124305 +0.562958 +0.230831 +0.079715 +1.33352 +- +- +0.002864 +0.067319 +0.27752 +0.203718 +0.075591 +1.77828 +- +- +- +0.031357 +0.120054 +0.158039 +0.068419 +3.16228 +- +- +- +0.001057 +0.013737 +0.022075 +0.036021 +5.62341 +- +- +- +- +0.000413 +0.000814 +0.001068 +10.0 +- +- +- +- +- +5e-06 +5e-06 +17.78279 +- +- +- +- +- +- +- +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.095162 +0.271108 +0.579404 +0.649946 +1.57957 +0.243368 +0.081882 +0.1 +0.095162 +0.271108 +0.56606 +0.641494 +1.567037 +0.243074 +0.081833 +0.17783 +0.095162 +0.271108 +0.529958 +0.623092 +1.539665 +0.242432 +0.081725 +0.31623 +0.095162 +0.271108 +0.45241 +0.565417 +1.453129 +0.240406 +0.081383 +0.45 +0.08906 +0.177033 +0.357556 +0.480232 +1.323479 +0.237381 +0.080871 +0.5 +0.047581 +0.135554 +0.31607 +0.441134 +1.263425 +0.235984 +0.080634 +0.56234 +- +0.084378 +0.264889 +0.392645 +1.185561 +0.234042 +0.080305 +0.75 +- +- +0.140336 +0.276699 +0.962617 +0.226873 +0.079082 +1.0 +- +- +0.02637 +0.175485 +0.692975 +0.214346 +0.076927 +1.33352 +- +- +- +0.097064 +0.392628 +0.192725 +0.073141 +1.77828 +- +- +- +0.033465 +0.155709 +0.156395 +0.066564 +3.16228 +- +- +- +- +0.007509 +0.03175 +0.03718 +5.62341 +- +- +- +- +4.7e-05 +0.000413 +0.001075 +10.0 +- +- +- +- +- +- +- +17.78279 +- +- +- +- +- +- +- +to the specified final width. This was not done out of necessity, but rather to ascertain weather the moving +mesh was useful in these problems. +On the difference between the full and S2 solutions, we point out that it can be seen in Figure 10 that +the S2 solution is not as accurate in resolving the source region after the source has been on for some time +then becomes more accurate after the source is off. This is not surprising since the source emits uncollided +particles in every direction while it is on. At later times, the two solutions are visually indistinguishable. +6.4 +Constant Cv problem with a Gaussian source +We include a problem with the same source and parameters as the last section but with a constant specific +heat so that the temperature to material energy density conversion is given by Eq. (12). We specify the the +dimensional specific heat to be Cv0 = 0.03 GJ · cm−3 · keV−1. While we have less certainty in forecasting +the behavior of these nonlinear results, we still expect geometric convergence since there are no sources of +nonsmoothness. +Like the Gaussian source in the linearized system, we specified a static mesh that evenly spans some +18 + +Table 4: Transport (top) and S2 (bottom) results for the material energy density, e, for the thin square +source constant Cv problem with x0 = 0.5, t0 = 10, and Cv0 = 0.03 GJ · cm−3 · keV−1. Convergence results +for these answers are plotted in Figures 5 and 6 +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.004837 +0.045121 +0.354022 +1.613529 +2.57461 +1.592549 +1.190296 +0.1 +0.004837 +0.045121 +0.350958 +1.601467 +2.568476 +1.591998 +1.190108 +0.17783 +0.004837 +0.045121 +0.343803 +1.573757 +2.554747 +1.590795 +1.189698 +0.31623 +0.004837 +0.044507 +0.316063 +1.47078 +2.507772 +1.586979 +1.188398 +0.45 +0.004705 +0.036765 +0.249325 +1.238666 +2.421019 +1.581228 +1.186445 +0.5 +0.002419 +0.022562 +0.183937 +1.025219 +2.361647 +1.578549 +1.185538 +0.56234 +5.1e-05 +0.006779 +0.108887 +0.759317 +2.280932 +1.5748 +1.184271 +0.75 +- +6.4e-05 +0.034842 +0.416175 +2.069946 +1.56071 +1.179537 +1.0 +- +- +0.006872 +0.214491 +1.68516 +1.535052 +1.171036 +1.33352 +- +- +0.000168 +0.094966 +1.028758 +1.487096 +1.155611 +1.77828 +- +- +- +0.032116 +0.471906 +1.391456 +1.127131 +3.16228 +- +- +- +0.000196 +0.049604 +0.471468 +0.954827 +5.62341 +- +- +- +- +0.001163 +0.019493 +0.082189 +10.0 +- +- +- +- +- +0.000113 +0.000487 +17.78279 +- +- +- +- +- +- +- +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.004837 +0.045121 +0.364106 +1.512533 +2.431781 +1.555683 +1.178842 +0.1 +0.004837 +0.045121 +0.357116 +1.497067 +2.426675 +1.555215 +1.178665 +0.17783 +0.004837 +0.045121 +0.343553 +1.462337 +2.415402 +1.554193 +1.178277 +0.31623 +0.004837 +0.045121 +0.299658 +1.343527 +2.378606 +1.550952 +1.177048 +0.45 +0.004796 +0.033907 +0.223636 +1.138096 +2.319758 +1.546073 +1.175203 +0.5 +0.002418 +0.02256 +0.183934 +1.031348 +2.290735 +1.543804 +1.174346 +0.56234 +- +0.00909 +0.135476 +0.89324 +2.251138 +1.54063 +1.173149 +0.75 +- +- +0.043524 +0.56287 +2.120765 +1.528729 +1.16868 +1.0 +- +- +0.001805 +0.289458 +1.885442 +1.507188 +1.160663 +1.33352 +- +- +- +0.106017 +1.298617 +1.467469 +1.146151 +1.77828 +- +- +- +0.018336 +0.530154 +1.391381 +1.119483 +3.16228 +- +- +- +- +0.023197 +0.536229 +0.964849 +5.62341 +- +- +- +- +3.1e-05 +0.005921 +0.056159 +10.0 +- +- +- +- +- +- +2.4e-05 +17.78279 +- +- +- +- +- +- +- +estimated width of the actual solution domain. The convergence results in Figures 11 and 12 show that +this was sufficient to achieve geometric convergence at all chosen times. We also see the phenomenon first +observed in Section 6.2 of extremely fast convergence for the scalar flux at early times. This is once again +the result of the scalar flux being most uncollided at these times, making solving for the collided portion +simpler. +During the time the source is on there is a large discrepancy between the 2 and transport solutions; this +discrepancy recedes after the source is turned off. Similar to the nonlinear square source, the solution is not +in equilibrium by t = 100. This leads us to draw the conclusion that equilibrium is not as impacted by the +nonsmoothness of the source as it is by the functional form of the specific heat, since the Su-Olson problem +with a nonsmooth source goes more quickly into equilibrium than this constant Cv smooth source problem. +We present the solutions in Tables 7 and 8. +19 + +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +64 cells, t = 0.1 +64 cells, t = 0.31623 +64 cells, t = 1.0 +64 cells, t = 3.16228 +128 cells, t = 10.0 +128 cells, t = 31.6228 +128 cells, t = 100.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +64 cells, t = 0.1 +64 cells, t = 0.31623 +64 cells, t = 1.0 +64 cells, t = 3.16228 +128 cells, t = 10.0 +128 cells, t = 31.6228 +128 cells, t = 100.0 +(b) Material energy density, e +Figure 11: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for the +optically thin (σa = 1 cm−1) constant Cv Gaussian source problem where x0 = 0.5, t0 = 10. The quadrature +order for all results is S16. All results were calculated with a static mesh and uncollided source treatment. +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +64 cells, t = 0.1 +64 cells, t = 0.31623 +64 cells, t = 1.0 +64 cells, t = 3.16228 +128 cells, t = 10.0 +128 cells, t = 31.6228 +128 cells, t = 100.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +64 cells, t = 0.1 +64 cells, t = 0.31623 +64 cells, t = 1.0 +64 cells, t = 3.16228 +128 cells, t = 10.0 +128 cells, t = 31.6228 +128 cells, t = 100.0 +(b) Material energy density, e +Figure 12: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for the +optically thin (σa = 1 cm−1) S2 constant Cv Gaussian source problem where x0 = 0.5, t0 = 10. All results +were calculated with a static mesh and uncollided source treatment. +20 + +Table 5: Transport (top) and S2 (bottom) results for the scalar flux, φ, for the thin Gaussian source Su-Olson +problem with x0 = 0.5, t0 = 10. RMSE values included in the last row of the S2 table are calculated from +the S2 benchmark from [6]. Convergence results for these answers are plotted in Figures 8 and 18. +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.094869 +0.264712 +0.571441 +1.053107 +1.956705 +0.610025 +0.316341 +0.1 +0.091217 +0.255181 +0.556247 +1.033176 +1.933017 +0.609634 +0.31629 +0.17783 +0.083721 +0.235541 +0.524582 +0.991429 +1.883176 +0.608783 +0.316178 +0.31623 +0.063836 +0.182853 +0.436813 +0.873994 +1.741091 +0.6061 +0.315826 +0.45 +0.042514 +0.125118 +0.334025 +0.732182 +1.564765 +0.602104 +0.315298 +0.5 +0.035215 +0.104949 +0.29568 +0.67763 +1.49511 +0.600263 +0.315054 +0.56234 +0.027081 +0.082142 +0.250049 +0.611096 +1.408369 +0.597705 +0.314714 +0.75 +0.010198 +0.033042 +0.137011 +0.433926 +1.163525 +0.588305 +0.313452 +1.0 +0.001799 +0.006564 +0.050023 +0.267122 +0.898484 +0.572018 +0.311224 +1.33352 +8.2e-05 +0.000371 +0.008965 +0.139587 +0.64098 +0.544297 +0.3073 +1.77828 +- +2e-06 +0.000387 +0.05756 +0.40731 +0.498676 +0.300446 +3.16228 +- +- +- +0.00137 +0.087854 +0.327865 +0.268818 +5.62341 +- +- +- +- +0.003363 +0.096556 +0.189497 +10.0 +- +- +- +- +- +0.003523 +0.063953 +17.78279 +- +- +- +- +- +- +0.002424 +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.00467 +0.040073 +0.240699 +0.727949 +1.572225 +0.594815 +0.314048 +0.1 +0.00449 +0.03858 +0.23398 +0.717893 +1.560265 +0.594488 +0.313999 +0.17783 +0.004119 +0.035509 +0.219946 +0.696571 +1.534799 +0.593775 +0.313893 +0.31623 +0.003137 +0.027317 +0.180837 +0.6345 +1.45974 +0.591526 +0.313557 +0.45 +0.002085 +0.018437 +0.134764 +0.554569 +1.360669 +0.58817 +0.313054 +0.5 +0.001726 +0.015365 +0.117566 +0.522032 +1.319396 +0.586621 +0.312821 +0.56234 +0.001326 +0.011918 +0.09716 +0.480715 +1.266036 +0.584468 +0.312496 +0.75 +0.000497 +0.004632 +0.047669 +0.359391 +1.101318 +0.576529 +0.311293 +1.0 +8.7e-05 +0.000863 +0.013277 +0.223358 +0.893342 +0.562683 +0.309166 +1.33352 +3e-06 +4.3e-05 +0.001308 +0.100634 +0.657392 +0.538848 +0.305419 +1.77828 +- +- +1.8e-05 +0.022908 +0.417809 +0.498866 +0.298865 +3.16228 +- +- +- +- +0.068504 +0.34004 +0.268475 +5.62341 +- +- +- +- +0.000121 +0.098551 +0.191126 +10.0 +- +- +- +- +- +0.001487 +0.064775 +17.78279 +- +- +- +- +- +- +0.001958 +RMSE +9.105e-10 +8.561e-10 +2.604e-09 +7.839e-09 +1.321e-08 +6.808e-09 +6.899e-05 +21 + +Table 6: Transport (top) and S2 (bottom) results for the material energy density, e, for the thin Gaussian +source Su-Olson problem with x0 = 0.5, t0 = 10. RMSE values included in the last row of the S2 table are +calculated from the S2 benchmark from [6]. Convergence results for these answers are plotted in Figures 8 +and 18. +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.00467 +0.040089 +0.245869 +0.833299 +1.847977 +0.623026 +0.318058 +0.1 +0.00449 +0.038593 +0.238404 +0.815919 +1.824555 +0.622608 +0.318006 +0.17783 +0.004119 +0.035517 +0.222913 +0.779578 +1.775289 +0.621695 +0.317892 +0.31623 +0.003137 +0.027313 +0.18051 +0.677838 +1.63499 +0.61882 +0.317534 +0.45 +0.002085 +0.018426 +0.132108 +0.556161 +1.461245 +0.614538 +0.316998 +0.5 +0.001726 +0.015355 +0.114504 +0.509791 +1.392751 +0.612566 +0.316749 +0.56234 +0.001326 +0.011908 +0.093967 +0.453643 +1.307592 +0.609828 +0.316404 +0.75 +0.000497 +0.004629 +0.04582 +0.307085 +1.068295 +0.59977 +0.315121 +1.0 +8.7e-05 +0.000865 +0.013544 +0.175481 +0.812011 +0.58237 +0.312856 +1.33352 +3e-06 +4.4e-05 +0.001743 +0.082634 +0.567439 +0.552834 +0.308867 +1.77828 +- +- +5e-05 +0.029302 +0.35102 +0.504445 +0.301903 +3.16228 +- +- +- +0.000297 +0.069582 +0.325807 +0.269788 +5.62341 +- +- +- +- +0.002245 +0.091971 +0.189454 +10.0 +- +- +- +- +- +0.003086 +0.063216 +17.78279 +- +- +- +- +- +- +0.002324 +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.00467 +0.040073 +0.240699 +0.727949 +1.572225 +0.594815 +0.314048 +0.1 +0.00449 +0.03858 +0.23398 +0.717893 +1.560265 +0.594488 +0.313999 +0.17783 +0.004119 +0.035509 +0.219946 +0.696571 +1.534799 +0.593775 +0.313893 +0.31623 +0.003137 +0.027317 +0.180837 +0.6345 +1.45974 +0.591526 +0.313557 +0.45 +0.002085 +0.018437 +0.134764 +0.554569 +1.360669 +0.58817 +0.313054 +0.5 +0.001726 +0.015365 +0.117566 +0.522032 +1.319396 +0.586621 +0.312821 +0.56234 +0.001326 +0.011918 +0.09716 +0.480715 +1.266036 +0.584468 +0.312496 +0.75 +0.000497 +0.004632 +0.047669 +0.359391 +1.101318 +0.576529 +0.311293 +1.0 +8.7e-05 +0.000863 +0.013277 +0.223358 +0.893342 +0.562683 +0.309166 +1.33352 +3e-06 +4.3e-05 +0.001308 +0.100634 +0.657392 +0.538848 +0.305419 +1.77828 +- +- +1.8e-05 +0.022908 +0.417809 +0.498866 +0.298865 +3.16228 +- +- +- +- +0.068504 +0.34004 +0.268475 +5.62341 +- +- +- +- +0.000121 +0.098551 +0.191126 +10.0 +- +- +- +- +- +0.001487 +0.064775 +17.78279 +- +- +- +- +- +- +0.001958 +RMSE +9.105e-10 +8.561e-10 +2.604e-09 +7.839e-09 +1.321e-08 +6.808e-09 +6.899e-05 +22 + +−2 +0 +2 +0.0 +0.2 +0.4 +S2 +Transport +(a) t = 0.1 +−2 +0 +2 +0.0 +0.2 +0.4 +0.6 +S2 +Transport +(b) t = 0.31623 +−2.5 +0.0 +2.5 +0.00 +0.25 +0.50 +0.75 +S2 +Transport +(c) t = 1 +−2.5 +0.0 +2.5 +0.00 +0.25 +0.50 +0.75 +S2 +Transport +(d) t = 3.16228 +−10 +0 +10 +0.0 +0.5 +1.0 +S2 +Transport +(e) t = 10 +−10 +0 +10 +0.0 +0.2 +0.4 +S2 +Transport +(f) t = 100 +Figure 13: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thin constant Cv +Gaussian source problem with x0 = 0.5, t0 = 10. Solid lines are radiation temperature φ +1/4, and dashed are +temperature, T. +23 + +Table 7: Transport (top) and S2 (bottom) results for the scalar flux, φ, for the thin Gaussian source constant +Cv problem with x0 = 0.5, t0 = 10, and Cv0 = 0.03 GJ · cm−3 · keV−1. Convergence results for these answers +are plotted in Figures 11 and 12. +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.094715 +0.260632 +0.501868 +0.653649 +1.639873 +0.218959 +0.069516 +0.1 +0.091069 +0.251252 +0.488575 +0.636913 +1.614763 +0.21862 +0.069466 +0.17783 +0.083585 +0.231922 +0.460879 +0.602409 +1.561824 +0.217881 +0.069356 +0.31623 +0.063732 +0.180063 +0.384158 +0.509408 +1.41001 +0.215547 +0.069011 +0.45 +0.042445 +0.123229 +0.294366 +0.405067 +1.219261 +0.212063 +0.068495 +0.5 +0.035158 +0.103372 +0.260874 +0.367198 +1.142975 +0.210455 +0.068256 +0.56234 +0.027037 +0.080916 +0.221011 +0.322683 +1.047037 +0.208219 +0.067923 +0.75 +0.010181 +0.032561 +0.122095 +0.213179 +0.768814 +0.19997 +0.066689 +1.0 +0.001796 +0.006473 +0.045352 +0.123297 +0.461235 +0.185567 +0.064514 +1.33352 +8.2e-05 +0.000367 +0.008352 +0.064418 +0.223392 +0.160718 +0.060695 +1.77828 +- +2e-06 +0.000371 +0.029558 +0.097211 +0.118847 +0.054065 +3.16228 +- +- +- +0.001165 +0.011091 +0.014263 +0.024308 +5.62341 +- +- +- +- +0.00034 +0.00057 +0.000634 +10.0 +- +- +- +- +- +4e-06 +3e-06 +17.78279 +- +- +- +- +- +- +- +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.094714 +0.260323 +0.477228 +0.539628 +1.295862 +0.197266 +0.066556 +0.1 +0.091067 +0.250998 +0.467159 +0.53082 +1.283232 +0.197001 +0.066511 +0.17783 +0.083584 +0.231775 +0.445829 +0.512305 +1.256308 +0.196423 +0.066411 +0.31623 +0.063733 +0.180133 +0.384035 +0.459709 +1.176709 +0.194598 +0.066097 +0.45 +0.042446 +0.123414 +0.305857 +0.394987 +1.071024 +0.191874 +0.065626 +0.5 +0.035159 +0.103564 +0.274806 +0.369646 +1.026763 +0.190617 +0.065409 +0.56234 +0.027038 +0.081091 +0.236322 +0.338318 +0.969319 +0.188869 +0.065106 +0.75 +0.010181 +0.032613 +0.132763 +0.251849 +0.790231 +0.18242 +0.063983 +1.0 +0.001796 +0.006438 +0.045179 +0.164176 +0.560306 +0.171164 +0.062005 +1.33352 +8.1e-05 +0.000354 +0.005814 +0.08995 +0.308784 +0.151783 +0.058534 +1.77828 +- +1e-06 +0.000114 +0.03286 +0.12204 +0.119388 +0.052522 +3.16228 +- +- +-0.0 +2e-06 +0.006114 +0.019949 +0.02605 +5.62341 +- +- +- +- +4.6e-05 +0.000264 +0.000564 +10.0 +- +- +- +- +- +- +- +17.78279 +- +- +- +- +- +- +- +24 + +Table 8: Transport (top) and S2 (bottom) results for the material energy density, e, for the thin Gaussian +source constant Cv problem with x0 = 0.5, t0 = 10, and Cv0 = 0.03 GJ · cm−3 · keV−1. Convergence results +for these answers are plotted in Figures 11 and 12 +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.004825 +0.044224 +0.323377 +1.4684 +2.455993 +1.518526 +1.133064 +0.1 +0.004638 +0.042571 +0.313295 +1.438115 +2.446023 +1.51794 +1.132861 +0.17783 +0.004255 +0.039172 +0.292399 +1.373348 +2.424567 +1.516657 +1.132417 +0.31623 +0.003241 +0.030112 +0.235379 +1.183116 +2.359381 +1.512586 +1.131013 +0.45 +0.002154 +0.020302 +0.170708 +0.944125 +2.267966 +1.506441 +1.128901 +0.5 +0.001783 +0.016913 +0.147332 +0.851718 +2.227558 +1.503576 +1.12792 +0.56234 +0.001369 +0.013111 +0.120189 +0.740141 +2.172608 +1.499564 +1.126549 +0.75 +0.000513 +0.005089 +0.057328 +0.458276 +1.96946 +1.484442 +1.121422 +1.0 +8.9e-05 +0.000949 +0.016339 +0.231272 +1.546952 +1.456735 +1.112196 +1.33352 +4e-06 +4.8e-05 +0.002002 +0.097311 +0.889883 +1.4042 +1.095386 +1.77828 +- +- +5.5e-05 +0.032461 +0.400128 +1.294119 +1.064084 +3.16228 +- +- +- +0.000322 +0.042204 +0.349552 +0.857807 +5.62341 +- +- +- +- +0.001039 +0.014856 +0.054666 +10.0 +- +- +- +- +- +8.8e-05 +0.000336 +17.78279 +- +- +- +- +- +- +- +x/t +0.1 +0.31623 +1.0 +3.16228 +10.0 +31.6228 +100.0 +0.01 +0.004825 +0.044206 +0.317475 +1.332131 +2.309445 +1.479441 +1.120796 +0.1 +0.004638 +0.042557 +0.308253 +1.31107 +2.303385 +1.478948 +1.120606 +0.17783 +0.004255 +0.039164 +0.289032 +1.265902 +2.29029 +1.47787 +1.120189 +0.31623 +0.003241 +0.030116 +0.235788 +1.131312 +2.250065 +1.474452 +1.118869 +0.45 +0.002154 +0.020312 +0.173764 +0.954656 +2.192602 +1.469301 +1.116886 +0.5 +0.001783 +0.016923 +0.150839 +0.882882 +2.166882 +1.466903 +1.115965 +0.56234 +0.001369 +0.013121 +0.123828 +0.79271 +2.13172 +1.463548 +1.114678 +0.75 +0.000513 +0.005092 +0.059398 +0.540414 +2.00314 +1.45095 +1.109868 +1.0 +8.9e-05 +0.000947 +0.016015 +0.293197 +1.73921 +1.428067 +1.101228 +1.33352 +4e-06 +4.7e-05 +0.001515 +0.113254 +1.125017 +1.385564 +1.085536 +1.77828 +- +- +2e-05 +0.022889 +0.453017 +1.302486 +1.05652 +3.16228 +- +- +- +- +0.021263 +0.375028 +0.880385 +5.62341 +- +- +- +- +3.7e-05 +0.00417 +0.03231 +10.0 +- +- +- +- +- +1e-06 +1.4e-05 +17.78279 +- +- +- +- +- +- +- +25 + +7 +Optically thick results +Here we include results for problems that we consider optically thick. By this, we mean that the source +width is far greater than a mean free path. We accomplish this by specifying, σa = 800 cm−1 and z0 < 1 +cm (z0 is the dimensional x0). In this section, we include results for linearized Gaussian and square sources +as well as a constant Cv Gaussian source problem. We do not include a constant Cv square source, since +our method could not resolve the nonlinear, nonequilibrium, and very sharp wave that the square source +induces. +Since in an optically thick problem results are of interest only after many mean free times, we give +results for τ ≈ 0.01, 0.1, and 1 ns. By these times, the uncollided source is negligible and any discontinuous +wavefronts have decayed. For these reasons, the problems in this section do not employ an uncollided source +or a moving mesh. +7.1 +Su-Olson problem with a square source +To keep the dimensional spatial domain manageable, we set l = +1 +800 in Eqs. (5) and (6). This makes the +dimensional and nondimensional domains the same, but stiffens the system. We are again using a square +source (Eq. (36)) with x0 = 0.5 and t0 = 10. We forgo the the use if an uncollided source since the evaluation +times of interest are long after the uncollided solution has decayed to zero. +For the mesh in this problem, only a static mesh was necessary for satisfactory convergence. We use a +initialization outlined in Section 6.1 but that the initial width δx is not set to a small number, but to a guess +of the solution width at the evaluation time and the edges never move. Essentially, the mesh is the same as +the initial mesh for the thin square source but covering a the whole domain. The Gauss-Legendre spacing +of the edges has the effect of concentrating static edges around the source edge which makes it more likely +that the region where the wavefront will be is resolved. +The initial guess for the solution width was important and was refined with each run increasing the +number of spatial divisions. Since negative solutions are possible in our DG formulation and more likely to +occur when there is a sharp wavefront, the temperature was calculated with T = sign(e)|e|1/4. +The solution plots for this problem (Figure 16) show that for the chosen times, the solution is in local +equilibrium. Unlike the selected thin square source solutions where a discontinuous wave travelling at the +wavespeed determines the speed the solution travels, here a wave resembling a nonlinear heat waves moves +outwards while the source is on. Geometric convergence, shown in Figures 14 and 15, is only possible since +the nonsmooth portions of the scalar flux have decayed to zero and the solution is in equilibrium, which has +the effect of smoothing the leading edge of the wavefront. +We also take not of the similarity between the transport and S2 solutions, apparent in the solution plots +and in Tables 9 and 10. This is expected from the optically thick results, since we saw the two solutions +converge at long times where there was equilibrium. +. +7.2 +Su-Olson problem with a Gaussian source +In order to provide consistent examples across optically thin and thick regimes and as a trial run for the +constant Cv thick Gaussian of the next section, we include here a optically thick Gaussian source. For this +and the nonlinear Gaussian, we specify the length parameter x0 = 0.375. The source duration, t0, is still 10 +and l = +1 +800. Once again, the source is given by Eq. (37) and we do not use an uncollided source. Like the +thin Gaussian sources, a moving mesh is not necessary. +Figure 19 show that like the optically thick square source, the solutions are in equilibrium during the +selected time window. There is however, no wavefront but the solution maintains Gaussian characteristics. +Like the square source, the transport and S2 solutions are very similar. +Geometric convergence of our +standard DG method is shown in Figures 17 and 18. We note that in this problem and the Su-Olson, square +source 128 spatial cells were required to achieve the desired rate of convergence, though the solution was +smooth. This was due to the small length scales. +26 + +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.3 +128 cells, t = 3.0 +128 cells, t = 30.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.3 +128 cells, t = 3.0 +128 cells, t = 30.0 +(b) Material energy density, e +Figure 14: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for the +optically thick (σa = 800 cm−1) Su-Olson square source problem where x0 = 0.5, t0 = 10. The quadrature +order for all results is S16. All results were calculated with a static mesh and standard source treatment. +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.3 +128 cells, t = 3.0 +128 cells, t = 30.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.3 +128 cells, t = 3.0 +128 cells, t = 30.0 +(b) Material energy density, e +Figure 15: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for the +optically thick (σa = 800 cm−1) S2 Su-Olson square source problem where x0 = 0.5, t0 = 10. All results +were calculated with a static mesh and standard source treatment. +27 + +−1 +0 +1 +0 +2 +4 +S2 +Transport +(a) t = 0.3 +−1 +0 +1 +0 +2 +4 +S2 +Transport +(b) t = 3 +−1 +0 +1 +0 +2 +4 +S2 +Transport +(c) t = 30 +Figure 16: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thick Su-Olson +square source problem with x0 = 0.5, t0 = 10. Solid lines are scalar flux, φ, and dashed are material energy +density, e. On the scale of this figure, the solid and dashed lines are coincident. +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.3 +128 cells, t = 3.0 +128 cells, t = 30.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.3 +128 cells, t = 3.0 +128 cells, t = 30.0 +(b) Material energy density, e +Figure 17: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for +the optically thick (σa = 800 cm−1) Su-Olson Gaussian source problem where x0 = 0.375, t0 = 10. The +quadrature order for all results is S16. All results were calculated with a static mesh and standard source +treatment. +28 + +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.3 +128 cells, t = 3.0 +128 cells, t = 30.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.3 +128 cells, t = 3.0 +128 cells, t = 30.0 +(b) Material energy density, e +Figure 18: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for +the optically thick (σa = 800 cm−1) S2 Su-Olson square Gaussian problem where x0 = 0.375, t0 = 10. All +results were calculated with a static mesh and standard source treatment. +−1 +0 +1 +0 +2 +4 +S2 +Transport +(a) t = 0.3 +−1 +0 +1 +0 +2 +4 +S2 +Transport +(b) t = 3 +−1 +0 +1 +0 +2 +4 +S2 +Transport +(c) t = 30 +Figure 19: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thick Su-Olson +Gaussian source problem with x0 = 0.375, t0 = 10. Solid lines are scalar flux, φ, and dashed are material +energy density, e. On the scale of this figure, the solid and dashed lines are coincident. +29 + +Table 9: Transport (top) and S2 (bottom) results for the scalar flux, φ, for the thick square source Su-Olson +problem with x0 = 0.5, t0 = 10. Convergence results for these answers are plotted in Figures 14 and 15. +x/t +0.3 +3.0 +30.0 +- +4.999998 +4.999998 +4.999957 +0.0579 +4.999998 +4.999998 +4.999802 +0.1158 +4.999998 +4.999998 +4.998522 +0.1737 +4.999998 +4.999998 +4.991207 +0.2316 +4.999998 +4.999998 +4.959099 +0.2895 +4.999998 +4.999998 +4.850729 +0.3474 +4.999998 +4.999957 +4.569407 +0.4053 +4.999998 +4.981594 +4.007694 +0.4632 +4.997551 +4.256673 +3.144975 +0.5211 +0.141706 +1.375262 +2.125719 +0.5789 +- +0.063798 +1.200785 +0.6368 +- +0.000274 +0.552639 +0.6947 +- +- +0.203932 +0.7526 +- +- +0.05963 +0.8105 +- +- +0.013701 +0.8684 +- +- +0.002459 +0.9263 +- +- +0.000343 +0.9842 +- +- +3.7e-05 +1.0421 +- +- +3e-06 +1.1 +- +- +- +x/t +0.3 +3.0 +30.0 +- +4.999999 +5.0 +4.999961 +0.0579 +4.999999 +5.0 +4.999807 +0.1158 +4.999999 +5.0 +4.998528 +0.1737 +4.999999 +5.0 +4.991218 +0.2316 +4.999999 +5.0 +4.959116 +0.2895 +4.999999 +4.999999 +4.850739 +0.3474 +4.999999 +4.999961 +4.56939 +0.4053 +4.999999 +4.981698 +4.007652 +0.4632 +4.997908 +4.256289 +3.144948 +0.5211 +0.141029 +1.375703 +2.125739 +0.5789 +- +0.063676 +1.200832 +0.6368 +- +0.000265 +0.552668 +0.6947 +- +- +0.203932 +0.7526 +- +- +0.059617 +0.8105 +- +- +0.013692 +0.8684 +- +- +0.002455 +0.9263 +- +- +0.000342 +0.9842 +- +- +3.7e-05 +1.0421 +- +- +3e-06 +1.1 +- +- +- +7.3 +Constant Cv Gaussian problem +Finally, we provide results for the constant Cv optically thick problem with a Gaussian source. Like the +linear version of this problem, x0 = 0.375, t0 = 10, and l = +1 +800. We choose our constant opacity to be the +same as we used for the optically thin case, Cv0 = 0.03. The source is again given by Eq/ (37) and the +uncollided solution is not used. Like the linear thick Gaussian, a static mesh is employed. +30 + +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.3 +128 cells, t = 3.0 +128 cells, t = 30.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.3 +128 cells, t = 3.0 +128 cells, t = 30.0 +(b) Material energy density, e +Figure 20: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for +the optically thick (σa = 800 cm−1) constant Cv Gaussian source problem where x0 = 0.375, t0 = 10. The +quadrature order for all results is S16. All results were calculated with a static mesh and standard source +treatment. +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.3 +128 cells, t = 3.0 +128 cells, t = 30.0 +(a) Radiation energy density, φ +2 +4 +8 +12 +M +10−14 +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +avg. |cn| +128 cells, t = 0.3 +128 cells, t = 3.0 +128 cells, t = 30.0 +(b) Material energy density, e +Figure 21: Log-linear scaled average value of the solution expansion coefficients (found by Eqs. (44)) for the +optically thick (σa = 800 cm−1) S2 constant Cv Gaussian problem where x0 = 0.375, t0 = 10. All results +were calculated with a static mesh and standard source treatment. +31 + +Table 10: Transport (top) and S2 (bottom) results for the material energy density, e, for the thick square +source Su-Olson problem with x0 = 0.5, t0 = 10. Convergence results for these answers are plotted in Figures +14 and 15. +- +0.3 +3.0 +30.0 +- +4.999998 +4.999998 +4.999957 +0.0579 +4.999998 +4.999998 +4.999802 +0.1158 +4.999998 +4.999998 +4.998522 +0.1737 +4.999998 +4.999998 +4.991208 +0.2316 +4.999998 +4.999998 +4.959105 +0.2895 +4.999998 +4.999998 +4.850742 +0.3474 +4.999998 +4.999957 +4.56943 +0.4053 +4.999998 +4.981624 +4.007719 +0.4632 +4.997609 +4.256925 +3.144988 +0.5211 +0.140405 +1.375054 +2.125712 +0.5789 +- +0.063722 +1.200762 +0.6368 +- +0.000273 +0.552615 +0.6947 +- +- +0.203916 +0.7526 +- +- +0.059622 +0.8105 +- +- +0.013699 +0.8684 +- +- +0.002458 +0.9263 +- +- +0.000343 +0.9842 +- +- +3.7e-05 +1.0421 +- +- +3e-06 +1.1 +- +- +- +- +0.3 +3.0 +30.0 +- +4.999999 +5.0 +4.999961 +0.0579 +4.999999 +5.0 +4.999807 +0.1158 +4.999999 +5.0 +4.998529 +0.1737 +4.999999 +5.0 +4.99122 +0.2316 +4.999999 +5.0 +4.959121 +0.2895 +4.999999 +4.999999 +4.850752 +0.3474 +4.999999 +4.999961 +4.569413 +0.4053 +4.999999 +4.981729 +4.007677 +0.4632 +4.997962 +4.256541 +3.144961 +0.5211 +0.139713 +1.375495 +2.125732 +0.5789 +- +0.063599 +1.200809 +0.6368 +- +0.000264 +0.552644 +0.6947 +- +- +0.203916 +0.7526 +- +- +0.05961 +0.8105 +- +- +0.013689 +0.8684 +- +- +0.002455 +0.9263 +- +- +0.000342 +0.9842 +- +- +3.6e-05 +1.0421 +- +- +3e-06 +1.1 +- +- +- +Although the Gaussian is slightly misshapen in the solution plots (Figure 22) when compared to the +linearized Gaussian, the solution is smooth like the linearized problem and spectral convergence is observed +in Figures 20 and 21. The scalar flux and material energy density are given in Tables 13 and 14. +32 + +Table 11: Transport (top) and S2 (bottom) results for the scalar flux, φ, for the thick Gaussian source +Su-Olson problem with x0 = 0.375, t0 = 10. Convergence results for these answers are plotted in Figures 17 +and 9. +x/t +0.3 +3.0 +30.0 +- +4.99565 +4.956221 +4.607285 +0.0842 +4.750452 +4.71669 +4.414224 +0.1684 +4.084736 +4.065342 +3.882236 +0.2526 +3.175988 +3.173439 +3.13421 +0.3368 +2.232954 +2.243553 +2.322698 +0.4211 +1.418754 +1.435689 +1.579266 +0.5053 +0.815508 +0.832457 +0.986082 +0.5895 +0.423872 +0.437156 +0.565183 +0.6737 +0.199217 +0.207914 +0.297361 +0.7579 +0.084665 +0.089558 +0.143615 +0.8421 +0.032536 +0.034938 +0.063669 +0.9263 +0.011306 +0.012344 +0.025911 +1.0105 +0.003552 +0.00395 +0.009679 +1.0947 +0.001009 +0.001144 +0.003319 +1.1789 +0.000259 +0.0003 +0.001044 +1.2632 +6e-05 +7.1e-05 +0.000301 +1.3474 +1.2e-05 +1.5e-05 +7.9e-05 +1.4316 +2e-06 +2e-06 +1.9e-05 +1.5158 +- +- +4e-06 +1.6 +- +- +- +x/t +0.3 +3.0 +30.0 +- +4.995653 +4.956229 +4.607284 +0.0842 +4.750456 +4.716697 +4.414223 +0.1684 +4.084739 +4.065349 +3.882238 +0.2526 +3.175991 +3.173444 +3.134214 +0.3368 +2.232956 +2.243557 +2.322703 +0.4211 +1.418755 +1.435691 +1.57927 +0.5053 +0.815508 +0.832459 +0.986085 +0.5895 +0.423872 +0.437157 +0.565185 +0.6737 +0.199218 +0.207914 +0.297362 +0.7579 +0.084665 +0.089558 +0.143614 +0.8421 +0.032536 +0.034938 +0.063669 +0.9263 +0.011306 +0.012344 +0.02591 +1.0105 +0.003552 +0.00395 +0.009679 +1.0947 +0.001009 +0.001144 +0.003319 +1.1789 +0.000259 +0.0003 +0.001044 +1.2632 +6e-05 +7.1e-05 +0.000301 +1.3474 +1.2e-05 +1.5e-05 +7.9e-05 +1.4316 +2e-06 +2e-06 +1.9e-05 +1.5158 +- +- +4e-06 +1.6 +- +- +- +33 + +Table 12: Transport (top) and S2 (bottom) results for the material energy density, e, for the thick Gaussian +source Su-Olson problem with x0 = 0.375, t0 = 10. Convergence results for these answers are plotted in +Figures 17 and 9. +x/t +0.3 +3.0 +30.0 +- +4.995668 +4.956239 +4.6073 +0.0842 +4.750468 +4.716705 +4.414236 +0.1684 +4.084745 +4.065351 +3.882244 +0.2526 +3.17599 +3.17344 +3.134212 +0.3368 +2.232949 +2.243548 +2.322695 +0.4211 +1.418746 +1.435681 +1.57926 +0.5053 +0.8155 +0.832449 +0.986075 +0.5895 +0.423866 +0.43715 +0.565178 +0.6737 +0.199213 +0.20791 +0.297357 +0.7579 +0.084663 +0.089556 +0.143612 +0.8421 +0.032535 +0.034937 +0.063668 +0.9263 +0.011305 +0.012343 +0.02591 +1.0105 +0.003552 +0.003949 +0.009679 +1.0947 +0.001009 +0.001144 +0.003319 +1.1789 +0.000259 +0.0003 +0.001044 +1.2632 +6e-05 +7.1e-05 +0.000301 +1.3474 +1.2e-05 +1.5e-05 +7.9e-05 +1.4316 +2e-06 +2e-06 +1.9e-05 +1.5158 +- +- +4e-06 +1.6 +- +- +- +x/t +0.3 +3.0 +30.0 +- +4.995672 +4.956247 +4.607298 +0.0842 +4.750472 +4.716712 +4.414236 +0.1684 +4.084748 +4.065358 +3.882246 +0.2526 +3.175992 +3.173446 +3.134216 +0.3368 +2.232951 +2.243552 +2.3227 +0.4211 +1.418747 +1.435684 +1.579264 +0.5053 +0.8155 +0.832451 +0.986078 +0.5895 +0.423866 +0.437151 +0.565179 +0.6737 +0.199214 +0.20791 +0.297357 +0.7579 +0.084663 +0.089556 +0.143611 +0.8421 +0.032535 +0.034937 +0.063667 +0.9263 +0.011305 +0.012343 +0.025909 +1.0105 +0.003552 +0.003949 +0.009679 +1.0947 +0.001009 +0.001144 +0.003319 +1.1789 +0.000259 +0.0003 +0.001044 +1.2632 +6e-05 +7.1e-05 +0.000301 +1.3474 +1.2e-05 +1.5e-05 +7.9e-05 +1.4316 +2e-06 +2e-06 +1.9e-05 +1.5158 +- +- +4e-06 +1.6 +- +- +- +34 + +−2 +0 +2 +0.0 +0.5 +1.0 +1.5 +S2 +Transport +(a) t = 0.3 +−2 +0 +2 +0.0 +0.5 +1.0 +1.5 +S2 +Transport +(b) t = 3 +−2 +0 +2 +0.0 +0.5 +1.0 +1.5 +S2 +Transport +(c) t = 30 +Figure 22: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thick constant Cv +Gaussian source problem with x0 = 0.375, t0 = 10. Solid lines are radiation temperature φ +1/4, and dashed +are temperature, T. On the scale of this figure, the solid and dashed lines are coincident. +Table 13: Transport (top) and S2 (bottom) results for the scalar flux, φ, for the thick Gaussian source +constant Cv problem with x0 = 0.375, t0 = 10, and Cv0 = 0.03 GJ · cm−3 · keV−1. Convergence results for +these answers are plotted in Figures 20 and 21. +x/t +0.3 +3.0 +30.0 +- +6.494763 +6.374101 +5.426524 +0.0789 +6.11596 +6.012864 +5.186732 +0.1579 +5.085009 +5.026915 +4.518059 +0.2368 +3.682628 +3.677866 +3.56146 +0.3158 +2.248727 +2.285336 +2.499432 +0.3947 +1.081952 +1.134228 +1.513531 +0.4737 +0.352326 +0.389958 +0.733651 +0.5526 +0.060476 +0.069031 +0.230211 +0.6316 +0.00502 +0.005255 +0.01702 +0.7105 +0.000253 +0.000255 +0.000274 +0.7895 +8e-06 +8e-06 +8e-06 +0.8684 +- +- +- +x/t +0.3 +3.0 +30.0 +- +6.494762 +6.374098 +5.42651 +0.0789 +6.115958 +6.012861 +5.18672 +0.1579 +5.085008 +5.026914 +4.518052 +0.2368 +3.682627 +3.677866 +3.561461 +0.3158 +2.248727 +2.285337 +2.499437 +0.3947 +1.081952 +1.134229 +1.513539 +0.4737 +0.352326 +0.389959 +0.733658 +0.5526 +0.060476 +0.069032 +0.230216 +0.6316 +0.00502 +0.005255 +0.017021 +0.7105 +0.000253 +0.000255 +0.000274 +0.7895 +8e-06 +8e-06 +8e-06 +0.8684 +- +- +- +35 + +Table 14: Transport (top) and S2 (bottom) results for the material energy density, e, for the thick Gaussian +source constant Cv problem with x0 = 0.375, t0 = 10, and Cv0 = 0.03GJ·cm−3 ·keV−1. Convergence results +for these answers are plotted in Figures 20 and 21. +x/t +0.3 +3.0 +30.0 +- +3.490028 +3.473704 +3.33671 +0.0737 +3.444608 +3.429805 +3.303993 +0.1474 +3.309364 +3.299069 +3.20642 +0.2211 +3.086882 +3.083965 +3.04554 +0.2947 +2.780372 +2.787696 +2.82358 +0.3684 +2.389727 +2.410536 +2.540796 +0.4421 +1.913502 +1.950669 +2.194284 +0.5158 +1.363511 +1.408883 +1.769126 +0.5895 +0.82657 +0.846776 +1.213573 +0.6632 +0.436854 +0.439279 +0.485835 +0.7368 +0.210529 +0.210694 +0.212429 +0.8105 +0.093598 +0.093606 +0.093685 +0.8842 +0.038508 +0.038508 +0.038511 +0.9579 +0.014665 +0.014665 +0.014665 +1.0316 +0.005169 +0.005169 +0.005169 +1.1053 +0.001686 +0.001686 +0.001686 +1.1789 +0.00051 +0.00051 +0.00051 +1.2526 +0.000142 +0.000142 +0.000142 +1.3263 +3.6e-05 +3.6e-05 +3.6e-05 +1.4 +8e-06 +8e-06 +8e-06 +x/t +0.3 +3.0 +30.0 +- +3.490028 +3.473704 +3.336708 +0.0737 +3.444608 +3.429805 +3.303991 +0.1474 +3.309364 +3.299069 +3.206419 +0.2211 +3.086882 +3.083965 +3.04554 +0.2947 +2.780372 +2.787696 +2.823581 +0.3684 +2.389727 +2.410537 +2.540799 +0.4421 +1.913502 +1.95067 +2.194288 +0.5158 +1.363511 +1.408884 +1.769132 +0.5895 +0.82657 +0.846774 +1.213586 +0.6632 +0.436853 +0.439278 +0.485758 +0.7368 +0.210529 +0.210693 +0.212427 +0.8105 +0.093598 +0.093606 +0.093684 +0.8842 +0.038508 +0.038508 +0.038511 +0.9579 +0.014665 +0.014665 +0.014665 +1.0316 +0.005169 +0.005169 +0.005169 +1.1053 +0.001686 +0.001686 +0.001686 +1.1789 +0.00051 +0.00051 +0.00051 +1.2526 +0.000142 +0.000142 +0.000142 +1.3263 +3.6e-05 +3.6e-05 +3.6e-05 +1.4 +8e-06 +8e-06 +8e-06 +36 + +8 +Conclusions +We have presented benchmark solutions to time dependent radiative transfer problems with two functional +forms for the specific heat in optically thin and thick media. These solutions will be useful to researchers +who seek a nonlinear radiative transfer benchmark for verification purposes, desire more digits of accuracy +for the Su-Olson type problem, or who intend to resolve a mean free path in the optically thick limit +with a transport code. Although discontinuous sources are inconvenient for DG methods and required a +complex mesh treatment to find accurate results we have provided solutions for square sources since they are +simple to simulate in numerical codes and are already implemented in codes that run the Su-Olson problem. +Researchers who implement the DG friendly Gaussian source that we have defined will be able to converge +to even more accurate results than for the square source. +To support our claim of benchmark quality results, we presented the convergence of the magnitude of +coefficients in the solution expansion for each result. While this does not necessarily guarantee the correctness +of the system being solved, it does provide confidence that the solution is converged to a particular value. +We further increased confidence in our solutions by running S2 benchmarks for the linearized problems to +high accuracy and comparing to the original published results in [8]. Furthermore, the nonlinear problems +were checked for systematic errors with a Sn code. +A +Uncollided solutions to the S2 transport equation +This section contains solutions to Eq. (41) for a square and a Gaussian source, which are used for the +uncollided source treatment in S2 problems. +A.1 +Gaussian source +With Eq. (37) as S in Eq. (41), the integral evaluates to, +φ +gs +u (x, t) = 1 +4 +√ +3πσe +3σ2 +4 − +√ +3x +� +erf +� +−3σ2 − 2t + 2t0 + 2 +√ +3x +2 +√ +3σ +� ++ +e2 +√ +3x +� +erf +�√ +3σ +2 ++ +t +√ +3σ + x +σ +� +− erf +� +3σ2 + 2t − 2t0 + 2 +√ +3x +2 +√ +3σ +�� ++ erf +� +3 +√ +3σ2 + 2 +√ +3t − 6x +6σ +�� +(56) +Where σ is the standard deviation. +A.2 +Square source +To evaluate the integral, Eq. (41) for a square source, Eq. (36), each possible case of integration limits +allowed by the step functions in the source must be considered. This will result in a piecewise function for +the uncollided scalar flux. If we define, +F(t, τ) = −1 +2 exp(−t + τ), +(57) +37 + +then, if t ≤ t0, +φss +u (x, t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +F +���� +max(0,τb) +max(0,τa) +(|x| > x0) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +2F +���� +min(t,t0) +0 +(t + +√ +3|x| ≤ +√ +3x0) +F +���� +τc +τa ++ 2F +���� +min(t,t0) +τc +� +t + +√ +3 |x| ≥ +√ +3 x0 +� +& +� +t − +√ +3(|x| + x0) > 0 +� +F +���� +max(0,τc) +0 ++ 2F +���� +min(t,t0) +max(0,τc) +� +t + +√ +3 |x| ≥ +√ +3 x0 +� +& +� +t − +√ +3(|x| + x0) ≤ 0 +� +(|x| ≤ x0) +(58) +or, if t > t0, +φss +u (x, t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +F +���� +min(τd,t0) +min(τe,t0) +x0 − +√ +3 +3 (t − t0) ≤ x ≤ x0 + +√ +3 +3 (t − t0) +F +���� +min(τd,t0) +min(τe,t0) +x > x0 + +√ +3 +3 (t − t0) +� +� +� +� +� +� +� +� +� +� +� +� +� +2F +���� +t0 +0 +t − +3 +√ +3(x0 − x) ≤ 0 +2F +���� +t0 +τd ++ F +���� +τd +0 +t − +3 +√ +3(x0 − x) > 0 and t − +3 +√ +3(x0 − x) < t0 +x < x0 − +√ +3 +3 (t − t0) +(59) +Where +τa = t − +√ +3(|x| + x0), +(60) +τb = t − +√ +3(|x| − x0), +(61) +τc = t − +√ +3(x0 − |x|), +(62) +τd = +����t − 3 +√ +3(x − x0) +���� ++ +, +(63) +and +τe = +����t − 3 +√ +3(x + x0) +���� ++ +. +(64) +| · | returns the positive part of its argument. +References +[1] J Stefan. Uber die beziehung zwischen der warmestrahlung und der temperatur, sitzungsberichte der +mathematisch-naturwissenschaftlichen classe der kaiserlichen. Akademie der Wissenschaften, 79:S–391, +1879. +[2] R. E. Marshak. Effect of radiation on shock wave behavior. The Physics of Fluids, 1(1), 1958. +38 + +[3] Albert G Petschek, Ralph E Williamson, and John K Wooten Jr. The penetration of radiation with +constant driving temperature. Technical report, Los Alamos Scientific Lab., N. Mex., 1960. +[4] William Bennett and Ryan G McClarren. Self-similar solutions for high-energy density radiative transfer +with separate ion and electron temperatures. Proceedings of the Royal Society A, 477(2249):20210119, +2021. +[5] GC Pomraning. The non-equilibrium marshak wave problem. Journal of Quantitative Spectroscopy +and Radiative Transfer, 21(3):249–261, 1979. +[6] Ryan G McClarren, James Paul Holloway, and Thomas A Brunner. Analytic p1 solutions for time- +dependent, thermal radiative transfer in several geometries. Journal of Quantitative Spectroscopy and +Radiative Transfer, 109(3):389–403, 2008. +[7] Barry D Ganapol and GC Pomraning. The non-equilibrium marshak wave problem: a transport theory +solution. Journal of Quantitative Spectroscopy and Radiative Transfer, 29(4):311–320, 1983. +[8] Bingjing Su and Gordon L. Olson. An analytical benchmark for non-equilibrium radiative transfer in +an isotropically scattering medium. Annals of Nuclear Energy, 24(13):1035–1055, 1997. +[9] Bingjing Su and Gordon L. Olson. Non-grey benchmark results for two temperature non-equilibrium +radiative transfer. Journal of Quantitative Spectroscopy and Radiative Transfer, 62(3):279–302, 1999. +[10] Su Bingjing and Gordon L Olson. Benchmark results for the non-equilibrium marshak diffusion problem. +Journal of Quantitative Spectroscopy and Radiative Transfer, 56(3):337–351, 1996. +[11] Ryan G McClarren and John G W¨ohlbier. Solutions for ion–electron–radiation coupling with radiation +and electron diffusion. Journal of Quantitative Spectroscopy and Radiative Transfer, 112(1):119–130, +2011. +[12] Ryan G. McClarren. Two-group radiative transfer benchmarks for the non-equilibrium diffusion model. +Journal of Computational and Theoretical Transport, 50(6-7):583–597, 2022. +[13] William Bennett and Ryan G. McClarren. Accurate solutions to time dependent transport problems +with a moving mesh and exact uncollided source treatment, 2022. +[14] B. D. Ganapol, R. S. Baker, J. A. Dahl, and Raymond E. Alcouffe. Homogeneous infinite media time- +dependent analytical benchmarks. Technical Report LA-UR-01-1854, Los Alamos National Laboratory, +2001. +[15] William Bennett and Ryan G McClarren. Benchmarks for infinite medium, time dependent transport +problems with isotropic scattering. arXiv preprint arXiv:2205.15783, 2022. +[16] Subrahmanyan Chandrasekhar. Radiative transfer. Dover Publications, 1960. +[17] J.E. Marsden and A. Tromba. Vector Calculus. Number pp. 1-93 in Vector Calculus. W. H. Freeman, +2003. +[18] Nico Schl¨omer, Nick Papior, Darius Arnold, Jan Blechta, and Rasmus Zetter. nschloe/quadpy: None, +2021. +[19] Pauli Virtanen, Ralf Gommers, et al. SciPy 1.0: fundamental algorithms for scientific computing in +Python. Nature Methods, 17(3):261–272, 2020. +39 + diff --git a/ztE0T4oBgHgl3EQftwFw/content/tmp_files/load_file.txt b/ztE0T4oBgHgl3EQftwFw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..093d7a1db5c38417a0504348cfe6ce4a00006a5e --- /dev/null +++ b/ztE0T4oBgHgl3EQftwFw/content/tmp_files/load_file.txt @@ -0,0 +1,3375 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf,len=3374 +page_content='Benchmark solutions for radiative transfer with a moving mesh and exact uncollided source treatments William Bennett Ryan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' McClarren January 9, 2023 Abstract The set of benchmark solutions used in the thermal radiative transfer community suffer some coverage gaps, in particular nonlinear, nonequilibrium problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Also, there are no nonequilibrium, optically thick benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' These shortcomings motivated the origination of a numerical method unfettered by preconditions of linearity and easily able to converge on smooth optically thick problems, a moving mesh Discontinuous Galerkin (DG) framework that utilizes an uncollided source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Having already proven this method on time dependent scattering transport problems, we present here solutions to non-equilibrium thermal radiative transfer problems for familiar linearized systems and more physical nonlinear systems in both optically thin and thick regimes, including both the full transport and the S2/P1 solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Geometric convergence is observed for smooth sources at all times and some nonsmooth sources at late times when there is local equilibrium and accurate solutions are achieved for step sources when the solution is not smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 1 Introduction The Stefan-Boltzmann law [1], which describes the relationship between radiation emitted from a material and its temperature as proportional to temperature to the fourth power, is to blame for the obdurate nonlinearity in high-energy density radiative heat transfer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For this reason, the extant analytic benchmarks with space and time dependence in this field are predicated on assumptions of linearity or equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' There are solutions that assume the radiation energy and internal energy of the material instantly equilibrate, inducing a Marshak Wave [2, 3, 4], and solutions for non-equilibrium problems that linearize the system in T 4 by invoking a form for the the material heat capacity that is proportional to temperature cubed, an innovation of Pomraning [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This technique of defining a heat capacity to linearize the system has been used to produce an abundance of solutions, including transport treatments for the P1 equations [6], full transport solutions with one speed [7, 8], and non-grey problems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' In the diffusion limit, benchmarks solutions have been provided for one temperature [10], three temperature [11], and a non-grey treatment [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' While these solutions are invaluable to code developers for verification, it is necessary to point out that there are certain drawbacks to using linear problems to verify codes whose purpose it is to solve nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' While ideally the numerical code in question would solve the fully nonlinear equations and implement a special equation of state when running these verification problems, there is nothing to prohibit the curators of these codes from simply solving a linearized system when the benchmark is being run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The result is the same in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Also, solutions to linear systems can be scaled to match benchmarks, unlike the unforgiving solutions to nonlinear systems, and the solution to the linearized equations equilibrates more quickly than a nonlinear problem as the temperature increases as a result of the special equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Solving a linear problem does not completely verify the functionality of a radiative transfer code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Although nonlinearity is an impediment for analytic methods, it is not necessarily a source of difficulty for spectral methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This was the impetus for our development of a moving mesh, uncollided source treatment Discontinuous Galerkin (DG) method for solving transport problems [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The time dependent cell edges, which we call a moving mesh, and the uncollided source treatment were added on to the DG 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='02596v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='CE] 6 Jan 2023 implementation because the transport equation with its finite wavespeeds admits discontinuities that inhibit DG methods from attaining their higher order convergence potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The moving mesh and uncollided source can present a smoother problem for the method to solve: the moving mesh by matching edges to moving wavefront discontinuities and the uncollided source by analytically resolving the most structured part of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' As documented in [13], we have already conducted extensive tests with this method on time-dependent transport problems, which allowed for a detailed analysis of the efficacy of the moving mesh and uncollided mesh for different source types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For example, for finite width, nonsmooth sources that induce a nonsmooth solution that is smoothed over time, the method proved the most beneficial when compared to a standard DG implementation, but displayed only algebraic error convergence, not the optimal geometric convergence that DG methods are capable of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For smooth, Gaussian sources however, we we able to achieve spectral convergence, but the importance of the moving mesh was diminished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' With an understanding of the effectiveness of this method on linear systems, we apply it to nonlinear radiative transfer problems and obtain results with accuracy comparable to an analytic solution, which is the stated intent of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Before attempting fully nonlinear problems however, we first apply our method the existing linear radiative transfer problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This will allow us to test our method on problems with known solutions and uncover deficiencies in a more forgiving arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For nonlinear problems, we can still gauge the precision of our solution by inspecting magnitude of the expansion coefficients and the accuracy by checking against existing numerical Sn solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We selected the Su and Olson transport benchmark [8] as an ideal verification solution for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Unfortunately, the results are not given to enough digits to fully demonstrate the effectiveness of our scheme and recalculation of these results is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Therefore, we rely on integration of a P1 version of this benchmark [6] to create solutions which are not necessarily as physically accurate as the full transport solution, but can be calculated out to more digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' There are no existing radiative transfer transport verification solutions for optically thick problems outside of the equilibrium diffusion limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' By optically thick, we mean that the source width or the support of an initial condition is orders of magnitude larger than a mean free path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Conversely, optically thin problems have source widths comparable to a mean free path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For a transport code to have sufficient coverage of verification problems, converging to a diffusion benchmark of a thick problem while not resolving a mean free path is a good test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' If the code resolves a mean free path, however, it will converge to the diffusion problem plus a transport correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' It is for the purpose of verifying this transport correction that we include transport solutions and the S2/P1 solutions for optically thick problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The remaining sections of the paper are organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Section 2 contains an introduction to our model equations, nondimensionalization, and derivation of the uncollided source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Our DG implementation is laid out briefly in Section 3, but a more detailed derivation is left to [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Section 4 is devoted to the calculation of S2 benchmarks and the corresponding uncollided solutions used in verifying our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Following this is a description of how the convergence of the error in the results section is calculated (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' then our results (Sections 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The results sections also contain specific details of the methods used in each problem and discussion of the solution characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 2 Equations We study non-equilbrium time dependent radiative heat transfer in an infinite, purely absorbing, stationary medium with an internal radiation source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The radiation transport and material balance equations for this system are, �1 v ∂ ∂τ + µ ∂ ∂z + σa � ψ(z, τ, µ) = σa �1 2avT(z, τ)4 � + 1 2S(z, τ), (1) ∂ ∂τ e(z, τ) = σa � φ(z, τ) − avT(z, τ)4� , (2) where the general form of the equation of state is, e = � T 0 dT ′ Cv(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (3) 2 The variables in these equations are, ψ, the angular flux or intensity, φ = � 1 −1dµ′ψ(x, t, µ′) the scalar flux, T, the temperature, and e, the material energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' ψ and φ have units of energy per area per time ([GJ·cm−2ns−1]) and e has units of energy density ([GJ·cm−3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' S is a source term with units of energy density per time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' µ ∈ [−1, 1] is the cosine of the particle direction with respect to the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' v is the particle velocity, which is the speed of light in a vacuum for our application, v = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='998 cm ns−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The radiation constant is a = 4σSB/v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0137225 GJ cm−3 keV−4, where σSB is the Stefan-Boltzman constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The absorption cross section, σa, is in units of inverse length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We seek a non-dimensionalization for these equations that is compatible with the non-dimensionalization given the in Su-Olson benchmark [9] and that may be used in optically thick problems without enlarging the non-dimensionalized length to accommodate the larger opacity, x = lσaz t = lvσaτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (4) l is a dimensionless scaling variable that is set to one for thin problems and a small number to offset the greater σa in optically thick problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Each equation is transformed into the new variables and divided by avσaT 4 H, where TH is the reference temperature, called the hohlraum temperature in previous work, � l ∂ ∂t + µl ∂ ∂x + 1 � ψ(x, t, µ) = ca �1 2T(x, t)4 � + 1 2Q(x, t), (5) l ∂ ∂te(x, t) = ca � φ(x, t) − T(x, t)4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (6) Our non-dimensional dependent variables are now, ψ(x, t) = ψ(x, t) avT 4 H φ(x, t) = φ(x, t) avT 4 H T(x, t) = T(x, t) TH e(x, t) = e(x, t) aT 4 H , (7) the non-dimensional source is, Q(x, t) = S(x, t) σaavT 4 H (8) and the absorption ratio is defined, ca = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (9) In this work, we consider two functional forms for Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' To solve the Su-Olson benchmark problem, we use the familiar form, Cv = αT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (10) which renders Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (6) linear in T 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' With the conventional choice of α = 4a, now eSU = T 4, (11) where the subscript “SU” indicates that this is the equation of state for the linear Su-Olson problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' While it is important for our investigation to solve these linear problems the novel aspect of this paper is results for nonlinear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For these, we choose a more physical, constant specific heat, Cv = Cv0, with units of of energy density per temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This choice renders e = Cv0T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' To find the relationship between the nondimensional variables with this equation of state, we define Cv0 = Cv0 aT 3 H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Now we can write eN = Cv0T, (12) where the subscript “N” indicates that this is our equation of state for the nonlinear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Other forms of Cv are certainly allowable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For instance, we could implement a specific heat that is dependent on a non-constant opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' However, this will be left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 Uncollided solutions In time dependent transport trials, we found that the deployment of an an uncollided source treatment, where using the solution to the equation, � l ∂ ∂t + µl ∂ ∂x + 1 � ψu(x, t, µ) = 1 2Q(x, t), (13) as a source term to solve for the collided flux is a significant boon for accuracy when the solution is not smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The Green’s solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (13) with l = 1 was provided by [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This solution is integrated for different source configurations in [15], including a square and a Gaussian source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Using these solutions, we can say that ψu is known and can be integrated analytically to find φu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For problems where l ̸= 1, a simple scaling is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For optically thick problems when l ≪ 1, the uncollided solution is not as useful since it has decayed to zero by the pertinent evaluation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' To solve for the remaining collided portion of the flux, we have the system, � l ∂ ∂t + µl ∂ ∂x + 1 � ψc(x, t, µ) = ca �1 2T(x, t)4 � (14) l ∂ ∂te(x, t) = ca � φc(x, t) + φu(x, t) − T(x, t)4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (15) In linear transport applications, it is possible to decompose the flux infinitely, not just into uncolllided and collided flux, but uncollided, first collided, second collided, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Even though the radiative transfer equations are nonlinear, we are able to use this linear solution technique since the uncollided flux has no interaction with the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' However, we cannot not further decompose the flux as we could in a linear transport problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Answers obtained with an “uncollided source” treatment refer to solutions to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (14) and (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' A “standard source” treatment refers to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The most useful source treatment used in a specific problem is determined by the behavior of the uncollided flux during the solution time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Tests run in [13] showed that integrating the uncollided source could require more computation time than a standard source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This is because the uncollided source is a complex function of space and more difficult to integrate with quadrature than the standard source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' As a rule, problems at times where the uncollided flux has not decayed enough to be a negligible portion of the flux are good candidates for an uncollided source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' In these problems, [13] showed increase in accuracy and rate of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' At times where the uncollided solution has decayed, the uncollided source treatment is not as helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' While the problems investigated in this paper are in purely absorbing media, the coupling between the material energy density and the radiation energy density acts as a scatterer in that it can smooth discontinuities over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For this reason, we expect that the insights derived from solving purely scattering transport problems with uncollided source treatments will extend to these purely absorbing radiative transfer problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 3 Moving Mesh DG spatial discretization Similar to the procedure in [13], we define a DG spatial discretization with a moving mesh to solve equations of the form (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We leave some of the details of the derivation to [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' To solve for the integral over µ to find the scalar flux, we discretize in angle via the method of discrete ordinates, where the solid angle µ ∈ [−1, 1] is discretized by choosing the points with a Gauss-Lobatto quadrature rule [16] for our full transport solution or, in the case of the S2 solution, a Gauss-Legendre rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' With the corresponding weights from our chosen quadrature, we can define the scalar flux as a weighted sum, φ ≈ N � n′=1 wn′ψ n′ , (16) where wn are the weights and ψn is the scalar flux evaluated at a given angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This choice makes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (5) and (6), 4 � l ∂ ∂t + µnl ∂ ∂x + 1 � ψ n(x, t) = ca �1 2T(x, t)4 � + 1 2Q(x, t) for n = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' N, (17) ∂ ∂te(x, t) = ca � N � n′=1 wn′ψn′ − T(x, t)4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (18) To discretize the spatial domain, we define K non-overlapping cells with time dependent edges xL(k, t) and xR(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' To allow simplifications to the coming weak form of the equations, we define a mapping variable x′(k, t) that maps x to [-1,1] inside a cell, x′(k, t) ≡ xL(k, t) + xR(k, t) − 2x xL(k, t) − xR(k, t) , k = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (19) Now we define an orthonormalized Legendre polynomial basis function in x′ for each cell, Bi,k(x′) = √2i + 1 � xR(k, t) − xL(k, t) Pi(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (20) Therefore, the weak solution of the angular flux in a cell for a given angle is ψ n(x, t) ≈ M � j=0 Bj,k(x′) un k,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (21) where u is an entry in our three dimensional solution matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Likewise, the solution for the energy density in a given cell is, e(x, t) ≈ M � j=0 Bj,k(x′) uN+1 k,j , (22) The standard DG procedure for finding the weak form of the equations involves multiplying each equation by a basis function, integrating over a cell, invoking integration by parts to shift the spatial derivative onto the basis function, and taking advantage of orthogonality to simplify the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Our moving mesh method is similar to this, but with the added step of invoking the Reynolds Transport Theorem [17] since our integration domain is time dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Leaving the general outline of this procedure to [13], we arrive at, d dtUn − GUn + � LUn �(surf) − µnLUn + 1 l Un = ca 2l H + 1 2lQ for n = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' N, (23) d dtUN+1 + RU surf − GU N+1 = ca l � N � n′=1 w′ nU n′ − H � , (24) where the time dependent solution vector is U n,k = [un k,0, un k,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=', un k,M]T , where M + 1 is the number of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We also define Li,j = � xR xL dx Bj,k(x′) dBi,k(x′) dx , (25) Gi,j = � xR xL dx Bj,k(x′) dBi(x′) dt , (26) Qi = � xR xL dx Bi,k(x′) Q(x, t), (27) Hi = � xR(k,t) xL(k,t) dx Bi(x′) T 4(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (28) 5 The numerical flux terms, which calculate the direction of flow of the solution with an upwinding scheme based on the relative velocity of a particle with the cell edges (LU)surf i = � µn − dxR dt � Bi,k(x′ = 1)ψ n+ − � µl − dxL dt � Bi,k(x′ = −1)ψ n−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (29) (RU)surf i = � −dxR dt � Bi,k(x′ = 1)e+ − � −dxL dt � Bi,k(x′ = −1)e−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (30) ψ l+ and ψ l− are found by evaluating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (21) and e+ and e− are found by evaluating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' If we choose to employ an uncollided source treatment, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (23) and (24) change slightly in that the source term Q disappears from the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (23) and ca l φ is added to the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (24) where, φu = � xR xL dx Bj,k(x′) φu(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (31) In this case, the numerical solution is for the collided flux, so it is necessary to add the uncollided flux at the final step to obtain the full solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The general solution procedure is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' First, parameters such as the number of basis functions, the number of spaces, and the Sn order are set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' A source is specified and depending on the source treatment, the uncollided solution or the standard source is integrated at each timestep with a standard Gaussian integrator with points equal to 2M + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The edges of the mesh are governed by a function designed to optimize the solution for the specific source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This function also returns velocities of the mesh edges in order to calculate the numerical flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The temperature balance terms are found by the equation of state and integrated in the same way as the source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The solver returns the coefficient arrays and the scalar flux and material energy are reconstructed via the expansions defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (21) and (22) and, depending on the source treatment, the uncollided flux is added onto the scalar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' To obtain solutions from our equations (23) and (24), we calculate the quadrature weights with the python package quadpy [18] and integrate the ODES in with a built in integrator from scipy [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Our python implementation can be found on Github1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 4 Benchmarks and uncollided solutions for the S2 radiative trans- fer equations In order to show more decimal places of accuracy in our linear problems than are given in [8], it was expedient to calculate our own analytic benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Since we already include results to each problem calculated with S2, we choose to verify our solver by using a S2/P1 benchmark given by [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This benchmark gives the analytic expression for the scalar flux and energy density solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (17) and (18) with N = 2, angles and Gauss-Legendre weighting, ([µ1, µ2] = [ −1 √ 3, 1 √ 3], and [w1, w2] = [1, 1]) when the source is a delta function in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The Green’s function given by [6] for φ = ψ1 + ψ2 for a delta function source at position s is, G(x, s, t) = v 2 √ 3e−t � � tI1 �� t2 − 3(x − s)2 � t2 − 3(x − s)2 Θ � t − √ 3|x − s| � + I0 �� t2 − 3(x − s)2 � δ � t − √ 3|x − s| � � � , (32) where Θ is a step function, δ is a Dirac delta function and I0 and I1 are modified Bessel functions of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The Green’s function for the material energy density is, GU(x, s, t) = √ 3 2 e−t � I0 �� t2 − 3(x − s)2 � Θ � t − √ 3|x − s| �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (33) 1www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='com/wbennett39/moving mesh radiative transfer 6 We choose to find solutions for a square source and a Gaussian source, to test our method on both smooth and nonsmooth problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Therefore, the solution to the integral, φss,gs = � ∞ −∞ ds � ∞ 0 dt′ Sss,gs(s, t′) G(x, s, t − t′) (34) gives the scalar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The energy density is likewise obtained by ess,gs = � ∞ −∞ ds � ∞ 0 dt′ Sss,gs(s, t′) Gu(x, s, t − t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (35) Where the subscript in the solution and the source is either “ss” for square source or “gs” for Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The source term is, Sss(x, t) = Θ(x0 − x)Θ(t0 − t), (36) for the square source, or Sgs(x, t) = exp �x2 x2 0 � Θ(t0 − t), (37) for the Gaussian source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Since finding a benchmark for an optically thick problem requires evaluating these integrals at extremely late times where the integrand is not well behaved, we only calculate S2 benchmarks for our thin problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 S2 uncollided solutions The uncollided solutions that we have utilized so far for the uncollided source treatment have been full transport solutions from [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We cannot use these to solve the S2 transport equation, since the two uncollided fluxes are not equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The full transport solutions are based on the assumption that the Sn order of the ODE’s sufficiently resolves the angular error and that the collided flux calculated with quadrature is a good approximation of the analytic integral over µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' N � n′=1 wn′ψ n′ ≈ � 1 −1 dµ′ ψ(x, t, µ′), (38) so that it is acceptable to employ the uncollided scalar flux found by integrating analytically the solution for the uncollided angular flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' In the S2 equations, the assumption of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (38) does not hold and the uncollided scalar flux must be found by numerical quadrature of the angular flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Therefore, the process for finding the uncollided scalar flux to use as a source in the S2 solutions to our radiative transfer problems is to find the Green’s solution for the angular flux, integrate that solution with quadrature, and then integrate again over the given source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The uncollided solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 13 with a delta function source (δ(x)δ(t)) is [14], ψu(x, t) = e−t 2t δ � µ − x t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (39) To find the S2 uncollided scalar flux, the integral is done by Gauss-Legendre quadrature with N = 2 to give the uncollided scalar flux, φ pl u (x, t) = e−t 2t � δ � − 1 √ 3 − x t � + δ � 1 √ 3 − x t �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (40) To finally find the uncollided scalar flux that corresponds to the benchmark solutions calculated with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (34), we integrate φ ss,gs u (x, t) = � ∞ −∞ ds � ∞ 0 dt′ φ pl u (x − s, t − t′) Sss,gs(s, t′), (41) where Sss,gs is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (36) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (41) are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 7 5 Error estimation methods In the problems presented, two methods are used to estimate the solution accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For problems with a benchmark solution, we use the root mean square error (RMSE) as our error metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This is calculated by, RMSE = � � � � N � i |yi − ˆyi|2 N , (42) where yi is either the calculated scalar flux or the calculated material energy density at a given node, ˆyi is the corresponding benchmark solution, and N is the total number of nodes in the computational solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For problems that demonstrate geometric spectral convergence, as M → ∞, the error can be modeled as ERROR = C exp(−c1M), (43) where M is the highest polynomial order of the basis and C and c1 are constants that could depend on the number of cells used in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This curve is a straight line on a logarithmic-linear scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For all of the problems in the following section, we plot the average of the absolute value of the coefficients in the solution expansion to characterize the solution convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We define the average value of the jth coefficient in the solution expansion, |cj| = �K k=1 |ak| K , (44) where j corresponds to the order of the Legendre polynomial in the basis and K is the number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' When characterizing the error of φ, since we are interested in the residual error of scalar flux, ak is the weighted average using the weights from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (16), ak = �N l′=1 wl′u(l′,k,j) �N l′=1 wl′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (45) For the material energy density, ak is, ak = u(N+1,k,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (46) 6 Optically thin results The results in this section are for problems where the source width is equal to a mean free path, meeting our definition of an optically thin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' These problems are characterized by solutions where the uncollided solution is a significant portion of the flux and travelling wavefronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Therefore, the problems in this section all use an uncollided source and the square sources whhich have travelling discontinuities employ a moving mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 Su-Olson problem with a square source We first replicate the the Su-Olson problem using the same square source originally presented in [8] with σa = 1 cm−1, the source width, x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 and the source duration t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The uncollided solution for this source has already been presented in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For the S2 treatment of this problem, the uncollided source is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The temperature is calculated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Some modifications were made to the original mesh function invented to solve the square source transport problem in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' In that mesh, the mesh edges inside the source never moved while the edges outside travelled outwards with the wavespeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This was done to resolve the static discontinuities at the source edge and the travelling discontinuities at the wavefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' In the original Su-Olson results, the source turns off at t0 = 10 and solutions are required long afterwards (t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228, 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' With our previous square source mesh, the edges would remain clustered around the source region long after the source has ceased to introduce nonsmoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This is not the optimal distribution of computational zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 8 Therefore, the mesh function used in this problem is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' If the mesh edges are defined as the vector, X(t) = � x0(t), x1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=', xK(t) � , (47) and initialized to be, if K 4 ≤ k ≤ 3K 4 , xk o = 1 2yj, (48) if k < K 4 , xk o = sk(δx) + 2x0 + δx 2 (49) if k > 3K 4 , xk o = sl(δx) − 2x0 − δx 2 (50) where yj are the Gauss-Lobatto evaluation points with N, the number of points, equal to K 2 + 1 numbered from 0 and sm are the Gauss-Lobatto evaluation points for N = K 4 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The indices j and m are equal to k − K 4 and k − K 2 + 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' δx is a small initial width, and K is always an even number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This initialization assigns one third of the edges to the source and the other two thirds to cover the rest of the solution domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Each subdomain is spanned by edges with Gauss-Lobatto spacing, which has the effect of concentrating cells near the source edges and the outgoing wavefronts, where discontinuities are most likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' As time progresses and the outside edges move outwards with the solution, their position is defined as, if t ≤ t0, xk(t) = xk o + xk xko × vt, (51) where v is the wavespeed, one for the transport problems and 1 √ 3 for the S2 problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The edge velocity is defined, if t ≤ t0, dxk dt = xk xko × vt, (52) Defining the edge positions and velocities this way preserves the relative spacing of the initialized edges, meaning that the edges are clustered at the source edges and the leading wavefronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' At later times when the source is off, the solution to a square source in an optically thin problem will behave more like the solution for a Gaussian source since the solution will become smoother without the source emitting uncollided particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Information flow is no longer dominated by the wavespeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The solution will be practically zero some distance from the origin that is much less than vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For instance, in the Su-Olson problem at t = 100 the solution is practically zero past x = ±30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For these reasons, when the source turns off, a constant acceleration will divert the trajectory of each edge so that at the final time, they are evenly spaced over a specified width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This width is an estimate to how far the solution will have traveled by the evaluation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We chose a constant acceleration instead of a instantaneous velocity change because the latter induced numerical errors which resulted in failure to converge to the benchmark solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The acceleration for each edge is found by, ck = 2 � dxk dt ���� t0 (t0 − tfinal) − xk ���� t0 + xk ���� tfinal � (t0 − tfinal)2 (53) where we find xk ���� tfinal by specifying that the final positions vector, X(tfinal), evenly span [−xf/2, xf/2] where xf is our estimate for the width of the solution domain at the evaluation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' With the acceleration defined, we calculate positions of the edges after the source has turned off with, if t > t0, xk(t) = 1 2ck (t − t0)2 + dxk dt ���� t0 (t − t0) + xk ���� t0 , (54) and the velocities, if t > t0, dxk dt = ck (t − t0) + dxk dt ���� t0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (55) 9 −10 0 10 x 0 10 20 30 t t0 Figure 1: Edge position for each edge in the thin square source mesh run out to tfinal = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 with 8 spaces, a wavespeed, v = 1 √ 3, and a final domain width, xf = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We refer to this method for governing the mesh edges and velocities as the “thin square source mesh”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' An example x vs t diagram of the mesh edges is given in Figure 1 for clarification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' After completing the necessary steps of defining a source, choosing a functional form for the temperature, and defining a mesh, we may present our results for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Tables 1 and 2 give our solutions for the same points and evaluation times as in Table 1 and 2 of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The convergence results for the coefficient expansions of these results are plotted in Figures 2 and 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The solutions at a few selected times are plotted in Figure 4 For each case, a moving mesh and the uncollided source was used except in the S2 solution at times greater than t0 since the S2 uncollided solution becomes sharp and difficult to resolve via quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' In these cases, the standard square source was integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The convergence results (Figures 2 and 3) show that for this problem, S2 is considerably smoother at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Twice as many spatial divisions were required at early times in the full transport solution 256 to achieve similar levels of convergence to the S2 solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Both cases exhibited similar behavior over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' At early times the significantly nonsmooth uncollided flux induced discontinuities in the material energy density and required far more spatial divisions to resolve, stiffening the problem and limiting the number of basis functions that could reasonably be used in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' After the source turned off, the solution smoothed and equilibrated locally (complete equilibrium is impossible with an infinite material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The solution became smoother and could easily be resolved with fewer spaces and more basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We also note that at later times after the source has turned off, the full transport and S2 solutions become more similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Since we claim to present benchmark quality results, we are obliged to discuss the accuracy of Tables 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' While [8] claims to converge their solution to four digits, the observant reader will see that in some cases only 3 digits match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Given that our solutions are converged beyond this point, we believe that our reported digits are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 10 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 256 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 256 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 256 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 256 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 128 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 32 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 32 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 256 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 256 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 256 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 256 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 128 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 32 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 32 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 2: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thin (σa = 1 cm−1) Su-Olson square source problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The quadrature order for all results is S256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a moving mesh and uncollided source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 128 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 128 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 32 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 32 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 128 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 128 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 32 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 32 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 3: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44))) for the optically thin (σa = 1 cm−1) S2 Su-Olson square source problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a moving mesh and uncollided source treatment except for the t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 and t = 100 cases where a standard source treatment was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='10 S2 Transport (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 S2 Transport (b) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 −1 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6 S2 Transport (c) t = 1 −2 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 S2 Transport (d) t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 −10 0 10 0 1 2 S2 Transport (e) t = 10 −20 0 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 S2 Transport (f) t = 100 Figure 4: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thin Su-Olson square source problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Solid lines are scalar flux, φ, and dashed are material energy density, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 Constant Cv problem with a square source This problem uses the same source as the problem of the last section but with a different functional form of the heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (12) our system becomes nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We choose Cv0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='03 GJ · cm−3 · keV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This value was chosen to see an appreciable change in temperature during the selected time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Now that we no longer have the convenient condition that e = T 4, the local equilibrium condition is not φ = e as in the Su-Olson problem but φ 1/4 = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For this reason, the solution plots for this problem and all subsequent constant Cv problems do not show scalar flux and material energy density but rather radiation temperature and material temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Though we can no longer rely on benchmark solutions for this problem,, we can be confident that our solution is converged by plotting the magnitude of the coefficients and check for systematic errors with a Sn solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Also, since the mesh method employed here is the same method described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1, we can be confident that the mesh is not introducing error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Solutions to this problem are plotted in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We note that the problem is not everywhere at equilibrium by t = 100 as the Su-Olson problem is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Also, we note that the solution does not travel as far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' In the Su-Olson problem, the specific heat is very small when temperature is small and increases with the cube of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This has the effect of attracting the solution to equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This effect is not present in a constant Cv case and there is less incentive for the solution to fall into local equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' It is also noteworthy that at very early times (t < 1) the scalar flux has not interacted with the material as much as in the Su-Olson problem and is mostly made up of the uncollided flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This is apparent in Figures 5a and 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Since the solution has not fully equilibrated at later times, the solutions are less smooth compared to the Su-Olson problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The repercussions of this can be observed by comparing the convergence results at late times for this problem in Figures 5 and 6 to the convergence results of the Su-Olson problem in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Also, the convergence results show that the material energy density is generally more nonsmooth than the scalar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Nevertheless, we are satisfied with the convergence of these results and present them in Tables 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 12 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 256 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 256 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 256 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 256 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 128 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 32 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 64 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 256 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 256 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 256 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 256 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 128 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 32 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 64 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 5: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thin (σa = 1 cm−1) constant Cv square source problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The quadrature order for all results is S256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a moving mesh and uncollided source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 128 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 128 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 32 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 32 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 128 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 128 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 32 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 32 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 6: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thin (σa = 1 cm−1) S2 constant Cv square source problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a moving mesh and uncollided source treatment except for the t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 and t = 100 cases where a standard source treatment was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 13 Table 1: Transport (top) and S2 (bottom) results for the scalar flux, φ for the thin square source Su-Olson problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' RMSE values included in the last row of the S2 table are calculated from the S2 benchmark from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Convergence results for these answers are plotted in Figures 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Bolded digits are those that agree with the published Su-Olson solution in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' x/t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='01 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='002009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='073828 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='78279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='002307 RMSE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='656e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='747e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='642e-06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='589e-07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='647e-07 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='157e-08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='128e-09 The difference between the full transport solution and our S2 result is also of interest, as it provides insight into the physical characteristics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We note that the two solutions only begin to look similar at later times as the solution equilibrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This tells us that the solution becomes less angularly dependent and better approximated by only two angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 Su-Olson problem with a Gaussian source Returning to the linearized Su-Olson problem, we consider a Gaussian source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Here the source is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 37 where the uncollided solution is taken from [15] for the full transport solution or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (56) for the S2 solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We set x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 and the source duration is still t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' In [13], smooth Gaussian sources allowed for geometric convergence of the solution at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We expect the same result in this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Since there are no discontinuities induced by nonsmoothness in the source, we are able to employ a far simpler mesh function than what was used for the thin square source problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We only guess the edge 14 Table 2: Transport (top) and S2 (bottom) results for the material energy density, e for the thin square source Su-Olson problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' RMSE values included in the last row of the S2 table are calculated from the S2 benchmark from [6].' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='052e-08 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='937e-08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='936e-09 of the problem domain and span the given space evenly with stationary edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The moving mesh was not used in this case because earlier tests in [13] revealed the mesh to be non-useful in smooth problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The uncollided solution however, was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We include Gaussian sources though they do not prove to be challenging enough problems to require the full application of our method because we can achieve very accurate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' While S256 was used for the full transport solutions for the thin square sources, we only use S64 on the Gaussian sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This choice is informed by tests run in [13] that showed that far fewer quadrature points are required to resolve the angular error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' With the temperature defined by Eq (11) we present solutions in Tables 5 and 6 with convergence results shown in Figures 8 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Solutions are shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' As illustrated in the aforementioned convergence plots, the problem converges geometrically even with a standard static mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We quickly note that for t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 and t = 100 in the S2 results in Figure 18, a moving mesh was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' In this case, the mesh moved with a constant speed from the initial width 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='4 S2 Transport (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6 S2 Transport (b) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 −1 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='75 S2 Transport (c) t = 1 −2 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 S2 Transport (d) t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 −10 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 S2 Transport (e) t = 10 −20 0 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='4 S2 Transport (f) t = 100 Figure 7: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thin constant Cv square source problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Solid lines are radiation temperature φ 1/4, and dashed are temperature, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 64 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 64 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 64 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 64 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 64 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 64 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 64 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 64 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 64 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 64 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 8: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thin (σa = 1 cm−1) Su-Olson Gaussian source problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The quadrature order for all results is S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a static mesh and uncollided source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 16 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 64 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 64 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 64 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 64 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 32 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 64 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 64 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 64 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 64 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 32 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 9: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thin (σa = 1 cm−1) S2 Su-Olson Gaussian source problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a moving mesh and uncollided source treatment except for the t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 and t = 100 cases where a standard source treatment was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The dashed lines represent solutions found with a moving mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' −2 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='05 S2 Transport (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 −2 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 S2 Transport (b) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6 S2 Transport (c) t = 1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 S2 Transport (d) t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 −10 0 10 0 1 2 S2 Transport (e) t = 10 −25 0 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 S2 Transport (f) t = 100 Figure 10: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thin Su-Olson Gaussian source problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Solid lines are scalar flux, φ, and dashed are material energy density, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 17 Table 3: Transport (top) and S2 (bottom) results for the scalar flux, φ, for the thin square source constant Cv problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10, and Cv0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='03 GJ · cm−3 · keV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Convergence results for these answers are plotted in Figures 5 and 6 x/t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='01 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='78279 to the specified final width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This was not done out of necessity, but rather to ascertain weather the moving mesh was useful in these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' On the difference between the full and S2 solutions, we point out that it can be seen in Figure 10 that the S2 solution is not as accurate in resolving the source region after the source has been on for some time then becomes more accurate after the source is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This is not surprising since the source emits uncollided particles in every direction while it is on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' At later times, the two solutions are visually indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='4 Constant Cv problem with a Gaussian source We include a problem with the same source and parameters as the last section but with a constant specific heat so that the temperature to material energy density conversion is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We specify the the dimensional specific heat to be Cv0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='03 GJ · cm−3 · keV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' While we have less certainty in forecasting the behavior of these nonlinear results, we still expect geometric convergence since there are no sources of nonsmoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Like the Gaussian source in the linearized system, we specified a static mesh that evenly spans some 18 Table 4: Transport (top) and S2 (bottom) results for the material energy density, e, for the thin square source constant Cv problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10, and Cv0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='03 GJ · cm−3 · keV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Convergence results for these answers are plotted in Figures 5 and 6 x/t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='023197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='536229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='964849 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='62341 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='005921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='056159 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='4e-05 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='78279 estimated width of the actual solution domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The convergence results in Figures 11 and 12 show that this was sufficient to achieve geometric convergence at all chosen times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We also see the phenomenon first observed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 of extremely fast convergence for the scalar flux at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This is once again the result of the scalar flux being most uncollided at these times, making solving for the collided portion simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' During the time the source is on there is a large discrepancy between the 2 and transport solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' this discrepancy recedes after the source is turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Similar to the nonlinear square source, the solution is not in equilibrium by t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This leads us to draw the conclusion that equilibrium is not as impacted by the nonsmoothness of the source as it is by the functional form of the specific heat, since the Su-Olson problem with a nonsmooth source goes more quickly into equilibrium than this constant Cv smooth source problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We present the solutions in Tables 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 19 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 64 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 64 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 128 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 128 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 64 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 64 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 128 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 128 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 11: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thin (σa = 1 cm−1) constant Cv Gaussian source problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The quadrature order for all results is S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a static mesh and uncollided source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 64 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 64 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 128 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 128 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 64 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='31623 64 cells, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 64 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='16228 128 cells, t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6228 128 cells, t = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 12: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thin (σa = 1 cm−1) S2 constant Cv Gaussian source problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a static mesh and uncollided source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 20 Table 5: Transport (top) and S2 (bottom) results for the scalar flux, φ, for the thin Gaussian source Su-Olson problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' RMSE values included in the last row of the S2 table are calculated from the S2 benchmark from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Convergence results for these answers are plotted in Figures 8 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' x/t 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='78279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='001958 RMSE 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='105e-10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='561e-10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='604e-09 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='839e-09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='321e-08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='808e-09 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='899e-05 21 Table 6: Transport (top) and S2 (bottom) results for the material energy density, e, for the thin Gaussian source Su-Olson problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' RMSE values included in the last row of the S2 table are calculated from the S2 benchmark from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Convergence results for these 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 S2 Transport (e) t = 10 −10 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='4 S2 Transport (f) t = 100 Figure 13: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thin constant Cv Gaussian source problem with x0 = 0.' metadata={'source': 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+page_content='03231 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 1e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='4e-05 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='78279 25 7 Optically thick results Here we include results for problems that we consider optically thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' By this, we mean that the source width is far greater than a mean free path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We accomplish this by specifying, σa = 800 cm−1 and z0 < 1 cm (z0 is the dimensional x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' In this section, we include results for linearized Gaussian and square sources as well as a constant Cv Gaussian source problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We do not include a constant Cv square source, since our method could not resolve the nonlinear, nonequilibrium, and very sharp wave that the square source induces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Since in an optically thick problem results are of interest only after many mean free times, we give results for τ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1, and 1 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' By these times, the uncollided source is negligible and any discontinuous wavefronts have decayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For these reasons, the problems in this section do not employ an uncollided source or a moving mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 Su-Olson problem with a square source To keep the dimensional spatial domain manageable, we set l = 1 800 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This makes the dimensional and nondimensional domains the same, but stiffens the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We are again using a square source (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (36)) with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 and t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We forgo the the use if an uncollided source since the evaluation times of interest are long after the uncollided solution has decayed to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For the mesh in this problem, only a static mesh was necessary for satisfactory convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We use a initialization outlined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 but that the initial width δx is not set to a small number, but to a guess of the solution width at the evaluation time and the edges never move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Essentially, the mesh is the same as the initial mesh for the thin square source but covering a the whole domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The Gauss-Legendre spacing of the edges has the effect of concentrating static edges around the source edge which makes it more likely that the region where the wavefront will be is resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The initial guess for the solution width was important and was refined with each run increasing the number of spatial divisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Since negative solutions are possible in our DG formulation and more likely to occur when there is a sharp wavefront, the temperature was calculated with T = sign(e)|e|1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The solution plots for this problem (Figure 16) show that for the chosen times, the solution is in local equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Unlike the selected thin square source solutions where a discontinuous wave travelling at the wavespeed determines the speed the solution travels, here a wave resembling a nonlinear heat waves moves outwards while the source is on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Geometric convergence, shown in Figures 14 and 15, is only possible since the nonsmooth portions of the scalar flux have decayed to zero and the solution is in equilibrium, which has the effect of smoothing the leading edge of the wavefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We also take not of the similarity between the transport and S2 solutions, apparent in the solution plots and in Tables 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This is expected from the optically thick results, since we saw the two solutions converge at long times where there was equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 Su-Olson problem with a Gaussian source In order to provide consistent examples across optically thin and thick regimes and as a trial run for the constant Cv thick Gaussian of the next section, we include here a optically thick Gaussian source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' For this and the nonlinear Gaussian, we specify the length parameter x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The source duration, t0, is still 10 and l = 1 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Once again, the source is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (37) and we do not use an uncollided source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Like the thin Gaussian sources, a moving mesh is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Figure 19 show that like the optically thick square source, the solutions are in equilibrium during the selected time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' There is however, no wavefront but the solution maintains Gaussian characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Like the square source, the transport and S2 solutions are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Geometric convergence of our standard DG method is shown in Figures 17 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We note that in this problem and the Su-Olson, square source 128 spatial cells were required to achieve the desired rate of convergence, though the solution was smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This was due to the small length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 26 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 14: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thick (σa = 800 cm−1) Su-Olson square source problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The quadrature order for all results is S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a static mesh and standard source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 15: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thick (σa = 800 cm−1) S2 Su-Olson square source problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a static mesh and standard source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 27 −1 0 1 0 2 4 S2 Transport (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 −1 0 1 0 2 4 S2 Transport (b) t = 3 −1 0 1 0 2 4 S2 Transport (c) t = 30 Figure 16: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thick Su-Olson square source problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Solid lines are scalar flux, φ, and dashed are material energy density, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' On the scale of this figure, the solid and dashed lines are coincident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 17: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thick (σa = 800 cm−1) Su-Olson Gaussian source problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='375, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The quadrature order for all results is S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a static mesh and standard source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 28 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 18: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thick (σa = 800 cm−1) S2 Su-Olson square Gaussian problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='375, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a static mesh and standard source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' −1 0 1 0 2 4 S2 Transport (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 −1 0 1 0 2 4 S2 Transport (b) t = 3 −1 0 1 0 2 4 S2 Transport (c) t = 30 Figure 19: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thick Su-Olson Gaussian source problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='375, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Solid lines are scalar flux, φ, and dashed are material energy density, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' On the scale of this figure, the solid and dashed lines are coincident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 29 Table 9: Transport (top) and S2 (bottom) results for the scalar flux, φ, for the thick square source Su-Olson problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Convergence results for these answers are plotted in Figures 14 and 15.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='9263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='000342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='9842 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='7e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0421 3e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 Constant Cv Gaussian problem Finally, we provide results for the constant Cv optically thick problem with a Gaussian source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Like the linear version of this problem, x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='375, t0 = 10, and l = 1 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We choose our constant opacity to be the same as we used for the optically thin case, Cv0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The source is again given by Eq/ (37) and the uncollided solution is not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Like the linear thick Gaussian, a static mesh is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 30 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 20: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thick (σa = 800 cm−1) constant Cv Gaussian source problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='375, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' The quadrature order for all results is S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a static mesh and standard source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (a) Radiation energy density, φ 2 4 8 12 M 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' |cn| 128 cells, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 128 cells, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 128 cells, t = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 (b) Material energy density, e Figure 21: Log-linear scaled average value of the solution expansion coefficients (found by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (44)) for the optically thick (σa = 800 cm−1) S2 constant Cv Gaussian problem where x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='375, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' All results were calculated with a static mesh and standard source treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 31 Table 10: Transport (top) and S2 (bottom) results for the material energy density, e, for the thick square source Su-Olson problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Convergence results for these answers are plotted in Figures 14 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 S2 Transport (a) t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 −2 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 S2 Transport (b) t = 3 −2 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='5 S2 Transport (c) t = 30 Figure 22: S2 (left of x = 0) and full transport (right of x = 0) solutions for the optically thick constant Cv Gaussian source problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='375, t0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Solid lines are radiation temperature φ 1/4, and dashed are temperature, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' On the scale of this figure, the solid and dashed lines are coincident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Table 13: Transport (top) and S2 (bottom) results for the scalar flux, φ, for the thick Gaussian source constant Cv problem with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='375, t0 = 10, and Cv0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='03 GJ · cm−3 · keV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Convergence results for these answers are plotted in Figures 20 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' x/t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='0 6.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='6e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='4 8e-06 8e-06 8e-06 36 8 Conclusions We have presented benchmark solutions to time dependent radiative transfer problems with two functional forms for the specific heat in optically thin and thick media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' These solutions will be useful to researchers who seek a nonlinear radiative transfer benchmark for verification purposes, desire more digits of accuracy for the Su-Olson type problem, or who intend to resolve a mean free path in the optically thick limit with a transport code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Although discontinuous sources are inconvenient for DG methods and required a complex mesh treatment to find accurate results we have provided solutions for square sources since they are simple to simulate in numerical codes and are already implemented in codes that run the Su-Olson problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Researchers who implement the DG friendly Gaussian source that we have defined will be able to converge to even more accurate results than for the square source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' To support our claim of benchmark quality results, we presented the convergence of the magnitude of coefficients in the solution expansion for each result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' While this does not necessarily guarantee the correctness of the system being solved, it does provide confidence that the solution is converged to a particular value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' We further increased confidence in our solutions by running S2 benchmarks for the linearized problems to high accuracy and comparing to the original published results in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Furthermore, the nonlinear problems were checked for systematic errors with a Sn code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' A Uncollided solutions to the S2 transport equation This section contains solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (41) for a square and a Gaussian source, which are used for the uncollided source treatment in S2 problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='1 Gaussian source With Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (37) as S in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (41), the integral evaluates to, φ gs u (x, t) = 1 4 √ 3πσe 3σ2 4 − √ 3x � erf � −3σ2 − 2t + 2t0 + 2 √ 3x 2 √ 3σ � + e2 √ 3x � erf �√ 3σ 2 + t √ 3σ + x σ � − erf � 3σ2 + 2t − 2t0 + 2 √ 3x 2 √ 3σ �� + erf � 3 √ 3σ2 + 2 √ 3t − 6x 6σ �� (56) Where σ is the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='2 Square source To evaluate the integral, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (41) for a square source, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (36), each possible case of integration limits allowed by the step functions in the source must be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' This will result in a piecewise function for the uncollided scalar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' If we define,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' F(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' τ) = −1 2 exp(−t + τ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (57) 37 then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' if t ≤ t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' φss u (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' t) = � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � F ���� max(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='τb) max(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='τa) (|x| > x0) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 2F ���� min(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='t0) 0 (t + √ 3|x| ≤ √ 3x0) F ���� τc τa + 2F ���� min(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='t0) τc � t + √ 3 |x| ≥ √ 3 x0 � & � t − √ 3(|x| + x0) > 0 � F ���� max(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='τc) 0 + 2F ���� min(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='t0) max(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='τc) � t + √ 3 |x| ≥ √ 3 x0 � & � t − √ 3(|x| + x0) ≤ 0 � (|x| ≤ x0) (58) or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' if t > t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' φss u (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' t) = � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � F ���� min(τd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='t0) min(τe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='t0) x0 − √ 3 3 (t − t0) ≤ x ≤ x0 + √ 3 3 (t − t0) F ���� min(τd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='t0) min(τe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content='t0) x > x0 + √ 3 3 (t − t0) � � � � � � � � � � � � � 2F ���� t0 0 t − 3 √ 3(x0 − x) ≤ 0 2F ���� t0 τd + F ���� τd 0 t − 3 √ 3(x0 − x) > 0 and t − 3 √ 3(x0 − x) < t0 x < x0 − √ 3 3 (t − t0) (59) Where τa = t − √ 3(|x| + x0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (60) τb = t − √ 3(|x| − x0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (61) τc = t − √ 3(x0 − |x|),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (62) τd = ����t − 3 √ 3(x − x0) ���� + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (63) and τe = ����t − 3 √ 3(x + x0) ���� + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' (64) | · | returns the positive part of its argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' References [1] J Stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Uber die beziehung zwischen der warmestrahlung und der temperatur, sitzungsberichte der mathematisch-naturwissenschaftlichen classe der kaiserlichen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' Akademie der Wissenschaften, 79:S–391, 1879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE0T4oBgHgl3EQftwFw/content/2301.02596v1.pdf'} +page_content=' [2] R.' 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